AGM-like Paraconsistent Partial Meet Abductive Expansion Operation

# AGM-like Paraconsistent Partial Meet Abductive Expansion Operation
Source: [https://arxiv.org/html/2607.09729](https://arxiv.org/html/2607.09729)
Ulisses Franceschi Eliano Institute of Philosophy and the Humanities \(IFCH\) and Centre for Logic, Epistemology and The History of Science \(CLE\), University of Campinas \(UNICAMP\) u225017@dac\.unicamp\.brThe author is a PhD student under the supervision of Marcelo Esteban Coniglio and supported by FAPESP \(Fundação de Amparo à Pesquisa do Estado de São Paulo\) – process2024/22555\-9\.

###### Abstract

In his 1996 doctoral thesis\[[19](https://arxiv.org/html/2607.09729#bib.bib19)\], Maurice Pagnucco created the firstAGM\-like abductive expansion operation\. Taking his operation as a basis, as well as a taxonomy—inspired by Atocha Aliseda\[[1](https://arxiv.org/html/2607.09729#bib.bib1)\]—responsible for highlighting and formalizing the main components of abductive reasoning, the main aim of this paper is to present a newparaconsistentAGM\-like abductive expansion operation—capable of assimilating contradictory explanatory hypotheses without trivialization and the consequent absurd epistemic state—, with its postulates and its transitively relationalpartial meetconstruction\. To a large extent, the formal development presented in this paper was only made possible by the recent creation of the paraconsistent logicRCbr\[[9](https://arxiv.org/html/2607.09729#bib.bib9)\], anLFI\(Logics of Formal Inconsistencies\) that establishes properties especially relevant to belief revision contexts, in particular, the ability to beself\-extensional—i\.e\., to satisfy thereplacementproperty\. This is the first of two papers: the paraconsistent abductive expansion operation announced here—which is part of a new system calledAGMpa​b​dp\_\{abd\}—despite bringing many interesting features, does not assign any relevant epistemic role to the paraconsistent operators of negation¬\\negand consistency∘\\circ\. Only in a second paper will an analogous paraconsistent abductive expansion operation—which is part of another new system,AGM∘a​b​d\\circ\_\{abd\}—be enhanced in this direction\. Nevertheless, to the best of my knowledge, the operation developed in this paper is the first of its kind in theAGMliterature\.

## 1Introduction

Charles Sanders Peirce is considered the founder of abduction as a third type of reasoning, independent of deduction and induction\. Abduction is the reasoning responsible for formulating and selecting hypotheses capable of explaining, albeit rudimentarily and conjecturally, intriguing and surprising facts\. Or, according to Fann, “\[…\] Peirce’s theory of abduction is concerned with the reasoning which starts from data and moves towards hypotheses”\[[12](https://arxiv.org/html/2607.09729#bib.bib12), p\.5\]\. Peirce’s theory of abduction is philosophically rich and is related to many other concepts in his philosophy\. For the purposes of this article, however, I would like to highlight four important aspects of abduction: \(i\) thecreative aspect, \(ii\) thelogical\-inferential aspect, \(iii\) itsdistinction from the other two types of reasoning, and \(iv\) theselective aspect\. The creative aspect of abduction \(i\) manifests itself when an inquiry agent, faced with a surprising fact, originally conceives hypotheses to explain it, as an instinctive act111Peirce, in fact, understood abduction as a kind ofguessing instinct\. In his words, abduction “\[…\] tries whatil lume naturale, which lit the footsteps of Galileo, can do\. It is really an appeal to instinct” \(CP 1\.630\) and abduction is “\[…\] nothing but guessing” \(CP 7\.219\)\., and, thus,expandsits initial informational repertoire with new conceptions\. In Peirce’s words, “Abduction is originary in respect to being the only kind of argument which starts a new idea” \(CP 2\.96\)222The acronym “CP” refers to theCollected Papersof Charles Sanders Peirce, which can be found in the references as\[[20](https://arxiv.org/html/2607.09729#bib.bib20)\]and\[[21](https://arxiv.org/html/2607.09729#bib.bib21)\]\.\. The creative originality of this type of reasoning places it at the center of the debate about the possibility of a logic of discovery, in Hans Reichenbach’s well\-known terms333The distinction betweencontext of discoveryandcontext of justificationis well known in the literature\. The author considered, for example, Einstein’s new theory of gravitation a great discovery, not because the physicist was lucky or had a hunch, but because “\[…\] the known facts indicate such a theory; i\.e\., that an inductive expansion of the known facts leads to the new theory\. This is precisely what distinguishes the great scientific discoverer from a clairvoyant”\[[22](https://arxiv.org/html/2607.09729#bib.bib22), p\.382\]\. Reichenbach, however, identified this type of reasoning involved in scientific discovery, like many authors of his time \(and today\), with induction\., something that goes beyond the scope of this article\. Despite being creative, abduction has a logical\-inferential aspect \(ii\), which seems paradoxical444This is a well\-known paradox in the specialized literature and is called theabduction paradox\. Unfortunately, this interesting topic will not be developed in this document\. To read more about this, including an excellent attempt to resolve it, see\[[26](https://arxiv.org/html/2607.09729#bib.bib26)\]\.\. The Peircean conception of inference555In Peirce’s words, “It must be remembered that abduction, although it is very little hampered by logical rules, nevertheless is logical inference, asserting its conclusion only problematically or conjecturally, it is true, but nevertheless having a perfectly definite logical form” \(CP 5\.188\)\., however, bears little or no resemblance to the notion of logical consequence, syntactical, or semantic, which has become the canonical standard in the orthodoxy of contemporary logic\. It is not possible to address Peirce’s inferentialism here, however, it is my understanding that an inference, in the Peircean general sense, but in particular the abductive one, is best represented by anAGM\-style belief change operation embedded in theabductive processdepicted in the taxonomy in Section[2](https://arxiv.org/html/2607.09729#S2)

Furthermore, it is important to keep in mind that Peircean abduction should not be confused with the other two types of reasoning \(iii\)\. The distinction between abduction and deduction is clearer and does not usually generate controversy, but abduction is often confused with induction\. It is not infrequently considered a type of induction666This confusion seems to have its origins in Gilbert Harman’s influential article,The Inference to the Best Explanation\. I think that Harman’s article, in light of Peirce’s original abduction theory, deserves much criticism \(and it is not uncommon in the literature\), but I will not go into it in this article\.\. In Peirce’s words, however, “I don’t think the adoption of a hypothesis on probation can properly be called induction; and yet it is reasoning, and though its security is low, its uberty is high” \(CP 8\.388\)\. Abduction, in turn, is the only truly ampliative inference, in Peirce’s most mature conception, and what differentiates it from induction is that the latter “\[…\] infers the existence of phenomena such as we have observed in cases which aresimilar, while hypothesis supposes something of adifferent kindfrom what we have directly observed, and frequently something which it would be impossible for us to observe directly” \(CP 2\.640\)\. While abduction is the “\[…\] first step of scientific reasoning” \(CP 7\.218\), and therefore seeks to formulate a new theory, induction, on the other hand, is only responsible for evaluating the experimental results as the final step of inquiry, without actually adding anything new\. Abduction also selects \(iv\) the “best” hypotheses from among the countless777The incountability of possible explanatory hypotheses, that is, “\[…\] the idea that data underdetermine theory \- that an infinite variety of alternative hypotheses can conform equally well to any finite body of empirical data”\[[23](https://arxiv.org/html/2607.09729#bib.bib23), p\.72\], seems to be a commonplace in the philosophy of science and Peirce seemed, in his time, to endorse such a view\. In his words, “\[…\] Think of what trillions of trillions of hypotheses might be made of which one only is true; and yet after two or three or at the very most a dozen guesses, the physicist hits pretty nearly on the correct hypothesis” \(CP 5\.172\)\.formulated ones\. The philosophical and scientific literature concerning the criteria that adequately apprehend the notion of “best” or more “preferable” in this context is vast\. The passage in CP 7\.220, in which Peirce cites the three criteria for selecting hypotheses, is well known\. They are: the hypotheses \(a\) must, in fact,explainthe surprising facts, \(b\) must be capable of being subjected toexperimental verification, and \(c\) must be conditioned by theprinciple of economy of research\. Each of these criteria requires considerable philosophical exposition, including the doctrine of Pragmatism, which is closely linked to abduction, a topic that, unfortunately, cannot be adequately addressed in this article\. In terms of contemporary logic \(but not Peircean conception of logic\), these criteria can be considered “extra\-logical”, and so I will refer to them in this paper, more generally, as thepragmatic and explanatory principlesof hypothesis selection\.

To the best of my knowledge, Peirce did not address the question of the possibility of the coexistence of contradictory hypotheses\. It seems to me, however, that they can indeed coexist in a wide variety of contexts, as argued by the authors, for example, in\[[5](https://arxiv.org/html/2607.09729#bib.bib5)\]and\[[24](https://arxiv.org/html/2607.09729#bib.bib24)\]\. A paraconsistent approach is therefore necessary to formally represent such situations\. The paraconsistent abductive expansion operation presented in this paper will be referred to as theAGMpa​b​dp\_\{abd\}abductive expansion operation, the system of which it is a component888In Testa’s work\[[27](https://arxiv.org/html/2607.09729#bib.bib27)\], later improved in\[[11](https://arxiv.org/html/2607.09729#bib.bib11)\], two systems were created:AGMppandAGM∘\\circ\. The former establishes paraconsistent contraction and revision operations in which no epistemic role is, in fact, assigned to the paraconsistent operators¬\\negand∘\\circ\. In the latter, the operators gain more epistemically relevant roles, especially the consistency operator∘\\circ, which confers theirrefutabilityof a given beliefφ\\varphi, whenφ∈Θ\\varphi\\in\\Thetaand∘φ∈Θ\\circ\\varphi\\in\\Theta, forΘ=C​n​\(Θ\)\\Theta=Cn\(\\Theta\)—in this case,φ\\varphiis said to beboldly acceptedinΘ\\Theta\. In turn, twoAGM\-like abductive systems, with their respective operations, postulates and constructions, are being developed in my ongoing PhD research, and the namesAGMpa​b​dp\_\{abd\}andAGM∘a​b​d\\circ\_\{abd\}were analogously adopted\. This paper concerns only the abductive expansion operation that is part of theAGMpa​b​dp\_\{abd\}system\.\. As is conventional in theAGMliterature, especially following the publication of Gärdenfors and Rott’s article\[[14](https://arxiv.org/html/2607.09729#bib.bib14)\]999As is well known in the literature, the criteria established by the authors are:consistency—which, in the classical case, is the same as non\-contradiction—,deductive closure,minimalityandepistemic entrenchment\.,AGMoperations should be guided byrationality criteria\. For theAGMpa​b​dp\_\{abd\}system the rationality criteria are the following:

\(i\)Non\-triviality: an epistemic state must bealwaysnon\-trivial;

\(ii\)Deductive closure: any belief deductively implied by other beliefs of an epistemic state must be included in the epistemic state;

\(iii\)Minimality: when discarding information, the maximum amount of information from the original state should be preserved;

\(iv\)Epistemic entrenchment: epistemically more valuable beliefs \(or “more entrenched”\) should be more resistant to change;

\(v\)Information acquisition: the agent is interested in acquiring new and valuable error\-free information;

\(v\.a\)Surprising fact: the agent, when faced with a surprising fact,alwaysinitiates anabductive processto elaborate and incorporate explanatory hypotheses;

\(v\.b\)Explanatory adequacy: the new hypotheses must beproperly explanatoryfor the surprising fact;

\(v\.c\)Informational balance: the agent must achieve a balance between the amount of new hypotheses to be incorporated and the risk of error, depending on their degree of caution or bouldness;

\(v\.d\)Abductive entrenchment: some hypotheses are more significant and relevant than others, and the agent selects them according toexplanatory and pragmatic principles;

Criterion \(i\) of non\-triviality differs from the classicalAGMsystem in at least two respects: first, the specific requirement ofnon\-triviality, rather thanconsistency\. This is a fundamental distinction in the paraconsistency literature, since, unlikeCPL\(Classical Propositional Logic\), in which the terms “inconsistency”, “trivialization” and “contradiction” are synonymous \(and formally equivalent\), these notions do not coincide in anLFI, as we shall see in Section[3](https://arxiv.org/html/2607.09729#S3)\. The second aspect concerns the term “always”\. Unlike the classicalAGMsystem, in which the expansion operation does not formally block initial trivial epistemic states, nor does it prohibit expansion towards triviality, in theAGMpa​b​dp\_\{abd\}abductive expansion there is no room for triviality\. Criterion \(ii\) of deductive closure remains the same as in classicalAGM, i\.e\., epistemic states will be modelled as deductively closed belief sets\. Criteria \(iii\) and \(iv\) are old acquaintances of theAGMliterature and are only present here to guide other operations of theAGMpa​b​dp\_\{abd\}system that do not strictly concern the paraconsistent abductive expansion operation, but which are worth keeping as general guiding criteria for the system101010Regarding criterion \(iii\) of minimality, three things should be noted\. 1\) In his inaugural thesis\[[19](https://arxiv.org/html/2607.09729#bib.bib19)\], Pagnucco also develops the abductive revision operation, based on the Levi Identity, but with the Levi saturated contraction, followed by his abductive expansion operation, i\.e\.,Θφ⊗=\(Θ∼​φ−\)φ⊕\\Theta^\{\\otimes\}\_\{\\varphi\}=\(\\Theta^\{\-\}\_\{\\mathord\{\\sim\}\\varphi\}\)^\{\\oplus\}\_\{\\varphi\}\. This is a simpler operation—quite limited due to the non\-monotonic nature of the abductive expansion operation and the impossibility of establishing an adequate inclusion relation between abductive expansion and revision \(see\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.165\-166\]\)—but which, because of the prior contraction operation, needs to be guided by the minimality criterion\. The same occurs in theAGMpa​b​dp\_\{abd\}system, with the difference that, since it is a paraconsistent system, anexternal revisionoperation—i\.e\., obtained by means of the Reverse Levi Identity, in which expansion occursbeforecontraction, i\.e\.,Θφ○⊗e=\(Θφ○⊕\)∼​φ−\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\otimes$\\crcr\}\}\}\_\{e\}\}\_\{\\varphi\}=\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\)^\{\-\}\_\{\\mathord\{\\sim\}\\varphi\}—can also be obtained, although suffering from the same \(and others\) limitations\. Unfortunately, revision operations are not the subject of this paper\. 2\) Moreover, Pagnucco also provides a semantic interpretation of minimality via possible worlds and Grove spheres \(see\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.114\-121\]\)\. I will not delve into this topic in this paper\. 3\) Furthermore, notoriously, the minimality criterion contrasts with criterion \(v\), which seems more appropriate for abductive expansion, as we shall see\. I also think that, specifically for abductive expansion, the minimality criterion should be subordinated to other foundations, of an “extra\-logical” nature, according topragmatic principlesof hypothesis selection and preference—especially those found in C\.S\. Peirce’s own work, such assimplicity, explanatory power, economy, experimental verification, etc\.—which are not within the scope of this paper\. This issue, however, will be taken up again in the second paper, concerning theAGM∘a​b​d\\circ\_\{abd\}system\.111111In the case of criterion \(iv\), as in the previous case, it is prudent to keep it, since it concerns the abductive revision operations—in fact, the contraction operation underlying them\. Epistemic entrenchment, however, will have no role in the abductive expansion operation proper\.\.

Criterion \(v\) is an inheritance from Pagnucco’s own system, insofar as the author reinterprets it from Isaac Levi’s work\[[16](https://arxiv.org/html/2607.09729#bib.bib16), p\.72\]\. This criterion indicates the balance, desired by an ideal rational agent, between the interest in acquiring new and valuable information, in this case through abduction, and the risk of falling into error121212It is important to note, however, that depending on the logic underlying the system, whether classical or paraconsistent, this criterion may mean different things\. On the one hand, as we shall see, paraconsistency allows the acquisition of new information in the abductive process in cases where the classical one does not\. On the other hand, the term “error”, in the classical case, means inconsistency, in the sense of contradiction\. In the paraconsistent case, the term “error” comes to be interpreted exclusively as triviality\. Contradiction, therefore, no longer enters the agent’s rational balance as an error and comes to hold an informational value that deserves due attention\.\. I consider this criterion, however, sufficiently general to encompass new criteria, which determinehowthe agent will carry out such informational acquisition\. Thus, criteria \(v\.a\) and \(v\.b\) capture important aspects of abductive reasoning proper that are very prominent in C\.S\. Peirce’s work, as we just discussed\. First, criterion \(v\.a\) tells us that abductive reasoning is a process that arises from a surprising fact, which impresses upon us a troubling doubt, and which needs to be explained\. As such, it is ampliative par excellence—in fact, theonlytruly ampliative reasoning\. The emphasis on the term “always” is therefore important, since there is no abductive reasoning that leaves the initial epistemic state unchanged\. Criterion \(v\.b\) minimally qualifies the hypotheses, accordingly to Peircean thought, in the sense of requiring them to be properly explanatory\. These two criteria justify the creation of a specific taxonomy for abduction, which will be addressed in Section[2](https://arxiv.org/html/2607.09729#S2), and, to a large extent, in addition to paraconsistency itself, differentiate the abductive expansion operation developed here from Pagnucco’s operation\.

Finally, criteria \(v\.c\) and \(v\.d\) can also be found, in a different exposition, in Pagnucco’s thesis\. Criterion \(v\.c\) tells us that a balance must be achieved between an extremely self\-confident agent, i\.e\., one who elaborates too many hypotheses and thus may more easily fall into error, and, on the other hand, a very sceptical agent, who does not run many risks of incorporating errors, but, to the same extent, does not elaborate enough hypotheses and therefore fails to incorporate relevant information\. The operation that performs this balance is precisely thepartial meetabductive expansion that will be developed in Section[6](https://arxiv.org/html/2607.09729#S6)131313In his thesis\[[19](https://arxiv.org/html/2607.09729#bib.bib19)\], Pagnucco also presentsmaxichoiceandfull meetabductive expansion operations, in line with the classicalAGMliterature\. The same operations can also be established in theAGMpa​b​dp\_\{abd\}system, with many of the same characteristics already found by Pagnucco, among others\. Unfortunately, it will not be possible to develop them in this paper\.\. Criterion \(v\.d\) attempts to capture the notions underlyingbetternessrelations between hypotheses, which are generally those used by the agent to select them\. As we just discussed, Peirce himself presented many “extra\-logical” criteria in this sense, thepragmatic and explanatory principlesof hypothesis selection\. Formally, however, there are several possible ways of representing such notions141414Pagnucco\[[19](https://arxiv.org/html/2607.09729#bib.bib19)\]also presented a construction calledabductive entrenchment, similar to the epistemic entrenchment of classicalAGM, but interested, in the opposite way, in sentences that lie outside the belief set and need to be incorporated\. The same construction can also be obtained in theAGMpa​b​dp\_\{abd\}system, with the appropriate formal changes\. Once again, this is a construction that was only made possible thanks to the properties of theRCbrlogic, which will be introduced in Section[3](https://arxiv.org/html/2607.09729#S3)\. The paraconsistent abductive expansion operation based on the abductive entrenchment order, however, will be the subject of another future paper\.\. In this paper, the relationality and transitivity imposed on thepartial meetselection function will fulfil this role\.

## 2A Taxonomy for Abducion

Inspired by the taxonomy for abduction proposed by Atocha Aliseda151515See\[[1](https://arxiv.org/html/2607.09729#bib.bib1), 46\-48\]\. It is important to note that Aliseda implements her abductive process specifically through abductive tableaux \(despite the author establishing some interesting connections between her tableaux and theAGMsystem \(see\[[1](https://arxiv.org/html/2607.09729#bib.bib1), Chapter 8\]\)\. The objective of this article, however, like Pagnucco’s, is to present an AGM\-style operation with its respectiveconstructionfor epistemic states expanded by explanatory hypotheses, not toconstructthe hypotheses themselves via tableaux, as proposed by Aliseda\. Furthermore, Aliseda proposes a ternary format for her inferential parameterΘ\|φ⇒α\\Theta~\|~\\varphi\\Rightarrow\\alpha\(see\[[1](https://arxiv.org/html/2607.09729#bib.bib1), p\.68\]\) and various abductive styles \(see\[[1](https://arxiv.org/html/2607.09729#bib.bib1), 74\]\)\. In the first case, Aliseda’s objective is to study the behavior of the well\-known structural rules of Gentzen\-style sequent calculus when applied to different kinds of premises \(as required by the Hempel\-Oppenheim DN model\), which is not my objective in this article\. In the second case, the adequate explanatory inference proposed here contains exactly the same restrictions, jointly, as the consistent and explanatory styles proposed by the author\. My inspiration in Aliseda’s work, however, is justified by recognizing, in her taxonomy, a very interesting theoretical conception general enough to address various implementations of abductive reasoning, provided it is thought of in terms of process, not merely as a simple logical form\., a taxonomy for abduction is a general systematization of three components of anabductive process: \(i\) thetriggers, \(ii\) anadequate explanatory inference, and \(iii\) theoutputs\. Let us therefore look at each of the components\.

\(i\) Triggers

The triggers represent the beginning of the abductive process, that is, they characterize theinitial surprisein the face of an unusual and surprising phenomenon or factφ\\varphi\. From a formal point of view and considering initially classical propositional logic𝕃=⟨ℒΣ,⊢⟩\\mathbb\{L\}=\\langle\\mathcal\{L\}\_\{\\Sigma\},\\vdash\\rangle, in witch formulas inℒ\\mathcal\{L\}are generated by classical propositional signatureΣ=\{∧,∨,→,∼,⊥,⊤\}\\Sigma=\\\{\\land,\\lor,\\to,\\mathord\{\\sim\},\\bot,\\top\\\}\. In this case, the simpler notation𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\}will be used\. LetΘ\\Thetabe a set of formulas representing agent’s previous theories or beliefs andΘ∪\{φ\}⊆ℒΣ\\Theta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\}\. The \(classical\) triggers for an abdutctive process are:

\(i\)Abductive novelty:φ\\varphiis a novelty ifΘ⊬𝕃φ\\Theta\\nvdash\_\{\\mathbb\{L\}\}\\varphiandΘ⊬𝕃∼​φ\\Theta\\nvdash\_\{\\mathbb\{L\}\}\\mathord\{\\sim\}\\varphi

\(ii\)Abductive anomaly:φ\\varphiis an anomaly ifΘ⊬𝕃φ\\Theta\\nvdash\_\{\\mathbb\{L\}\}\\varphiandΘ⊢𝕃∼φ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\mathord\{\\sim\}\\varphi

The requirementΘ⊬𝕃φ\\Theta\\nvdash\_\{\\mathbb\{L\}\}\\varphiis the same in both triggers\. It is justified, in the abduction context, because, ifΘ⊢𝕃φ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\varphi, there is no surprise and, thus, no abducive reasoning\. The difference between both triggers, however, is in the fact that, in an abductive novelty, the agent don’t have any initial clue about the observed fact, and in the abductive anomaly the fact is inconsistent—in a classical sense, contradictory—with his previous beliefs\.

\(ii\) Adequate explanatory inference

Let us consider the following definition\.

###### Definition 2\.1\(Abductive explanation\)\.

Let𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\}andΘ∪\{φ\}∪\{α\}⊆ℒΣ\\Theta\\cup\\\{\\varphi\\\}\\cup\\\{\\alpha\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\}\. The sentenceα\\alphais an abductive explanation ofφ\\varphiwith respect toΘ\\Thetaif and only if:

\(i\)Θ∪\{α\}⊢𝕃φ\\Theta\\cup\\\{\\alpha\\\}\\vdash\_\{\\mathbb\{L\}\}\\varphi;

\(ii\)Θ∪\{α\}⊬𝕃⊥\\Theta\\cup\\\{\\alpha\\\}\\nvdash\_\{\\mathbb\{L\}\}\\bot;

\(iii\)Θ⊬𝕃φ\\Theta\\nvdash\_\{\\mathbb\{L\}\}\\varphi;

\(iv\)\{α\}⊬𝕃φ\\\{\\alpha\\\}\\nvdash\_\{\\mathbb\{L\}\}\\varphi\.

