Probabilistic Extension of Neuro-Symbolic AGI Robots based on Belnap's Typed Intensional FOL

# Probabilistic Extension of Neuro-Symbolic AGI Robots based on Belnap’s Typed Intensional FOL
Source: [https://arxiv.org/html/2607.13073](https://arxiv.org/html/2607.13073)
11institutetext:ISRST, Tallahassee, FL, USA
11email:majk\.1234@yahoo\.com###### Abstract

Neuro\-symbolic AI based onI​F​O​LBIFOL\_\{B\}is a way to combine neural learning and symbolic reasoning to overcome limitations of purely neural systems \(like lack of interpretability and logical structure\) with formal logical machinery for self\-reference\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]\. In this paper we expand the cognitive power ofI​F​O​LBIFOL\_\{B\}by using the probability computation for the currently unknown sentences, based on Nilsson’s probability structure for theI​F​O​LBIFOL\_\{B\}\. We introduce the global symmetry transformation that preserves the current knowledge database𝒦\\mathcal\{K\}and logical deduction, and the local one used for real\-time decisions about concrete \(sub\)problems that involve only a very strict subset ofI​F​O​LBIFOL\_\{B\}predicates\. The computation of probability density functionK​IKIin both cases, based on the Shannon’s maximum information entropy, is provided by neural networks of this probabilistic neuro\-symbolic AGI\.

## 1Introduction to IFOL\-based Proposal of Self\-awareness

This theory on robot self\-awareness is rooted in the development of Intensional First\-Order Logic \(IFOL\)\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\. We argue that true ”Strong AI” or ”autoepistemic” robots require a symbolic architecture that allows them to reason about their own internal states as distinct from the external objects they perceive\. This introduction is a short presentation of the my approach to AGI \(Strong\-AI\) for a new generation of intelligent robots, recently published in the papers\[[4](https://arxiv.org/html/2607.13073#bib.bib4)\]and\[[5](https://arxiv.org/html/2607.13073#bib.bib5)\]\. Neuro\-symbolic AI attempts to integrate neural and symbolic architectures in a manner that addresses strengths and weaknesses of each, in a complementary fashion, in order to support robust strong AI capable of reasoning, learning, and cognitive modeling\. In this approach to AGI I considered the Intensional First Order Logic\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]as a symbolic architecture of modern robots, able to use natural languages to communicate with humans and to reason about their own knowledge with self\-reference and abstraction language property\. In what follows we will consider the 4\-valued typedI​F​O​LBIFOL\_\{B\}based on Belnap’s bilattice of truth\-valuesX=ℬ4=\{f,t,⊥,⊤\}X=\\mathcal\{B\}\_\{4\}=\\\{f,t,\\bot,\\top\\\}introduced in\[[6](https://arxiv.org/html/2607.13073#bib.bib6),[32](https://arxiv.org/html/2607.13073#bib.bib32)\]and denoted byI​F​O​LBIFOL\_\{B\}\.

Intensional entities \(or concepts\) are such things as Propositions, Relations and Properties \(PRP\)\. What make them ”intensional” is that they violate the principle of extensionality; the principle that extensional equivalence implies identity\. All \(or most\) of these intensional entities have been classified at one time or another as kinds of*Universals*\[[7](https://arxiv.org/html/2607.13073#bib.bib7)\], in the case of many\-valuedI​F​O​LBIFOL\_\{B\},DI=D1\+D2\+D3\+…D\_\{I\}=D\_\{1\}\+D\_\{2\}\+D\_\{3\}\+\.\.\.\(with propositions \(L\-concepts\)D1D\_\{1\}, and relational conceptsDnD\_\{n\},n≥2n\\geq 2\), and*particulars*D0D\_\{0\}\[[8](https://arxiv.org/html/2607.13073#bib.bib8)\], which define the PRP domain𝒟=D0\+DI\\mathcal\{D\}=D\_\{0\}\+D\_\{I\}, with intensional mapping from the set of FOL formulaeℒ\\mathcal\{L\}to these intensional concepts

I:ℒ→𝒟I:\\mathcal\{L\}\\rightarrow\\mathcal\{D\}Ifϕ​\(x\)\\phi\(\\textbf\{x\}\)is an open formula \(virtual predicate with a list \(a tuple\) of free variables inx=\(x1,…,xn\)\\textbf\{x\}=\(x\_\{1\},\.\.\.,x\_\{n\}\)\), thenI​\(ϕ​\(x\)\)∈DnI\(\\phi\(\\textbf\{x\}\)\)\\in D\_\{n\}is a n\-ary concept\. This concept mapping can be extended to the homomorphismI:𝒜​𝔅F​O​L→𝒜​𝔅i​n​tI:\\mathcal\{A\}\\mathfrak\{B\}\_\{FOL\}\\rightarrow\\mathcal\{A\}\\mathfrak\{B\}\_\{int\}, between the FOL syntax algebra to the algebra of concepts\. Thus, for a given 4\-valued Herbrand baseHHofI​F​O​LBIFOL\_\{B\}andX=ℬ4=\{f,t,⊥,⊤\}X=\\mathcal\{B\}\_\{4\}=\\\{f,t,\\bot,\\top\\\}the set of Belnap’s truth\-values \(false, true, unknown and inconsistent, respectively\) and a Herbrand interpretation that must satisfy all built\-in predicates as well\)

v:H→Xv:H\\rightarrow X\(1\)and its unique extension to all sentencesv∗:ℒ0→Xv^\{\*\}:\\mathcal\{L\}\_\{0\}\\rightarrow X, withv∗∈ℐM​Vv^\{\*\}\\in\\mathcal\{I\}\_\{MV\}of the set of all well\-defined Herbrand interpretations \(that respects the built\-in predicates in Definition 4 in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\]\)\. Note that the set of all well\-defined Herbrand interpretationsℐH\\mathcal\{I\}\_\{H\}is only a subset of functions inXHX^\{H\}, because we have the built\-in predicates as well \(from Definition 4 in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\]\),

ℐH⊂XHw​i​t​hℐM​V=\{v∗:ℒ0→X\|v∈ℐH\}\\mathcal\{I\}\_\{H\}\\subset X^\{H\}\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ with\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\mathcal\{I\}\_\{MV\}=\\\{v^\{\*\}:\\mathcal\{L\}\_\{0\}\\rightarrow X\|v\\in\\mathcal\{I\}\_\{H\}\\\}\(2\)wherev∗v^\{\*\}is the extension of Herbrand interpretationvvto all sentencesℒ0\\mathcal\{L\}\_\{0\}of our typedI​F​O​LBIFOL\_\{B\}\.

Each extensional interpretationhhassigns to the intensional elements of𝒟\\mathcal\{D\}an appropriate extension: in the case of particularsu∈D0u\\in D\_\{0\},h0​\(u\)∈D0h\_\{0\}\(u\)\\in D\_\{0\}, such that for each logic valuea∈X⊂D0a\\in X\\subset D\_\{0\},h0​\(a\)=ah\_\{0\}\(a\)=a\. Thus, we have the particular’s mappingh0:D0→D0h\_\{0\}:D\_\{0\}\\rightarrow D\_\{0\}and more generally \(here\+′′\{\}^\{\\prime\}\+^\{\\prime\}is considered as disjoint union\) from\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\],

h=∑i∈ℕhi:𝒟⟶D0\+∑i≥1ℜ​𝔪ih=\\sum\_\{i\\in\\mathbb\{N\}\}h\_\{i\}:\\mathcal\{D\}\\longrightarrow D\_\{0\}\+\\sum\_\{i\\geq 1\}\\mathfrak\{Rm\}\_\{i\}\(3\)whereℜ​𝔪i\\mathfrak\{Rm\}\_\{i\}are i\-ary relations and whereh1h\_\{1\}assigns to each L\-conceptu∈D1u\\in D\_\{1\}\(of a sentence with truth\-valuea∈Xa\\in X\), a relation composed by the single tupleh1​\(u\)=\{a\}h\_\{1\}\(u\)=\\\{a\\\}ifa≠⊥a\\neq\\bot\),∅\\penalty 10000\\ \\emptysetotherwise, andhi:Di→ℜ​𝔪ih\_\{i\}:D\_\{i\}\\rightarrow\\mathfrak\{Rm\}\_\{i\}, fori≥2i\\geq 2, that assigns a m\-extension to non\-sentence concepts \(obtained, for example, an\(i−1\)\(i\-1\)\-ary predicate, so that the lastii\-th column ofℜ​𝔪i\\mathfrak\{Rm\}\_\{i\}is a truth\-value of a ground atom of this predicate\. In this way, the relationℜ​𝔪i\\mathfrak\{Rm\}\_\{i\}represents the set of tuples of ground atoms of a given predicate both with their truth\-valuesa∈Xa\\in X\.

The two\-steps interpretation ofI​F​O​LBIFOL\_\{B\}based on two homomorphisms, fixed intensionalII, and extensionalization mappinghh, is provided by commutative diagram in Corollary 3 in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\], which defines the MV\-interpretation

IB∗=h∘II^\{\*\}\_\{B\}=h\\circ I\.

So we can define the following set of different ”possible worlds” for this many\-valuedI​F​O​LBIFOL\_\{B\}\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\]based on Herbrand interpretations \([1](https://arxiv.org/html/2607.13073#S1.E1)\) above:

𝒲=\{IB∗\|v∗∈ℐM​V\}\\mathcal\{W\}=\\\{I^\{\*\}\_\{B\}\\penalty 10000\\ \|\\penalty 10000\\ v^\{\*\}\\in\\mathcal\{I\}\_\{MV\}\\\}\\penalty 10000\\ \\penalty 10000\\\(4\)such that for each Herbrand interpretationv∈ℐH⊂XHv\\in\\mathcal\{I\}\_\{H\}\\subset X^\{H\}, we have a unique MV\-interpretationIB∗=h∘II^\{\*\}\_\{B\}=h\\circ I, that is, the bijections

i​sH:ℐH≃𝒲a​n​di​sM​V:ℐM​V≃𝒲is\_\{H\}:\\mathcal\{I\}\_\{H\}\\penalty 10000\\ \\simeq\\mathcal\{W\}\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ and\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ is\_\{MV\}:\\mathcal\{I\}\_\{MV\}\\penalty 10000\\ \\simeq\\mathcal\{W\}\(5\)withIB∗=i​sM​V​\(v∗\)=i​sH​\(v\)I^\{\*\}\_\{B\}=is\_\{MV\}\(v^\{\*\}\)=is\_\{H\}\(v\)andv∗=i​sM​V−1∘i​sH​\(v\)v^\{\*\}=is^\{\-1\}\_\{MV\}\\circ is\_\{H\}\(v\), that is, from the fact thatIIis fixed intensional interpretation, each possible world is fundamentally an extensionalization functionhh, and we denote byℰi​n\\mathcal\{E\}\_\{in\}the set of these extensionalization functions that respects all built\-in predicates, with bijection\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\]

i​si​n:𝒲≃ℰi​nis\_\{in\}:\\mathcal\{W\}\\simeq\\mathcal\{E\}\_\{in\}\(6\)The extensions of the IFOL concepts change in time \(the robot’s knowledge\), so that we can use for specification ofhhthe time\-index as their ordered representation\.