Conditions \(i\) and \(ii\) are quite common in the literature\. However, for the explanatory aspect of a hypothesisα\\alpha, ifΘ∪\{α\}⊢𝕃φ\\Theta\\cup\\\{\\alpha\\\}\\vdash\_\{\\mathbb\{L\}\}\\varphiandΘ∪\{α\}⊢𝕃⊥\\Theta\\cup\\\{\\alpha\\\}\\vdash\_\{\\mathbb\{L\}\}\\botare necessary161616It is well known in the literature, since Aristotle’sPosterior Analytics, that deduction is not sufficient for explanation, but it is necessary \(see\[[3](https://arxiv.org/html/2607.09729#bib.bib3)\]e\[[2](https://arxiv.org/html/2607.09729#bib.bib2)\]\)\. There are many other intensional aspects to consider whenexplanatory relevanceandexplanatory/causal directionare in question\. They simply can not be captured by deduction alone\. The problem of scientific explanation has survived through time until it was revived in the 20th century by Hampel and Oppenheim and their well\-known DN\-model\[[15](https://arxiv.org/html/2607.09729#bib.bib15)\]\. If not the most successful contemporary approach to scientific explanation, it has certainly become paradigmatic, especially in the formal logic field\. Of course, there are many others contemporary approaches in philosophy that even disconsider deduction as a necessary condition for explanation, for instance,\[[25](https://arxiv.org/html/2607.09729#bib.bib25)\]and\[[28](https://arxiv.org/html/2607.09729#bib.bib28)\]\.condition for explanation, in Hempelian terms at least, they are not sufficient171717Of course, philosophically, Hempel and Oppenheim\[[15](https://arxiv.org/html/2607.09729#bib.bib15)\]are very distant from Peircean philosophy and his Pragmatism\. The philosophical notions of cause, laws of nature and theories \(and, of course, explanation\) are very different between these two schools of thought\. Several 20th\-century logical positivists and philosophers of science, including Hempel and Oppenheim themselves, were influenced by previous interpretations, dating back to the 19th century, of the Humean philosphy that recognized him as a causal regularist\. I cannot go further on this topic, but it is important to say that, when the main motivation is the logical formalization of the explanation \(thus disregarding strictly statistical and computational approaches\), the work of Hempel and Oppenheim seems to be, to a large extent, inescapable\.\. I claim that the restrictionsΘ⊢𝕃φ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\varphiand\{α\}⊢𝕃φ\\\{\\alpha\\\}\\vdash\_\{\\mathbb\{L\}\}\\varphimust also be considered asminimumconditions to achievesome181818Note that Hempel and Oppenheim give us many logical and empirical conditions to guarantee explanatory relevance \(see\[[15](https://arxiv.org/html/2607.09729#bib.bib15), 247\-249\]\)\. I think that their definition ofpotential explanans\(\[[15](https://arxiv.org/html/2607.09729#bib.bib15), 277\-278\]can be considered a desirable general goal for all those who desire a good logical\-deductive representation of explanation, at least when abduction and its potential hypotheses are in question\. Unfortunately, Classical Propositional Logic, used predominantly in AGM\-like implementations, makes this complete adequacy to their definition unfeasible, as the authors require quantified first\-order language to represent laws or background theories\. For the purposes of this paper, however, I think these proposed restrictions are good enough\.explanatory relevance over the factφ\\varphi: the appropriate explanation forφ\\varphimust be an interplay of the background theoriesΘ\\Thetatogether with the hypothesisα\\alpha, andα\\alphacan not manifests itself as a total self\-explanation191919A self\-explanation is an empirical and logical problem that emerges when deduction is underneath\. LetΘ\\Thetabe background theories andφ\\varphianexplanandum\. So, due to the deduction logical properties, it is always possible to create explanations of the kindφ⊢φ\\varphi\\vdash\\varphi\. If reflexivity is a desirable property for deduction, certainly it is not for explanation\., witch is, in this context, represents the blocking of the reflexivity property, present in all Tarskian deductive logic\.

\(iii\) Outputs

If the triggers are the beginning of the process, the outputs are the end, the final result of the process\. In the case of the formal system presented by Aliseda\[[1](https://arxiv.org/html/2607.09729#bib.bib1)\], based on abductive tableaux responsible for “generating” or “constructing” the abductive hypotheses, the outputs of the process are the multiple abductive hypothesesα\\alphathemselves\. Differently, as in Pagnucco’s case\[[19](https://arxiv.org/html/2607.09729#bib.bib19)\], my proposal is to represent abductive reasoning by means ofAGM\-like operation\. Thus, a belief change operation is a function that takes an initial epistemic state, together with a sentence of the language—in this case, a sentence representing the phenomenon to be explained—and returns multiple possible expanded epistemic states, proportionally to the number of available explanatory hypotheses\. In other words, the outputs of the abductive process, instead of the abductive hypotheses, due to the nature ofAGM\-style operations, are the expanded epistemic states themselves relative to them202020It is important to emphasize, therefore, that unlike the abductive tableau obtained by Aliseda \(and also the paraconsistent abductive tableaux found in\[[5](https://arxiv.org/html/2607.09729#bib.bib5)\]and\[[24](https://arxiv.org/html/2607.09729#bib.bib24)\]\), which is responsible for “constructing” the hypotheses—which can be interpreted as a formal attempt to represent the “creative” aspect of abduction—the paraconsistent abductive expansion operation presented in this paper already starts from constructed hypotheses, as can be noted in the very definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)in Section[5](https://arxiv.org/html/2607.09729#S5)\. In my view, therefore, these are complementary formal implementations, not competing ones\.\.

## 3The RCbr Paraconsistent Logic

RCbris a recently developedLFI\[[9](https://arxiv.org/html/2607.09729#bib.bib9)\]specifically designed to serve as the underlying logic for the paraconsistent belief revisionAGM∘\\circsystem212121As already mentioned in footnote[8](https://arxiv.org/html/2607.09729#footnote8), theAGM∘\\circis a paraconsistent belief revision system first developed in\[[27](https://arxiv.org/html/2607.09729#bib.bib27)\]and extensively improved in\[[11](https://arxiv.org/html/2607.09729#bib.bib11)\]and\[[9](https://arxiv.org/html/2607.09729#bib.bib9)\]\. This system can establish four new AGM\-style epistemic attitudes—besides the three well\-known from standardAGM—relative to the agent’s beliefstrengthand contradictoriness, a new AGM\-like paraconsistent contraction operation, and other paraconsistent internal and \(now possible\) external revision operations\. In\[[27](https://arxiv.org/html/2607.09729#bib.bib27)\], due to generalLFIproperties, only paraconsistent non\-extensionalpartial meetcontraction and revision operations were developed, and in\[[11](https://arxiv.org/html/2607.09729#bib.bib11)\], the authors obtained anextensionalversion of the same operations\. Only withRCbrcould an epistemic entrenchment order be established and proven to satisfy the paraconsistent contraction postulates\.\. To describeRCbr’s syntax, semantics, and general properties, some brief background is necessary\. The main idea underlyingLFIs is the distinction between trivialization and contradiction, where explosion—due to the classicalex falsoprinciple—occurs only in acontrolledmanner when the contradictory proposition in question is marked as consistent by means of the consistency operator∘\\circ\. Thus, inLFIs:

φ,¬φ⊬L​F​Iψ\\varphi,\\neg\\varphi\\nvdash\_\{LFI\}\\psi, but∘φ,φ,¬φ⊢L​F​Iψ\\circ\\varphi,\\varphi,\\neg\\varphi\\vdash\_\{LFI\}\\psi

It is well known in the paraconsistency literature222222The most comprehensive guide toLFIs—particularly tombCand many of its extensions—can be found in\[[7](https://arxiv.org/html/2607.09729#bib.bib7)\]\.thatmbCis the weakestLFI\. Therefore, let us consider a standard and supraclassical Tarskian logic232323Definitions[A\.2](https://arxiv.org/html/2607.09729#A1.Thmtheorem2)and[A\.3](https://arxiv.org/html/2607.09729#A1.Thmtheorem3)found in Appendix[A](https://arxiv.org/html/2607.09729#A1)\. As so, thembClogic also satisfies fundamental rules and properties ofCPL, such as thededuction theorem,disjunction of premises, andproof by cases, as well asd​e​l​t​adelta\-saturated properties and Lindenbaum\-Łoś Theorem\. See properties[A\.4](https://arxiv.org/html/2607.09729#A1.Thmtheorem4)and[A\.6](https://arxiv.org/html/2607.09729#A1.Thmtheorem6)and theorem[A\.8](https://arxiv.org/html/2607.09729#A1.Thmtheorem8)in Appendix[A](https://arxiv.org/html/2607.09729#A1)\.mbC=⟨ℒΣ∘,⊢⟩=\\langle\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\},\\vdash\\rangleover the languageℒΣ∘\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\}generated by the signatureΣ∘=\{∧,∨,→,¬,∘\}\\Sigma\_\{\\circ\}=\\\{\\land,\\lor,\\to,\\neg,\\circ\\\}, where¬\\negdenotes the paraconsistent negation and∘\\circthe primitive consistency operator\. Let us also define⊥ψ=defψ∧¬ψ∧∘ψ\\bot\_\{\\psi\}=\_\{\\text\{def\}\}\\psi\\land\\neg\\psi\\land\\circ\\psias thefalsumparticle and∼ψφ=def\(φ→⊥ψ\)\\sim\_\{\\psi\}\\\!\\varphi=\_\{\\text\{def\}\}\(\\varphi\\to\\bot\_\{\\psi\}\)as the defined classical \(strong\) negation\. Consider the following Hilbert\-style axiomatic scheme:

\(1\)AllCPL\+basic axioms for∧,∨,→\\land,\\lor,\\toplusModus Ponens

\(2\)φ∨¬φ\\varphi\\lor\\neg\\varphi

\(3\)\(b​c​1bc1\)∘φ→\(φ→\(¬φ→ψ\)\)\\circ\\varphi\\to\(\\varphi\\to\(\\neg\\varphi\\to\\psi\)\)

The derivation inmbCis defined as usual242424See\[[7](https://arxiv.org/html/2607.09729#bib.bib7), p\.34\]\.\. Three main characteristics deserve attention: \(i\) the absence of the principle of non\-contradiction, \(ii\) the excluded middle axiom 2, which is formulated using the paraconsistent negation¬\\neg, and \(iii\) the axiom 3 \(b​c​1bc1\), which governs the behavior of the consistency operator∘\\circ\. Semantic valuations for the classical connectives operate as usual, while specific non\-deterministic valuations for¬\\negand∘\\circare provided below:

\(v​N​e​gvNeg\)v​\(¬φ\)=0⟹v​\(φ\)=1v\(\\neg\\varphi\)=0\\implies v\(\\varphi\)=1

\(v​C​o​nvCon\)v\(∘φ\)=1⟹v\(φ\)=0v\(\\circ\\varphi\)=1\\implies v\(\\varphi\)=0\\;orv​\(¬φ\)=0\\;v\(\\neg\\varphi\)=0

ThembCfundamental properties252525The most notable properties ofmbCare: \(i\) a contradiction implies inconsistency, but not vice versa, that is,φ∧¬φ⊢mbC¬∘φ\\varphi\\land\\neg\\varphi\\vdash\_\{\\textbf\{mbC\}\}\\neg\\circ\\varphi, but¬∘φ⊬mbCφ∧¬φ\\neg\\circ\\varphi\\nvdash\_\{\\textbf\{mbC\}\}\\varphi\\land\\neg\\varphi; \(ii\) consistency implies non\-contradiction, but not vice versa, that is,∘φ⊢mbC¬\(φ∧¬φ\)\\circ\\varphi\\vdash\_\{\\textbf\{mbC\}\}\\neg\(\\varphi\\land\\neg\\varphi\), but¬\(φ∧¬φ\)⊬mbC∘φ\\neg\(\\varphi\\land\\neg\\varphi\)\\nvdash\_\{\\textbf\{mbC\}\}\\circ\\varphi; finally, since most classical equivalences involving negation do not hold in general,mbC—like mostLFIs—is notself\-extensional; that is, it does not satisfy thereplacementproperty, which we will examine shortly\.and its soundness and completeness theorems are well\-established262626See\[[7](https://arxiv.org/html/2607.09729#bib.bib7), p\.36\-38\]\.\. There are manymbCextensions in the literature\. The following extensions and axioms—where those on the right incorporate the adjacent ones on the left—are relevant for our purposes:

mbC⊂\\subsetmbCciw⊂\\subsetCbr⊂\\subsetRCbrmbC axioms\+\+\(ciw\)∘φ∨\(φ∧¬φ\)\\circ\\varphi\\lor\(\\varphi\\land\\neg\\varphi\)\+\+\(ce\)φ→¬¬φ\\varphi\\to\\neg\\neg\\varphi\(cf\)¬¬φ→φ\\neg\\neg\\varphi\\to\\varphi\+\+replacement

Table 1:RCbraxiomatic hierarchy\.ThembCciwlogic is the minimal extension ofmbCthat guarantees that the truth values ofφ\\varphiand¬φ\\neg\\varphicompletely determine the truth value of∘φ\\circ\\varphi272727Other importantmbCciwproperties are: \(i\) strongly denying a formula is equivalent to weakly denying it when the formula is considered consistent, that is,∼φ≡mbCciw\\;\\sim\\\!\\varphi\\equiv\_\{\\textbf\{mbCciw\}\}∘φ∧¬φ\\circ\\varphi\\land\\neg\\varphi; and \(ii\) consistency operator can be defined in terms of other connectives, that is,∘φ≡mbCciw∼\(φ∧¬φ\)\\circ\\varphi\\equiv\_\{\\textbf\{mbCciw\}\}\\;\\sim\\\!\(\\varphi\\land\\neg\\varphi\)\.\. The extensionCbrincorporates both double negation axioms and thus satisfiesφ≡Cbr¬¬φ\\varphi\\equiv\_\{\\textbf\{Cbr\}\}\\neg\\neg\\varphi\. The semantics forCbris obtained through an algebraic swap structure\-based three\-valued non\-deterministic matrix \(Nmatrix\) as follows282828The algebraization and the single finite matrix characterization of logical systems is often desirable\. However, the problem of algebraizingLFIs, in the sense of Blok and Pigozzi\[[4](https://arxiv.org/html/2607.09729#bib.bib4)\], and obtain its respective single finite matrix characterization is well\-known\. Although algebraization in this sense can be obtained for a whole class ofLFIs based on a three\-valued finite matrix \(see\[[7](https://arxiv.org/html/2607.09729#bib.bib7), Section 4\.4\]\), in general, manyLFIs cannot be algebraitized in this sense or characterized by a single finite deterministic matrix \(see\[[7](https://arxiv.org/html/2607.09729#bib.bib7), Sec\. 4\.2\]\)—includingmbCand all its extensions discussed in this document\. Nevertheless, they can be characterized by a single finitenon\-deterministicmatrix, as demonstrated through a class of multialgebras calledswap structures\(see\[[7](https://arxiv.org/html/2607.09729#bib.bib7), Ch\. 6\],\[[8](https://arxiv.org/html/2607.09729#bib.bib8)\], and\[[10](https://arxiv.org/html/2607.09729#bib.bib10)\]\)\. In the article in question\[[9](https://arxiv.org/html/2607.09729#bib.bib9)\], therefore, the authors use these previously obtained results to derive the following Nmatrix\. I will not detail such methods here, as I believe assuming the obtained results as valid is sufficient for this exposition\.:

###### Definition 3\.1\(Nmatrix forCbr\)\.

LetℳC​b​r\\mathcal\{M\}\_\{Cbr\}be a three\-valued non\-deterministic matrix ⟨𝒯,𝒟,\{∧^,∨^,→^,¬^,∘^\}⟩\\langle\\mathcal\{T\},\\mathcal\{D\},\\\{\\hat\{\\land\},\\hat\{\\lor\},\\hat\{\\to\},\\hat\{\\neg\},\\hat\{\\circ\}\\\}\\rangleover the signatureΣ∘=\{∧,∨,→,¬,∘\}\\Sigma\_\{\\circ\}=\\\{\\land,\\lor,\\to,\\neg,\\circ\\\}with domain𝒯=\{1,12,0\}\\mathcal\{T\}=\\\{1,\\frac\{1\}\{2\},0\\\}and set of designated values𝒟=\{1,12\}\\mathcal\{D\}=\\\{1,\\frac\{1\}\{2\}\\\}, such that the truth tables associated with each connective are as follows:

ℳC​b​r\\mathcal\{M\}\_\{Cbr\}∧^\\hat\{\\land\}112\\frac\{1\}\{2\}0∨^\\hat\{\\lor\}112\\frac\{1\}\{2\}0→^\\hat\{\\to\}112\\frac\{1\}\{2\}0¬^\\hat\{\\neg\}∘^\\hat\{\\circ\}1𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}\{0\}1𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}1𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}\{0\}1\{0\}1𝒟\\mathcal\{D\}12\\frac\{1\}\{2\}𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}\{0\}12\\frac\{1\}\{2\}𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}12\\frac\{1\}\{2\}𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}\{0\}12\\frac\{1\}\{2\}\{12\}\\\{\\frac\{1\}\{2\}\\\}12\\frac\{1\}\{2\}\{0\}0\{0\}\{0\}\{0\}0𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}\{0\}0𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}𝒟\\mathcal\{D\}0\{1\}0𝒟\\mathcal\{D\}

Table 2:NmatrixℳC​b​r\\mathcal\{M\}\_\{Cbr\}forCbrconnectives\.Given thatΓ⊧ℳC​b​rφ\\Gamma\\models\_\{\\mathcal\{M\}\_\{Cbr\}\}\\varphiif for every valuationvvin the NmatrixℳC​b​r\\mathcal\{M\}\_\{Cbr\}292929Wherev:𝔉​𝔪→𝒯v:\\mathfrak\{Fm\}\\to\\mathcal\{T\}is considered a valuation inℳC​b​r\\mathcal\{M\}\_\{Cbr\}ifv​\(φ​\#​ψ\)∈v​\(φ\)​\#^​v​\(ψ\)v\(\\varphi\\\#\\psi\)\\in v\(\\varphi\)\\;\\hat\{\\\#\}\\;v\(\\psi\)for\#∈\{∨,∧,→\}\\\#\\in\\\{\\lor,\\land,\\to\\\}andv​\(\#​φ\)∈\#^​v​\(φ\)v\(\\\#\\varphi\)\\in\\hat\{\\\#\}v\(\\varphi\)for\#∈\{¬,∘\}\\\#\\in\\\{\\neg,\\circ\\\}\., it holds that for everyψ∈Γ\\psi\\in\\Gamma, ifv​\(ψ\)∈𝒟v\(\\psi\)\\in\\mathcal\{D\}, thenv​\(φ\)∈𝒟v\(\\varphi\)\\in\\mathcal\{D\}, the soundness and completeness theorems forCbrcan be established303030Follows from\[[8](https://arxiv.org/html/2607.09729#bib.bib8), Sec\. 6\.4\]\.\. All these conditions are necessary—yet still not sufficient—forRCbrto satisfy three crucial properties for epistemic dynamics313131Properties \(i\) and \(ii\) are desirable insofar as it is reasonable that an epistemic agent who considers a belief consistent should also consider its negation consistent; moreover, if the agent considers two beliefs logically equivalent, their consistency status should also be equivalent\. Thereplacementproperty, in turn, is a final necessary technical requirement to makeRCbrsuitable for an extensionalAGMoperation, as well as uniform substitutions\.:

###### Properties 3\.2\.

\(i\)∘φ≡∘¬φ\\circ\\varphi\\equiv\\circ\\neg\\varphi;

\(ii\)Ifφ≡ψ\\varphi\\equiv\\psiand¬φ≡¬ψ\\neg\\varphi\\equiv\\neg\\psi, then∘φ≡∘ψ\\circ\\varphi\\equiv\\circ\\psi\.

\(iii\)\(Replacement\) Given formulasφi\\varphi\_\{i\}andψi\\psi\_\{i\}, for1⩽i⩽n1\\leqslant i\\leqslant n, such thatφ1≡ψ1,…,φn≡ψn\\varphi\_\{1\}\\equiv\\psi\_\{1\},\.\.\.,\\varphi\_\{n\}\\equiv\\psi\_\{n\}, thenγ​\(φ1,…,φn\)≡γ​\(ψ1,…,ψn\)\\gamma\(\\varphi\_\{1\},\.\.\.,\\varphi\_\{n\}\)\\equiv\\gamma\(\\psi\_\{1\},\.\.\.,\\psi\_\{n\}\), for any formulaγ​\(a1,…,an\)\\gamma\(a\_\{1\},\.\.\.,a\_\{n\}\)\.

Observing NmatrixℳC​b​r\\mathcal\{M\}\_\{Cbr\}, it is clear thatCbrsatisfies \(i\) and \(ii\), but it is notself\-extensional, i\.e\., does not satisfies \(iii\)\. In\[[6](https://arxiv.org/html/2607.09729#bib.bib6)\], the authors present a class of self\-extensionalLFIs—denotedR𝕃\\mathbb\{L\}, where𝕃\\mathbb\{L\}is someLFI—, the weakest of which isRmbC323232The authors consider several self\-extensionalLFIs and their respectiveBALFImodels, but the results can be generalized\. In\[[9](https://arxiv.org/html/2607.09729#bib.bib9)\], the logicsRCieandRCbr—which are the self\-extensional versions ofCieandCbr, respectively—are considered\. For the purposes of this document, onlyRCbrwill be considered, and thus the following definitions and results will be adapted accordingly\.\. Replacement satisfiability is obtained, briefly, as follows\. First, two newglobalinference rules—that is, rules valid only for theorems—and a new derivation notion are introduced333333These inferential rules and the derivation notion are defined similarly to the necessitation rule in modal logic\. They should be read as: ifφ↔ψ\\varphi\\leftrightarrow\\psiis a theorem, then\#​φ↔\#​ψ\\\#\\varphi\\leftrightarrow\\\#\\psiis a theorem, for\#∈\{¬,∘\}\\\#\\in\\\{\\neg,\\circ\\\}\. A more detailed version of the derivation definition can be found in\[[6](https://arxiv.org/html/2607.09729#bib.bib6), p\.6\]\. However, a shorter version, sufficient for our purposes, is available in\[[9](https://arxiv.org/html/2607.09729#bib.bib9), p\.8\]\.:

φ↔ψ¬φ↔¬ψ\\dfrac\{\\varphi\\leftrightarrow\\psi\}\{\\neg\\varphi\\leftrightarrow\\neg\\psi\}\(R¬\)\(R\_\{\\neg\}\)φ↔ψ∘φ↔∘ψ\\dfrac\{\\varphi\\leftrightarrow\\psi\}\{\\circ\\varphi\\leftrightarrow\\circ\\psi\}\(R∘\)\(R\_\{\\circ\}\)###### Definition 3\.3\.

We say thatφ\\varphiis derivable inR𝕃\\mathbb\{L\}, written⊢R​𝕃φ\\vdash\_\{\\text\{R\}\\mathbb\{L\}\}\\varphi, if there is a derivation inR𝕃\\mathbb\{L\}in the usual sense\. On the other hand,φ\\varphiis derivable inR𝕃\\mathbb\{L\}from a set of premisesΓ\\Gamma, writtenΓ⊢R​𝕃φ\\Gamma\\vdash\_\{\\text\{R\}\\mathbb\{L\}\}\\varphi, is either⊢R​𝕃φ\\vdash\_\{\\text\{R\}\\mathbb\{L\}\}\\varphior there existsφ1,…,φn∈Γ\\varphi\_\{1\},\.\.\.,\\varphi\_\{n\}\\in\\Gamma, such that⊢R​𝕃\(φ1∧…∧φn\)→φ\\vdash\_\{\\text\{R\}\\mathbb\{L\}\}\(\\varphi\_\{1\}\\land\.\.\.\\land\\varphi\_\{n\}\)\\to\\varphi\.

The algebraization ofR𝕃\\mathbb\{L\}in the Lindenbaum\-Tarski sense consists of expansions of Boolean algebras through the addition of operators¬~\\tilde\{\\neg\}and∘~\\tilde\{\\circ\}\. The definition ofBALFI\(Boolean Algebra withLFIOperators\) for the classR𝕃\\mathbb\{L\}, with𝕃=\\mathbb\{L\}=\{mbC,mbCciw,Cbr\}, and the corresponding semantic logical consequence is given as follows343434We denote the algebraic operations by⊓\\sqcap,⊔\\sqcup,−\-, and⇒\\Rightarrowformeet,join,complement, andimplication, respectively\. As is usual in logical algebraization, valuations are expressed via homomorphismsv:ℒΣ∘→𝔅v:\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\}\\to\\mathfrak\{B\}as follows:v​\(\#​φ\)=\#~​v​\(φ\)v\(\\\#\\varphi\)=\\tilde\{\\\#\}\\;v\(\\varphi\)for\#∈\{¬,∘\}\\\#\\in\\\{\\neg,\\circ\\\},v\(∼φ\)=−v​\(φ\)v\(\\sim\\\!\\varphi\)=\-v\(\\varphi\), andv​\(φ​\#​ψ\)=v​\(φ\)​\#~​v​\(ψ\)v\(\\varphi\\\#\\psi\)=v\(\\varphi\)\\;\\tilde\{\\\#\}\\;v\(\\psi\)for\#∈\{∨,∧,→\}\\\#\\in\\\{\\lor,\\land,\\to\\\}and\#~∈\{⊔,⊓,⇒\}\\tilde\{\\\#\}\\in\\\{\\sqcup,\\sqcap,\\Rightarrow\\\}, respectively\.:

###### Definition 3\.4\(BALFI\)\.

A Boolean algebra withLFIoperators \(BALFI\) is an algebra𝔅=⟨A,⊓,⊔,−,⇒,¬~,∘~,0,1⟩\\mathfrak\{B\}=\\langle A,\\sqcap,\\sqcup,\-,\\Rightarrow,\\tilde\{\\neg\},\\tilde\{\\circ\},0,1\\rangle, obtained by expanding a Boolean algebra𝔄=⟨A,⊓,⊔,−,⇒,0,1⟩\\mathfrak\{A\}=\\langle A,\\sqcap,\\sqcup,\-,\\Rightarrow,0,1\\ranglewith the unary operators¬~,∘~\\tilde\{\\neg\},\\tilde\{\\circ\}\. For allx∈Ax\\in A: a BALFI forRmbCis an algebra𝔅\\mathfrak\{B\}such thatx⊔¬x~=1x\\sqcup\\tilde\{\\neg x\}=1andx⊓¬~​x⊓∘~​x=0x\\sqcap\\tilde\{\\neg\}x\\sqcap\\tilde\{\\circ\}x=0; A BALFI forRmbCciwis a BALFI forRmbCsuch that∘~​x=−\(x⊓¬~​x\)\\tilde\{\\circ\}x=\-\(x\\sqcap\\tilde\{\\neg\}x\)\. A BALFI forRCbris a BALFI forRmbCciwsuch that¬~​¬~​x=x\\tilde\{\\neg\}\\tilde\{\\neg\}x=x\. Given𝕃=\\mathbb\{L\}=mbC,mbCciw,Cbr, the class of BALFIs forR𝕃\\mathbb\{L\}is denoted𝔹​\(R​𝕃\)\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)\.

###### Definition 3\.5\(Logical consequence inBALFIs\)\.

Let𝔅∈𝔹​\(R​𝕃\)\\mathfrak\{B\}\\in\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)\. \(i\) We say thatφ\\varphiis valid in𝔅\\mathfrak\{B\}, that is,⊧𝔅φ\\models\_\{\\mathfrak\{B\}\}\\varphi, ifv​\(φ\)=1v\(\\varphi\)=1for every homomorphismv:ℒΣ∘→𝔅v:\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\}\\to\\mathfrak\{B\}; \(ii\) A formulaφ\\varphiis valid in𝔹​\(R​𝕃\)\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\), that is,⊧𝔹​\(R​𝕃\)\\models\_\{\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)\}, if it is valid in every𝔅∈𝔹​\(R​𝕃\)\\mathfrak\{B\}\\in\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\); \(iii\) For everyΓ∪\{φ\}⊆ℒΣ∘\\Gamma\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\}, we say thatφ\\varphiis a consequence ofΓ\\Gammain𝔹​\(R​𝕃\)\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\), that is,Γ⊧𝔹​\(R​𝕃\)φ\\Gamma\\models\_\{\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)\}\\varphi, if eitherφ\\varphiis valid in𝔹​\(R​𝕃\)\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)or there existsφ1,…,φn∈Γ\\varphi\_\{1\},\.\.\.,\\varphi\_\{n\}\\in\\Gamma, such that\(φ1∧…∧φn\)→φ\(\\varphi\_\{1\}\\land\.\.\.\\land\\varphi\_\{n\}\)\\to\\varphiis valid in𝔹​\(R​𝕃\)\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)\.

The soundness and completeness theorem—that is, for everyΓ∪\{φ\}⊆ℒΣ∘\\Gamma\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\},Γ⊢R​𝕃φ\\Gamma\\vdash\_\{R\\mathbb\{L\}\}\\varphiif and only ifΓ⊧𝔹​\(R​𝕃\)φ\\Gamma\\models\_\{\\mathbb\{B\}\(\\text\{R\}\\mathbb\{L\}\)\}\\varphi—is obtained353535See\[[6](https://arxiv.org/html/2607.09729#bib.bib6), p\.8\]and\[[9](https://arxiv.org/html/2607.09729#bib.bib9), p\.9\]\.\. Since in logicCbrthe property[3\.2](https://arxiv.org/html/2607.09729#S3.Thmtheorem2)\(ii\) holds, the inference rule\(R∘\)\(R\_\{\\circ\}\)is superfluous; that is, it can be derived in the self\-extensional logicRCbr\. Therefore,RCbr=Cbr\+\(R¬\)\+\(R\_\{\\neg\}\)\. In\[[9](https://arxiv.org/html/2607.09729#bib.bib9), p\.9\-18\], the authors present an interesting algebraic model𝔅=𝔹​\(RCbr\)\\mathfrak\{B\}=\\mathbb\{B\}\(\\text\{\{RCbr\}\}\)that validates all its axioms while remaining a paraconsistentLFI\. Furthermore—and importantly—𝔅\\mathfrak\{B\}is a countermodel for⊧̸R​C​b​r∘∘φ\\not\\models\_\{RCbr\}\\circ\\\!\\circ\\\!\\varphi,⊧̸R​C​b​r∘φ→∘∘φ\\not\\models\_\{RCbr\}\\circ\\varphi\\to\\circ\\\!\\circ\\\!\\varphi,⊧̸R​C​b​r\(φ∧∘φ\)→∘∘φ\\not\\models\_\{RCbr\}\(\\varphi\\land\\circ\\varphi\)\\to\\circ\\\!\\circ\\\!\\varphi, and⊧̸R​C​b​r\(¬φ∧∘φ\)→∘∘φ\\not\\models\_\{RCbr\}\(\\neg\\varphi\\land\\circ\\varphi\)\\to\\circ\\\!\\circ\\\!\\varphi, which makes theRCbrlogic ideal for belief revision purposes363636Without these countermodel, theAGM∘\\circsystem would be unable to overturn beliefs, since they would automatically be considered strongly accepted\. See\[[11](https://arxiv.org/html/2607.09729#bib.bib11)\]for more specific information aboutAGM∘\\circsystem and its epistemic attitudes\.\.

## 4AGMpa​b​dp\_\{abd\}system preliminaries

Thanks to the special properties of the logicRCbr, theAGMpa​b​dp\_\{abd\}system was developed to allow bothCPLandRCbras underlying logics, without substantial changes to the statements, definitions and proofs373737This is possible mainly due to the following reasons: the guarantee of substitutions between formulas, the satisfiability of classical properties, and the non\-direct use of the paraconsistent connectives¬\\negand∘\\circin its postulates and constructions\. As we have seen, the logicRCbr, as an extension ofmbC, is a standard Tarskian logic \(definition[A\.2](https://arxiv.org/html/2607.09729#A1.Thmtheorem2)\) and supraclassical \(definition[A\.3](https://arxiv.org/html/2607.09729#A1.Thmtheorem3)\)\. Thus, it satisfies both structural substitutions and fundamental rules and properties ofCPL, such asmodus ponens, thededuction theorem,disjunction of premisesandproof by cases\(see properties[A\.4](https://arxiv.org/html/2607.09729#A1.Thmtheorem4)\)\. Moreover, the logicRCbris self\-extensional, i\.e\., it satisfies thereplacementproperty, which naturally allows substitutions between logically equivalent formulas—a particularly important feature for theextensionalitypostulate and its developments, as we shall see\. Furthermore, substitutions at the metalogical level, according to well\-known conventions in set theory and commonly used in manipulating the properties of the Tarskian consequence operatorC​nCn, are evidently natural\.\. This quite special feature makes paraconsistency optional, should it be convenient to overcome problematic scenarios in the classical world, as we shall see below\. With this in mind, for reasons of simplification, the notation𝕃∈\\mathbb\{L\}\\in\{CPL,RCbr\} and the subscripts⊢𝕃\\vdash\_\{\\mathbb\{L\}\},⊬𝕃\\nvdash\_\{\\mathbb\{L\}\},≡𝕃\\equiv\_\{\\mathbb\{L\}\}andC​n𝕃Cn\_\{\\mathbb\{L\}\}, used many times throughout this article, should be considered a metatheoretical indicator, such that each statement, definition, theorem and proof, when pertaining to both logics, is presented only once, with𝕃\\mathbb\{L\}being a parameter that can be instantiated as eitherCPLorRCbr\. In both cases, we have that𝕃=⟨ℒΣ∗,C​n𝕃⟩\\mathbb\{L\}=\\langle\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\},Cn\_\{\\mathbb\{L\}\}\\rangleis a standard and supraclassical Tarskian logic, where, on the one hand, in the case of𝕃=\\mathbb\{L\}=\{CPL\}, we should consider the languageℒΣ∗\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}as generated by the signatureΣ∗=\{∧,∨,→,∼,⊥,⊤\}\\Sigma\_\{\*\}=\\\{\\land,\\lor,\\to,\\mathord\{\\sim\},\\bot,\\top\\\}, where∼\\mathord\{\\sim\}is the primitive classical negation\. On the other hand, when𝕃=\\mathbb\{L\}=\{RCbr\}, we should consider the languageℒΣ∗\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}as generated by the signatureΣ∗=\{∧,∨,→,¬,∘\}\\Sigma\_\{\*\}=\\\{\\land,\\lor,\\to,\\neg,\\circ\\\}—the same used in the exposition ofmbCs in Section[3](https://arxiv.org/html/2607.09729#S3)\. In this case, thefalsumparticle and the strong negation will be defined by⊥ψ=defψ∧¬ψ∧∘ψ\\bot\_\{\\psi\}=\_\{\\text\{def\}\}\\psi\\land\\neg\\psi\\land\\circ\\psiand \(ii\)∼ψ​φ=def\(φ→⊥ψ\)\\mathord\{\\sim\}\_\{\\psi\}\\varphi=\_\{\\text\{def\}\}\(\\varphi\\to\\bot\_\{\\psi\}\)\.

At this point, I present some important definitions for the following sections:

###### Definition 4\.1\.

Let𝕃∈\{CPL,RCbr\}\\mathbb\{L\}\\in\\\{\\textbf\{CPL\},\\textbf\{RCbr\}\\\}\. The setT​h​\(𝕃\)Th\(\\mathbb\{L\}\)of all theories of the logic𝕃\\mathbb\{L\}is given by:

T​h​\(𝕃\)=\{Θ⊆ℒΣ∗:Θ=C​n𝕃​\(Θ\)\}\.Th\(\\mathbb\{L\}\)=\\\{\\Theta\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\\ :\\ \\Theta=Cn\_\{\\mathbb\{L\}\}\(\\Theta\)\\\}\.

###### Definition 4\.2\(Belief set\)\.

A setΘ⊆ℒΣ∗\\Theta\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}of sentences is a belief set if, and only if,Θ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\)\.

SinceΘ\\Thetais a belief set, the \(classical\) triggers presented in Section[2](https://arxiv.org/html/2607.09729#S2)can be defined as follows:

###### Definition 4\.3\(Classical abductive process triggers\)\.

Let𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\},Θ∪\{φ\}⊆ℒΣ∗\\Theta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}be a set of formulas andΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\)\.φ\\varphiis a trigger of the abductive process with respect toΘ\\Thetaif:

\(i\)Abductive novelty:φ∉Θ\\varphi\\notin\\Thetaand∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta;

\(ii\)Abductive anomaly:φ∉Θ\\varphi\\notin\\Thetaand∼​φ∈Θ\\mathord\{\\sim\}\\varphi\\in\\Theta\.

Classically, the abductive process, with its triggers and its operationsΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}andΘφ○⊗i\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\otimes$\\crcr\}\}\}\_\{i\}\}\_\{\\varphi\}of abductive expansion and \(internal\) revisionAGMpa​b​dp\_\{abd\}, respectively, as well as other components of the taxonomy, can be better detailed visually according to Figure[1](https://arxiv.org/html/2607.09729#S4.F1)\. Let us recall, however, that only the abductive expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}is the focus of this paper:

![Refer to caption](https://arxiv.org/html/2607.09729v1/figures/processo_abdutivo_AGMpabd_classico_ingles.png)Figure 1:The classical abductive process and the AGMpa​b​dp\_\{abd\}operations\.In order to make the exposition and the formal proofs ahead more simplified, as we shall see, the relation between each of these triggers and their respective operations is established by means ofappropriate pairs\. Thus, in the classical case, we have the following definitions\.

###### Definition 4\.4\(Appropriate pair for classicalAGMpa​b​dp\_\{abd\}abductive expansion\)\.

Let𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\},Θ∪\{φ\}⊆ℒΣ∗\\Theta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}be a set of formulas andΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\)\. We say that\(Θ,φ\)\(\\Theta,\\varphi\)—and its substitutions—is an appropriate pair for classicalAGMpa​b​dp\_\{abd\}abductive expansion—notation\(Θ,φ\)𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}—ifφ\\varphiis an abductive novelty trigger \(definition[4\.3](https://arxiv.org/html/2607.09729#S4.Thmtheorem3)\(i\)\), i\.e\., ifφ∉Θ\\varphi\\notin\\Thetaand∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\.

Let us now consider the paraconsistent context\. When𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\}, the richness of the language allows us to consider, combinatorially, forφ∉Θ\\varphi\\notin\\Theta, four triggers, according to the following definition\.

###### Definition 4\.5\(Paraconsistent abductive process triggers\)\.

Let𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\},Θ∪\{φ\}⊆ℒΣ∗\\Theta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}be a set of formulas andΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\)\.φ\\varphiis a trigger of the abductive process with respect toΘ\\Thetaif:

\(i\)Strong abductive novelty:φ∉Θ\\varphi\\notin\\Theta,∘φ∉Θ\\circ\\varphi\\notin\\Thetaand¬φ∉Θ\\neg\\varphi\\notin\\Theta;

\(ii\)Weak abductive novelty:φ∉Θ\\varphi\\notin\\Theta,∘φ∈Θ\\circ\\varphi\\in\\Thetaand¬φ∉Θ\\neg\\varphi\\notin\\Theta;

\(iii\)Weak abductive anomaly:φ∉Θ\\varphi\\notin\\Theta,∘φ∉Θ\\circ\\varphi\\notin\\Thetaand¬φ∈Θ\\neg\\varphi\\in\\Theta;

\(iv\)Strong abductive anomaly:φ∉Θ\\varphi\\notin\\Theta,∘φ∈Θ\\circ\\varphi\\in\\Thetaand¬φ∈Θ\\neg\\varphi\\in\\Theta\.

Thus, analogously to the classical perspective, Figure[2](https://arxiv.org/html/2607.09729#S4.F2)details the abductive process in light of the new triggers and their respectiveAGMpa​b​dp\_\{abd\}operations\.

![Refer to caption](https://arxiv.org/html/2607.09729v1/figures/processo_abdutivo_AGMpabd_paraconsistente_ingles.png)Figure 2:The paraconsistent abductive process and the AGMpa​b​dp\_\{abd\}operations\.Unlike the classical case, therefore, the increased expressive power of the language ofRCbr—and of anyLFI, in fact—allows us to consider four more sophisticated and interesting interpretations of the initial surprise when faced with a surprising fact\. The definitions ofappropriate pairsfor the respective operations, in the paraconsistent case, introduce new elements and require some care\. Note that the first three triggers—strong and weak abductive novelties and weak abductive anomaly—can initiate an abductive expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. So, we may define:

###### Definition 4\.6\(Appropriate pair for paraconsistentAGMpa​b​dp\_\{abd\}abductive expansion\)\.

Let𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\},Θ∪\{φ\}⊆ℒΣ∗\\Theta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}be a set of formulas andΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\)\. We say that\(Θ,φ\)\(\\Theta,\\varphi\)—and its substitutions—is an appropriate pair for paraconsistent abductive expansion—notation\(Θ,φ\)∘𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{\\circ\}—if \(i\)φ\\varphiis a strong abductive novelty trigger forΘ\\Thetaor \(ii\)φ\\varphiis a weak abductive novelty trigger forΘ\\Thetaor \(iii\)φ\\varphiis a weak abductive anomaly trigger forΘ\\Theta\(definition[4\.5](https://arxiv.org/html/2607.09729#S4.Thmtheorem5)\)\.

However, since the logicRCbris an extension ofmbCciw, as we have seen, and∼φ≡mbCciw∘φ∧¬φ\\mathord\{\\sim\}\\varphi\\equiv\_\{\\textbf\{mbCciw\}\}\\circ\\varphi\\land\\neg\\varphi\(footnote[27](https://arxiv.org/html/2607.09729#footnote27)\), as well asΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), then the following observation conveniently shows that, when using strong negation, the strong abductive novelty, weak abductive novelty and weak abductive anomaly triggers in theAGMpa​b​dp\_\{abd\}system “collapse” into the classical case383838The differentiated treatment for each of these cases, within the paraconsistent abductive expansion operation, is precisely the proposal of theAGM∘a​b​d\\circ\_\{abd\}system to be presented in the second paper\.\.

This means that\(Θ,φ\)𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}denotes an appropriate pair for abductive expansion in the classical case—that is, when𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\}is the underlying logic—and\(Θ,φ\)∘𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{\\circ\}denotes an appropriate pair for paraconsistent abductive expansion—namely, when𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\}is the underlying logic\. However, since both are defined by \(i\)φ∉Θ\\varphi\\notin\\Thetaand \(ii\)∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta, they can be used interchangeably in statements and formal proofs\. By convention and for notational simplification, therefore, the following more general definition will be adopted\.

###### Definition 4\.8\(Appropriate pair forAGMpa​b​dp\_\{abd\}abductive expansion\)\.

Let𝕃∈\{CPL,RCbr\}\\mathbb\{L\}\\in\\\{\\textbf\{CPL\},\\textbf\{RCbr\}\\\},Θ∪\{φ\}⊆ℒΣ∗\\Theta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}be a set of formulas andΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\)\.\(Θ,φ\)\(\\Theta,\\varphi\)is an appropriate pair forAGMpa​b​dp\_\{abd\}abductive expansion \(classical or paraconsistent\)—notation\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}—if \(i\)φ∉Θ\\varphi\\notin\\Thetaand \(ii\)∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\.

## 5AGMpa​b​dp\_\{abd\}abductive expansion \- postulates

Before presenting the postulates of the operation, as a first step, I propose to formalize the following taxonomic component, the adequate explanatory inference, described in Section[2](https://arxiv.org/html/2607.09729#S2), accordingly to the rationality criterion of theexplanatory adequacy, with its respective requirements, as follows:

###### Definition 5\.1\.

Let𝕃∈\{CPL,RCbr\}\\mathbb\{L\}\\in\\\{\\textbf\{CPL\},\\textbf\{RCbr\}\\\},Θ∪\{α\}∪\{φ\}⊆ℒΣ∗\\Theta\\cup\\\{\\alpha\\\}\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}andφ∉Θ\\varphi\\notin\\Theta\. The setH​\(Θ,φ\)H\(\\Theta,\\varphi\), given by:

H\(Θ,φ\)=\{α∈ℒΣ∗:\(i\)Θ∪\{α\}⊢𝕃φ,\(ii\)Θ∪\{α\}⊬𝕃⊥and\(iii\)\{α\}⊬𝕃φ\}H\(\\Theta,\\varphi\)=\\\{\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\\ :\(i\)\\ \\Theta\\cup\\\{\\alpha\\\}\\vdash\_\{\\mathbb\{L\}\}\\varphi,\\ \(ii\)\\ \\Theta\\cup\\\{\\alpha\\\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\\ \\mbox\{ and \}\\ \(iii\)\\ \\\{\\alpha\\\}\\nvdash\_\{\\mathbb\{L\}\}\\varphi\\\}is the set of abductive hypotheses for\(Θ,φ\)\(\\Theta,\\varphi\)\.

The following \(desirable\) results, in line with the rationality criterion of thesurprising fact, can be easily demonstrated\.

###### Lemma 5\.2\(Partial self\-explanation\)\.

If\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case \(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\), then there always exists someα∈ℒΣ∗\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}that satisfies criteria \(i\)\-\(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\. Hence,H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\.

Thanks to the logicRCbr, the following result can be obtained, whether in the classical or the paraconsistent case:

###### Lemma 5\.4\.

Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and\(Θ,ψ\)p𝔸​ℙ​𝔼\(\\Theta,\\psi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\. LetH​\(Θ,φ\)H\(\\Theta,\\varphi\)andH​\(Θ,ψ\)H\(\\Theta,\\psi\)be the sets of abductive hypotheses forφ\\varphiandψ\\psi, respectively\. It is the case that if⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenH​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\)\.

Having presented the preliminary formal notions of the system, the postulates that characterize theAGMpa​b​dp\_\{abd\}abductive expansion operation are given below\.

###### Definition 5\.5\(Postulates forAGMpa​b​dp\_\{abd\}abductive expansion\)\.

Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\) andH​\(Θ,φ\)H\(\\Theta,\\varphi\)be the set of abductive hypotheses for\(Θ,φ\)\(\\Theta,\\varphi\)\(definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\)\. AnAGMpa​b​dp\_\{abd\}abductive expansion ofΘ\\Thetabyφ\\varphi—denotedΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}—is a function○⊕:Th\(𝕃\)×ℒΣ∗→Th\(𝕃\)\\;\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}:Th\(\\mathbb\{L\}\)\\times\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\\to Th\(\\mathbb\{L\}\)from pairs of belief sets and sentences of the language to belief sets that satisfies the following postulates:

\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}is a belief set \(closure\)

\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\)Θ⊂Θφ○⊕\\Theta\\subset\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\(inclusion\)

\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)Θφ○⊕∩H​\(Θ,φ\)≠∅\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\(success\)

\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)Θφ○⊕⊬𝕃⊥\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\(non\-triviality\)

\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)If⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenΘφ○⊕=Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\(extensionality\)

\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)Θφ○⊕⊆C​n𝕃​\(Θφ∨ψ○⊕∪\{φ\}\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\varphi\\\}\)\(supplementary 1\)

\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)If∼​φ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, thenΘφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\(supplementary 2\)

The first feature to be noted is the requirement that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}be an appropriate pair for abductive expansion—that is,φ∉Θ\\varphi\\notin\\Thetaand∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Thetaand their substitutions—applied to the definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of the operation itself, as a prerequisite to the postulates\. This is very relevant, because, in accordance with the rationality criteria and with the proposed taxonomy, the function○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}underlying the operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}cannot even be invoked in contexts where it is notably the case thatφ∈Θ\\varphi\\in\\Thetaor∼​φ∈Θ\\mathord\{\\sim\}\\varphi\\in\\Theta\. For, in that case, we have an ill\-defined function and, therefore, inappropriate for any purpose\. Postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)is well known and requires no further consideration\. Postulate\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\)tells us thatΘ\\Thetais apropersubset ofΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. In other words, the operation does not considerΘφ○⊕=Θ\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Thetaas a legitimate expansion, which is fully justified by the rationality criterion of thesurprising fact\. Postulate\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)of success, in turn, guarantees that there is at least one abductive hypothesisα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\)in the expanded belief set\. Notably, this postulate authorizes multiple possible expansions, according to the number of hypotheses present in the setH​\(Θ,φ\)H\(\\Theta,\\varphi\)393939It is important to note the many distinctions, up to this point, in relation to the postulates proposed by Pagnucco\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.103\-105\]\. First, the author allowsφ∈Θ\\varphi\\in\\Thetaand, therefore, thatΘφ⊕=Θ\\Theta^\{\\oplus\}\_\{\\varphi\}=\\Theta\. As discussed earlier, this is a situation that I believe is not desirable, whether in the context of abduction or in the context of explanation\. According to observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6)below, as in Pagnucco’s system, it is the case thatφ∈Θφ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. However, in theAGMpa​b​dp\_\{abd\}abductive expansion operation,φ\\varphi, the phenomenon to be explained, is found exclusively in the “ring”Θφ○⊕∖Θ\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\setminus\\Theta, effectively characterizing an expansion of the initial epistemic state\. Moreover, unlike definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)ofH​\(Θ,φ\)H\(\\Theta,\\varphi\), Pagnucco’s notion of explanation\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.79\]does not block\{α\}⊢φ\\\{\\alpha\\\}\\vdash\\varphi, something which, as discussed in Section[2](https://arxiv.org/html/2607.09729#S2), I consider inappropriate\. Finally, Pagnucco’slimited successpostulate, despite being quite elegant, depends on a very specific condition being satisfied: the adopted language must befinite—see definition of abductive expansion\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.102\]and theorem 5\.2\.1\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.105\]—a requirement that will not be adopted here\.\. With these three postulates and, evidently, considering\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}, as the definition requires, we obtain the following observation:

In accordance, once again, with the rationality criterion of thesurprising fact—and unlike Pagnucco’s system—afailurepostulate is not necessary, as the following results indicate\. This is, therefore, an always successful operation, given\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\.