In reflective languages, reification data is causally connected to the related reified aspect such that a modification to one of them affects the other; by using intensional FOL the robots can formalize also the natural language expressions ”I see the blue color” by a predicate ”See\(I,blue color\)” where the sense of the ground term ”I” \(*Self*, me\)111Self in a sense which implies that all our activities are controlled by powerful creatures inside ourselves, who do our thinking and feeling for us\.for a robot is the name of the main working coordination program which activate all other algorithms \(neuro\-symbolic AI subprograms\) like visual recognition of color of the object in focus\. But also the auto\-conscience sentence like ”I know that I see the blue color” by using abstracting operators ”⋖\_⋗\\lessdot\\\_\\gtrdot” of intensional FOL, expressed by the predicate ”Know\(I,⋖\\lessdotSee\(I, blue color\)⋗\\gtrdot\)”, etc… Ifϕ​\(x\)\\phi\(\\textbf\{x\}\)is a virtual predicate with a list \(a tuple\) of free variables inx=\(x1,…,xn\)\\textbf\{x\}=\(x\_\{1\},\.\.\.,x\_\{n\}\)andα\\alphais its subset of*distinct*variables, then⋖ϕ​\(x\)⋗αβ\\lessdot\\phi\(\\textbf\{x\}\)\\gtrdot\_\{\\alpha\}^\{\\beta\}is a term, whereβ\\betais the remaining set of free variables inx\. The externally quantifiable variables are the*free*variables not inα\\alpha\. Whenn=0,⋖ϕ⋗n=0,\\penalty 10000\\ \\lessdot\\phi\\gtrdotis a term which denotes a proposition, forn≥1n\\geq 1it denotes a n\-ary concept\.

###### Definition 1

Intensional abstraction convention:

From the fact that we can use any permutation of the variables in a given virtual predicate, we introduce the convention that

⋖ϕ\(x\)⋗αβisatermobtainedfromvirtualpredicateϕ\(x\)\\lessdot\\phi\(\\textbf\{x\}\)\\gtrdot\_\{\\alpha\}^\{\\beta\}\\penalty 10000\\ \\penalty 10000\\ is\\penalty 10000\\ a\\penalty 10000\\ term\\penalty 10000\\ obtained\\penalty 10000\\ from\\penalty 10000\\ virtual\\penalty 10000\\ predicate\\penalty 10000\\ \\penalty 10000\\ \\phi\(\\textbf\{x\}\)\(7\)ifα\\alphaisnot emptysuch thatα​⋃β\\alpha\\bigcup\\betais the set of all variables in the list \(tuple of variables\)x=\(x1,…,xn\)\\textbf\{x\}=\(x\_\{1\},\.\.\.,x\_\{n\}\)of the virtual predicate \(an open logic formula\)ϕ\\phi, andα​⋂β=∅\\alpha\\bigcap\\beta=\\emptyset, so that\|α\|\+\|β\|=\|x\|=n\|\\alpha\|\+\|\\beta\|=\|\\textbf\{x\}\|=n\. Only the variables inβ\\beta\(which are the only free variables of this term\), can be quantified\. Ifβ\\betais empty then⋖ϕ​\(x\)⋗α\\lessdot\\phi\(\\textbf\{x\}\)\\gtrdot\_\{\\alpha\}is a*ground term*\. Ifϕ\\phiis a sentence and hence bothα\\alphaandβ\\betaare empty, we write simply⋖ϕ⋗\\lessdot\\phi\\gtrdotfor this ground term\.

More about this general definition of abstract terms can be find in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\. In this paper we will use the most simple cases of ground terms⋖ϕ⋗\\lessdot\\phi\\gtrdot, whereϕ\\phiis a sentence\.

By using the intensional mappingII, we are able to extend the simple assignment to variables to all \(also abstracted\) terms:

###### Definition 2

An assignmentg:𝒱→𝒟g:\\mathcal\{V\}\\rightarrow\\mathcal\{D\}forb variables in𝒱\\mathcal\{V\}is applied only to free variables in terms and formulae\. Such an assignmentg∈𝒟𝒱g\\in\\mathcal\{D\}^\{\\mathcal\{V\}\}can be recursively uniquely extended into the assignmentg∗:𝒯→𝒟g^\{\*\}:\\mathcal\{T\}\\rightarrow\\mathcal\{D\}, where𝒯\\mathcal\{T\}denotes the set of all terms \(hereIIis an intensional interpretation of this FOL, as explained in what follows\), by :

1. 1\.g∗​\(t\)=g​\(x\)∈𝒟g^\{\*\}\(t\)=g\(x\)\\in\\mathcal\{D\}if the termttis a variablex∈𝒱x\\in\\mathcal\{V\}\.
2. 2\.g∗​\(t\)=I​\(c\)∈𝒟g^\{\*\}\(t\)=I\(c\)\\in\\mathcal\{D\}if the termttis a constant \(nullary functional symbol\)c∈Pc\\in P\.
3. 3\.Ifttis an abstracted term obtained for an open formulaϕi\\phi\_\{i\},⋖ϕi​\(xi\)⋗αiβi\\lessdot\\phi\_\{i\}\(\\textbf\{x\}\_\{i\}\)\\gtrdot\_\{\\alpha\_\{i\}\}^\{\\beta\_\{i\}\}, then we must restrict the assignment tog∈𝒟βig\\in\\mathcal\{D\}^\{\\beta\_\{i\}\}and to obtain recursive definition \(when alsoϕi​\(xi\)\\phi\_\{i\}\(\\textbf\{x\}\_\{i\}\)contains abstracted terms: g∗\(⋖ϕi\(xi\)⋗αiβi\)=d​e​f\{I\(ϕi\(xi\)\)∈D\|αi\|\+1,ifβiis emptyI\(ϕi\(xi\)\[βi/g\(βi\)\]\)∈D\|αi\|\+1,otherwiseg^\{\*\}\(\\lessdot\\phi\_\{i\}\(\\textbf\{x\}\_\{i\}\)\\gtrdot\_\{\\alpha\_\{i\}\}^\{\\beta\_\{i\}\}\)=\_\{def\}\\left\\\{\\begin\{array\}\[\]\{ll\}I\(\\phi\_\{i\}\(\\textbf\{x\}\_\{i\}\)\)\\penalty 10000\\ \\penalty 10000\\ \\in D\_\{\|\\alpha\_\{i\}\|\+1\},&\\hbox\{if $\\beta\_\{i\}$ is empty\}\\\\ I\(\\phi\_\{i\}\(\\textbf\{x\}\_\{i\}\)\[\\beta\_\{i\}/g\(\\beta\_\{i\}\)\]\)\\penalty 10000\\ \\penalty 10000\\ \\in D\_\{\|\\alpha\_\{i\}\|\+1\},&\\hbox\{otherwise\}\\end\{array\}\\right\.\(8\)whereg\(β\)=g\(\{y1,\.\.,ym\}\)=\{g\(y1\),…,g\(ym\)\}g\(\\beta\)=g\(\\\{y\_\{1\},\.\.,y\_\{m\}\\\}\)=\\\{g\(y\_\{1\}\),\.\.\.,g\(y\_\{m\}\)\\\}and\[β/g​\(β\)\]\[\\beta/g\(\\beta\)\]is a uniform replacement of each i\-th variable in the setβ\\betawith the i\-th constant in the setg​\(β\)g\(\\beta\)\. Notice thatα\\alphais the set of all free variables in the formulaϕ​\[β/g​\(β\)\]\\phi\[\\beta/g\(\\beta\)\]\.
4. 4\.Ift=⋖ϕi⋗\\penalty 10000\\ t=\\lessdot\\phi\_\{i\}\\gtrdotis an abstracted term obtained from a sentenceϕi\\phi\_\{i\}then g∗\(⋖ϕi⋗\)=I\(ϕi\)∈D0g^\{\*\}\(\\lessdot\\phi\_\{i\}\\gtrdot\)=I\(\\phi\_\{i\}\)\\in D\_\{0\}\.222This case 4 is the particular case 3 when the tuple of variablesxi\\textbf\{x\}\_\{i\}is empty and henceβi\\beta\_\{i\}andαi\\alpha\_\{i\}are empty sets of variables with\|αi\|=0\|\\alpha\_\{i\}\|=0\.

By introduction of the abstraction operators with autoepistemic capacities, suported by theK​n​o​wKnow\(meta\)predicate, we do not use more a*pure*logical deduction of the standard FOL, but a kind of autoepistemic deduction\[[9](https://arxiv.org/html/2607.13073#bib.bib9),[10](https://arxiv.org/html/2607.13073#bib.bib10)\]with a proper set of new axioms forI​F​O​LBIFOL\_\{B\}provided in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\]\. It has been demonstrated that in such a minimal intensional enrichment of standard \(extensional\) FOL, we obtain exactly the Montague’s definition of the intension \(see Proposition 5 in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\)\.

We recall that each robot’s extensionalitation functionhhin \([3](https://arxiv.org/html/2607.13073#S1.E3)\) is indexed by the time\-instance\. Clearly, the robots knowledge changes in time and hence determines the extensionalization functionhhin any given instance of time, based on robots experiences\. Thus, as for humans, also the robot’s knowledge and logic is a kind of temporal logic, and evolves with time\. Note that the explicit \(conscious\) robot’s knowledge in actual worldℏ\\hbarhere is represented by the ground atoms of theK​n​o​wKnowpredicate, for a given assignments of variables in𝒱\\mathcal\{V\},g:𝒱→𝒟g:\\mathcal\{V\}\\rightarrow\\mathcal\{D\},

Know\(y1,y2,⋖ψ\(x\)⋗αβ\)/g=Know\(g∗\(y1\),g∗\(y2\),g∗\(⋖ψ\(x\)⋗αβ\)\)Know\(y\_\{1\},y\_\{2\},\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\_\{\\alpha\}\)/g=Know\(g^\{\*\}\(y\_\{1\}\),g^\{\*\}\(y\_\{2\}\),g^\{\*\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\_\{\\alpha\}\)\)\(9\)with\{y1,y2\}​⋃β​⋃α⊆𝒱\\\{y\_\{1\},y\_\{2\}\\\}\\bigcup\\beta\\bigcup\\alpha\\subseteq\\mathcal\{V\}, such thatg∗​\(y1\)=i​n​p​r​e​s​e​n​tg^\{\*\}\(y\_\{1\}\)=in\\penalty 10000\\ presentandg∗​\(y2\)=Ig^\{\*\}\(y\_\{2\}\)=\\textbf\{I\}\(the robot itself\), for the extended assignmentsg∗:𝒯→𝒟g^\{\*\}:\\mathcal\{T\}\\rightarrow\\mathcal\{D\}\(from Definition 17 in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\), where the set of terms𝒯\\mathcal\{T\}ofI​F​O​LBIFOL\_\{B\}is composed by the set𝒱\\mathcal\{V\}of variables used in the set of predicates ofI​F​O​LBIFOL\_\{B\}, by the set of constants and*abstracted terms*⋖ψ\(x\)⋗αβ\)\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\_\{\\alpha\}\), so that in the actual worldℏ\\hbar, the known fact \([9](https://arxiv.org/html/2607.13073#S1.E9)\) for robot becomes

Know\(y1,y2,⋖ψ\(x\)⋗αβ\)/g=Know\(inpresent,I,I\(ψ\[β/g\(β\)\]\)\)Know\(y\_\{1\},y\_\{2\},\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\_\{\\alpha\}\)/g=Know\(in\\penalty 10000\\ present,\\textbf\{I\},I\(\\psi\[\\beta/g\(\\beta\)\]\)\) which is true in actual word, that is, from proposition \(intensional concept\) u=I​\(K​n​o​w​\(i​n​p​r​e​s​e​n​t,I,I​\(ψ​\[β/g​\(β\)\]\)\)\)∈D0u=I\(Know\(in\\penalty 10000\\ present,\\textbf\{I\},I\(\\psi\[\\beta/g\(\\beta\)\]\)\)\)\\in D\_\{0\}, we obtain the truth value ℏ​\(u\)=ℏ​\(I​\(K​n​o​w​\(i​n​p​r​e​s​e​n​t,I,I​\(ψ​\[β/g​\(β\)\]\)\)\)\)=t\\hbar\(u\)=\\hbar\(I\(Know\(in\\penalty 10000\\ present,\\textbf\{I\},I\(\\psi\[\\beta/g\(\\beta\)\]\)\)\)\)=t\. Note that for the assignmentsg:𝒱→𝒟g:\\mathcal\{V\}\\rightarrow\\mathcal\{D\}, such thatg​\(y1\)=i​n​f​u​t​u​r​eg\(y\_\{1\}\)=in\\penalty 10000\\ futureandg​\(y2\)g\(y\_\{2\}\)we consider robot’s hypothetical knowledge in future, while in the cases wheng​\(y1\)=i​n​p​a​s​tg\(y\_\{1\}\)=in\\penalty 10000\\ pastwe consider what was robot’s knowledge in the past\. So, the robots current knowledge \(ground atoms of predicateK​n​o​wKnow\) is directly derived from its experiences \(based on its neuro\-system processes that robot is using\) in an analog way as human brain does:

- •As an activation \(under robot’s attention\) of its neuro\-system process, as a consequence of some human command to execute some particular job\.
- •As an activation of some process under current attention of robot, which is part of some complex plan of robot’s activities connected with its general objectives and services\.

Remark: We consider that only robot’s experiences \(under robot’s attention\) are transformed into the ground atoms of the many\-valuedK​n​o​wKnowpredicate, and the required \(by robot\) deductions from them \(by using general many\-valued deduction\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]extended by the three epistemic axioms\) are transformed into ground atoms ofK​n​o​wKnowpredicate, and hence are saved in robot’s temporary memory as a part of robot’s*conscience*\. Some background process \(unconscious for the robot\) would successively transform these temporary memory knowledge into permanent robot’s knowledge as it happen for humans\. By such fixing by humans of robot’s unconciseness part with active semantics \(which can not be modified by robots and their live experience\) of all significant for human robot’s concepts and their properties, we will obtain ethically confident and socially safe and non danger robots \(controlled by public human ethical security organizations for the production of robots with general strong\-AI capabilities\)\. □\\square IFOL\-based approach is part of a general neuro\-symbolic AI paradigm, where researchers aim to blend:Neural methods\(e\.g\., deep learning\) for pattern recognition and learning from data, andsymbolic logicfor structured knowledge representation and reasoning\. This broader field \(which includes work at IBM Research, MIT, and in academic surveys\) seeks to build AI systems that can both*learn from experience*and*reason abstractly*\. These ideas contribute to ongoing discussions in AI about*symbol grounding*,*logical inference*, and*self\-referential reasoning*— all important for Strong AI research\.

Key aspects of this theory include:

1. 1\.Formalizing the ”Self”\(I\): We propose that for a robot, the ground term ”I” \(or ”me”, Self\) acts as the name of its main working coordination program\. This master program activates all other sub\-programs, such as visual recognition or motor control, allowing the robot to represent expressions likeS​e​e​\(I,b​l​u​e​c​o​l​o​r\)See\(\\textbf\{I\},bluecolor\)in its logical framework\.
2. 2\.Neuro\-Symbolic Integration: We advocate for a ”dual\-process” model inspired by Daniel Kahneman’s System 1 and System 2\. \-System 1 \(Neural\): Handles fast, unconscious pattern recognition \(e\.g\., deep learning\)\. \-System 2 \(Symbolic\): Uses IFOL \(Intensional FOL\) for slow, explicit, and deliberative thinking, such as planning and logical deduction\.
3. 3\.Intensional Abstraction and Self\-Reference: Using intensional abstraction, a robot can treat its own knowledge \(propositions\) as ”individuals” within its logic\. This allows the robot to perform self\-referential reasoning—literally thinking about its own thoughts—which we identify as a prerequisite for self\-awareness\.
4. 4\.Grounding through Experience: We argue that robots must ”ground” their language concepts by associating them with their own sensory\-motor experiences\. For example, the sense of the word ”blue” isn’t just a label, but is grounded in the robot’s specific internal neural experience of processing that color\.

We addresseslogical omniscience—the unrealistic assumption that an agent automatically knows all logical consequences of its beliefs—by usingintensional abstractionto shift from ”extensional” to ”intensional” reasoning\. This work is detailed in the book, Intensional First\-Order Logic\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]: From AI to New SQL Big Data, which outlines these principles as a path toward a new generation of Strong\-AI robots\. In standard modal logics \(like S5\), knowledge is closed under logical implication, meaning if an agent knowsAA, it must instantly know allBBwhereA→BA\\rightarrow B\. We solve this by re\-engineering how propositions are represented:

- •Reification of Propositions: Through intensional abstraction, predicates and sentences are ”reified”—treated as individual objects \(intensional entities\) within the same domain as physical objects\.
- •Decoupling Truth from Meaning: In this framework, two propositions can be extensionally equivalent \(true in all the same possible worlds\) but intensionally distinct\. For example, a robot might know ”Triangle A is equilateral” without yet knowing ”Triangle A is equiangular,” because those two concepts are different intensional ”individuals” that require a specific computational inference step to link\.

This approach allows for Strong\-AI robots that can reason about their own knowledge \(autoepistemic reasoning\) as a finite, step\-by\-step process, mirroring human cognitive limitations rather than possessing infinite mathematical foresight\.

In our framework, autoepistemic reasoning is the ability of a robot to reason about its own state of knowledge and belief as if they were objects in the world\. Unlike standard AI, which might ”know” a fact without ”knowing that it knows,” our Strong\-AI Autoepistemic Robots use specialized logical structures to achieve formal self\-reflection\.

1. 1\.The ”Temporal Know” Predicate: We implement autoepistemic capabilities by introducing a specific temporal ”*Know*” predicate\. This allows the robot to handle complex internal axioms: \-Reflexive Axiom: The robot understands that if it knows something, that thing must be part of its internal truth model\. \-Positive Introspection: The robot can derive that ”I know that I knowAA”\. \-Distributive Axiom: The robot can apply its knowledge across logical implications \(e\.g\., if it knowsAAand knowsA⇒BA\\Rightarrow B, it can reason its way toBB\)\.
2. 2\.Grounding through Neuronal Experience: A central part of our theory is that a robot’s ”internal” language must be grounded in its own hardware experiences: \-Mining the ”Sense”: The robot associates high\-level logical concepts with the specific firing patterns of its neural architectures\. \-Subjective Knowledge: Because this grounding is unique to the robot’s own sensors and processors, its autoepistemic reasoning is truly ”self\-centered”—it reasons based on how it specifically perceives the world\.
3. 3\.Handling Inconsistency: Traditional logic often breaks down when faced with a contradiction \(the ”explosion principle”\)\. Our Autoepistemic Logic\[[6](https://arxiv.org/html/2607.13073#bib.bib6),[32](https://arxiv.org/html/2607.13073#bib.bib32)\]is designed to be many\-valued, based on Belnap’s bilattice, meaning the robot can: \- Reason with incomplete or inconsistent information without crashing\. \- Revise its beliefs when new ”experiences” from its neural layer contradict its previous symbolic ”knowledge”\. \- Support the Knowledge Assumption Closure\.
4. 4\.Language as a Tool for Self\-Reference: We argue that natural language is inherently ”many\-sorted” and intensional\. By giving robots an IFOL architecture, they don’t just process strings of text; they use language as a symbolic coordination tool to label and manipulate their own internal programs and knowledge states

In this research, bridging the gap between neural networks \(sub\-symbolic\) and symbolic logic is not just about making them work side\-by\-side; it is about creating a formal translation layer where the robot’s ”internal feelings” \(neuron firings\) become ”meaningful concepts” \(logical terms\)\. We achieve this via a Dual\-System Architecture grounded in Intensional First\-Order Logic:

1. 1\.System 1: The ”Neuronal Grounding” Layer; System 1 consists of deep learning and neural networks\. For us, this layer is responsible for pattern recognition and sensory\-motor coordination\. \-The Output: Instead of just outputting a label like ”Blue,” the neural network produces a specific internal state \(a vector or firing pattern\)\. \-The Problem: On its own, the neural network doesn’t ”understand” the concept; it just reacts\. This is where the gap exists\.
2. 2\.The Bridge: Intensional Abstraction: This is our ”secret sauce\.” We use intensional abstraction to turn the complex activity of the neural network into a ”Logic Object\.” \-Mapping: Each distinct neural pattern is mapped to a specific intensional entity in the IFOL\. \-Meaning vs\. Reference: The ”meaning” \(intension\) of a word like heavy is the specific neural experience of the robot’s motors straining\. The ”reference” \(extension\) is the actual physical object being lifted\. \-Self\-Labeling: The robot uses autoepistemic reasoning to say, ”I am currently experiencing neural state Y, which I have mapped to the concept ’Heavy’\.”
3. 3\.System 2: The ”Autoepistemic Logic” Layer: Once the neural patterns are abstracted into logical symbols, System 2 takes over\. This is the deliberative, slow\-thinking part of the brain\. \-High\-Level Planning: Because these symbols are now part of a formal logicI​F​O​LBIFOL\_\{B\}, the robot can use them, both with probabilistic computations, in complex ”If\-Then” scenarios that neural networks struggle with\. \-Predefined set of axioms and prohibitions: Any kind of reasoning results and plans must satisfy these axioms before the executions of actions derived from such plans\. In this way the robots can not become danger for the humans and the environment in which they are active\. \-Feedback Loop: System 2 can ”query” System 1\. For example, the symbolic layer might ask, ”Check the visual sensors again; does that object match the ’Bird’ intension?” This forces the neural network to re\-process data based on logical needs\.
4. 4\.Solving the ”Black Box” Problem: One of the most interesting aspects of this bridge is Explainability\. \- In standard neural networks, we don’t know why a robot chose an action\. \- In this framework, because every neural state is linked to an intensional logic term, the robot can provide a symbolic trace of its reasoning: ”I performed Action A because my neural sensors triggered the ’Danger’ intension, which my logic defines as a state to be avoided”\.

Summary Table:The neuro\-symbolic’s Bridge Sistem 1 \(Neural\)The Bridge \(Intensional\)System 2 \(Symbolic\)FunctionFast, reactive sensingAbstraction&\\&GroundingSlow, logical planningData TypeVectors / TensorsIntensional EntitiesLogic FormulasRolePerceives the worldMaps neurons to symbolsReasons about perceptionsSelf\-awarenessUnconsciousThe ”I” links the twoConscious autoepistemicthought

In this paper we introduce more expressive 4\-valued

I​F​O​LBIFOL\_\{B\}, based on Belnap’s bilattice, by probabilistic features as well, in order to make neuro\-symbolic learning and many\-valued deductions \(as humans\) also in the presence of unknown and inconsistent \(contradictory\) sentences\[[6](https://arxiv.org/html/2607.13073#bib.bib6),[32](https://arxiv.org/html/2607.13073#bib.bib32)\], because is such cases the standard 2\-valued FOL is unable to work well \(deduces absolutely anything\)\. In this paper we will extend this truth\-many\-valued neuro\-symbolic framework also to the probabilistic learning inside this

I​F​O​LBIFOL\_\{B\}useful also for the simulation, planing and verification of different strategies in order that the AGI\-robot’s would be able to achieve their objectives and adaptive goal\-directed behaviour under varying circumstances\. With this extension, the robots will be able to reason both with algebraic many\-valued method \(as epistemic logic\) and also to manage uncertainty by*probabilistic logic*and symbol\-guided*common\-sense*obtained by deep neural learning in the same formal

I​F​O​LBIFOL\_\{B\}cognitive model\. Such AGI\-robots cognitive model is a general highly unified neuro\-symbolic model\.

It is passed twenty years from my proposed a fairly ambitious revision in that time of the earlier probabilistic logic programming \(PLP\) frameworks developed by V\. S\. Subrahmanian and collaborators\[[13](https://arxiv.org/html/2607.13073#bib.bib13),[15](https://arxiv.org/html/2607.13073#bib.bib15)\], where I tried to address exactly the kinds of their semantic weaknesses: problematic fixpoint semantics, interval ambiguity, mismatch between syntax and possible\-world semantics, temporal inconsistency, and many\-valued logical complications\. In particular, I criticized implicit temporal semantics, unclear Herbrand interpretation structure, interval annotations without explicit world semantics, overly syntactic fixpoint constructions\. I argued that time should be explicitly represented, probabilistic worlds should remain classical possible worlds, probability should be defined over sets of temporal Herbrand interpretations\. This is actually a fairly deep semantic correction which effectively reduces temporal probabilistic logic to classical logic over expanded temporal predicates with classical model theory, accepted by the next developments of this revision, especially in today’s main\-stream of distributive semantics for PLP \(modern PLP research mostly evolved toward distribution semantics, weighted logic, probabilistic graphical models, and differentiable probabilistic inference\)\.

Earlier systems often treated time as metadata, annotation, external modal structure, while in my revision I instead internalized time directly into Herbrand models, predicates and possible worlds, by restoring semantic clarity: each possible world becomes an ordinary temporal interpretation, and probability distributions are defined over classical logical models\. Rather than relying primarily on interval probability propagation and many\-valued fixpoint operators, this revision has been philosophically important: probability should remain measure\-theoretic and logic should remain 2\-valued at the world level, which aligns more closely with standard probability theory, modal semantics, Kripke semantics, and probabilistic model theory\. I explicitly criticized the idea that probabilistic truth should be treated as ordinary many\-valued logic, by adopting Nils Nilsson probability distributions over possible worlds, instead of embedding probabilities directly as many\-valued truth values: The probabilities are not truth values themselves, they are properties of propositions across worlds\. This direction is actually philosophically closer to Nilsson, Sato, and important researchers: Fabrizio Riguzzi, Luc De Raedt, Angelika Kimmig and Terrance Swift, with key systems ProbLog, PRISM and LPADs\. A major semantic consolidation paper was by Riguzzi&\\&Swift on well\-defined distribution semantics\[[26](https://arxiv.org/html/2607.13073#bib.bib26)\]\.

My revision reflected a deeper foundational issue: Is probability a generalized truth value, or a measure over classical worlds? This is a profound distinction, with the answer: worlds remain classical and probability is meta\-level structure over worlds\. That position is philosophically closer to Kolmogorov probability, modal realism and stochastic semantics, than, fir example, to fuzzy logic\. This resembles modal logic, intensional semantics\[[25](https://arxiv.org/html/2607.13073#bib.bib25),[1](https://arxiv.org/html/2607.13073#bib.bib1)\]with possible\-world semantics and Montague\-style semantics\. Here I will extend it\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]also to 4\-valued \(Belnap’s bilattice\) logic for AGI robots\.

## 2Nilsson’s Structures and Reasoning about Probabilities

The probability theory is a well\-studied branch of mathematics, in order to carry out formal reasoning about probability\. Thus, it is important to have a logic, both for computation of probabilities and for reasoning about probabilities, with a well\-defined syntax and semantics\. Both current approaches, based on Nilsson’s probability structures/logics, and on linear inequalities in order to reason about probabilities, have some weak points \(Section 2\.2 in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\)\.

We will show that the logic for reasoning*about*probabilities can be naturally embedded into a 4\-valued intensional FOL with intensional abstraction, by avoiding current ad\-hoc system composed of*two different*2\-valued logics: one for the classical propositional logic at lower\-level, and a new one at higher\-level for probabilistic constraints with probabilistic variables\.

The main motivation for an introduction of the intensionality in the probabilistic\-theory of the propositional logic is based on the desire to have the*full*logical embedding of the probability into the First\-Order Logic \(FOL\), with a clear difference from the classic concept of truth of the logic formulae and the concept of their probabilities\. In this way we are able to replace the ad\-hoc syntax and semantics, used in current practice for Probabilistic Logic Programs\[[11](https://arxiv.org/html/2607.13073#bib.bib11),[12](https://arxiv.org/html/2607.13073#bib.bib12),[13](https://arxiv.org/html/2607.13073#bib.bib13)\]and probabilistic deduction\[[14](https://arxiv.org/html/2607.13073#bib.bib14)\], by the standard syntax and semantics used for the FOL where the probabilistic\-theory properties are expressed simply by the particular constraints on their interpretations and models\.

In this section we will consider the probabilistic semantics for the propositional logic only \(it can be easily extended to predicate logics as well\)\[[16](https://arxiv.org/html/2607.13073#bib.bib16),[17](https://arxiv.org/html/2607.13073#bib.bib17)\]with a fixed finite setP=\{p1,…,pn\}P=\\\{p\_\{1\},\.\.\.,p\_\{n\}\\\}of primitive propositions, which can be thought of as corresponding to basic probabilistic events\. The setℒ​\(P\)\\mathcal\{L\}\(P\)of the propositional formulae is the closure ofPPunder the Boolean operations for conjunction and negation,∧\\wedgeand¬\\neg, that is, it is the set of all formulae of the*propositional*logic\(P,\{∧,¬\}\)\(P,\\\{\\wedge,\\neg\\\}\)\.

In order to give the probabilistic semantics to such formulae, we first need to review briefly the probability theory \(see, for example,\[[18](https://arxiv.org/html/2607.13073#bib.bib18),[19](https://arxiv.org/html/2607.13073#bib.bib19)\]\):

###### Definition 3

A probability space\(S,𝒳,μ\)\(S,\\mathcal\{X\},\\mu\)consists of a setSS, called the sample space, aσ\\sigma\-algebra𝒳\\mathcal\{X\}of subsets ofSS\(i\.e\., a set of subsets ofSScontainingSSand closed under complementation and countable union, but not necessarily consisting of all subsets ofSS\) whose elements are called measurable sets, and a probability measureμ:𝒳→\[0,1\]\\mu:\\mathcal\{X\}\\rightarrow\[0,1\]where\[0,1\]\[0,1\]is the closed interval of reals from 0 to 1\. This mapping satisfies Kolmogorov axioms\[[20](https://arxiv.org/html/2607.13073#bib.bib20)\]:

A\.1μ​\(Y\)≥0\\penalty 10000\\ \\mu\(Y\)\\geq 0for allY∈𝒳Y\\in\\mathcal\{X\}\.