###### Lemma 5\.7\.

Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}andH​\(Θ,φ\)H\(\\Theta,\\varphi\)be the set of abductive hypotheses for\(Θ,φ\)\(\\Theta,\\varphi\)\. Ifα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), then it is the case thatα∨φ∈H​\(Θ,φ\)\\alpha\\lor\\varphi\\in H\(\\Theta,\\varphi\)\.

###### Lemma 5\.8\.

IfΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5), then, forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θφ○⊕\\alpha\\lor\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\.

###### Corollary 5\.9\.

On the one hand, ifΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5), then, by lemma[5\.8](https://arxiv.org/html/2607.09729#S5.Thmtheorem8), forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θφ○⊕\\alpha\\lor\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. On the other hand, by lemma[5\.7](https://arxiv.org/html/2607.09729#S5.Thmtheorem7), if\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case—a requirement of definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of the operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}—then it is also the case thatα∨φ∈H​\(Θ,φ\)\\alpha\\lor\\varphi\\in H\(\\Theta,\\varphi\)\. Thus, it is always the case thatα∨φ∈Θφ○⊕∩H​\(Θ,φ\)\\alpha\\lor\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap H\(\\Theta,\\varphi\)\. Hence, these conditions are sufficient for a failure postulate not to be necessary\.

Postulate\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)evidently expresses a distinct behavior of the operation depending on the logic underlying theAGMpa​b​dp\_\{abd\}system\. If𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\}, theAGMpa​b​dp\_\{abd\}abductive expansion operation prevents expansion to contradiction\. On the other hand, in the case where𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\}, the postulate blocks expansion to triviality, while allowing contradictions—hence its paraconsistent feature\. Moreover, since\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is an initial requirement,Θ≠Θ⊥\\Theta\\neq\\Theta\_\{\\bot\}, which satisfies the rationality criterion ofnon\-triviality, i\.e\., there is no room for triviality from the beginning to the end of the abductive expansion process\. Postulate\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)guarantees thePrinciple of Irrelevance of Syntax\. Note that this postulate is a direct consequence of lemma[5\.4](https://arxiv.org/html/2607.09729#S5.Thmtheorem4)—that is, if⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenH​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\)\. Classically, this postulate is obvious404040It is quite interesting to note that, in his system, Pagnucco presents thestrong extensionalitypostulate—ifΘ⊢φ↔ψ\\Theta\\vdash\\varphi\\leftrightarrow\\psi, thenΘφ⊕=Θψ⊕\\Theta^\{\\oplus\}\_\{\\varphi\}=\\Theta^\{\\oplus\}\_\{\\psi\}\. In theAGMpa​b​dp\_\{abd\}system, however, the strong extensionality postulate cannot be considered, whether in the classical or the paraconsistent case, due to observation[B\.1](https://arxiv.org/html/2607.09729#A2.Thmtheorem1)\. In its brief proof \(found in Appendix[B](https://arxiv.org/html/2607.09729#A2)\), it becomes evident that ifΘ⊢𝕃φ↔ψ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, but⊬𝕃φ↔ψ\\nvdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenψ∈H​\(Θ,φ\)\\psi\\in H\(\\Theta,\\varphi\)andφ∈H​\(Θ,ψ\)\\varphi\\in H\(\\Theta,\\psi\), butψ∉H​\(Θ,ψ\)\\psi\\notin H\(\\Theta,\\psi\)andφ∉H​\(Θ,φ\)\\varphi\\notin H\(\\Theta,\\varphi\)\. This means that, if we opted for the strong extensionality postulate instead of\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\), the consequentΘφ○⊕=Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}would make the postulate ill\-defined\. Note that ifΘ⊢𝕃φ↔ψ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, but⊬𝕃φ↔ψ\\nvdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, in anAGMpa​b​dp\_\{abd\}abductive expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}, it may be the case, by the success postulate\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\), thatψ∈Θφ○⊕∩H​\(Θ,φ\)\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap H\(\\Theta,\\varphi\)\. However, certainlyψ∉Θψ○⊕∩H​\(Θ,ψ\)\\psi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\cap H\(\\Theta,\\psi\)\. This situation does not occur if the requirement is the weak extensionality postulate\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)\(note that if⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, then, by lemma[5\.4](https://arxiv.org/html/2607.09729#S5.Thmtheorem4),H​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\), since\{φ,ψ\}∉H​\(Θ,φ\)\\\{\\varphi,\\psi\\\}\\notin H\(\\Theta,\\varphi\)and, similarly,\{φ,ψ\}∉H​\(Θ,ψ\)\\\{\\varphi,\\psi\\\}\\notin H\(\\Theta,\\psi\)\)\. To a large extent, this is a feature imposed by criterion \(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)of the set of abductive hypotheses, absent in Pagnucco’s system\., but in the paraconsistent context, this postulate is only possible because the logicRCbrsatisfies thereplacementproperty\. Finally, the supplementary postulates\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)and\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\), as well as their properties414141See properties[B\.2](https://arxiv.org/html/2607.09729#A2.Thmtheorem2)and[B\.3](https://arxiv.org/html/2607.09729#A2.Thmtheorem3)in Appendix[B](https://arxiv.org/html/2607.09729#A2)\. The proofs are, with small adjustments, the same as those obtained by Pagnucco\., are, strictly and curiously, the same as in Pagnucco’s system424242It should be borne in mind, however, that, unlike that system, given definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of the postulates, the iterated abductive expansion operationΘφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}requires, by substitution,\(Θ,φ∨ψ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\\lor\\psi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\. Therefore, when the operationΘφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}is invoked, we haveφ∨ψ∉Θ\\varphi\\lor\\psi\\notin\\Thetaand∼​\(φ∨ψ\)∉Θ\\mathord\{\\sim\}\(\\varphi\\lor\\psi\)\\notin\\Theta\. Considering that, for𝕃∈\{CPL,RCbr\}\\mathbb\{L\}\\in\\\{\\textbf\{CPL\},\\textbf\{RCbr\}\\\}, De Morgan’s Laws hold for classical \(strong\) negation∼\\mathord\{\\sim\}and thatΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), then we evidently have∼​φ∧∼​ψ∉Θ\\mathord\{\\sim\}\\varphi\\land\\mathord\{\\sim\}\\psi\\notin\\Theta\.\.

### 5\.1Distinctions between classical and paraconsistentAGMpa​b​dp\_\{abd\}abductive expansions

At this point, it is necessary to establish some important behavioral distinctions in the abductive expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}, for the cases𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\}and𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\}, without, however, compromising the formal proofs\. Similarly to the interesting examples brought in the article\[[5](https://arxiv.org/html/2607.09729#bib.bib5), p\.325\-326\]\(and also revisited in\[[24](https://arxiv.org/html/2607.09729#bib.bib24), p\.25\-27\]\)434343It is important to highlight some important distinctions between these works and the system developed in this paper\. First, the authors, in both works, focus on paraconsistent abductive tableaux, not onAGM\-style abductive operations, with their respective postulates and constructions\. Similarly, the set of initial theories or previous beliefs in those works is not closed under logical consequences, as is the case of theAGMpa​b​dp\_\{abd\}system\. Moreover, there are subtle but important differences between the notion of explanation adopted in those works \(see, for example,\[[24](https://arxiv.org/html/2607.09729#bib.bib24), p\.24\-25\]\) and the notion adopted in this paper \(Subsection[2](https://arxiv.org/html/2607.09729#S2)\), which I will not go into here\. No less important, the paraconsistent logics used for the construction of their abductive tableaux arembC, in the case of the first article, andLETk, in the case of the second\. The examples presented, however, with the appropriate modifications, serve perfectly for the purposes of this exposition\., the abductive expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}behaves, in some cases, identically in the classical and paraconsistent cases—example \(i\) below—but, in other cases, quite differently—examples \(ii\)\-\(iv\) below\. After all, a richer language, in this case the language ofRCbr, also enables the representation of richer epistemic changes and explanations\.

Let us first consider \(i\) cases where the result of the operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}is the same for the classical and paraconsistent cases\. Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and\{ψ→φ,δ→φ\}⊆Θ\\\{\\psi\\to\\varphi,\\delta\\to\\varphi\\\}\\subseteq\\Theta\. Naturally,ψ,δ∈H​\(Θ,φ\)\\psi,\\delta\\in H\(\\Theta,\\varphi\)—by conditions \(i\)\-\(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\. By postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)and observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), we have that\{ψ→φ,δ→φ,ψ,φ\}⊆Θφ○⊕\\\{\\psi\\to\\varphi,\\delta\\to\\varphi,\\psi,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\},\{ψ→φ,δ→φ,δ,φ\}⊆Θφ○⊕\\\{\\psi\\to\\varphi,\\delta\\to\\varphi,\\delta,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}and\{ψ→φ,δ→φ,ψ,δ,φ\}⊆Θφ○⊕\\\{\\psi\\to\\varphi,\\delta\\to\\varphi,\\psi,\\delta,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\(as well as their logical consequences\) are all possible abductive expansions, both in the classical and in the paraconsistent cases\. The same occurs for\{ψ→δ,δ→φ\}⊆Θ\\\{\\psi\\to\\delta,\\delta\\to\\varphi\\\}\\subseteq\\Theta\. The indistinction between both expansions is clearly linked to the fact that there are no paraconsistent operators¬\\negand∘\\circinvolved in the operations\.

On the other hand, let us consider three cases where the adoption of paraconsistency alters the behavior of the operation\. Consider \(ii\)\(Θ,φ\)∘𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{\\circ\}\(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\) and\{ψ→φ,¬ψ\}⊆Θ\\\{\\psi\\to\\varphi,\\neg\\psi\\\}\\subseteq\\Theta\. Since, in this case,𝕃=\{RCbr\}\\mathbb\{L\}=\\\{\\textbf\{RCbr\}\\\}, we have thatψ∈H​\(Θ,φ\)\\psi\\in H\(\\Theta,\\varphi\)—that is, unlike the classical case,ψ\\psibecomes a possible abductive hypothesis for explainingφ\\varphi, according to conditions \(i\)\-\(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)—and, notably, by postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)and observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), we have that\{ψ→φ,¬ψ,ψ,φ\}⊆Θφ○⊕\\\{\\psi\\to\\varphi,\\neg\\psi,\\psi,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}becomes a possible paraconsistent abductive expansion, insofar asΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}comes to accept contradictory hypotheses444444This example is called by the authors, in\[[5](https://arxiv.org/html/2607.09729#bib.bib5), p\.325\]and\[[24](https://arxiv.org/html/2607.09729#bib.bib24), p\.26\], as “Impossible explanations explained\.” Indeed, givenΘ\\Theta, we have a new explanationψ\\psifor the phenomenonφ\\varphithat would be impossible in the classical case, since, if𝕃=\{CPL\}\\mathbb\{L\}=\\\{\\textbf\{CPL\}\\\}, thenΘφ○⊕=Θ⊥\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Theta\_\{\\bot\}\.\. In the classical case, this expansion would evidently be blocked by postulate\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)\. Note, moreover, that by postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\),Θφ○⊕∈T​h​\(𝕃\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\in Th\(\\mathbb\{L\}\), so, sinceψ∧¬ψ⊢mbCciw¬∘ψ\\psi\\land\\neg\\psi\\vdash\_\{\\textbf\{mbCciw\}\}\\neg\\circ\\psi454545In fact, this derivation is already valid inmbC\., andRCbris an extension ofmbCciw, we have that¬∘ψ∈Θφ○⊕\\neg\\circ\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. In other words,ψ\\psibecomes an abductive hypothesis forφ\\varphi, at the cost of the agent inevitably assuming its inconsistency\.

Now let us consider the following case \(iii\) from the classical point of view, initially\. Let\(Θ,φ\)𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\(definition[4\.4](https://arxiv.org/html/2607.09729#S4.Thmtheorem4)\) and\{φ→ψ,φ→∼​ψ\}⊆Θ\\\{\\varphi\\to\\psi,\\varphi\\to\\mathord\{\\sim\}\\psi\\\}\\subseteq\\Theta\. In this case, sinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), because\(Θ,φ\)\(\\Theta,\\varphi\)is an appropriate pair for abductive expansion and, byreductio,\{φ→ψ,φ→∼ψ\}⊢CPL∼φ\\\{\\varphi\\to\\psi,\\varphi\\to\\mathord\{\\sim\}\\psi\\\}\\vdash\_\{\\textbf\{CPL\}\}\\mathord\{\\sim\}\\varphi, then the agent would be forced to \(strongly\) negateφ\\varphi, i\.e\.,∼​φ∈Θ\\mathord\{\\sim\}\\varphi\\in\\Theta\. But this \(strongly\) contradicts the initial assumption that\(Θ,φ\)𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\. Hence, the belief setΘ\\Thetawould be rejected from the start by the operation\. Even if the expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}were authorized, since∼​φ∈Θ\\mathord\{\\sim\}\\varphi\\in\\Theta, given postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\), and, by observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6),φ∈Θφ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}, thenΘφ○⊕⊢⊥\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\vdash\\bot, which violates postulate\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)\. On the other hand, let us reformulate the example for the paraconsistent case\. Let\(Θ,φ\)∘𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{\\circ\}\(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\) and\{φ→ψ,φ→¬ψ\}⊆Θ\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi\\\}\\subseteq\\Theta\. Since\{φ→ψ,φ→¬ψ\}⊬RCbr¬φ\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi\\\}\\nvdash\_\{\\textbf\{RCbr\}\}\\neg\\varphi\(and also\{φ→ψ,φ→¬ψ\}⊬RCbr∼​φ\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi\\\}\\nvdash\_\{\\textbf\{RCbr\}\}\\mathord\{\\sim\}\\varphi\), we have an initial belief set that is not blocked initially by the operation\. In fact, given postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)and observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6),\{φ→ψ,φ→¬ψ,φ\}⊆Θφ○⊕\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}becomes a perfectly admissible operation464646Recalling that, according to lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2), if\(Θ,φ\)∘𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{\\circ\}, thenH​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\. In this case, given conditions \(i\)\-\(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1),\(φ→ψ\)→φ∈H​\(Θ,φ\)\(\\varphi\\to\\psi\)\\to\\varphi\\in H\(\\Theta,\\varphi\), for example, may be an abductive hypothesis for explainingφ\\varphi\.\. In other words, paraconsistency and the richness of the language ofRCbrallow us to work with initial belief sets that, in the classical case, would be immediately rejected by the operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. It is important to note, however, that if the agent initially judgesψ\\psito be consistent, i\.e\.,\{φ→ψ,φ→¬ψ,∘ψ\}⊆Θ\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi,\\circ\\psi\\\}\\subseteq\\Theta, then the abductive expansion operation fails474747The initial belief setΘ\\Theta, however, is not rejected from the start for the operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}as in the classical case, even though\{φ→ψ,φ→¬ψ,∘ψ\}⊢RCbr¬φ\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi,\\circ\\psi\\\}\\vdash\_\{\\textbf\{RCbr\}\}\\neg\\varphi, according toreductiorules valid inmbC\. In this case, sinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), we obtain\{φ→ψ,φ→¬ψ,∘ψ,¬φ\}⊆Θ\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi,\\circ\\psi,\\neg\\varphi\\\}\\subseteq\\Theta\. This is a case ofweak abductive anomaly\(definition[4\.5](https://arxiv.org/html/2607.09729#S4.Thmtheorem5)\), which may be subject to the expansion operationΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}normally\.: by observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6),φ∈Θφ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. By postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)and\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\), we obtain\{φ→ψ,φ→¬ψ,∘ψ,φ\}⊆Θφ○⊕\\\{\\varphi\\to\\psi,\\varphi\\to\\neg\\psi,\\circ\\psi,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}which, naturally, by postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)again, results inΘφ○⊕=Θ⊥\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Theta\_\{\\bot\}, which violates postulate\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)\. The same occurs in the case where∘¬ψ∈Θ\\circ\\neg\\psi\\in\\Theta, since, recall, inRCbr,∘ψ≡∘¬ψ\\circ\\psi\\equiv\\circ\\neg\\psi\(property[3\.2](https://arxiv.org/html/2607.09729#S3.Thmtheorem2), p\.[3\.2](https://arxiv.org/html/2607.09729#S3.Thmtheorem2)\)\.

Finally, consider example \(iv\)\(Θ,φ\)∘𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{\\circ\}\(definition[4\.6](https://arxiv.org/html/2607.09729#S4.Thmtheorem6)\) and\{φ∨ψ,¬ψ\}⊆Θ\\\{\\varphi\\lor\\psi,\\neg\\psi\\\}\\subseteq\\Theta\. Note that, in the classical case—that is,\(Θ,φ\)𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}and\{φ∨ψ,∼​ψ\}⊆Θ\\\{\\varphi\\lor\\psi,\\mathord\{\\sim\}\\psi\\\}\\subseteq\\Theta—there is a situation similar to the previous example, namely, sinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), thenφ∈Θ\\varphi\\in\\Theta—becauseφ∨ψ,∼ψ⊢CPLφ\\varphi\\lor\\psi,\\mathord\{\\sim\}\\psi\\vdash\_\{\\textbf\{CPL\}\}\\varphi—which violates the conditions of\(Θ,φ\)𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\. In other words, the pair\(Θ,φ\)\(\\Theta,\\varphi\)is not appropriate for classicalAGMpa​b​dp\_\{abd\}abductive expansion and is thus rejected beforehand\. In the paraconsistent case, however, sinceΘ,∘ψ⊢𝕃φ\\Theta,\\circ\\psi\\vdash\_\{\\mathbb\{L\}\}\\varphi—becauseφ∨ψ,¬ψ,∘ψ⊢RCbrφ\\varphi\\lor\\psi,\\neg\\psi,\\circ\\psi\\vdash\_\{\\textbf\{RCbr\}\}\\varphi484848It is also the case thatφ∨ψ,∘ψ⊢mbC¬ψ→φ\\varphi\\lor\\psi,\\circ\\psi\\vdash\_\{\\textbf\{mbC\}\}\\neg\\psi\\to\\varphi\. By theDeduction Theorem, we obtainφ∨ψ,∘ψ,¬ψ⊢mbCφ\\varphi\\lor\\psi,\\circ\\psi,\\neg\\psi\\vdash\_\{\\textbf\{mbC\}\}\\varphi\. Evidently, therefore, this is also the case inmbCciwandRCbr\.—,Θ,∘ψ⊬𝕃⊥\\Theta,\\circ\\psi\\nvdash\_\{\\mathbb\{L\}\}\\botand∘ψ⊬𝕃φ\\circ\\psi\\nvdash\_\{\\mathbb\{L\}\}\\varphi, according to conditions \(i\)\-\(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1), we have that∘ψ∈H\(Θ,φ\)\\circ\\psi\\in H\(\\Theta,\\varphi\)—that is,∘ψ\\circ\\psibecomes an abductive hypothesis for explainingφ\\varphi—as well as∘¬ψ\\circ\\neg\\psi, since, again, inRCbr,∘ψ≡∘¬ψ\\circ\\psi\\equiv\\circ\\neg\\psi—a novelty evidently inconceivable in the classical case\. Thus, by postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)and observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), we have that\{φ∨ψ,¬ψ,∘ψ,φ\}⊆Θφ○⊕\\\{\\varphi\\lor\\psi,\\neg\\psi,\\circ\\psi,\\varphi\\\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}becomes an admissible abductive expansion494949This is a formal consequence of the paraconsistentAGMpa​b​dp\_\{abd\}abductive expansion operation, but its epistemic interpretation is naturally controversial\. A sentence marked with the consistency operator∘\\circby the agent should more adequately represent a more consolidated theory, not a hypothesis of a conjectural nature\. At least with respect to theAGMpa​b​dp\_\{abd\}system, however, this feature is acceptable, insofar as it fulfills the function of establishing formal differences between the classical and paraconsistent cases\.\.

## 6AGMpa​b​dp\_\{abd\}abductive expansion \- construction

Analogously to the classicalAGMsystem and similarly to Pagnucco’s system \(see\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.108\]\), let us consider the following definitions\.

###### Definition 6\.1\.

Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\) andH​\(Θ,φ\)H\(\\Theta,\\varphi\)be the set of abductive hypotheses for\(Θ,φ\)\(\\Theta,\\varphi\)\(definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\)\. A setΘ′\\Theta^\{\\prime\}is a maximal non\-trivialAGMpa​b​dp\_\{abd\}superset ofΘ\\Thetawith respect toφ\\varphiif, and only if:

\(i\)Θ⊂Θ′\\Theta\\subset\\Theta^\{\\prime\}

\(ii\)Θ′∩H​\(Θ,φ\)≠∅\\Theta^\{\\prime\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset

\(iii\)⊥∉Cn𝕃\(Θ′\)\\bot\\notin Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)

\(iv\) There is noΘ′′⊃Θ′\\Theta^\{\\prime\\prime\}\\supset\\Theta^\{\\prime\}that satisfies \(i\), \(ii\) and \(iii\)\.

###### Definition 6\.2\(Surplus set\)\.

The surplus set ofΘ\\Thetawith respect toφ\\varphi, denotedΘ⊤φ\\Theta\\top\\varphi, is the set of all maximal non\-trivialAGMpa​b​dp\_\{abd\}supersets ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\) such that:

Θ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiif, and only if,Θ′\\Theta^\{\\prime\}satisfies conditions \(i\)\-\(iv\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\.

The surplus setΘ⊤φ\\Theta\\top\\varphi, therefore, contains the whole family of maximal non\-trivial supersets satisfying \(i\)\-\(iv\)\. As in the case of the postulates, there is the initial requirement that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and conditions \(i\)\-\(iii\) are analogous to postulates\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\)\-\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)\. Requirement \(iv\) merely guarantees the maximality of the supersetsΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. Just as the operation○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}defined by the postulates does not require a failure postulate \(see Corollary[A\.7](https://arxiv.org/html/2607.09729#A1.Thmtheorem7)\), since it always expands the initial epistemic state, the following results can also be obtained here:

###### Lemma 6\.5\.

LetΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphibe a maximal non\-trivialAGMpa​b​dp\_\{abd\}superset ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\)\. Forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\.

###### Corollary 6\.6\.

On the one hand, by lemma[6\.5](https://arxiv.org/html/2607.09729#S6.Thmtheorem5), forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′⊆Θ⊤φ\\Theta^\{\\prime\}\\subseteq\\Theta\\top\\varphi\. On the other hand, by lemma[5\.7](https://arxiv.org/html/2607.09729#S5.Thmtheorem7), if it is the case that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}—a requirement of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)—then it is also the case thatα∨φ∈H​\(Θ,φ\)\\alpha\\lor\\varphi\\in H\(\\Theta,\\varphi\)\. Thus, we have thatα∨φ∈Θ′∩H​\(Θ,φ\)\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}\\cap H\(\\Theta,\\varphi\)for anyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiandα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\)\.

###### Lemma 6\.7\.

LetΘ⊤φ\\Theta\\top\\varphibe the surplus set ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\)\. It is the case thatΘ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptyset\.

###### Corollary 6\.8\.

By lemma[6\.5](https://arxiv.org/html/2607.09729#S6.Thmtheorem5), forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. Therefore,α∨φ∈⋂Θ⊤φ\\alpha\\lor\\varphi\\in\\bigcap\\Theta\\top\\varphi\.

Let us now consider the following well\-known definitions from theAGMliterature—evidently, definitions that disregard, thanks to Lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7), the caseΘ⊤φ=∅\\Theta\\top\\varphi=\\emptyset:

###### Definition 6\.9\(Partial meet selection functionγ\\gamma\)\.

LetΘ⊤φ\\Theta\\top\\varphibe the surplus set ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\)\. A partial meet selection function is a functionγ:T​h​\(𝕃\)×ℒΣ∗→℘​\(T​h​\(𝕃\)\)∖\{∅\}\\gamma:Th\(\\mathbb\{L\}\)\\times\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\\to\\wp\(Th\(\\mathbb\{L\}\)\)\\setminus\\\{\\emptyset\\\}such that, for everyΘ\\Thetaandφ\\varphisuch that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}, it selects some elements ofΘ⊤φ\\Theta\\top\\varphi, i\.e\.:

γ​\(Θ,φ\)⊆Θ⊤φ\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi

###### Definition 6\.10\(AGMpa​b​dp\_\{abd\}partial meetabductive expansion operation\)\.

We define○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}as aAGMpa​b​dp\_\{abd\}partial meet abductive expansion operation ofΘ\\Thetawith respect toφ\\varphiby:

Θφ○⊕=⋂γ​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\(\\Theta,\\varphi\)

As expected, the following more relevant result can also be obtained in this system:

###### Theorem 6\.11\.

For everyΘ\\Thetaandφ\\varphisuch that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\},○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}is aAGMpa​b​dp\_\{abd\}partial meet abductive expansion operation ofΘ\\Thetawith respect toφ\\varphi—definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)—if, and only if,○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of abductive expansion ofΘ\\Thetawith respect toφ\\varphi\.