A\.2μ​\(S\)=1\\penalty 10000\\ \\mu\(S\)=1\.

A\.3μ​\(⋃i≥1Yi\)=∑i≥1μ​\(Yi\)\\penalty 10000\\ \\mu\(\\bigcup\_\{i\\geq 1\}Y\_\{i\}\)=\\sum\_\{i\\geq 1\}\\mu\(Y\_\{i\}\), ifYi\\penalty 10000\\ Y\_\{i\}’s are nonempty pairwise disjoint members of𝒳\\penalty 10000\\ \\mathcal\{X\}\. We define a probability density function,K​I=μ∘i​nKI=\\mu\\circ in, wherei​n:S↪𝒫​\(S\)in:S\\hookrightarrow\\mathcal\{P\}\(S\)is an inclusion such thati​n​\(s\)=\{s\}in\(s\)=\\\{s\\\}\.

Theμ​\(\{s\}\)=K​I​\(s\)\\mu\(\\\{s\\\}\)=KI\(s\)is the value of probability in a single point of spacess\.

The property A\.3 is called*countable additivity*for the probabilities in a spaceSS\. In the case when𝒳\\penalty 10000\\ \\mathcal\{X\}is finite set, then we can simplify property A\.3 above to

A\.3’μ​\(Z​⋃Y\)=μ​\(Z\)\+μ​\(Y\)\\penalty 10000\\ \\mu\(Z\\bigcup Y\)=\\mu\(Z\)\+\\mu\(Y\), ifZZandYYare disjoint members of𝒳\\penalty 10000\\ \\mathcal\{X\}, or, equivalently, to the following axiom:

A\.3”μ​\(Z\)=μ​\(Z​⋂Y\)\+μ​\(Z​⋂Y¯\)\\penalty 10000\\ \\mu\(Z\)=\\mu\(Z\\bigcap Y\)\+\\mu\(Z\\bigcap\\overline\{Y\}\), whereY¯\\overline\{Y\}is the compliment ofYYinSS, so thatμ​\(Y¯\)=1−μ​\(Y\)\\mu\(\\overline\{Y\}\)=1\-\\mu\(Y\)\.

In what follows we will consider only finite sample spaceSS, so that𝒳=𝒫​\(S\)\\penalty 10000\\ \\mathcal\{X\}=\\mathcal\{P\}\(S\)\. Thus, in our case of a finite setSSwe obtain, form A\.1 and A\.2, that for anyY∈𝒫​\(S\)Y\\in\\mathcal\{P\}\(S\),

μ​\(Y\)=∑s∈Yμ​\(\{s\}\)=∑s∈YK​I​\(s\)\\penalty 10000\\ \\mu\(Y\)=\\sum\_\{s\\in Y\}\\mu\(\\\{s\\\}\)=\\sum\_\{s\\in Y\}KI\(s\)\(10\)Based on the work of Nilsson in\[[16](https://arxiv.org/html/2607.13073#bib.bib16)\]we can define for a given propositional logic with a finite set of primitive propositionsPPthe sample spaceS=2PS=\\textbf\{2\}^\{P\}, where

2=\{f,t\}⊂X=ℬ4=\{f,t,⊥,⊤\}\\textbf\{2\}=\\\{f,t\\\}\\subset X=\\mathcal\{B\}\_\{4\}=\\\{f,t,\\bot,\\top\\\}\(11\)is the set of truth values of standard 2\-valued logic, so that the probability space is equal to the Nilsson’s structureN=\(2P,𝒫​\(2P\),μ\)N=\(\\textbf\{2\}^\{P\},\\mathcal\{P\}\(\\textbf\{2\}^\{P\}\),\\mu\)\.

In his work \(page 2, line 4\-6 in\[[16](https://arxiv.org/html/2607.13073#bib.bib16)\]\) Nilsson considered a Probabilistic Logic ”in which the truth values of sentences can range between0and11\. The truth value of a sentence in*probabilistic logic*is taken to be the probability of that sentence in ordinary first\-order logic\.” That is, he considered this logic as a kind of a*many\-valued*\(shown in Section 2\.2\.1 in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\), but not a compositional truth\-valued, logic\. But in his paper he did not defined the formal syntax and semantics for such a probabilistic logic, but only the matrix equations where the probability of a sentenceϕ∈ℒ​\(P\)\\phi\\in\\mathcal\{L\}\(P\)is the sum of the probabilities of the sets of possible worlds \(equal to the setS=2PS=\\textbf\{2\}^\{P\}\) in which that sentence is*true*\. So that he assigns*two*different logic values to each sentenceϕ\\phi: one is its probability value and another is a classic 2\-valued truth value in a given possible worldv∈𝒲=S=2Pv\\in\\mathcal\{W\}=S=\\textbf\{2\}^\{P\}\.

The*logic*inadequacy of this seminal work\[[16](https://arxiv.org/html/2607.13073#bib.bib16)\]of Nilsson is also considered in\[[21](https://arxiv.org/html/2607.13073#bib.bib21)\], by extending this Nilsson’s structure into a more general*probability structure*M=\(2P,𝒫​\(2P\),μ,π\)M=\(\\textbf\{2\}^\{P\},\\mathcal\{P\}\(\\textbf\{2\}^\{P\}\),\\mu,\\pi\), whereπ\\piassociates with eachs∈S=2Ps\\in S=\\textbf\{2\}^\{P\}the truth assignmentπ​\(s\):P→2\\pi\(s\):P\\rightarrow\\textbf\{2\}\. However, in our case whenS=2PS=\\textbf\{2\}^\{P\},π\\piis just an identity, so not necessary, and we consider eachssas a truth valuations=v:P→\{f,t\}s=v:P\\rightarrow\\\{f,t\\\}which can be uniquely extended to the truth assignmentv∗v^\{\*\}to all formulae inℒ​\(P\)\\mathcal\{L\}\(P\), by taking the usual rules of propositional logic \(the unique homomorphic extension to all formulae\), and we can associate to each propositional formulaϕ∈ℒ​\(P\)\\phi\\in\\mathcal\{L\}\(P\)the setϕM\\phi^\{M\}consisting of all statess∈Ss\\in Swhere the sentenceϕ\\phiis true, so that

‖ϕ‖=\{v∈𝒲=2P\|v∗​\(ϕ\)=t\}\\penalty 10000\\ \\\|\\phi\\\|=\\\{v\\in\\mathcal\{W\}=\\textbf\{2\}^\{P\}\\penalty 10000\\ \|\\penalty 10000\\ v^\{\*\}\(\\phi\)=t\\\}\\penalty 10000\\ \\penalty 10000\\\(12\)But, differently from Nilsson, in\[[21](https://arxiv.org/html/2607.13073#bib.bib21)\]the authors did not define a many\-valued propositional logic, but a kind of 2\-valued logic based on probabilistic constraints\. They denoted bywN​\(ϕ\)w\_\{N\}\(\\phi\)the*weight*or*probability*ofϕ\\phiin Nilsson structureNN, correspondent to the valueμ​\(‖ϕ‖\)\\mu\(\\\|\\phi\\\|\), so that the basic probabilistic 2\-valued constraint can be defined by expressionsc1≤wN​\(ϕ\)c\_\{1\}\\leq w\_\{N\}\(\\phi\)andwN​\(ϕ\)≤c2w\_\{N\}\(\\phi\)\\leq c\_\{2\}for given constantsc1,c2∈\[0,1\]c\_\{1\},c\_\{2\}\\in\[0,1\]\. They expected their logic to be used for reasoning*about*probabilities\. But, again, they did not defined a unique logic, but*two different*logics: one for the classical propositional logicℒ​\(P\)\\mathcal\{L\}\(P\), and a new one for 2\-valued probabilistic constraints obtained from the basic probabilistic formulae above and Boolean operators∧\\wedgeand¬\\neg\. They did not consider the introduced symbolwNw\_\{N\}as a formal functional symbol for a mappingwN:ℒ​\(P\)→\[0,1\]w\_\{N\}:\\mathcal\{L\}\(P\)\\rightarrow\[0,1\], such that for any propositional formulaϕ∈ℒ​\(P\)\\phi\\in\\mathcal\{L\}\(P\), with𝒲=S=2P\\mathcal\{W\}=S=\\textbf\{2\}^\{P\}, the*probability to be true*of sentenceϕ\\phiis

wN​\(ϕ\)=μ​\(‖ϕ‖\)=∑v∈‖ϕ‖K​I​\(v\)w\_\{N\}\(\\phi\)=\\mu\(\\\|\\phi\\\|\)=\\sum\_\{v\\in\\\|\\phi\\\|\}KI\(v\)\. Instead of this intuitive meaning forwNw\_\{N\}they considered each expressionwN​\(ϕ\)w\_\{N\}\(\\phi\)as a particular probabilistic term \(more precisely, as a structured probabilistic*variable*over the domain of values in\[0,1\]\[0,1\]\)\.

It seams that such a dichotomy and difficulty to have*a unique*2\-valued probabilistic logic, both for an original propositional formulae inℒ​\(P\)\\mathcal\{L\}\(P\)and for the probabilistic constraints, is based on the fact that if we considerwNw\_\{N\}as a function with one argument then it has to be formally represented as a binary predicatewN​\(ϕ,a\)w\_\{N\}\(\\phi,a\)\(for the graph of this function\) where the first argument is a formula and the second is its resulting probability value\. Consequently, a constraint ”the probability ofϕ\\phito be true is less or equal tocc”, has to be formally expressed by the logic formulawN​\(ϕ,a\)∧≤\(a,c\)w\_\{N\}\(\\phi,a\)\\wedge\\leq\(a,c\)\(here we use a symbol≤\\leqas a built\-in rigid binary predicate where≤\(a,c\)\\leq\(a,c\)is equivalent toa≤ca\\leq c\), which is a*second\-order*syntax becauseϕ\\phiis a logic*formula*in such a unified logic language\. That is, the problem of obtaining the unique logical framework for probabilistic logic comes out with the necessity of a*reification*feature of this logic language, analogously to the case of the intensional semantics for RDF data structures\[[22](https://arxiv.org/html/2607.13073#bib.bib22),[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\.