In order to capture the notion that the functionγ\\gammaselects those maximal non\-trivial supersets ofΘ\\Thetathat do not implyφ\\varphi, which areworth retainingat least as much as any other—since they are considered “more important” or “better”—in thepartial meetabductive expansion operation of theAGMpa​b​dp\_\{abd\}system, an ordering is established among the elementsΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. Just as in the classicalAGMsystem and in Pagnucco’s system505050According to\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.112\]\., a transitively relationalAGMpa​b​dp\_\{abd\}partial meetabductive expansion operation—which also encompasses postulates \(Θ○⊕​6\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\) and \(Θ○⊕​7\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)—can also be established here\. Let us therefore consider the following definition\.

###### Definition 6\.12\(Marking\-off identity\)\.

γ⩽\(Θ,φ\)=\{Θ′∈Θ⊤φ:Θ′′⩽Θ′\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)=\\\{\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi:\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}for everyΘ′′∈Θ⊤φ\}\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\\\}\.

The relationality of⩽\\leqslantis already given in definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)itself, and the transitivity of⩽\\leqslantis notably conferred upon it by assuming the property that ifΘ′⩽Θ′′\\Theta^\{\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}andΘ′′⩽Θ′′′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\\prime\\prime\}, thenΘ′⩽Θ′′′\\Theta^\{\\prime\}\\leqslant\\Theta^\{\\prime\\prime\\prime\}\. No further property needs to be imposed on the relation⩽\\leqslantat this point for the following lemmas to be demonstrated515151I emphasize that, since the supplementary postulates\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)and\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)are strictly the same as in theAGMpa​b​dp\_\{abd\}system, the lemmas[B\.7](https://arxiv.org/html/2607.09729#A2.Thmtheorem7),[B\.8](https://arxiv.org/html/2607.09729#A2.Thmtheorem8)and[B\.9](https://arxiv.org/html/2607.09729#A2.Thmtheorem9), reproduced in Appendix[A](https://arxiv.org/html/2607.09729#A1), are analogous, with some differences and adjustments, to those proved by Pagnucco\[[19](https://arxiv.org/html/2607.09729#bib.bib19), lemmas B\.5, B\.6 and B\.7, p\.226\-227\]\.\.

###### Lemma 6\.13\.

Any relational partial meet abductive expansion function, i\.e\.,Θφ○⊕=⋂γ⩽​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\(definitions[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)and[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\), satisfies postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)\.

###### Lemma 6\.14\.

Any transitively relational partial meet abductive expansion function \(definitions[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)and[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\) satisfies postulate\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)\.

Finally, as the most important result of this paper, the following theorem can be demonstrated, both in the classical and in the paraconsistent cases\.

###### Theorem 6\.15\.

For everyΘ\\Thetaandφ\\varphisuch that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\},○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}is a transitively relationalAGMpa​b​dp\_\{abd\}partial meet abductive expansion operation \(definitions[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)and[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\) if, and only if,○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of abductive expansion onΘ\\Theta\.

## 7Conclusion and Future Work

In this paper, the paraconsistentAGM\-style abductive expansion operation was developed—to the best of my knowledge, the first of its kind in theAGMliterature\. Specifically, its postulates and its transitively relationalpartial meetconstruction\. This operation was elaborated from rationality criteria specifically designed to satisfy some important philosophical demands of both abductive reasoning, especially those found in the thought of C\.S\. Peirce, its creator, and explanation; both commonly neglected by the formal logic literature\. A general taxonomy, inspired by Aliseda\[[1](https://arxiv.org/html/2607.09729#bib.bib1)\], which attempts to capture the processual aspect of abductive reasoning—from the initial surprise in the face of a surprising phenomenon, which demands adequately explanatory hypotheses, to the expanded epistemic states—was designed accordingly\. Thus, the abductive expansion operation developed in this paper, even in its classical aspect, although formally based on the abductive expansion operation originally created by Pagnucco\[[19](https://arxiv.org/html/2607.09729#bib.bib19)\], implements a different philosophical perspective: for the author, abduction, as a mere logical form525252Definition found in\[[19](https://arxiv.org/html/2607.09729#bib.bib19), 79\]\., adds interesting features to theAGMsystem535353This view is in accordance with Pagnucco’s proposal and can be easily noted in the very term “within” in the title of his thesis: “The role of abductive reasoning within the process of belief revision”\.\. On the other hand, for me, it is theAGMsystem that adds some interesting formal aspects to abduction—or rather, abductive reasoning is a philosophically complex inferential process, and some of these aspects can be formally captured by the interaction between the taxonomy and theAGMoperation\. Moreover, and as the most relevant formal development, thanks to the logicRCbr, the abductive expansion operation developed in this paper has a paraconsistent aspect, capable of providing unique capabilities to the operation, as seen in Subsection[5\.1](https://arxiv.org/html/2607.09729#S5.SS1)\.

The content presented in this paper is part of an ongoing PhD thesis\. Thus, the paraconsistent abductive expansion operation is part of a broader system calledAGMpa​b​dp\_\{abd\}\. Other operations, such as internal and external revision—the latter being possible only due to paraconsistency—can be obtained, with many limitations545454The same limitations were obtained by Pagnucco, in the sense that only a few simple postulates can be obtained, since the non\-monotonicity of the abductive expansion operation entails that an adequate inclusion relation between the expansion and revision operations, as occurs in classicalAGMrevision, cannot be fully obtained\. See\[[19](https://arxiv.org/html/2607.09729#bib.bib19), p\.165\-166\]\.\. Moreover, a construction based on anabductive entrenchmentorder, also with its paraconsistent aspect, thanks to the underlying logicRCbr, can be implemented\. A new system calledAGM∘a​b​d\\circ\_\{abd\}—which will be the subject of a second paper—is also being developed to deal, in a more granular and particular way, with the triggers of abductive strong novelty, weak novelty and weak anomaly \(definition[4\.5](https://arxiv.org/html/2607.09729#S4.Thmtheorem5)and figure[2](https://arxiv.org/html/2607.09729#S4.F2)\), assigning differentiated roles to the paraconsistent operators¬\\negand∘\\circ\. Two other future works, associated with these systems, will also be developed\. They concern the dynamics of preferences among hypotheses, contradictory or not—specifically in the association between the logicRCbrand the modal logicDBL\(Dynamic Betterness Logicof Fenrong Liu\[[17](https://arxiv.org/html/2607.09729#bib.bib17)\]\)—and the non\-monotonicity of the paraconsistent abductive expansion operation—similarly to the results obtained by Makinson and Gärdenfors,\[[18](https://arxiv.org/html/2607.09729#bib.bib18)\]and\[[13](https://arxiv.org/html/2607.09729#bib.bib13)\], for classicalAGMrevision\.

## Appendix ASupplementary Materials

### A\.1Tarskian Logic Properties

Some general properties about Tarskian logic that will be useful in many proofs\. As usually, we should also assume thatφ↔ψ\\varphi\\leftrightarrow\\psiis the same as\(φ→ψ\)∧\(ψ→φ\)\(\\varphi\\to\\psi\)\\land\(\\psi\\to\\varphi\)andφ≡ψ\\varphi\\equiv\\psiis the same as⊢φ↔ψ\\vdash\\varphi\\leftrightarrow\\psi\.

###### Definition A\.1\(Tarskian Logic\)\.

A logic𝕃=⟨ℒΣ,C​n⟩\\mathbb\{L\}=\\langle\\mathcal\{L\}\_\{\\Sigma\},Cn\\rangledefined over a languageℒΣ\\mathcal\{L\}\_\{\\Sigma\}and a consequence relationC​nCn\- or⊢\\vdash\- is Tarskian if it satisfies the following three properties555555I believe it is important to maintain both notations, since they can be frequently found inAGMliterature and they will be used interchangeably in this paper, considering that the Tarskian consequence operatorC​n:℘​\(ℒΣ\)→℘​\(ℒΣ\)Cn:\\wp\(\\mathcal\{L\}\_\{\\Sigma\}\)\\to\\wp\(\\mathcal\{L\}\_\{\\Sigma\}\)is defined as follows:C​n​\(Γ\)=\{φ∈ℒΣ:Γ⊢φ\}Cn\(\\Gamma\)=\\\{\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\}:\\Gamma\\vdash\\varphi\\\}\. In particular, therefore,⊢φ\\vdash\\varphiis the same asφ∈C​n​\(∅\)\\varphi\\in Cn\(\\emptyset\)\., for allΓ∪Δ∪\{φ\}⊆ℒΣ\\Gamma\\cup\\Delta\\cup\\\{\\varphi\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\}:

\(i\)Γ⊆C​n​\(Γ\)\\Gamma\\subseteq Cn\(\\Gamma\)\(inclusion\)

\(ii\)C​n​\(C​n​\(Γ\)\)⊆C​n​\(Γ\)Cn\(Cn\(\\Gamma\)\)\\subseteq Cn\(\\Gamma\)\(iteration\)

\(iii\) IfΓ⊆Δ\\Gamma\\subseteq\\Delta, thenC​n​\(Γ\)⊆C​n​\(Δ\)Cn\(\\Gamma\)\\subseteq Cn\(\\Delta\)\(monotonicity\)

or

\(i\) Ifφ∈Γ\\varphi\\in\\Gamma, thenΓ⊢φ\\Gamma\\vdash\\varphi\(reflexivity\)

\(ii\) IfΔ⊢φ\\Delta\\vdash\\varphiandΓ⊢ψ\\Gamma\\vdash\\psifor anyψ∈Δ\\psi\\in\\Delta, thenΓ⊢φ\\Gamma\\vdash\\varphi\(cut\)

\(iii\) IfΓ⊢φ\\Gamma\\vdash\\varphiandΓ⊆Δ\\Gamma\\subseteq\\Delta, thenΔ⊢φ\\Delta\\vdash\\varphi\(monotonicity\)

###### Definition A\.2\(Standard logic\)\.

A logic𝕃\\mathbb\{L\}is considered standard if it is Tarskian, finitary, and structural, that is, if it satisfies, in addition to the three properties described in Definition[A\.1](https://arxiv.org/html/2607.09729#A1.Thmtheorem1), two other properties, respectively:

\(iv\) IfΓ⊢φ\\Gamma\\vdash\\varphi, then there exists a finite subsetΓ0⊆Γ\\Gamma\_\{0\}\\subseteq\\Gammasuch thatΓ0⊢φ\\Gamma\_\{0\}\\vdash\\varphi\(compactness\);

\(v\) IfΓ⊢φ\\Gamma\\vdash\\varphi, thenσ​\[Γ\]⊢σ​\[φ\]\\sigma\[\\Gamma\]\\vdash\\sigma\[\\varphi\]for every substitutionσ\\sigmaof formulas for variables \(substitution\)\.

###### Definition A\.3\(Supraclassical Tarskian logic\)\.

A Tarskian logic𝕃=⟨ℒΣ,C​n⟩\\mathbb\{L\}=\\langle\\mathcal\{L\}\_\{\\Sigma\},Cn\\rangleis considered supraclassical if, given a classical consequence operatorCC​P​LC\_\{CPL\}, it is the case thatCC​P​L​\(Γ\)⊆C​n​\(Γ\)C\_\{CPL\}\(\\Gamma\)\\subseteq Cn\(\\Gamma\)for every set of formulasΓ\\Gamma\. That is, for every formulaφ\\varphithat can be classically derived fromΓ\\Gamma, i\.e\.,Γ⊢C​P​Lφ\\Gamma\\vdash\_\{CPL\}\\varphi, thenφ∈C​n​\(Γ\)\\varphi\\in Cn\(\\Gamma\)\.

###### Properties A\.4\(Valid properties in a supraclassical Tarskian logic𝕃\\mathbb\{L\}565656IncludingCPLandmbC\(and its extensions\)\. Of course, in a classical notation, the operator¬\\negfound in \(iii\) must be read as the strong classical negation, but, in the case osmbCand its extensions,¬\\negis the paraconsistent negation\. Property \(iii\) can be obtained, sinceψ∨¬ψ\\psi\\lor\\neg\\psiis an axiom in thembcaxiomatic scheme\.\)\.

\(i\)Γ∪\{ψ\}⊢φ\\Gamma\\cup\\\{\\psi\\\}\\vdash\\varphiif and only ifΓ⊢ψ→φ\\Gamma\\vdash\\psi\\to\\varphi\(Deduction Theorem\);

\(ii\) IfΓ∪\{ψ\}⊢φ\\Gamma\\cup\\\{\\psi\\\}\\vdash\\varphiandΓ∪\{δ\}⊢φ\\Gamma\\cup\\\{\\delta\\\}\\vdash\\varphi, thenΓ∪\{ψ∨δ\}⊢φ\\Gamma\\cup\\\{\\psi\\lor\\delta\\\}\\vdash\\varphi\(Disjunction of Premises\);

\(iii\) IfΓ∪\{ψ\}⊢φ\\Gamma\\cup\\\{\\psi\\\}\\vdash\\varphiandΓ∪\{¬ψ\}⊢φ\\Gamma\\cup\\\{\\neg\\psi\\\}\\vdash\\varphi, thenΓ⊢φ\\Gamma\\vdash\\varphi\(Proof by Cases\)\.

The following definitions and properties are well\-known in literature\.

###### Definition A\.5\(δ\\delta\-saturated set\)\.

For some Tarskian logic𝕃\\mathbb\{L\}over the languageℒΣ\\mathcal\{L\}\_\{\\Sigma\}, letΓ∪\{δ\}⊆ℒΣ\\Gamma\\cup\\\{\\delta\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\}\. We say thatΓ¯\\bar\{\\Gamma\}is a maximal consistent set with respect toδ\\deltain𝕃\\mathbb\{L\}\- orδ\\delta\-saturated \- if it satisfies the following properties:

\(i\)Γ¯⊬δ\\bar\{\\Gamma\}\\nvdash\\delta;

\(ii\) ifψ∉Γ¯\\psi\\not\\in\\bar\{\\Gamma\}thenΓ¯,ψ⊢δ\\bar\{\\Gamma\},\\psi\\vdash\\delta\.

###### Properties A\.6\.

LetΓ¯\\bar\{\\Gamma\}be aδ\\delta\-saturated set \(Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\)\. Then,Γ¯\\bar\{\\Gamma\}satisfies the following properties575757Again, includingCPLandmbC\(and its extensions\)\. When consideringCPL, the operator¬\\negmust be interpreted as classical strong negation\. In the case ofmbCand its extensions, on the other hand,¬\\negmust be interpreted as paraconsistent negation\. It is important to emphasize that, in the case ofCPL, property \(ii\) can be verified in both directions, i\.e\.,ψ∉Γ¯\\psi\\notin\\bar\{\\Gamma\}if and only if¬ψ∈Γ¯\\neg\\psi\\in\\bar\{\\Gamma\}\. To verify theonly ifdirection, it suffices to suppose¬ψ∈Γ¯\\neg\\psi\\in\\bar\{\\Gamma\}and, byreductio, supposeψ∈Γ¯\\psi\\in\\bar\{\\Gamma\}\. Since, byex falso,ψ,¬ψ⊢δ\\psi,\\neg\\psi\\vdash\\delta, then we haveΓ¯⊢δ\\bar\{\\Gamma\}\\vdash\\delta, a contradiction with the very definition ofδ\\delta\-saturated\. Hence,ψ∉Γ¯\\psi\\notin\\bar\{\\Gamma\}\. I choose, however, to present property \(ii\) in this way, with only one direction of the conditional, because, thus described, it \(and the corollary[A\.7](https://arxiv.org/html/2607.09729#A1.Thmtheorem7)below, by the way\) is valid in the paraconsistent logicmbC—and, therefore, in its extensions—even with the paraconsistent negation¬\\neg\. The proof is identical, since the rule ofproof by cases—property[A\.4](https://arxiv.org/html/2607.09729#A1.Thmtheorem4)\(iii\)—holds inmbC\.:

\(i\)Γ¯\\bar\{\\Gamma\}is a set closed under logical consequences;

\(ii\) ifψ∉Γ¯\\psi\\notin\\bar\{\\Gamma\}, then¬ψ∈Γ¯\\neg\\psi\\in\\bar\{\\Gamma\}\.

###### Proof\.

\(i\) We need to show thatΓ¯⊢ψ\\bar\{\\Gamma\}\\vdash\\psiif and only ifψ∈Γ¯\\psi\\in\\bar\{\\Gamma\}\. \(If:\) Ifψ∈Γ¯\\psi\\in\\bar\{\\Gamma\}, since𝕃\\mathbb\{L\}is a Tarskian logic, byinclusion, we directly have thatΓ¯⊢ψ\\bar\{\\Gamma\}\\vdash\\psi\. \(Only if:\) We need to show that ifΓ¯⊢ψ\\bar\{\\Gamma\}\\vdash\\psi, thenψ∈Γ¯\\psi\\in\\bar\{\\Gamma\}\. Bycontraposition, this is equivalent to showing that ifψ∉Γ¯\\psi\\notin\\bar\{\\Gamma\}, thenΓ¯⊬ψ\\bar\{\\Gamma\}\\nvdash\\psi\. Suppose, therefore, thatψ∉Γ¯\\psi\\notin\\bar\{\\Gamma\}\. By Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\(ii\),Γ¯,ψ⊢δ\\bar\{\\Gamma\},\\psi\\vdash\\delta\. Byreductio, let us assume thatΓ¯⊢ψ\\bar\{\\Gamma\}\\vdash\\psi\. In this case,Γ¯⊢δ\\bar\{\\Gamma\}\\vdash\\delta, a contradiction with Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\(i\)\. Therefore,Γ¯⊬ψ\\bar\{\\Gamma\}\\nvdash\\psi\. \(ii\) Suppose thatψ∉Γ¯\\psi\\notin\\bar\{\\Gamma\}\. By Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\(ii\),Γ¯,ψ⊢δ\\bar\{\\Gamma\},\\psi\\vdash\\delta\. In this case, suppose, byreductio, that¬ψ∉Γ¯\\neg\\psi\\notin\\bar\{\\Gamma\}\. Again, by Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\(ii\),Γ¯,¬ψ⊢δ\\bar\{\\Gamma\},\\neg\\psi\\vdash\\delta\. We therefore have, byproof by cases, thatΓ¯⊢δ\\bar\{\\Gamma\}\\vdash\\delta, a contradiction with Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\(i\)\. Hence,¬ψ∈Γ¯\\neg\\psi\\in\\bar\{\\Gamma\}\. ∎

###### Corollary A\.7\.

LetΓ¯\\bar\{\\Gamma\}be aδ\\delta\-saturated set\. Sinceδ∉Γ¯\\delta\\notin\\bar\{\\Gamma\}, by Proposition[A\.6](https://arxiv.org/html/2607.09729#A1.Thmtheorem6), then¬δ∈Γ¯\\neg\\delta\\in\\bar\{\\Gamma\}\.

###### Theorem A\.8\(Lindenbaum\-Łoś Theorem585858The proof of this theorem can be found in\[[29](https://arxiv.org/html/2607.09729#bib.bib29), p\.54, Theorem 22\.2\]and, in an adapted version,\[[7](https://arxiv.org/html/2607.09729#bib.bib7), p\.37, Theorem 2\.2\.6\]\.\)\.

Let𝕃\\mathbb\{L\}be a finitary Tarskian logic over the languageℒΣ\\mathcal\{L\}\_\{\\Sigma\}\. LetΓ∪\{δ\}⊆ℒΣ\\Gamma\\cup\\\{\\delta\\\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\}such thatΓ⊬δ\\Gamma\\nvdash\\delta\. There exists a setΔ\\Deltasuch thatΓ⊆Δ⊂ℒΣ\\Gamma\\subseteq\\Delta\\subset\\mathcal\{L\}\_\{\\Sigma\}withΔ\\Deltamaximally consistent \(non\-trivial\) with respect toδ∈ℒΣ\\delta\\in\\mathcal\{L\}\_\{\\Sigma\}\- that is,δ\\delta\-saturated \(Definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\)\.

## Appendix BSections[4](https://arxiv.org/html/2607.09729#S4),[5](https://arxiv.org/html/2607.09729#S5)and[6](https://arxiv.org/html/2607.09729#S6)proofs

Observation[4\.7](https://arxiv.org/html/2607.09729#S4.Thmtheorem7):It is the case that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\) if and only ifφ∉Θ\\varphi\\notin\\Thetaand∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\.

###### Proof\.

\(If:\) By definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8),\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, equivalently, if: \(i\)φ∉Θ\\varphi\\notin\\Thetaand \(ii\) \(∘φ∉Θ\\circ\\varphi\\notin\\Thetaand¬φ∉Θ\\neg\\varphi\\notin\\Theta\) or \(∘φ∈Θ\\circ\\varphi\\in\\Thetaand¬φ∉Θ\\neg\\varphi\\notin\\Theta\) or \(∘φ∉Θ\\circ\\varphi\\notin\\Thetaand¬φ∈Θ\\neg\\varphi\\in\\Theta\)\. From this, it follows that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case if \(i\)φ∉Θ\\varphi\\notin\\Thetaand \(ii\) it is not the case that∘φ∈Θ\\circ\\varphi\\in\\Thetaand¬φ∈Θ\\neg\\varphi\\in\\Theta\. SinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), then\(∘φ∧¬φ\)∉Θ\(\\circ\\varphi\\land\\neg\\varphi\)\\notin\\Theta\. Since∼φ≡𝕃∘φ∧¬φ\\mathord\{\\sim\}\\varphi\\equiv\_\{\\mathbb\{L\}\}\\circ\\varphi\\land\\neg\\varphi, then∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\. \(Only if:\) Considerφ∉Θ\\varphi\\notin\\Thetaand∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\. Since∼φ≡𝕃∘φ∧¬φ\\mathord\{\\sim\}\\varphi\\equiv\_\{\\mathbb\{L\}\}\\circ\\varphi\\land\\neg\\varphi, then\(∘φ∧¬φ\)∉Θ\(\\circ\\varphi\\land\\neg\\varphi\)\\notin\\Theta\. SinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), then we have three possibilities:∘φ∉Θ\\circ\\varphi\\notin\\Thetaor¬φ∉Θ\\neg\\varphi\\notin\\Thetaor \(∘φ∉Θ\\circ\\varphi\\notin\\Thetaand¬φ∉Θ\\neg\\varphi\\notin\\Theta\), which exactly reflects the conditions of definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\. ∎

Lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2)\(Partial self\-explanation\):If\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case \(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\), then there always exists someα∈ℒΣ∗\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}that satisfies criteria \(i\)\-\(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\. Hence,H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\.

###### Proof\.

Suppose\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case—that is,φ∉Θ\\varphi\\notin\\Thetaand∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\(definition[4\.8](https://arxiv.org/html/2607.09729#S4.Thmtheorem8)\)\. SinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), thenΘ≠∅\\Theta\\neq\\emptyset\. Consider, therefore, arbitraryδ,α∈ℒΣ∗\\delta,\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}such thatδ∈Θ\\delta\\in\\Thetaandα=δ→φ\\alpha=\\delta\\to\\varphi\. We need to show thatα\\alphasatisfies criteria \(i\)\-\(iii\) of definition[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2)\. Since, bymodus ponens,Θ∪\{δ→φ\}⊢𝕃φ\\Theta\\cup\\\{\\delta\\to\\varphi\\\}\\vdash\_\{\\mathbb\{L\}\}\\varphi, thenα\\alphasatisfies criterion \(i\)\. Since\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, then∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\. Thus,Θ∪\{δ→φ\}⊬𝕃⊥\\Theta\\cup\\\{\\delta\\to\\varphi\\\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\. Soα\\alphasatisfies criterion \(ii\)\. Finally,δ→φ⊬𝕃φ\\delta\\to\\varphi\\nvdash\_\{\\mathbb\{L\}\}\\varphiand condition \(iii\) is satisfied\. Therefore, if\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, then there always exists anα∈ℒΣ∘\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\\circ\}\}such thatα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\)\. Hence,H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\. ∎

Observation[5\.3](https://arxiv.org/html/2607.09729#S5.Thmtheorem3):LetH​\(Θ,φ\)H\(\\Theta,\\varphi\)be the set of abductive hypotheses for the pair\(Θ,φ\)\(\\Theta,\\varphi\)\(definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\)\. It is the case thatΘ∩H​\(Θ,φ\)=∅\\Theta\\cap H\(\\Theta,\\varphi\)=\\emptyset\.

###### Proof\.

From definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1), we have thatφ∉Θ\\varphi\\notin\\Theta\. Suppose, byreductio, thatΘ∩H​\(Θ,φ\)≠∅\\Theta\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\. In this case, from condition \(i\)Θ,α⊢𝕃φ\\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\varphi, we infer thatΘ⊢𝕃φ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\varphi\. SinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), thenφ∈Θ\\varphi\\in\\Theta, a contradiction\. Hence, as required,Θ∩H​\(Θ,φ\)=∅\\Theta\\cap H\(\\Theta,\\varphi\)=\\emptyset\. ∎

Lemma[5\.4](https://arxiv.org/html/2607.09729#S5.Thmtheorem4):Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and\(Θ,ψ\)p𝔸​ℙ​𝔼\(\\Theta,\\psi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\. LetH​\(Θ,φ\)H\(\\Theta,\\varphi\)andH​\(Θ,ψ\)H\(\\Theta,\\psi\)be the sets of abductive hypotheses forφ\\varphiandψ\\psi, respectively\. It is the case that if⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenH​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\)\.

###### Proof\.

Since\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and\(Θ,ψ\)p𝔸​ℙ​𝔼\(\\Theta,\\psi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}are the case, then, by lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2),H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptysetandH​\(Θ,ψ\)≠∅H\(\\Theta,\\psi\)\\neq\\emptyset\. This condition, therefore, does not need to be considered\. Consider, therefore,⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi\. We need to show that \(1\)H​\(Θ,φ\)⊆H​\(Θ,ψ\)H\(\\Theta,\\varphi\)\\subseteq H\(\\Theta,\\psi\)and \(2\)H​\(Θ,ψ\)⊆H​\(Θ,φ\)H\(\\Theta,\\psi\)\\subseteq H\(\\Theta,\\varphi\):

\(1\)H​\(Θ,φ\)⊆H​\(Θ,ψ\)H\(\\Theta,\\varphi\)\\subseteq H\(\\Theta,\\psi\)\. We need to show that ifα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), thenα∈H​\(Θ,ψ\)\\alpha\\in H\(\\Theta,\\psi\)\. Ifα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), then, by the three conditions of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1),\(i\)Θ,α⊢𝕃φ,\(ii\)Θ,α⊬𝕃⊥and\(iii\)α⊬𝕃φ\(i\)\\ \\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\varphi,\\ \(ii\)\\ \\Theta,\\alpha\\nvdash\_\{\\mathbb\{L\}\}\\bot\\ \\mbox\{ and \}\\ \(iii\)\\ \\alpha\\nvdash\_\{\\mathbb\{L\}\}\\varphi\. Since⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenφ\\varphiandψ\\psican be substituted in any context—including in the paraconsistent case, sinceRCbrsatisfies thereplacementproperty\. In particular, therefore, it is also the case that\(i\)Θ,α⊢𝕃ψ,\(ii\)Θ,α⊬𝕃⊥and\(iii\)α⊬𝕃ψ\(i\)\\ \\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\psi,\\ \(ii\)\\ \\Theta,\\alpha\\nvdash\_\{\\mathbb\{L\}\}\\bot\\ \\mbox\{ and \}\\ \(iii\)\\ \\alpha\\nvdash\_\{\\mathbb\{L\}\}\\psifor anyφ,ψ∈ℒΣ∗\\varphi,\\psi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\. Hence,α∈H​\(Θ,ψ\)\\alpha\\in H\(\\Theta,\\psi\)\.

\(2\)H​\(Θ,ψ\)⊆H​\(Θ,φ\)H\(\\Theta,\\psi\)\\subseteq H\(\\Theta,\\varphi\)\. Exactly the same reasoning as in \(1\)\.

Therefore,H​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\)\. ∎

###### Proof\.

ConsiderΘ⊢𝕃φ↔ψ\\Theta\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psiand⊬𝕃φ↔ψ\\nvdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi\. SinceΘ∈T​h​\(𝕃\)\\Theta\\in Th\(\\mathbb\{L\}\), thenφ↔ψ∈Θ\\varphi\\leftrightarrow\\psi\\in\\Theta\. Thus, by definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1),\(i\)Θ,ψ⊢𝕃φ,\(ii\)Θ,ψ⊬𝕃⊥and\(iii\)ψ⊬𝕃φ\(i\)\\ \\Theta,\\psi\\vdash\_\{\\mathbb\{L\}\}\\varphi,\\ \(ii\)\\ \\Theta,\\psi\\nvdash\_\{\\mathbb\{L\}\}\\bot\\ \\mbox\{ and \}\\ \(iii\)\\ \\psi\\nvdash\_\{\\mathbb\{L\}\}\\varphi\. Hence,ψ∈H​\(Θ,φ\)\\psi\\in H\(\\Theta,\\varphi\)\. Butφ∉H​\(Θ,φ\)\\varphi\\notin H\(\\Theta,\\varphi\)sinceφ⊢𝕃φ\\varphi\\vdash\_\{\\mathbb\{L\}\}\\varphi, which violates criterion \(iii\)\. The same reasoning can be employed to conclude thatφ∈H​\(Θ,ψ\)\\varphi\\in H\(\\Theta,\\psi\), butψ∉H​\(Θ,ψ\)\\psi\\notin H\(\\Theta,\\psi\)\. Therefore,H​\(Θ,φ\)≠H​\(Θ,ψ\)H\(\\Theta,\\varphi\)\\neq H\(\\Theta,\\psi\)\. ∎

Observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6):IfΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5), thenφ∈Θφ○⊕∖Θ\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\setminus\\Theta\.

###### Proof\.

Since, by definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5),\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, by lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2),H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\. By postulate\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\), therefore, there must be at least one abductive hypothesisα∈Θφ○⊕\\alpha\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. Since, by postulate\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\),Θ⊂Θφ○⊕\\Theta\\subset\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}and by postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)Θφ○⊕∈T​h​\(𝕃\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\in Th\(\\mathbb\{L\}\), thenφ∈Θφ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\(since, by condition \(i\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)ofH​\(Θ,φ\)H\(\\Theta,\\varphi\), it is the case thatΘ,α⊢𝕃φ\\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\varphi\)\. Since, again,\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, thenφ∉Θ\\varphi\\notin\\Theta\. Hence,φ∈Θφ○⊕∖Θ\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\setminus\\Theta\. ∎

Lemma[5\.7](https://arxiv.org/html/2607.09729#S5.Thmtheorem7)Let\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}andH​\(Θ,φ\)H\(\\Theta,\\varphi\)be the set of abductive hypotheses for\(Θ,φ\)\(\\Theta,\\varphi\)\. Ifα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), then it is the case thatα∨φ∈H​\(Θ,φ\)\\alpha\\lor\\varphi\\in H\(\\Theta,\\varphi\)\.

###### Proof\.

By lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2),H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\. Therefore, there is anα∈ℒΣ∗\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}such thatα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\)\. In these terms, we need to show thatα∨φ\\alpha\\lor\\varphisatisfies conditions \(i\), \(ii\) and \(iii\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)\. Given that, by condition \(i\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1),Θ,α⊢𝕃φ\\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\varphi, and it is the case thatΘ,φ⊢𝕃φ\\Theta,\\varphi\\vdash\_\{\\mathbb\{L\}\}\\varphi, bydisjunction of premises—a rule valid inmbCand, therefore, inRCbr—we have thatΘ∪\{α∨φ\}⊢𝕃φ\\Theta\\cup\\\{\\alpha\\lor\\varphi\\\}\\vdash\_\{\\mathbb\{L\}\}\\varphi\(i\)\. Suppose now, byreductio, thatΘ∪\{α∨φ\}⊢𝕃⊥\\Theta\\cup\\\{\\alpha\\lor\\varphi\\\}\\vdash\_\{\\mathbb\{L\}\}\\bot\. Since\{α\}⊢𝕃α∨φ\\\{\\alpha\\\}\\vdash\_\{\\mathbb\{L\}\}\\alpha\\lor\\varphi, then \(considering, evidently, thatRCbr, like anyLFI, is a standard Tarskian logic, according to definition[A\.2](https://arxiv.org/html/2607.09729#A1.Thmtheorem2), p\.[A\.2](https://arxiv.org/html/2607.09729#A1.Thmtheorem2)\), bycut, we infer thatΘ,α⊢𝕃⊥\\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\bot, a contradiction with condition \(ii\)\. Hence,Θ∪\{α∨φ\}⊬𝕃⊥\\Theta\\cup\\\{\\alpha\\lor\\varphi\\\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\(ii\)\. Finally, it is the case that\{α∨φ\}⊬𝕃φ\\\{\\alpha\\lor\\varphi\\\}\\nvdash\_\{\\mathbb\{L\}\}\\varphi\(iii\)\. Therefore,α∨φ∈H​\(Θ,φ\)\\alpha\\lor\\varphi\\in H\(\\Theta,\\varphi\), as required\. ∎

Lemma[5\.8](https://arxiv.org/html/2607.09729#S5.Thmtheorem8)IfΘφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5), then, forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θφ○⊕\\alpha\\lor\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\.

###### Proof\.

Since, by definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5),\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, then, by lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2),H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\. Therefore, there is anα∈ℒΣ∗\\alpha\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}such thatα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\)\. By lemma[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), we have thatφ∈Θφ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. Since, by postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\),Θφ○⊕∈T​h​\(𝕃\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\in Th\(\\mathbb\{L\}\)and it is the case thatφ⊢𝕃α∨φ\\varphi\\vdash\_\{\\mathbb\{L\}\}\\alpha\\lor\\varphi, then, forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θφ○⊕\\alpha\\lor\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. ∎

###### Properties B\.2\.

The following properties are consequences of postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)in the presence of\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\):

\(i\)Θφ○⊕∩Θψ○⊕⊆Θφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}

\(ii\) Ifφ∈Θφ∨ψ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, thenΘφ○⊕⊆Θφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}

###### Proof\.

\(i\)Θφ○⊕∩Θψ○⊕⊆Θφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}:

We need to show that ifδ∈Θφ○⊕∩Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}, thenδ∈Θφ∨ψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Ifδ∈Θφ○⊕∩Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}, thenδ∈Θφ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}andδ∈Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\. Consider the first case\. By postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\),δ∈C​n𝕃​\(Θφ∨ψ○⊕∪\{φ\}\)\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\varphi\\\}\)and, by thededuction theorem, we obtainφ→δ∈C​n𝕃​\(Θφ∨ψ○⊕\)\\varphi\\to\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)\. By postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\), we have \(a\)φ→δ∈Θφ∨ψ○⊕\\varphi\\to\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Analogously, from the second case whereδ∈Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}, we have, by\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\),δ∈C​n𝕃​\(Θφ∨ψ○⊕∪\{ψ\}\)\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\psi\\\}\)and, with the same reasoning, we obtain \(b\)ψ→δ∈Θφ∨ψ○⊕\\psi\\to\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. From \(a\) and \(b\) and, again,\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\), we obtain\(φ→δ\)∧\(ψ→δ\)∈C​n𝕃​\(Θφ∨ψ○⊕\)\(\\varphi\\to\\delta\)\\land\(\\psi\\to\\delta\)\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)\. Bydisjunction of premises, we have\(φ∨ψ\)→δ∈C​n𝕃​\(Θφ∨ψ○⊕\)\(\\varphi\\lor\\psi\)\\to\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)\. By observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), it is the case thatφ∨ψ∈Θφ∨ψ○⊕\\varphi\\lor\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Therefore, by\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)andmodus ponens, we haveδ∈Θφ∨ψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, as desired\.

\(ii\) Ifφ∈Θφ∨ψ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, thenΘφ○⊕⊆Θφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}:

Supposeφ∈Θφ∨ψ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. From this, we have thatΘφ∨ψ○⊕∪\{φ\}=Θφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\varphi\\\}=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Bymonotonicityand\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\), we haveC​n𝕃​\(Θφ∨ψ○⊕∪\{φ\}\)=C​n𝕃​\(Θφ∨ψ○⊕\)=Θφ∨ψ○⊕Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\varphi\\\}\)=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. SinceΘφ○⊕⊆C​n𝕃​\(Θφ∨ψ○⊕∪\{φ\}\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\varphi\\\}\), by\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\), then, as required,Θφ○⊕⊆Θφ∨ψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\.

∎

###### Properties B\.3\.

The following properties are consequences of postulate\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)in the presence of\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\):

\(i\)Θφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}orΘφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}

\(ii\) Ifφ∉Θφ∨ψ○⊕\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, thenΘφ∨ψ○⊕∈Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}

###### Proof\.

\(i\)Θφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}orΘφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}:

By observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), we have thatφ∨ψ∈Θφ∨ψ○⊕\\varphi\\lor\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. By postulate\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\), we therefore have that∼​\(φ∨ψ\)∉Θφ∨ψ○⊕\\mathord\{\\sim\}\(\\varphi\\lor\\psi\)\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. By\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\), it is also the case that∼​φ∧∼​ψ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\land\\mathord\{\\sim\}\\psi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. From this, we have that it is not the case that both∼​φ∈Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}and∼​ψ∈Θφ∨ψ○⊕\\mathord\{\\sim\}\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\(otherwise, by postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\), we would have∼​φ∧∼​ψ∈Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\land\\mathord\{\\sim\}\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, which contradicts∼​φ∧∼​ψ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\land\\mathord\{\\sim\}\\psi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)\. Therefore, either∼​φ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}or∼​ψ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\psi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Hence, by postulate\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\), we obtainΘφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}orΘφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}, as required\.

\(ii\) Ifφ∉Θφ∨ψ○⊕\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, thenΘφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}:

Assumeφ∉Θφ∨ψ○⊕\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. By observation[5\.6](https://arxiv.org/html/2607.09729#S5.Thmtheorem6), we have thatφ∨ψ∈Θφ∨ψ○⊕\\varphi\\lor\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Thus, suppose byreductiothat∼​ψ∈Θφ∨ψ○⊕\\mathord\{\\sim\}\\psi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. In this case, sinceφ∨ψ,∼ψ⊢𝕃φ\\varphi\\lor\\psi,\\mathord\{\\sim\}\\psi\\vdash\_\{\\mathbb\{L\}\}\\varphi, then, by postulate\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\),φ∈Θφ∨ψ○⊕\\varphi\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, a contradiction with the initial assumption\. Therefore,∼​ψ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\psi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Hence, by postulate\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\), we obtain, as required,Θφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\.

∎

Observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3):AnyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\) is a belief set\.

###### Proof\.

We need to show that, for anyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, it is the case thatΘ′=C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\. Suppose, byreductio, that there is someΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphisuch thatΘ′≠C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}\\neq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\. Byinclusion, we know thatΘ′⊆C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), but, by thereductioassumption, we haveΘ′⊂C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}\\subset Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\(a\)\. Note thatC​n𝕃​\(Θ′\)Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)satisfies conditions \(i\), \(ii\) and \(iii\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1), since \(i\)Θ⊂Θ′⊂C​n𝕃​\(Θ′\)\\Theta\\subset\\Theta^\{\\prime\}\\subset Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), \(ii\) sinceΘ′⊂C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}\\subset Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), evidently, by set theory,C​n𝕃​\(Θ′\)∩H​\(Θ,φ\)≠∅Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset, and \(iii\) directly,⊥∉Cn𝕃\(Θ′\)\\bot\\notin Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\. By condition \(iv\), however, there is noΘ′′⊃Θ′\\Theta^\{\\prime\\prime\}\\supset\\Theta^\{\\prime\}that satisfies \(i\), \(ii\) and \(iii\)\. Therefore, we have a contradiction with \(a\)\. As required,Θ′=C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\. ∎

Observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4):LetΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphibe a maximal non\-trivialAGMpa​b​dp\_\{abd\}superset ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\)\. It is the case thatφ∈Θ′∖Θ\\varphi\\in\\Theta^\{\\prime\}\\setminus\\Thetafor everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\.

###### Proof\.

By definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1), it is the case that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\. Thus, by lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2),H​\(Θ,φ\)≠∅H\(\\Theta,\\varphi\)\\neq\\emptyset\. By condition \(ii\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1), therefore, there must be at least one abductive hypothesisα∈Θ′\\alpha\\in\\Theta^\{\\prime\}\. Since, by condition \(i\),Θ⊂Θ′\\Theta\\subset\\Theta^\{\\prime\}and by lemma[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3), it is the case thatΘ′\\Theta^\{\\prime\}is a belief set, thenφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\(since, by condition \(i\) of definition[5\.1](https://arxiv.org/html/2607.09729#S5.Thmtheorem1)ofH​\(Θ,φ\)H\(\\Theta,\\varphi\), it is the case thatΘ,α⊢𝕃φ\\Theta,\\alpha\\vdash\_\{\\mathbb\{L\}\}\\varphi\)\. Since, again,\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}is the case, thenφ∉Θ\\varphi\\notin\\Theta\. Hence,φ∈Θ′∖Θ\\varphi\\in\\Theta^\{\\prime\}\\setminus\\Thetafor everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. ∎

Lemma[6\.5](https://arxiv.org/html/2607.09729#S6.Thmtheorem5):LetΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphibe a maximal non\-trivialAGMpa​b​dp\_\{abd\}superset ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\)\. Forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\.

###### Proof\.

By observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4),φ∈Θ′∖Θ\\varphi\\in\\Theta^\{\\prime\}\\setminus\\Theta, so, in particular,φ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\. Since, by observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3),Θ′=C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), and it is the case thatφ⊢𝕃α∨φ\\varphi\\vdash\_\{\\mathbb\{L\}\}\\alpha\\lor\\varphiforα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), thenα∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′⊆Θ⊤φ\\Theta^\{\\prime\}\\subseteq\\Theta\\top\\varphi\. ∎

Lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7):LetΘ⊤φ\\Theta\\top\\varphibe the surplus set ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\)\. It is the case thatΘ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptyset\.

###### Proof\.

By definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1), it is the case that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}\. Then,Θ\\Thetais a belief set,Θ≠∅\\Theta\\neq\\emptyset,Θ≠Θ⊥\\Theta\\neq\\Theta\_\{\\bot\}and∼​φ∉Θ\\mathord\{\\sim\}\\varphi\\notin\\Theta\. Thus,Θ∪\{φ\}⊬𝕃⊥\\Theta\\cup\\\{\\varphi\\\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\. Since the setΘ∪\{φ\}\\Theta\\cup\\\{\\varphi\\\}is consistent, then, by theorem[A\.8](https://arxiv.org/html/2607.09729#A1.Thmtheorem8), of Lindenbaum\-Łoś, there is a⊥\\bot\-saturated set, i\.e\., a maximal non\-trivial setΘ′\\Theta^\{\\prime\}with respect to⊥\\bot, such thatφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\. Thus, taking conditions \(i\), \(ii\), \(iii\) and \(iv\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1), we evidently have that \(i\)Θ⊂Θ′\\Theta\\subset\\Theta^\{\\prime\}; by corollary[6\.6](https://arxiv.org/html/2607.09729#S6.Thmtheorem6)and by the fact that maximal sets are closed under logical consequences, we have thatα∨φ∈Θ′∩H​\(Θ,φ\)\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}\\cap H\(\\Theta,\\varphi\), forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\)and, therefore, \(ii\)Θ′∩H​\(Θ,φ\)≠∅\\Theta^\{\\prime\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset; finally, by the Lindenbaum\-Łoś theorem itself, we have thatΘ′\\Theta^\{\\prime\}is consistent, that is, \(iii\)Θ′⊬𝕃⊥\\Theta^\{\\prime\}\\nvdash\_\{\\mathbb\{L\}\}\\bot, and that it is maximal, i\.e\., \(iv\) there is noΘ′′⊃Θ′\\Theta^\{\\prime\\prime\}\\supset\\Theta^\{\\prime\}that satisfies \(i\), \(ii\) and \(iii\)\. Therefore, we can conclude that there is always aΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, so thatΘ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptyset\. ∎

###### Lemma B\.4\.

LetΘ⊤φ\\Theta\\top\\varphiandΘ⊤ψ\\Theta\\top\\psibe the surplus sets ofΘ\\Thetawith respect toφ\\varphiandψ\\psi, respectively \(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\)\. It is the case that if⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenΘ⊤φ=Θ⊤ψ\\Theta\\top\\varphi=\\Theta\\top\\psi\.

###### Proof\.

Since, by definition,\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and\(Θ,ψ\)p𝔸​ℙ​𝔼\(\\Theta,\\psi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}are the case, then, by lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7), it is always the case thatΘ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptysetandΘ⊤ψ≠∅\\Theta\\top\\psi\\neq\\emptyset\. Therefore, this condition does not need to be demonstrated\. Consider⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi\. By lemma[5\.4](https://arxiv.org/html/2607.09729#S5.Thmtheorem4), it is the case thatH​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\)\. Now consider anyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, i\.e\., aΘ′\\Theta^\{\\prime\}that satisfies conditions \(i\)\-\(iv\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\. SinceH​\(Θ,φ\)=H​\(Θ,ψ\)H\(\\Theta,\\varphi\)=H\(\\Theta,\\psi\), then, considering in particular condition \(ii\),Θ′∩H​\(Θ,φ\)=Θ′∩H​\(Θ,ψ\)\\Theta^\{\\prime\}\\cap H\(\\Theta,\\varphi\)=\\Theta^\{\\prime\}\\cap H\(\\Theta,\\psi\)\. In other words,Θ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psi\. Hence,Θ⊤φ=Θ⊤ψ\\Theta\\top\\varphi=\\Theta\\top\\psi\. ∎

###### Proof\.

By definition[6\.9](https://arxiv.org/html/2607.09729#S6.Thmtheorem9),γ​\(Θ,φ\)⊆Θ⊤φ\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphiand, by lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7),Θ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptyset\. Hence,⋂γ​\(Θ,φ\)≠∅\\bigcap\\gamma\(\\Theta,\\varphi\)\\neq\\emptysetand this condition does not need to be demonstrated\. Thus, we need to show that⋂γ​\(Θ,φ\)=C​n𝕃​\(⋂γ​\(Θ,φ\)\)\\bigcap\\gamma\(\\Theta,\\varphi\)=Cn\_\{\\mathbb\{L\}\}\(\\bigcap\\gamma\(\\Theta,\\varphi\)\)\. By observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3), everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiis a belief set, i\.e\.,Θ′=C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\. Thus, we have thatγ​\(Θ,φ\)=\{Θ′,Θ′′,…\}\\gamma\(\\Theta,\\varphi\)=\\\{\\Theta^\{\\prime\},\\Theta^\{\\prime\\prime\},\.\.\.\\\}\. Consequently,⋂γ​\(Θ,φ\)=⋂\{Θ′,Θ′′,…\}=Θ′∩Θ′′∩…\\bigcap\\gamma\(\\Theta,\\varphi\)=\\bigcap\\\{\\Theta^\{\\prime\},\\Theta^\{\\prime\\prime\},\.\.\.\\\}=\\Theta^\{\\prime\}\\cap\\Theta^\{\\prime\\prime\}\\cap\.\.\.\. Since deductive closure underC​nCnis preserved under intersections, thenC​n𝕃​\(⋂γ​\(Θ,φ\)\)=C​n𝕃​\(Θ′\)∩C​n𝕃​\(Θ′′\)∩…=Θ′∩Θ′′∩…=⋂γ​\(Θ,φ\)Cn\_\{\\mathbb\{L\}\}\(\\bigcap\\gamma\(\\Theta,\\varphi\)\)=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\\cap Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\\prime\}\)\\cap\.\.\.=\\Theta^\{\\prime\}\\cap\\Theta^\{\\prime\\prime\}\\cap\.\.\.=\\bigcap\\gamma\(\\Theta,\\varphi\), as required\. ∎

Theorem[6\.11](https://arxiv.org/html/2607.09729#S6.Thmtheorem11):For everyΘ\\Thetaandφ\\varphisuch that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\},○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}is anAGMpa​b​dp\_\{abd\}partial meet abductive expansion operation ofΘ\\Thetawith respect toφ\\varphi—definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)—if and only if○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of abductive expansion ofΘ\\Thetawith respect toφ\\varphi\.

###### Proof\.

Due to lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7)and corollary[5\.9](https://arxiv.org/html/2607.09729#S5.Thmtheorem9), the function○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}, on both sides, always obtains a non\-empty set\. Hence, such conditions do not need to be considered\.

\(Construction⇒\\Rightarrowpostulates\):

\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}is a belief set\.

By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10),Θφ○⊕=⋂γ​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\(\\Theta,\\varphi\)and by observation[B\.5](https://arxiv.org/html/2607.09729#A2.Thmtheorem5),⋂γ​\(Θ,φ\)\\bigcap\\gamma\(\\Theta,\\varphi\)is a belief set\. Therefore,Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}is a belief set\.

\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\)Θ⊂Θφ○⊕\\Theta\\subset\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\.

By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10),Θφ○⊕=⋂γ​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\(\\Theta,\\varphi\)and by definition[6\.9](https://arxiv.org/html/2607.09729#S6.Thmtheorem9),γ​\(Θ,φ\)⊆Θ⊤φ\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi\. Since, by condition \(i\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1), for anyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, it is the case thatΘ⊂Θ′\\Theta\\subset\\Theta^\{\\prime\}, thenΘ⊂⋂γ​\(Θ,φ\)\\Theta\\subset\\bigcap\\gamma\(\\Theta,\\varphi\)\. Thus,Θ⊂Θφ○⊕\\Theta\\subset\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\.