Consequently, we need a logic\[[25](https://arxiv.org/html/2607.13073#bib.bib25)\]which is able to deal directly with reification of logic formulae, that transforms a propositional formulaeϕ∈ℒ​\(P\)\\phi\\in\\mathcal\{L\}\(P\)into an abstracted*term*, denoted by⋖ϕ⋗\\lessdot\\phi\\gtrdot\. By this approach the expressionwN\(⋖ϕ⋗,a\)∧≤\(a,c\)w\_\{N\}\(\\lessdot\\phi\\gtrdot,a\)\\wedge\\leq\(a,c\)remains to be an ordinary first\-order formula\. In fact, if⋖ϕ⋗\\lessdot\\phi\\gtrdotis translated into non\-sentence ”thatϕ\\phi”, then the first\-order formula above corresponds to the sentence ”the probability`that`ϕ\\phiis true is less than or equal tocc”\.

Such approach has been used\[[23](https://arxiv.org/html/2607.13073#bib.bib23),[2](https://arxiv.org/html/2607.13073#bib.bib2)\]for the reduction of temporal probabilistic databases into constraint logic programs and to apply the interval PSAT in order to find the models of such interval\-based probabilistic*programs*\. In our case, for selfindependent neuro\-symbolic AGI robots, we do not need to write the probabilistic*programs*for them, but only to provide a general self\-reasoning about the probabilities of their 4\-valued sentences\. So, in next Section we will show how the probabilistic reasoning can be embedded in the 4\-valuedI​F​O​LBIFOL\_\{B\}based on Belnap’s bilattice of truth\-values inX=ℬ4=\{f,t,⊥,⊤\}X=\\mathcal\{B\}\_\{4\}=\\\{f,t,\\bot,\\top\\\}\.

## 3Neuro\-symbolic AGI Robots Reasoning about Probabilities of Uncertain Events

There are numerous proposals for probabilistic logics\. Very roughly, they can be categorized into two different classes: those logics that attempt to make a probabilistic extension to logical entailment, such as Markov logic networks, and those that attempt to address the problems of uncertainty and lack of evidence \(evidentiary logics\)\.

For AGI robots we considered theI​F​O​LBIFOL\_\{B\}with 4\-valued Belnap’s bilattice of truth\-values inX=ℬ4=\{f,t,⊥,⊤\}X=\\mathcal\{B\}\_\{4\}=\\\{f,t,\\bot,\\top\\\}with knowledge ordering as well\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\], where the value ”unknown” is the bottom value, the sentences with this value are indeed unknown facts, that is, the missed knowledge in the AGI robots\.

Thus, these unknown facts are not part of the robot’s knowledge database, and by learning through input and experiences, the robot’s knowledge would be naturally expanded over time\. Consequently, this phenomena has been represented by the Closed Knowledge Assumption and Logic Inference\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\], here for unknown facts, because of lack of evidence inside non\-probabilisticI​F​O​LBIFOL\_\{B\}and its autoepistemic deductive system enable to establish some of the*known*logic states in\{t,f,⊤\}⊂X\\\{t,f,\\top\\\}\\subset Xof the facts \(to be true, false or inconsistent\) the robot has to assign to such facts the ”absolute uncertainty”, that is the truth\-value⊥\\bot\(*unknown*logic value\)\. So, the minimization of this uncertainty can be obtained by introducing the probabilistic computation \(by using Nilsson’s structures with Kolmogorov’s axioms\) for these unknown facts, to obtain*what is the probability*of a currently unknown fact to be true, false or inconsistent\.

We recall that the extensionalization functionsh∈ℰi​nh\\in\\mathcal\{E\}\_\{in\}inI​F​O​LBIFOL\_\{B\}, are given in the disjoint union from \([3](https://arxiv.org/html/2607.13073#S1.E3)\),

h=∑i∈ℕhi:𝒟⟶D0\+∑i≥1ℜ​𝔪ih=\\sum\_\{i\\in\\mathbb\{N\}\}h\_\{i\}:\\mathcal\{D\}\\longrightarrow D\_\{0\}\+\\sum\_\{i\\geq 1\}\\mathfrak\{Rm\}\_\{i\}Thus, the intensions can be seen as*names*of abstract or concrete entities, while the extensions correspond to various rules that these entities play in different worlds we use the bijective mapping \([6](https://arxiv.org/html/2607.13073#S1.E6)\),i​si​n:𝒲→ℰi​nis\_\{in\}:\\mathcal\{W\}\\rightarrow\\mathcal\{E\}\_\{in\}, whereℰi​n\\mathcal\{E\}\_\{in\}is the set of all extensionalization functions that respect all built\-in predicates, and the ”set of possible worlds”𝒲=\{h∘I\|h∈ℰi​n\}\\mathcal\{W\}=\\\{h\\circ I\|h\\in\\mathcal\{E\}\_\{in\}\\\}\. So, from the commutative truth\-diagram for the setℒ0\\mathcal\{L\}\_\{0\}of all sentences of this logicℒi​n\\mathcal\{L\}\_\{in\}, provided by Theorem 1 in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\], we obtain thatIB∗=h∘II^\{\*\}\_\{B\}=h\\circ I, so that from Definition 15 in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\], given an assignmentg:𝒱→𝒟g:\\mathcal\{V\}\\rightarrow\\mathcal\{D\}, for any sentenceϕ/g∈ℒ0\\phi/g\\in\\mathcal\{L\}\_\{0\},

\(h∘I\)​\(ϕ/g\)=IB∗​\(ϕ/g\)=\{\{v∗​\(ϕ/g\)\}∈ℜ​𝔪,ifv∗​\(ϕ/g\)≠⊥∅,otherwise\(h\\circ I\)\(\\phi/g\)=I^\{\*\}\_\{B\}\(\\phi/g\)=\\left\\\{\\begin\{array\}\[\]\{ll\}\\\{v^\{\*\}\(\\phi/g\)\\\}\\in\\mathfrak\{Rm\},&\\hbox\{if $\\penalty 10000\\ \\penalty 10000\\ v^\{\*\}\(\\phi/g\)\\neq\\bot$\}\\\\ \\emptyset,&\\hbox\{otherwise\}\\end\{array\}\\right\.\(13\)Consequently, we are able to represent the whole*AGI robot’s knowledge Database*of its n\-ary concepts,n≥1n\\geq 1, in each fixed instance of time \(it is an analog to traditional relational Database with standard 2\-valued FOL\) as follows:

###### Definition 4

AGI Robot’s Current Atomic Knowledge Database333This definition is different from the definition in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\], in order to be able also to derive the current Herbrand model of robot’s knowledgev:H→Xv:H\\rightarrow X, whereHHis the Herbrand base for all predicates inPP, Note that the autoepistemic predicateK​n​o​wKnowis a meta\-èredicate, so thatK​n​o​w∉PKnow\\notin P\.: For a given instance of time, the current AGI robot knowledge is defined by its current MV\-interpretationIB∗\\textbf\{I\}^\{\*\}\_\{B\}\(that represents robot’s current worldw∈𝒲w\\in\\mathcal\{W\}\) defines the extension of robot’s knowledge Database𝒦\\mathcal\{K\}by:

𝒦=\{Rpik=IB∗\(pik\(x1,…,xk\)\|pik∈P,forxi∈𝒱,1≤i≤k\}\\penalty 10000\\ \\penalty 10000\\ \\mathcal\{K\}\\penalty 10000\\ =\\penalty 10000\\ \\\{R\_\{p\_\{i\}^\{k\}\}=\\textbf\{I\}^\{\*\}\_\{B\}\(p\_\{i\}^\{k\}\(x\_\{1\},\.\.\.,x\_\{k\}\)\\penalty 10000\\ \|\\penalty 10000\\ p\_\{i\}^\{k\}\\in P,\\penalty 10000\\ for\\penalty 10000\\ x\_\{i\}\\in\\mathcal\{V\},1\\leq i\\leq k\\\}\(14\)whereIB∗\\textbf\{I\}^\{\*\}\_\{B\}is the current MV\-interpretation \(a function fromℒ\\mathcal\{L\}toℜ​𝔪\\mathfrak\{Rm\}\) and𝒱\\mathcal\{V\}the set of variables andRpikR\_\{p\_\{i\}^\{k\}\}is the\(k\+1\)\(k\+1\)\-ary relation obtained for the k\-ary predicatepikp\_\{i\}^\{k\}\.

Consequently, the atomic knowledge database is the subset of current metaknowledge of a robot is the set of relations, that is,

𝒦⊂\{R\|R∈Im\(IB∗\),forar\(R\)≥2\}\\mathcal\{K\}\\subset\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\\{R\\penalty 10000\\ \|\\penalty 10000\\ R\\in Im\(\\textbf\{I\}^\{\*\}\_\{B\}\),\\penalty 10000\\ for\\penalty 10000\\ ar\(R\)\\geq 2\\\}\(15\)where for each tuple of ground terms,d=\(t1,…,tk,a\)∈Rpik∈𝒦\\textbf\{d\}=\(t\_\{1\},\.\.\.,t\_\{k\},a\)\\in R\_\{p\_\{i\}^\{k\}\}\\in\\mathcal\{K\}, of relation with arityk\+1≥2k\+1\\geq 2, the current truth\-value of this, for robot known, fact is equal to last value of this tuple,a=πk\+1​\(d\)∈\{f,⊤,t\}a=\\pi\_\{k\+1\}\(\\textbf\{d\}\)\\in\\\{f,\\top,t\\\}with truth\-orderingf<⊤<tf<\\top<t\. So, for any ground instance of each robot’s \(real or virtual\) predicate \(and corresponding intensional concept\), robot is able to know the level of truth in a given instance of time\. Thus, given current atomic knowledge database𝒦\\mathcal\{K\}, we are able to derive from it the current Herbrand modelv:H→Xv:H\\rightarrow Xand its extensionv∗v^\{\*\}to all sentences, for robot’s knowledge by, for any ground atompik​\(t1,…,tk\)∈Hp\_\{i\}^\{k\}\(t\_\{1\},\.\.\.,t\_\{k\}\)\\in H, with the ground termstit\_\{i\}, for1≤i≤k1\\leq i\\leq k,

v​\(pik​\(t1,…,tk\)\)=\{a∈\{f,⊤,t\},if\(t1,…,tk,a\)∈Rpik∈𝒦⊥,otherwisev\(p\_\{i\}^\{k\}\(t\_\{1\},\.\.\.,t\_\{k\}\)\)=\\left\\\{\\begin\{array\}\[\]\{ll\}a\\in\\\{f,\\top,t\\\},&\\hbox\{if $\\penalty 10000\\ \\penalty 10000\\ \(t\_\{1\},\.\.\.,t\_\{k\},a\)\\in R\_\{p\_\{i\}^\{k\}\}\\in\\mathcal\{K\}$\}\\\\ \\bot,&\\hbox\{otherwise\}\\end\{array\}\\right\.\(16\)Thus, from current atomic knowledge database𝒦\\mathcal\{K\}, we are able to derive current Herbrand modelvv, its extension to all sentencesv∗v^\{\*\}and thence the current possible world \(MV\-model in Definition[4](https://arxiv.org/html/2607.13073#Thmdefinition4)\)IB∗=h∘I∈𝒲\\textbf\{I\}^\{\*\}\_\{B\}=h\\circ I\\in\\mathcal\{W\}\.