\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)Θφ○⊕∩H​\(Θ,φ\)≠∅\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\.

By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10),Θφ○⊕=⋂γ​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\(\\Theta,\\varphi\)and by definition[6\.9](https://arxiv.org/html/2607.09729#S6.Thmtheorem9),γ​\(Θ,φ\)⊆Θ⊤φ\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi\. By lemma[6\.5](https://arxiv.org/html/2607.09729#S6.Thmtheorem5), forα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. In particular, therefore,α∨φ∈Θ′\\alpha\\lor\\varphi\\in\\Theta^\{\\prime\}for everyΘ′∈γ​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\(\\Theta,\\varphi\)\. By lemma[5\.7](https://arxiv.org/html/2607.09729#S5.Thmtheorem7), ifα∈H​\(Θ,φ\)\\alpha\\in H\(\\Theta,\\varphi\), it is the case thatα∨φ∈H​\(Θ,φ\)\\alpha\\lor\\varphi\\in H\(\\Theta,\\varphi\)\. Therefore,α∨φ∈⋂γ​\(Θ,φ\)∩H​\(Θ,φ\)≠∅\\alpha\\lor\\varphi\\in\\bigcap\\gamma\(\\Theta,\\varphi\)\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\. Hence,Θφ○⊕∩H​\(Θ,φ\)≠∅\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\.

\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\)Θφ○⊕⊬𝕃⊥\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\.

By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10),Θφ○⊕=⋂γ​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\(\\Theta,\\varphi\)and by definition[6\.9](https://arxiv.org/html/2607.09729#S6.Thmtheorem9),γ​\(Θ,φ\)⊆Θ⊤φ\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi\. Suppose, byreductio, that⋂γ\(Θ,φ\)⊢𝕃⊥\\bigcap\\gamma\(\\Theta,\\varphi\)\\vdash\_\{\\mathbb\{L\}\}\\bot\. In this case, for\{Θ′,Θ′′,…\}⊆γ​\(Θ,φ\)\\\{\\Theta^\{\\prime\},\\Theta^\{\\prime\\prime\},\.\.\.\\\}\\subseteq\\gamma\(\\Theta,\\varphi\), we have thatΘ′∩Θ′′∩…⊢𝕃⊥\\Theta^\{\\prime\}\\cap\\Theta^\{\\prime\\prime\}\\cap\.\.\.\\vdash\_\{\\mathbb\{L\}\}\\bot\. By observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3), for everyΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi—in particular, for everyΘ′∈γ​\(Θ,φ\)⊆Θ⊤φ\\Theta^\{\\prime\}\\in\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi—it is the case thatΘ′=C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\. Then, we have thatCn𝕃\(Θ′\)∩Cn𝕃\(Θ′′\)∩…⊢𝕃⊥Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\)\\cap Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\\prime\}\)\\cap\.\.\.\\vdash\_\{\\mathbb\{L\}\}\\bot\. Thus, for allΘ′∈γ​\(Θ,φ\)⊆Θ⊤φ\\Theta^\{\\prime\}\\in\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi, we have that⊥∈Cn𝕃\(Θ′\)\\bot\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), a contradiction with condition \(iii\) of definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)\. Therefore,⋂γ​\(Θ,φ\)⊬𝕃⊥\\bigcap\\gamma\(\\Theta,\\varphi\)\\nvdash\_\{\\mathbb\{L\}\}\\botand, as required,Θφ○⊕⊬𝕃⊥\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\nvdash\_\{\\mathbb\{L\}\}\\bot\.

\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)If⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi, thenΘφ○⊕=Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\.

By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10),Θφ○⊕=⋂γ​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\(\\Theta,\\varphi\)\. Suppose⊢𝕃φ↔ψ\\vdash\_\{\\mathbb\{L\}\}\\varphi\\leftrightarrow\\psi\. Then, by lemma[B\.4](https://arxiv.org/html/2607.09729#A2.Thmtheorem4),Θ⊤φ=Θ⊤ψ\\Theta\\top\\varphi=\\Theta\\top\\psi\. Since, by definition[6\.9](https://arxiv.org/html/2607.09729#S6.Thmtheorem9),γ​\(Θ,φ\)⊆Θ⊤φ\\gamma\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphiandγ\\gammais a function, thenγ​\(Θ,φ\)=γ​\(Θ,ψ\)\\gamma\(\\Theta,\\varphi\)=\\gamma\(\\Theta,\\psi\)\. Therefore,⋂γ​\(Θ,φ\)=⋂γ​\(Θ,ψ\)\\bigcap\\gamma\(\\Theta,\\varphi\)=\\bigcap\\gamma\(\\Theta,\\psi\)\. Finally, then,Θφ○⊕=Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\.

\(Postulates⇒\\Rightarrowconstruction\):

Let○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}be a function that satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)\. Letγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}be the following function:

γ○⊕​\(Θ,φ\)=\{Θ′:Θ′∈Θ⊤φ​and​Θφ○⊕⊆Θ′\}\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)=\\\{\\Theta^\{\\prime\}:\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\ \\text\{and\}\\ \\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}\\\}
We need to show that \(1\)γ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}is a selection function \(definition[6\.9](https://arxiv.org/html/2607.09729#S6.Thmtheorem9)\) and \(2\) thatΘφ○⊕=⋂γ○⊕​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\):

\(1\)γ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}is a selection function:

We have thatγ○⊕​\(Θ,φ\)⊆Θ⊤φ\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphiis immediate from the construction\. By lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7),Θ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptyset\. By postulates\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\),\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\)and\(Θ○⊕​4\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}4\), and conditions \(i\)\-\(iii\) of definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2), we have thatΘφ○⊕⊆Θ′∈Θ⊤φ\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}\\in\\Theta\\top\\varphifor someΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. By the definition ofγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\},Θ′∈γ○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\. Hence,γ○⊕​\(Θ,φ\)≠∅\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\neq\\emptyset\.

\(2\)Θφ○⊕=⋂γ○⊕​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\):

Θφ○⊕⊆⋂γ○⊕​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)is directly satisfied by the definition ofγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}, sinceΘφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}for everyΘ′∈γ○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\. It is therefore only necessary to show that⋂γ○⊕​\(Θ,φ\)⊆Θφ○⊕\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. Suppose, byreductio, that there exists aδ∈⋂γ○⊕​\(Θ,φ\)\\delta\\in\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\), butδ∉Θφ○⊕\\delta\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. By the definition ofγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}, the setsΘ′\\Theta^\{\\prime\}belonging toγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}must satisfy: \(a\)Θ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiand \(b\)Θφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}\. First, let us analyze \(b\)\. By the Lindenbaum\-Łoś theorem[A\.8](https://arxiv.org/html/2607.09729#A1.Thmtheorem8), ifδ∉Θφ○⊕\\delta\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\(thereductioassumption\), then there exists aδ\\delta\-saturated setΘ¯\\bar\{\\Theta\}\(definition[A\.5](https://arxiv.org/html/2607.09729#A1.Thmtheorem5)\) such thatΘφ○⊕⊆Θ¯⊆ℒΣ∗\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\bar\{\\Theta\}\\subseteq\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\. Therefore, theδ\\delta\-saturated setΘ¯\\bar\{\\Theta\}satisfies condition \(b\)\. Now let us analyze condition \(a\) forΘ¯\\bar\{\\Theta\}according to criteria \(i\)\-\(iv\) of definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\. By postulate\(Θ○⊕​2\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}2\), we directly haveΘ⊂Θφ○⊕⊆Θ¯\\Theta\\subset\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\bar\{\\Theta\}and thus \(i\)Θ⊂Θ¯\\Theta\\subset\\bar\{\\Theta\}\. By postulate\(Θ○⊕​3\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}3\), we haveΘφ○⊕∩H​\(Θ,φ\)≠∅\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\. SinceΘφ○⊕⊆Θ¯\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\bar\{\\Theta\}, then \(ii\)Θ¯∩H​\(Θ,φ\)≠∅\\bar\{\\Theta\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset\. SinceΘ¯\\bar\{\\Theta\}isδ\\delta\-saturated, then \(iii\)⊥∉Cn𝕃\(Θ¯\)\\bot\\notin Cn\_\{\\mathbb\{L\}\}\(\\bar\{\\Theta\}\)\. Finally, sinceΘ¯\\bar\{\\Theta\}is maximal, then \(iv\) there is noΘ′′⊃Θ¯\\Theta^\{\\prime\\prime\}\\supset\\bar\{\\Theta\}that satisfies \(i\), \(ii\) and \(iii\) \(i\.e\., for anyΘ′′⊃Θ¯\\Theta^\{\\prime\\prime\}\\supset\\bar\{\\Theta\},⊥∈Cn𝕃\(Θ′′\)\\bot\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\\prime\}\)\)\. Hence,Θ¯∈Θ⊤φ\\bar\{\\Theta\}\\in\\Theta\\top\\varphi, which satisfies condition \(a\)\. Thus,Θ¯∈γ○⊕​\(Θ,φ\)\\bar\{\\Theta\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\. However, by the initial assumption,δ∈⋂γ○⊕​\(Θ,φ\)\\delta\\in\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\), thereforeδ∈Θ¯\\delta\\in\\bar\{\\Theta\}\. A contradiction with the fact thatΘ¯\\bar\{\\Theta\}isδ\\delta\-saturated\. Hence,Θφ○⊕=⋂γ○⊕​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)as required\.

∎

###### Proof\.

By definitions[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)and[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2),\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}and\(Θ,ψ\)p𝔸​ℙ​𝔼\(\\Theta,\\psi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\}are the case\. SupposeΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiandψ∈Θ′\\psi\\in\\Theta^\{\\prime\}\. SinceΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, then: \(i\)Θ⊂Θ′\\Theta\\subset\\Theta^\{\\prime\}, \(ii\)Θ′∩H​\(Θ,φ\)≠∅\\Theta^\{\\prime\}\\cap H\(\\Theta,\\varphi\)\\neq\\emptyset, \(iii\)⊥∉Cn𝕃\(Θ′\)\\bot\\notin Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), and \(iv\)Θ′\\Theta^\{\\prime\}is maximal\. Sinceψ∈Θ′\\psi\\in\\Theta^\{\\prime\}, by Lemma[6\.5](https://arxiv.org/html/2607.09729#S6.Thmtheorem5), we have thatα∨ψ∈Θ′\\alpha\\lor\\psi\\in\\Theta^\{\\prime\}for someα∈H​\(Θ,ψ\)\\alpha\\in H\(\\Theta,\\psi\)\(considering, by Lemma[5\.2](https://arxiv.org/html/2607.09729#S5.Thmtheorem2), thatH​\(Θ,ψ\)≠∅H\(\\Theta,\\psi\)\\neq\\emptyset\)\. By Lemma[5\.7](https://arxiv.org/html/2607.09729#S5.Thmtheorem7), we have thatα∨ψ∈H​\(Θ,ψ\)\\alpha\\lor\\psi\\in H\(\\Theta,\\psi\)\. Therefore,α∨ψ∈Θ′∩H​\(Θ,ψ\)\\alpha\\lor\\psi\\in\\Theta^\{\\prime\}\\cap H\(\\Theta,\\psi\), soΘ′∩H​\(Θ,ψ\)≠∅\\Theta^\{\\prime\}\\cap H\(\\Theta,\\psi\)\\neq\\emptyset\. Thus,Θ′\\Theta^\{\\prime\}satisfies conditions \(i\)\-\(iv\) of Definition[6\.1](https://arxiv.org/html/2607.09729#S6.Thmtheorem1)forψ\\psi, i\.e\.,Θ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psi\. ∎

###### Lemma B\.7\.

LetΘ⊤φ\\Theta\\top\\varphi,Θ⊤ψ\\Theta\\top\\psiandΘ⊤φ∨ψ\\Theta\\top\\varphi\\lor\\psibe surplus sets ofΘ\\Thetawith respect toφ\\varphi,ψ\\psiandφ∨ψ\\varphi\\lor\\psi, respectively \(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\)\. It is the case thatΘ⊤φ∨ψ=Θ⊤φ∪Θ⊤ψ\\Theta\\top\\varphi\\lor\\psi=\\Theta\\top\\varphi\\cup\\Theta\\top\\psi\.

###### Proof\.

By lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7),Θ⊤φ≠∅\\Theta\\top\\varphi\\neq\\emptyset,Θ⊤ψ≠∅\\Theta\\top\\psi\\neq\\emptysetandΘ⊤φ∨ψ≠∅\\Theta\\top\\varphi\\lor\\psi\\neq\\emptyset\. \(1\)Θ⊤φ∨ψ⊆Θ⊤φ∪Θ⊤ψ\\Theta\\top\\varphi\\lor\\psi\\subseteq\\Theta\\top\\varphi\\cup\\Theta\\top\\psi:

IfΘ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi, thenΘ′∈Θ⊤φ∪Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\cup\\Theta\\top\\psi\. SupposeΘ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\. By observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4), it is the case thatφ∨ψ∈Θ′\\varphi\\lor\\psi\\in\\Theta^\{\\prime\}\. SinceΘ′\\Theta^\{\\prime\}is maximal, ifφ∉Θ′\\varphi\\notin\\Theta^\{\\prime\}, then∼​φ∈Θ′\\mathord\{\\sim\}\\varphi\\in\\Theta^\{\\prime\}\. In this case, since, by observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3),Θ′=C​n𝕃​\(Θ′\)\\Theta^\{\\prime\}=Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\prime\}\), thenψ∈Θ′\\psi\\in\\Theta^\{\\prime\}\. Analogously, ifψ∉Θ′\\psi\\notin\\Theta^\{\\prime\}, then∼​ψ∈Θ′\\mathord\{\\sim\}\\psi\\in\\Theta^\{\\prime\}andφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\. Therefore, eitherφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}orψ∈Θ′\\psi\\in\\Theta^\{\\prime\}\. By observation[B\.6](https://arxiv.org/html/2607.09729#A2.Thmtheorem6), eitherΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiorΘ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psi\. Hence, as required,Θ′∈Θ⊤φ∪Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\cup\\Theta\\top\\psi\.

\(2\)Θ⊤φ∪Θ⊤ψ⊆Θ⊤φ∨ψ\\Theta\\top\\varphi\\cup\\Theta\\top\\psi\\subseteq\\Theta\\top\\varphi\\lor\\psi:

IfΘ′∈Θ⊤φ∪Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\cup\\Theta\\top\\psi, thenΘ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\. SupposeΘ′∈Θ⊤φ∪Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\cup\\Theta\\top\\psi\. ThenΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiorΘ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psi\. Take the first disjunct\. IfΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, then, by observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4),φ∈Θ′\\varphi\\in\\Theta^\{\\prime\}and by observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3),φ∨ψ∈Θ′\\varphi\\lor\\psi\\in\\Theta^\{\\prime\}\. In this case, by observation[B\.6](https://arxiv.org/html/2607.09729#A2.Thmtheorem6),Θ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\. The same reasoning can be applied to the second disjunctΘ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psi\. Hence, as required,Θ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\.

∎

###### Lemma B\.8\.

LetΘ⊤φ\\Theta\\top\\varphibe the surplus set ofΘ\\Thetawith respect toφ\\varphi\(definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\) andγ⩽\\gamma\_\{\\leqslant\}be a selection function established by the \(relational and transitive\) marking\-off identity⩽\\leqslant\(definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\)\. It is the case that ifΘ⊤φ∩γ⩽​\(Θ,φ∨ψ\)≠∅\\Theta\\top\\varphi\\cap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\neq\\emptyset, thenγ⩽​\(Θ,φ\)⊆γ⩽​\(Θ,φ∨ψ\)\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\.

###### Proof\.

ConsiderΘ⊤φ∩γ⩽​\(Θ,φ∨ψ\)≠∅\\Theta\\top\\varphi\\cap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\neq\\emptyset\. We need to show that ifΘ′∈γ⩽​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\), thenΘ′∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. SupposeΘ′∈γ⩽​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)and, byreductio,Θ′∉γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\notin\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. By definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)and part \(2\) of lemma[B\.7](https://arxiv.org/html/2607.09729#A2.Thmtheorem7), we have thatΘ′∈γ⩽​\(Θ,φ\)⊆Θ⊤φ⊆Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi\\subseteq\\Theta\\top\\varphi\\lor\\psi\. Therefore, we have thatΘ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi, but, by thereductiohypothesis,Θ′∉γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\notin\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. In this case, by definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)of relational selection function, there existsΘ′′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\\lor\\psisuch thatΘ′<Θ′′\\Theta^\{\\prime\}<\\Theta^\{\\prime\\prime\}\(strictly better\), i\.e\.,Θ′⩽Θ′′\\Theta^\{\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}butΘ′′⩽̸Θ′\\Theta^\{\\prime\\prime\}\\nleqslant\\Theta^\{\\prime\}\(a\)\. Since, by the initial hypothesis,Θ⊤φ∩γ⩽​\(Θ,φ∨ψ\)≠∅\\Theta\\top\\varphi\\cap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\neq\\emptyset, then there is at least oneΘ\#∈Θ⊤φ\\Theta^\{\\\#\}\\in\\Theta\\top\\varphisuch thatΘ\#∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\\#\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. SinceΘ\#\\Theta^\{\\\#\}is⩽\\leqslant\-best inΘ⊤φ∨ψ\\Theta\\top\\varphi\\lor\\psi\(i\.e\.,Θ\#∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\\#\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\), we have thatΘ′′⩽Θ\#\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\\#\}\(b\)\. SinceΘ′∈γ⩽​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\(i\.e\.,Θ′\\Theta^\{\\prime\}is⩽\\leqslant\-best inΘ⊤φ\\Theta\\top\\varphi\) andΘ\#∈Θ⊤φ\\Theta^\{\\\#\}\\in\\Theta\\top\\varphi, we have thatΘ\#⩽Θ′\\Theta^\{\\\#\}\\leqslant\\Theta^\{\\prime\}\(c\)\. By the transitivity of⩽\\leqslant, we have, from \(b\) and \(c\),Θ′′⩽Θ\#⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\\#\}\\leqslant\\Theta^\{\\prime\}\. A contradiction with \(a\)\. Hence,Θ′∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. ∎

###### Lemma B\.9\.

Letγ⩽\\gamma\_\{\\leqslant\}be a selection function established by the marking\-off identity⩽\\leqslant\(definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\)\. Ifγ⩽​\(Θ,φ\)⊆γ⩽​\(Θ,ψ\)\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\), then⋂γ⩽​\(Θ,ψ\)⊆⋂γ⩽​\(Θ,φ\)\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\)\\subseteq\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\.

###### Proof\.

Considerγ⩽​\(Θ,φ\)⊆γ⩽​\(Θ,ψ\)\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\)\. We need to show that ifδ∈⋂γ⩽​\(Θ,ψ\)\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\), thenδ∈⋂γ⩽​\(Θ,φ\)\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\. Supposeδ∈⋂γ⩽​\(Θ,ψ\)\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\)\. Then,δ∈Θ′\\delta\\in\\Theta^\{\\prime\}for everyΘ′∈γ⩽​\(Θ,ψ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\)\. Sinceγ⩽​\(Θ,φ\)⊆γ⩽​\(Θ,ψ\)\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\gamma\_\{\\leqslant\}\(\\Theta,\\psi\), thenδ∈Θ′\\delta\\in\\Theta^\{\\prime\}for everyΘ′∈γ⩽​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\. Therefore,δ∈⋂γ⩽​\(Θ,φ\)\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\. ∎

Lemma[6\.13](https://arxiv.org/html/2607.09729#S6.Thmtheorem13):Any relational partial meet abductive expansion function, i\.e\.,Θφ○⊕=⋂γ⩽​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\(definitions[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)and[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\), satisfies postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)\.

###### Proof\.

Let○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}be a relationalAGMpa​b​dp\_\{abd\}partial meet abductive expansion function\. We need to show that○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}satisfies postulate\(Θ○⊕​6\)​Θφ○⊕⊆C​n𝕃​\(Θφ∨ψ○⊕∪\{φ\}\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\)\\ \\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\varphi\\\}\)\. By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10), it is the case thatΘφ○⊕=⋂γ⩽​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)and thatΘφ∨ψ○⊕=⋂γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Therefore, according to postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\), we need to show that⋂γ⩽​\(Θ,φ\)⊆C​n𝕃​\(⋂γ⩽​\(Θ,φ∨ψ\)∪\{φ\}\)\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\cup\\\{\\varphi\\\}\)\. Suppose, then, thatδ∈⋂γ⩽​\(Θ,φ\)\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\. We need to show thatδ∈C​n𝕃​\(⋂γ⩽​\(Θ,φ∨ψ\)∪\{φ\}\)\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\cup\\\{\\varphi\\\}\)\. To do this, consider anyΘ′∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. By definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)of the marking\-off identity,Θ′\\Theta^\{\\prime\}is⩽\\leqslant\-best inΘ⊤φ∨ψ\\Theta\\top\\varphi\\lor\\psi, i\.e\.,Θ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}for everyΘ′′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\(a\)\. It is also the case that eitherφ∉Θ′\\varphi\\notin\\Theta^\{\\prime\}orφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\. Consider the first disjunct\. In this case, sinceΘ′\\Theta^\{\\prime\}is maximal, then∼​φ∈Θ′\\mathord\{\\sim\}\\varphi\\in\\Theta^\{\\prime\}\. Since, by observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3),Θ′∈T​h​\(𝕃\)\\Theta^\{\\prime\}\\in Th\(\\mathbb\{L\}\), then∼​φ∨δ∈Θ′\\mathord\{\\sim\}\\varphi\\lor\\delta\\in\\Theta^\{\\prime\}and, since⊢𝕃∼φ∨δ↔φ→δ\\vdash\_\{\\mathbb\{L\}\}\\mathord\{\\sim\}\\varphi\\lor\\delta\\leftrightarrow\\varphi\\to\\delta, thenφ→δ∈Θ′\\varphi\\to\\delta\\in\\Theta^\{\\prime\}\(b\)\. In the case of the second disjunct,φ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\. SinceΘ′∈γ⩽​\(Θ,φ∨ψ\)⊆Θ⊤φ∨ψ\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\subseteq\\Theta\\top\\varphi\\lor\\psiandφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}, then by observation[B\.6](https://arxiv.org/html/2607.09729#A2.Thmtheorem6)\(applied withφ∨ψ\\varphi\\lor\\psiin the role ofφ\\varphiandφ\\varphiin the role ofψ\\psi\), we have thatΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. Consider some arbitraryΘ\#∈Θ⊤φ\\Theta^\{\\\#\}\\in\\Theta\\top\\varphi\. Since, by lemma[B\.7](https://arxiv.org/html/2607.09729#A2.Thmtheorem7),Θ⊤φ⊆Θ⊤φ∨ψ\\Theta\\top\\varphi\\subseteq\\Theta\\top\\varphi\\lor\\psi, thenΘ\#∈Θ⊤φ∨ψ\\Theta^\{\\\#\}\\in\\Theta\\top\\varphi\\lor\\psi\. Considering \(a\), we have, by definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12), thatΘ\#⩽Θ′\\Theta^\{\\\#\}\\leqslant\\Theta^\{\\prime\}\. Therefore, sinceΘ\#⩽Θ′\\Theta^\{\\\#\}\\leqslant\\Theta^\{\\prime\}for everyΘ\#∈Θ⊤φ\\Theta^\{\\\#\}\\in\\Theta\\top\\varphi\(andΘ\#\\Theta^\{\\\#\}was taken arbitrarily\), we have thatΘ′\\Theta^\{\\prime\}is⩽\\leqslant\-best inΘ⊤φ\\Theta\\top\\varphi, i\.e\.,Θ′∈γ⩽​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\. Since, by hypothesis,δ∈⋂γ⩽​\(Θ,φ\)\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\), andΘ′∈γ⩽​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\), thenδ∈Θ′\\delta\\in\\Theta^\{\\prime\}\. Thus, sinceΘ′∈T​h​\(𝕃\)\\Theta^\{\\prime\}\\in Th\(\\mathbb\{L\}\), thenφ→δ∈Θ′\\varphi\\to\\delta\\in\\Theta^\{\\prime\}\(c\)\. From \(b\) and \(c\), we have that in both disjuncts, it is the case thatφ→δ∈Θ′\\varphi\\to\\delta\\in\\Theta^\{\\prime\}for everyΘ′∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Therefore,φ→δ∈⋂γ⩽​\(Θ,φ∨ψ\)\\varphi\\to\\delta\\in\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Byinclusion, we have thatφ→δ∈C​n𝕃​\(⋂γ⩽​\(Θ,φ∨ψ\)\)\\varphi\\to\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\)and, by thededuction theorem, we have, as required, thatδ∈C​n𝕃​\(⋂γ⩽​\(Θ,φ∨ψ\)∪\{φ\}\)\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\cup\\\{\\varphi\\\}\)\. ∎

Lemma[6\.14](https://arxiv.org/html/2607.09729#S6.Thmtheorem14):Any transitively relational partial meet abductive expansion function \(definitions[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)and[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\) satisfies postulate\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)\.