It was demonstrated that the knowledge database𝒦\\mathcal\{K\}satisfies the following assumption444Here presented in corrected version w\.r\.t\. that in\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]\.:

###### Definition 5

Closed Knowledge Assumption\(CKA\): The CKA for the many\-sorted Intensional FOL based on Belnap’s 4\-valued bilattice of truth\-values is defined, for every Herbrand interpretationv:H→Xv:H\\rightarrow Xand assignmentg:𝒱→𝒟g:\\mathcal\{V\}\\rightarrow\\mathcal\{D\}, as follows: For each k\-ary virtual predicateϕ​\(x1,…,xk\)\\phi\(x\_\{1\},\.\.\.,x\_\{k\}\),k≥1k\\geq 1,

v∗​\(ϕ​\(x1,…,xk\)/g\)≠⊥v^\{\*\}\(\\phi\(x\_\{1\},\.\.\.,x\_\{k\}\)/g\)\\neq\\bot\\penalty 10000\\ \\penalty 10000\\iff\(g​\(x1\),…,g​\(xk\)\)∈π−i​\(IB∗​\(ϕ​\(x1,…,xk\)\)\)\\penalty 10000\\ \\penalty 10000\\ \(g\(x\_\{1\}\),\.\.\.,g\(x\_\{k\}\)\)\\in\\pi\_\{\-i\}\(I\_\{B\}^\{\*\}\(\\phi\(x\_\{1\},\.\.\.,x\_\{k\}\)\)\)\.

The*intensional abstract terms*are ”that\-clauses” used for*reification*features\[[24](https://arxiv.org/html/2607.13073#bib.bib24)\]ofI​F​O​LBIFOL\_\{B\}so that, for a logic formulaϕ​\(x\)\\phi\(\\textbf\{x\}\)and assignmentg∈𝒟𝒱g\\in\\mathcal\{D\}^\{\\mathcal\{V\}\}of variables in𝒱\\mathcal\{V\}, ”thatϕ\\phi”, from Definition[1](https://arxiv.org/html/2607.13073#Thmdefinition1)is denoted by the ground abstracted term⋖ϕ​\(x\)​\[β/g​\(β\)\]⋗α\\lessdot\\phi\(\\textbf\{x\}\)\[\\beta/g\(\\beta\)\]\\gtrdot\_\{\\alpha\}\(ifϕ\\phiis a sentence then bothα\\alphaandβ\\betaare empty\)\. Hence, the sentence ”the probability that a sentenceϕ\\phihas a truth\-valueaais less then or equal toc1c\_\{1\}” can be expressed by the first\-order logic ground atomwN\(⋖ϕ⋗,a,c\)∧≤\(c,c1\)w\_\{N\}\(\\lessdot\\phi\\gtrdot,a,c\)\\wedge\\leq\(c,c\_\{1\}\), where≤\\leqis the binary built\-in predicate with standard denotationc≤c1c\\leq c\_\{1\}, while ”the probability thatϕ\\phihas a truth\-valueaais equal tocc” is denoted by the ground atomwN\(⋖ϕ⋗,a,c\)w\_\{N\}\(\\lessdot\\phi\\gtrdot,a,c\)with the special ternary predicatewNw\_\{N\}which first argument is from an sentence abstracted term, the second argument is for the truth\-value of this sentence and third argument is the probability that it is so\. So, we introduce the following built\-in concepts \(for Definition 9 in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\]\) of typedI​F​O​LBIFOL\_\{B\}:

1. 1\.”truth values:sX\\penalty 10000\\ s\_\{X\}”, introduced in\[[6](https://arxiv.org/html/2607.13073#bib.bib6)\], that is a concepts inD2D\_\{2\}such that forX⊂D0X\\subset D\_\{0\}, its extension ish2\(truth values:sX\)=\{\(a,t\)\|a,t∈X\}h\_\{2\}\(\\textbf\{truth values\}:\\penalty 10000\\ s\_\{X\}\)=\\\{\(a,t\)\\penalty 10000\\ \|\\penalty 10000\\ a,t\\in X\\\}\. We will use the special typed variablexXx\_\{X\}for the set of truth values inX=ℬ4=\{f,t,⊥,⊤\}X=\\mathcal\{B\}\_\{4\}=\\\{f,t,\\bot,\\top\\\}, so that for each assignmentgg,g​\(xX\)∈Xg\(x\_\{X\}\)\\in X\.
2. 2\.”probability:sp\\penalty 10000\\ s\_\{p\}”, that is a concepts inD2D\_\{2\}such that its extension ish2\(probability:sp\)=\[0,1\]×\{t\}h\_\{2\}\(\\textbf\{probability\}:\\penalty 10000\\ s\_\{p\}\)=\[0,1\]\\times\\\{t\\\}, where\[0,1\]\[0,1\]is interval of positive reals used for the probabilities andt∈Xt\\in Xthe true logic value\. We will use the special typed variablexpx\_\{p\}for the probabilities, so that for each assignmentgg,g​\(xp\)∈\[0,1\]g\(x\_\{p\}\)\\in\[0,1\]\.
3. 3\.for each finite ordered calendar interval of time\[tI,tF\]τ\[\\textbf\{t\}\_\{I\},\\textbf\{t\}\_\{F\}\]\_\{\\tau\}with initialtI\\textbf\{t\}\_\{I\}time\-instance and finaltF\\textbf\{t\}\_\{F\}time\-instance, of sortssτs\_\{\\tau\}\) with a given granularityτ∈\{y​e​a​r,y​e​a​r:m​o​n​t​h,y​e​a​r:m​o​n​t​h:d​a​y,y​e​a​r:m​o​n​t​h:d​a​y:o​u​r,…\}\\tau\\in\\\{year,year:month,year:month:day,year:month:day:our,\.\.\.\\\}, we can have a particular calendar\-concept inD2D\_\{2\}, ”calendar:sτs\_\{\\tau\}”, such that its finite extension ish2\(calendar:sτ\)=\[tI,tF\]τ×\{t\}h\_\{2\}\(\\textbf\{calendar\}:\\penalty 10000\\ s\_\{\\tau\}\)=\[\\textbf\{t\}\_\{I\},\\textbf\{t\}\_\{F\}\]\_\{\\tau\}\\times\\\{t\\\}witht∈Xt\\in Xthe true logic value\. We will use the special typed variablesxτx\_\{\\tau\}for the time\-instances of these calendars, so that for each assignmentgg,t=g​\(xτ\)∈\[tI,tF\]τ\\textbf\{t\}=g\(x\_\{\\tau\}\)\\in\[\\textbf\{t\}\_\{I\},\\textbf\{t\}\_\{F\}\]\_\{\\tau\}\. Each predicatepi∈Pp\_\{i\}\\in Pthat has exactly one attribute of sortsτs\_\{\\tau\}is called an ”atomic event” predicate\.

Thus, in this version ofI​F​O​LBIFOL\_\{B\}we will have the following three meta\-predicates \(two\-valued predicates that represent’s the semantically very specific knowledge about properties of standard domain\-based predicates inPP\(the predicates inPPhave no the typed variablexsx\_\{s\}of sort ’nested sentence’, provided in Definition of static sorts in\[[3](https://arxiv.org/html/2607.13073#bib.bib3)\]\):

- •4\-valued Knowledge predicateK​n​o​wKnowused for autoepistemic deduction\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]\.
- •2\-valued \(true or false\) Probabilistic predicatewNw\_\{N\}with the first argument has the sort of ’nested sentence’ and with free\-variable atomwN\(⋖ψ\(x\)⋗β,xX,xp\)w\_\{N\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\},x\_\{X\},x\_\{p\}\)where the formulaψ\\psiis composed by predicates inPP\. Ifψ\\psiis composed also by ”atomic event” predicates, thenwN\(⋖ψ\(x\)⋗β,xX,xp\)w\_\{N\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\},x\_\{X\},x\_\{p\}\)is called Event probabilistic predicate”\.

so thatP​⋂\{K​n​o​w,wN,e\}P\\bigcap\\\{Know,w\_\{N\},\\textbf\{e\}\\\}is empty set\. This separation of standard and meta\-predicates is based on the fact that the truth\-value of meta\-predicate ground atoms depends on the truth\-values of the*another sentence*used in the abstract term of the typed variablexsx\_\{s\}\. In effect, from Definition 14 in\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\], we have that the computation of the truth\-value of the 4\-valuedK​n​o​wKnowground atoms is done as this atom is a particular logic sentence \(formula\) and not ground atom of Herbrand base:

v∗\(Know\(t1,t2,⋖ψ\(x\)⋗β\)/g\)=v∗\(ψ\(x\)/g\)∈X,andforIB∗=isM​V\(v∗\)IB∗\(Know\(t1,t2,⋖ψ\(x\)⋗β\)/g\)=IB∗\(ψ\(x\)/g\)v^\{\*\}\(Know\(t\_\{1\},t\_\{2\},\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\)/g\)=v^\{\*\}\(\\psi\(\\textbf\{x\}\)/g\)\\in X,\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ and\\penalty 10000\\ for\\penalty 10000\\ \\penalty 10000\\ I\_\{B\}^\{\*\}=is\_\{MV\}\(v^\{\*\}\)\\\\ I\_\{B\}^\{\*\}\(Know\(t\_\{1\},t\_\{2\},\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\)/g\)=I\_\{B\}^\{\*\}\(\\psi\(\\textbf\{x\}\)/g\)\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\\(17\)We have shown by Lemma 1 in\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]that, during time\-evolution for time\-instancesti≤ti\+1\\textbf\{t\}\_\{i\}\\leq\\textbf\{t\}\_\{i\+1\}of robot’s knowledge, it is always satisfied that

vi∗≼kvi\+1∗v^\{\*\}\_\{i\}\\preccurlyeq\_\{k\}v^\{\*\}\_\{i\+1\}\(18\)that is, we have a monotonic increments of knowledge, initially most \(non built\-in predicate\) ground atoms are unknown, and that during learning processes in future, the \(non built\-in predicate\) ground atoms can change their truth\-value in this knowledge\-monotonic way: 1\.⊥↦f\\bot\\mapsto fof⊥↦t\\bot\\mapsto t; 2\.f↦⊤f\\mapsto\\top; 3\.t↦⊤t\\mapsto\\top; 4\.⊤⁣↦⁣⊤\\top\\mapsto\\top\. Thus, for the Herbrand base \(from Definition 4 in\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]\)