###### Proof\.

Let○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}be a transitively relational partial meet abductive expansion function\. We need to show that○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}satisfies postulate\(Θ○⊕​7\)​If​∼​φ∉Θφ∨ψ○⊕\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)\\ \\text\{If \}\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, thenΘφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. Suppose∼​φ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. By definition[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10), it is the case thatΘφ○⊕=⋂γ⩽​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)and thatΘφ∨ψ○⊕=⋂γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Since, by assumption,∼​φ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}, then∼​φ∉⋂γ⩽​\(Θ,φ∨ψ\)\\mathord\{\\sim\}\\varphi\\notin\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Thus, there is aΘ′∈γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\in\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)such that∼​φ∉Θ′\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\prime\}\. SinceΘ′\\Theta^\{\\prime\}is maximal, thenφ∈Θ′\\varphi\\in\\Theta^\{\\prime\}\. By observation[B\.6](https://arxiv.org/html/2607.09729#A2.Thmtheorem6), we have thatΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiand, therefore, thatΘ′∈Θ⊤φ∩γ⩽​\(Θ,φ∨ψ\)\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\cap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Thus,Θ⊤φ∩γ⩽​\(Θ,φ∨ψ\)≠∅\\Theta\\top\\varphi\\cap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\neq\\emptysetand, by lemma[B\.8](https://arxiv.org/html/2607.09729#A2.Thmtheorem8), we have thatγ⩽​\(Θ,φ\)⊆γ⩽​\(Θ,φ∨ψ\)\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\. Therefore, by lemma[B\.9](https://arxiv.org/html/2607.09729#A2.Thmtheorem9), we have that⋂γ⩽​\(Θ,φ∨ψ\)⊆⋂γ⩽​\(Θ,φ\)\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\subseteq\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)\. Hence, as desired,Θφ∨ψ○⊕=⋂γ⩽​\(Θ,φ∨ψ\)⊆⋂γ⩽​\(Θ,φ\)=Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}=\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\\lor\\psi\)\\subseteq\\bigcap\\gamma\_\{\\leqslant\}\(\\Theta,\\varphi\)=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. ∎

Theorem[6\.15](https://arxiv.org/html/2607.09729#S6.Thmtheorem15):For everyΘ\\Thetaandφ\\varphisuch that\(Θ,φ\)p𝔸​ℙ​𝔼\(\\Theta,\\varphi\)^\{\\scriptscriptstyle\{\\mathbb\{APE\}\}\}\_\{p\},○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}is a transitively relationalAGMpa​b​dp\_\{abd\}partial meet abductive expansion operation \(definitions[6\.10](https://arxiv.org/html/2607.09729#S6.Thmtheorem10)and[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\) if, and only if,○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}satisfies postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5)of abductive expansion onΘ\\Theta\.

###### Proof\.

Due to lemma[6\.7](https://arxiv.org/html/2607.09729#S6.Thmtheorem7)and corollary[5\.9](https://arxiv.org/html/2607.09729#S5.Thmtheorem9), the function○⊕\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\bigcirc$\\cr$\\oplus$\\crcr\}\}\}, on both sides, always obtains a non\-empty set\. Hence, such conditions do not need to be considered\.

\(Construction⇒\\Rightarrowpostulates\):

Result already obtained by theorem[6\.11](https://arxiv.org/html/2607.09729#S6.Thmtheorem11)and lemmas[6\.13](https://arxiv.org/html/2607.09729#S6.Thmtheorem13)and[6\.14](https://arxiv.org/html/2607.09729#S6.Thmtheorem14)\.

\(Postulates⇒\\Rightarrowconstruction\):

Theorem[6\.11](https://arxiv.org/html/2607.09729#S6.Thmtheorem11)—postulates⇒\\Rightarrowconstruction direction—already shows that, assuming postulates\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\)\-\(Θ○⊕​5\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}5\)from definition[5\.5](https://arxiv.org/html/2607.09729#S5.Thmtheorem5),γ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}is a selection function and thatΘφ○⊕=⋂γ○⊕​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\. Thus, we need to show thatγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}is transitively relational\.

We define⩽\\leqslanton all maximal non\-trivial supersets ofΘ\\Thetaas follows:

For allΘ′,Θ′′∈T​h​\(𝕃\)\\Theta^\{\\prime\},\\Theta^\{\\prime\\prime\}\\in Th\(\\mathbb\{L\}\),Θ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}if and only if the following three conditions are satisfied:

\(i\)Θ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphifor someφ∈ℒΣ∗\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\};

\(ii\)Θ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiandΘφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}for someφ∈ℒΣ∗\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\};

\(iii\) For everyφ∈ℒΣ∗\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}, ifΘ′,Θ′′∈Θ⊤φ\\Theta^\{\\prime\},\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphiandΘφ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\\prime\}, thenΘφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}\.

We then define the relational selection functionγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}as the completion ofγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}:

γ⩽○⊕​\(Θ,φ\)=\{Θ′∈Θ⊤φ:⋂γ○⊕​\(Θ,φ\)⊆Θ′\}​for all​φ∈ℒΣ∗\.\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)=\\\{\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi:\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\subseteq\\Theta^\{\\prime\}\\\}\\text\{ for all \}\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\.
Thus defined, the functionγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}is a selection function—sinceγ⩽○⊕​\(Θ,φ\)⊆γ○⊕​\(Θ,φ\)⊆Θ⊤φ\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphiandγ⩽○⊕​\(Θ,φ\)≠∅\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)\\neq\\emptyset—and determines the samepartial meetabductive expansion operation asγ○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}—since⋂γ⩽○⊕​\(Θ,φ\)=⋂γ○⊕​\(Θ,φ\)=Θφ○⊕\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)=\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)=\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. With this in mind, we need to show that:

\(1\) The relation⩽\\leqslantis relational with respect toγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}, i\.e\., it satisfies the Marking\-Off Identity \(definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\);

\(2\) The relation⩽\\leqslantis transitive with respect toγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}, for allφ∈ℒΣ∗\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\.

\(1\) The relation⩽\\leqslantis relational with respect toγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(satisfies the Marking\-Off Identity\)595959Recall definition[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\(p\.[6\.12](https://arxiv.org/html/2607.09729#S6.Thmtheorem12)\)\. The Marking\-Off Identity with respect toγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}is given by:γ⩽○⊕​\(Θ,φ\)=\{Θ′∈Θ⊤φ:Θ′′⩽Θ′​for all​Θ′′∈Θ⊤φ\}\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)=\\\{\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi:\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}\\text\{ for all \}\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\\\}\.: To show that⩽\\leqslantsatisfies the Marking\-Off Identity, we need to verify two cases: \(1A\) ifΘ′∈γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)andΘ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi, thenΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}; \(1B\) ifΘ′∉γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\notin\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)andΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, thenΘ′≺Θ′′\\Theta^\{\\prime\}\\prec\\Theta^\{\\prime\\prime\}for someΘ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\. To prove \(1A\), supposeΘ′∈γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)andΘ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\. We need to show thatΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}, according to conditions \(i\)\-\(iii\) of⩽\\leqslantestablished above\. By assumption, condition \(i\)Θ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphifor someφ\\varphiis immediately satisfied\. SinceΘ′∈γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\), then, by the definition ofγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}, it is the case thatΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiand⋂γ○⊕​\(Θ,φ\)⊆Θ′\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\subseteq\\Theta^\{\\prime\}, withΘφ○⊕=⋂γ○⊕​\(Θ,φ\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}=\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\. Hence, condition \(ii\)Θ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphiandΘφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}for someφ\\varphiis also satisfied\. It remains to verify condition \(iii\)\. Letψ∈ℒΣ∗\\psi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}and supposeΘ′′,Θ′∈Θ⊤ψ\\Theta^\{\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\psiandΘψ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\\prime\}—the antecedent of condition \(iii\) in the definition of⩽\\leqslant\. To obtainΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}, we must therefore prove its consequent, i\.e\., thatΘψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\}\. More precisely, we need to show that ifδ∈Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}, thenδ∈Θ′\\delta\\in\\Theta^\{\\prime\}\. To do this, consider proposition[B\.3](https://arxiv.org/html/2607.09729#A2.Thmtheorem3)\(i\)Θφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}orΘφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\. Taking the first disjunct and the recently obtained condition \(ii\), we haveΘφ∨ψ○⊕⊆Θφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}forΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. Taking the second disjunct, the assumption thatΘψ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\\prime\}and condition \(i\) just obtained, we getΘφ∨ψ○⊕⊆Θψ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\\prime\}forΘ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\. In this latter case, however, we have that∼​φ∉Θφ∨ψ○⊕\\mathord\{\\sim\}\\varphi\\notin\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\(otherwise,Θ′′∉Θ⊤φ\\Theta^\{\\prime\\prime\}\\notin\\Theta\\top\\varphi, since, by Observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4),φ∈Θ′′\\varphi\\in\\Theta^\{\\prime\\prime\}and thus we would violate definition[6\.2](https://arxiv.org/html/2607.09729#S6.Thmtheorem2)\(iii\)\)\. Thus, by postulate\(Θ○⊕​7\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}7\), we haveΘφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\. Hence, from both disjuncts, we obtainΘφ∨ψ○⊕⊆Θφ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\}forΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\. Let us call this result \(a\)\. Finally, to show that ifδ∈Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}, thenδ∈Θ′\\delta\\in\\Theta^\{\\prime\}, supposeδ∈Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\. By postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\), it is the case thatΘψ○⊕⊆C​n𝕃​\(Θφ∨ψ○⊕∪\{ψ\}\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\psi\\\}\), soδ∈C​n𝕃​\(Θφ∨ψ○⊕∪\{ψ\}\)\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\psi\\\}\)\. By thededuction theorem, we haveψ→δ∈C​n𝕃​\(Θφ∨ψ○⊕\)\\psi\\to\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)and, by\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\),ψ→δ∈Θφ∨ψ○⊕\\psi\\to\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Thus, by result \(a\), we haveψ→δ∈Θ′\\psi\\to\\delta\\in\\Theta^\{\\prime\}\. But, by assumption,Θ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psiand, by Observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4),ψ∈Θ′\\psi\\in\\Theta^\{\\prime\}\. Therefore, by Observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3)andmodus ponens, we haveδ∈Θ′\\delta\\in\\Theta^\{\\prime\}\. Hence, it is the case thatΘψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\}and, thus, condition \(iii\) of the definition of⩽\\leqslantis finally satisfied and, as required,Θ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}\. That is, case \(1A\) is proven\. Consider case \(1B\), i\.e\., ifΘ′∉γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\notin\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)andΘ′∈Θ⊤φ\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi, thenΘ′≺Θ′′\\Theta^\{\\prime\}\\prec\\Theta^\{\\prime\\prime\}for someΘ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\(i\.e\.,Θ′⩽Θ′′\\Theta^\{\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}, butΘ′′⋠Θ′\\Theta^\{\\prime\\prime\}\\npreceq\\Theta^\{\\prime\}\)\. Let, therefore,Θ′′∈γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)\. Consider only condition \(iii\) of the definition of⩽\\leqslant\. Sinceγ⩽○⊕​\(Θ,φ\)⊆Θ⊤φ\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)\\subseteq\\Theta\\top\\varphi, we haveΘ′,Θ′′∈Θ⊤φ\\Theta^\{\\prime\},\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\. SinceΘ′′∈γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\\prime\}\\in\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\), by the definition of the completion functionγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}, we have⋂γ○⊕​\(Θ,φ\)⊆Θ′′\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\subseteq\\Theta^\{\\prime\\prime\}and, therefore,Θφ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\\prime\}\. ButΘ′∉γ⩽○⊕​\(Θ,φ\)\\Theta^\{\\prime\}\\notin\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}\(\\Theta,\\varphi\)\. Therefore, by the definition of the completion functionγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}, we have⋂γ○⊕​\(Θ,φ\)⊈Θ′\\bigcap\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\(\\Theta,\\varphi\)\\nsubseteq\\Theta^\{\\prime\}and, therefore,Θφ○⊕⊈Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\nsubseteq\\Theta^\{\\prime\}, which fails to satisfy condition \(iii\) of the definition of⩽\\leqslant\. Hence,Θ′⩽Θ′′\\Theta^\{\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}, butΘ′′⋠Θ′\\Theta^\{\\prime\\prime\}\\npreceq\\Theta^\{\\prime\}, as required\.

\(2\) The relation⩽\\leqslantis transitive with respect toγ⩽○⊕\\gamma^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\leqslant\}, for allφ∈ℒΣ∗\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\.

SupposeΘ′′′⩽Θ′′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}andΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}\(Assumption A\)\. We need to show thatΘ′′′⩽Θ′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}, that is, that conditions \(i\)\-\(iii\) of the definition of⩽\\leqslantare satisfied to obtainΘ′′′⩽Θ′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}\. SinceΘ′′′⩽Θ′′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}, we have thatΘ′′′∈Θ⊤δ\\Theta^\{\\prime\\prime\\prime\}\\in\\Theta\\top\\delta, for someδ∈ℒΣ∗\\delta\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\. Condition \(i\) is therefore satisfied\. Similarly, sinceΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}, we have thatΘ′∈Θ⊤δ\\Theta^\{\\prime\}\\in\\Theta\\top\\deltaandΘδ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\delta\}\\subseteq\\Theta^\{\\prime\}, for someδ∈ℒΣ∗\\delta\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\. Hence, condition \(ii\) is also satisfied\. We must therefore verify condition \(iii\) of the definition of⩽\\leqslantforΘ′′′⩽Θ′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}\. Suppose, accordingly, as per condition \(iii\), forψ∈ℒΣ∗\\psi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}, thatΘ′′′,Θ′∈Θ⊤ψ\\Theta^\{\\prime\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\psiandΘψ○⊕⊆Θ′′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\\prime\\prime\}\(Assumption B\)\. We need to show thatΘψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\}\. By lemma[B\.7](https://arxiv.org/html/2607.09729#A2.Thmtheorem7), we have thatΘ⊤φ∨ψ=Θ⊤φ∪Θ⊤ψ\\Theta\\top\\varphi\\lor\\psi=\\Theta\\top\\varphi\\cup\\Theta\\top\\psi\. Therefore, since by Assumption B,Θ′′′,Θ′∈Θ⊤ψ\\Theta^\{\\prime\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\psi, we have thatΘ′′′,Θ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\. Since, by Assumption A, we haveΘ′′′⩽Θ′′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}, then, by condition \(ii\) of the definition of⩽\\leqslant, we have thatΘ′′∈Θ⊤φ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphiandΘφ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\\prime\}for someφ∈ℒΣ∗\\varphi\\in\\mathcal\{L\}\_\{\\Sigma\_\{\*\}\}\(result \(b\)\)\. Therefore, it is also the case thatΘ′′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\. Thus, we haveΘ′′′,Θ′′,Θ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\\prime\},\\Theta^\{\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi\(result \(c\)\)\. By proposition[B\.3](https://arxiv.org/html/2607.09729#A2.Thmtheorem3)\(i\), we have thatΘφ∨ψ○⊕⊆Θφ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}orΘφ∨ψ○⊕⊆Θψ○⊕\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\. Taking the first disjunct and result \(b\), we haveΘφ∨ψ○⊕⊆Θφ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\}\\subseteq\\Theta^\{\\prime\\prime\}and, therefore,Θφ∨ψ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\prime\\prime\}\. Since, by Assumption A, we haveΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}, and by result \(c\), we haveΘ′′,Θ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi, then, by condition \(iii\) of the definition of⩽\\leqslant, we obtainΘφ∨ψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\prime\}\. Similarly, taking the second disjunct and Assumption B, we haveΘφ∨ψ○⊕⊆Θψ○⊕⊆Θ′′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\\prime\\prime\}\. Since, by Assumption A,Θ′′′⩽Θ′′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\\prime\}and, by result \(c\),Θ′′′,Θ′′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\\prime\},\\Theta^\{\\prime\\prime\}\\in\\Theta\\top\\varphi\\lor\\psi, then, by condition \(iii\) of the definition of⩽\\leqslant, we obtainΘφ∨ψ○⊕⊆Θ′′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\prime\\prime\}\. But, once again, sinceΘ′′⩽Θ′\\Theta^\{\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}by Assumption A, andΘ′′,Θ′∈Θ⊤φ∨ψ\\Theta^\{\\prime\\prime\},\\Theta^\{\\prime\}\\in\\Theta\\top\\varphi\\lor\\psiby result \(c\), then, by condition \(iii\) of⩽\\leqslant, we haveΘφ∨ψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\prime\}\. Therefore, from either disjunct, it is the case thatΘφ∨ψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\subseteq\\Theta^\{\\prime\}\(result \(d\)\)\. Finally, to show thatΘψ○⊕⊆Θ′\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq\\Theta^\{\\prime\}, supposeδ∈Θψ○⊕\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\. We need to show thatδ∈Θ′\\delta\\in\\Theta^\{\\prime\}\. By postulate\(Θ○⊕​6\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}6\), we haveΘψ○⊕⊆C​n𝕃​\(Θφ∨ψ○⊕∪\{ψ\}\)\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\psi\}\\subseteq Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\psi\\\}\)\. Therefore,δ∈C​n𝕃​\(Θφ∨ψ○⊕∪\{ψ\}\)\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\\cup\\\{\\psi\\\}\)and, by thededuction theorem,ψ→δ∈C​n𝕃​\(Θφ∨ψ○⊕\)\\psi\\to\\delta\\in Cn\_\{\\mathbb\{L\}\}\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\)and, by\(Θ○⊕​1\)\(\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}1\),ψ→δ∈Θφ∨ψ○⊕\\psi\\to\\delta\\in\\Theta^\{\\mathbin\{\\vtop\{\\halign\{\#\\cr$\\scriptstyle\\bigcirc$\\cr$\\scriptstyle\\oplus$\\crcr\}\}\}\}\_\{\\varphi\\lor\\psi\}\. Hence, by result \(d\),ψ→δ∈Θ′\\psi\\to\\delta\\in\\Theta^\{\\prime\}\. Since, by Assumption B,Θ′∈Θ⊤ψ\\Theta^\{\\prime\}\\in\\Theta\\top\\psi, then, by observation[6\.4](https://arxiv.org/html/2607.09729#S6.Thmtheorem4),ψ∈Θ′\\psi\\in\\Theta^\{\\prime\}\. Therefore, by observation[6\.3](https://arxiv.org/html/2607.09729#S6.Thmtheorem3)andmodus ponens, we haveδ∈Θ′\\delta\\in\\Theta^\{\\prime\}, as required\. Thus, condition \(iii\) of⩽\\leqslantforΘ′′′⩽Θ′\\Theta^\{\\prime\\prime\\prime\}\\leqslant\\Theta^\{\\prime\}is satisfied and, consequently, we obtain the transitivity of⩽\\leqslant\. ∎

## References

- \[1\]Atocha Aliseda\.Abductive Reasoning: Logical Investigations into Discovery and Explanations, volume 330 ofSynthese Library\.Springer, Holanda, 1 edition, 2006\.Edited by Vincent F\. Hendricks and John Simons\.
- \[2\]Lucas Angioni\.Os seis requisitos das premissas da demonstração científica em aristóteles\.Manuscrito – Revista Internacional de Filosofia, 35\(1\):7–60, 2012\.
- \[3\]Lucas Angioni\.Demonstração, silogismo e causalidade\.Lógica e Ciência em Aristóteles, 2014\.
- \[4\]Willem J\. Blok and Don Pigozzi\.Algebraizable Logics, volume 77 ofMemoirs of the American Mathematical Society\.American Mathematical Society, Providence, RI, USA, 1989\.Number 396\.
- \[5\]Juliana Bueno\-Soler, Walter Carnielli, Marcelo E\. Coniglio, and Abilio Rodrigues Filho\.Formal \(in\)consistency, abduction and modalities\.In Lorenzo Magnani and Tommaso Bertolotti, editors,Springer Handbook of Model\-Based Science, pages 315–335\. Springer International Publishing, Cham, 2017\.
- \[6\]Walter Carnielli, Marcelo E\. Coniglio, and David Fuenmayor\.Logics of formal inconsistency enriched with replacement: An algebraic and modal account\.Review of Symbolic Logic, 15\(3\):771–806, 2022\.
- \[7\]Walter A\. Carnielli and Marcelo E\. Coniglio\.Paraconsistent Logic: Consistency, Contradiction and Negation, volume 40 ofLogic, Epistemology, and the Unity of Science\.Springer International Publishing, 1 edition, 2016\.
- \[8\]Marcelo E\. Coniglio\.Ivlev\-like modal logics of formal inconsistency obtained by fibring swap structures\.Studia Logica, 2024\.
- \[9\]Marcelo E\. Coniglio, Martín Figallo, and Rafael R\. Testa\.Paraconsistent belief revision: A replacement\-enriched LFI for epistemic entrenchment\.Studia Logica, 2026\.
- \[10\]Marcelo E\. Coniglio, Aldo Figallo\-Orellano, and Ana Claudia Golzio\.Non\-deterministic algebraization of logics by swap structures\.Logic Journal of the IGPL, 28\(5\):1021–1059, 2020\.
- \[11\]Marcelo E\. Coniglio, Márcio M\. Ribeiro, and Rafael R\. Testa\.AGM\-like paraconsistent belief change\.Logic Journal of the IGPL, 25\(4\):632–672, 2017\.
- \[12\]Kuang Tih Fann\.Peirce´s Theory of Abducion\.Martinus Nijhojf, 1 edition, 1970\.
- \[13\]Peter Gärdenfors and David Makinson\.Nonmonotonic inference based on expectations\.Artificial Intelligence, 65\(2\):197–245, 1994\.
- \[14\]Peter Gärdenfors and Hans Rott\.Belief Revision, volume 4, pages 35–132\.Oxford University Press, 04 1995\.
- \[15\]Carl G\. Hempel and Paul Oppenheim\.Studies in the Logic of Explanation, pages 245–290\.The Free Press, New York, 1965 \[1948\]\.
- \[16\]Isaac Levi\.The Fixation of Belief and its Undoing: Changing Beliefs Through Inquiry\.Cambridge University Press, New York, NY, USA, 1991\.
- \[17\]Fenrong Liu\.Reasoning about Preference Dynamics, volume 354 ofSynthese Library\.Springer Dordrecht, 01 2011\.
- \[18\]David Makinson\.General Patterns in Nonmonotonic Reasoning, volume 3, pages 35–110\.Oxford University Press, 03 1994\.
- \[19\]Maurice Pagnucco\.The Role of Abductive Reasoning within the Process of Belief Revision\.PhD thesis, Sydney University, Sydney, Austrália, 1996\.
- \[20\]Charles Sanders Peirce\.Collected Papers of Charles Sanders Peirce, volume I\-VI\.Harvard University Press, Cambridge, MA, 1931\.Edited by Charles Hartshorne and Paul Weiss\.
- \[21\]Charles Sanders Peirce\.Collected Papers of Charles Sanders Peirce, volume VII\-VIII\.Harvard University Press, Cambridge, MA, 1958\.Edited by Arthur W\. Burks\.
- \[22\]Hans Reichenbach\.Experience and Prediction\.Phoenix Books\. University of Chicago Press, 5 edition, 1957\.
- \[23\]Nicholas Rescher\.Peirce and the economy of research\.Philosophy of Science, 43\(1\):71–98, 1976\.
- \[24\]A\. Rodrigues, M\. E\. Coniglio, H\. Antunes, J\. Bueno\-Soler, and W\. Carnielli\.Paraconsistency, evidence, and abduction\.In Lorenzo Magnani, editor,Handbook of Abductive Cognition, pages 1–38\. Springer International Publishing, Cham, 2022\.
- \[25\]Wesley C\. Salmon\.Scientific Explanation and the Causal Structure of the World\.Princeton University Press, 1984\.
- \[26\]Lucia Santaella\.Abduction: The logic of guessing\.Semiotica, 2005\(153 \- 1/4\):175–198, 2005\.
- \[27\]Rafael Rodrigues Testa\.Paraconsistent Belief Revision based on a Formal Consistency Operator\.PhD thesis, UNICAMP \- University of Campinas, Campinas, SP, Brazil, 2014\.
- \[28\]Bas Van Fraassen\.The Scientific Image\.Oxford University Press, New York, 1980\.Edited by L\. Jonathan Cohen\.
- \[29\]Ryszard Wójcicki\.Lectures on Propositional Calculi\.Pub\. House of the Polish Academy of Sciences, Ossolineum \[Poland\], 1984\.