H=\{pik\(t1,\.\.,tk\)\|pik∈P\\penalty 10000\\ \\penalty 10000\\ H=\\\{p\_\{i\}^\{k\}\(t\_\{1\},\.\.,t\_\{k\}\)\\penalty 10000\\ \|\\penalty 10000\\ p\_\{i\}^\{k\}\\in Pandt1,…,tkt\_\{1\},\.\.\.,t\_\{k\}are ground terms\}\\\} and the current Herbrand interpretationv:H→X\\textbf\{v\}:H\\rightarrow X,*for still unknown atoms*A∈HA\\in Hwithv​\(A\)=⊥\\textbf\{v\}\(A\)=\\bot\(note that such atoms can note be of built\-in predicates that have invariant truth\-value true or false\) we can use the probabilistic methods to establish with which probability they can have any truth\-value inXX, and to use this knowledge for another logical deductions\.

Consequently, in order that AGI robots be able to reason about probabilities of the sentences using the syntax of the FOL with the set of predicate symbolsPPof current robot’s knowledge \(robot would be able to increment this set with new predicates as well, based on its extended in time knowledge about external world\) we have the sample space of Definition[3](https://arxiv.org/html/2607.13073#Thmdefinition3), from \([2](https://arxiv.org/html/2607.13073#S1.E2)\),S⊆ℐH⊂XHS\\subseteq\\mathcal\{I\}\_\{H\}\\subset X^\{H\}whereHHis Herbrand base of this logicI​F​O​LBIFOL\_\{B\}with Belnap’s bilattice of truth\-values inXX\. And we impose the following constraints for the probability densityK​I=μ∘i​n:S→\[0,1\]KI=\\mu\\circ in:S\\rightarrow\[0,1\]:

###### Definition 6

Probability Density Constraints: For any current Herbrand interpretationv:H→X\\textbf\{v\}:H\\rightarrow X, the robot’s current knowledge database𝒦\\mathcal\{K\}ofI​F​O​LBIFOL\_\{B\}\(from Definition[4](https://arxiv.org/html/2607.13073#Thmdefinition4)\) must*remain invariant*by introducing the probability space\(S,𝒫​\(S\),μ\)\(S,\\mathcal\{P\}\(S\),\\mu\), in Definition[3](https://arxiv.org/html/2607.13073#Thmdefinition3)\. That is, for every atomA∈HA\\in Hsuch thatu=v​\(A\)≠⊥u=\\textbf\{v\}\(A\)\\neq\\bot, the probability thatAAhas the truth\-valueuumust be equal to 1, i\.e\.,

∀A∈H\.\(ifu=v\(A\)≠⊥then∑v∈S,v​\(A\)=uKI\(v\)=1\)\\forall A\\in H\.\(\\penalty 10000\\ if\\penalty 10000\\ \\penalty 10000\\ u=\\textbf\{v\}\(A\)\\neq\\bot\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ then\\penalty 10000\\ \\penalty 10000\\ \\sum\_\{v\\in S,v\(A\)=u\}KI\(v\)=1\)\(19\)

Thus, the action of introducing the probability computation inI​F​O​LBIFOL\_\{B\}will have the effects only to diminish the uncertainty of unknown facts, for which we can now compute*the probability*to have any truth\-value inXX\. Remark: This is a particular case of Noether’s conservation theorem applied to robot’s cognitive system: the \(global\)*symmetry transformation*of robot’s cognitive power by action of introduction of the probability space\(S,𝒫​\(S\),μ\)\(S,\\mathcal\{P\}\(S\),\\mu\)preserves its knowledge database𝒦\\mathcal\{K\}\. This symmetry is*global*because is valid for all current Herbrand base \(that is to all predicates inPPof robot’s knowledge base: so computationally it has relevant cost to be applied in real\-time robot’s processes\. However, we are able to consider less computationally expensive symmetry transformation, that can be done dynamically by robot to resolve some decisions about concrete problems that involve a very restricted subset of it predicates, relevant to such problems: such dynamic symmetries will be called a*local symmetries*in what follows\. □\\square So, in what follows we will extend the globally consistent and empirically satisfactory unification of classic probability theory and standard \(two\-valued\) first\-order logic that is suitable for inductive reasoning, developed by Gaifman and Snir\[[35](https://arxiv.org/html/2607.13073#bib.bib35),[36](https://arxiv.org/html/2607.13073#bib.bib36)\], to our more sophisticated Belnap’s based 4\-valued typedI​F​O​LBIFOL\_\{B\}used for humanoid AGI robots\.

###### Definition 7

Global Current Nilsson’s Structure forI​F​O​LBIFOL\_\{B\}: Letv:H→X\\textbf\{v\}:H\\rightarrow Xbe the current Herbrand interpretation of AGI robot with Belnap’s based 4\-valued typedI​F​O​LBIFOL\_\{B\}, and corresponding current possible worldIB∗=i​sH​\(v\)∈𝒲\\textbf\{I\}\_\{B\}^\{\*\}=is\_\{H\}\(\\textbf\{v\}\)\\in\\mathcal\{W\}\. Then we define the current Nilsson’s probability structure\(S,𝒫​\(S\),μ\)\(S,\\mathcal\{P\}\(S\),\\mu\)satisfying Kolmogorov axioms in Definition[3](https://arxiv.org/html/2607.13073#Thmdefinition3), such that the sample setSSis defined as a subset ofℐH\\mathcal\{I\}\_\{H\}in \([2](https://arxiv.org/html/2607.13073#S1.E2)\), by

S=d​e​f\{v∈ℐH⊂XH\|foreveryA∈H,v\(A\)=v\(A\)ifv\(A\)≠⊥\}S=\_\{def\}\\\{v\\in\\mathcal\{I\}\_\{H\}\\subset X^\{H\}\|\\penalty 10000\\ \\penalty 10000\\ for\\penalty 10000\\ every\\penalty 10000\\ \\penalty 10000\\ A\\in H,v\(A\)=\\textbf\{v\}\(A\)\\penalty 10000\\ \\penalty 10000\\ if\\penalty 10000\\ \\textbf\{v\}\(A\)\\neq\\bot\\\}\(20\)so that for anyv∈Sv\\in S, and ground atomA∈HA\\in Hsuch thatv​\(A\)=⊥\\textbf\{v\}\(A\)=\\bot, we can have thatv​\(A\)∈Xv\(A\)\\in Xcan have the value different from⊥\\botas well\.

Note that this sample setSSin \([20](https://arxiv.org/html/2607.13073#S3.E20)\) is defined only for current possible world \(and current knowledge base𝒦\\mathcal\{K\}, and does not modify the logic truth\-value of ground atomsA∈HA\\in Hfor whichv​\(A\)=⊥\\textbf\{v\}\(A\)=\\botor currently unknown sentencesϕ\\phiwithv∗​\(ϕ\)=⊥\\textbf\{v\}^\{\*\}\(\\phi\)=\\bot, but only offers the capacity to compute their*probability*to have any truth value inXX\. This global current Nilsson’s probability structure has the following important properties:

###### Corollary 1

The global current Nilsson’s probability structure with sample setSSin \([20](https://arxiv.org/html/2607.13073#S3.E20)\) satisfy the probability density constraints in Definition[6](https://arxiv.org/html/2607.13073#Thmdefinition6)\. Thus, it is consistent with current robot’s knowledge database𝒦\\mathcal\{K\}\(in Definition[4](https://arxiv.org/html/2607.13073#Thmdefinition4)\), consistent with Kolmogorow probability axioms and logical deduction, and allows inductive reasoning and confirmation of universally quantified hypotheses\.

Proof: It is enough to show the consistency with current robot’s knowledge database𝒦\\mathcal\{K\}\. In fact, for each ground atomA∈HA\\in Hfor which the current truth\-value in𝒦\\mathcal\{K\}isu=v​\(A\)≠⊥u=\\textbf\{v\}\(A\)\\neq\\bot, we have that the probability density constraint \([19](https://arxiv.org/html/2607.13073#S3.E19)\) is satisfied\. That is, the probabilitypup\_\{u\}that this atom has the truth\-valueuuis equal to 1:

pu=∑v∈S,v​\(A\)=uK​I​\(v\)p\_\{u\}=\\sum\_\{v\\in S,v\(A\)=u\}KI\(v\)

=∑v∈SK​I​\(v\)=\\sum\_\{v\\in S\}KI\(v\)from \([20](https://arxiv.org/html/2607.13073#S3.E20)\)

=μ​\(S\)=\\mu\(S\)

=1=1\. which preserves robot’s current knowledge database and all logical deductions from them\. □\\square Consequently, for new predicatewNw\_\{N\}\(not inPP\), we obtain that for any assignmentggand formulaψ​\(x\)\\psi\(\\textbf\{x\}\)composed by predicates \(only\) inPP, if, for a given assignmentgg, in current worldv∗​\(ψ​\(x\)/g\)=⊥\\textbf\{v\}^\{\*\}\(\\psi\(\\textbf\{x\}\)/g\)=\\bot, then we can compute in this current world \(current knowledge database𝒦\\mathcal\{K\}\)

v∗\(wN\(⋖ψ\(x\)⋗β/g,g\(xX\),g\(xp\)\)\)==\{t,ifg​\(xp\)=∑v∈S,v∗​\(ψ​\(x\)/g\)=g​\(xX\)K​I​\(v\)f,otherwise\\textbf\{v\}^\{\*\}\(w\_\{N\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}/g,g\(x\_\{X\}\),g\(x\_\{p\}\)\)\)=\\\\ =\\left\\\{\\begin\{array\}\[\]\{ll\}t,&\\hbox\{if $\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ g\(x\_\{p\}\)=\\sum\_\{v\\in S,v^\{\*\}\(\\psi\(\\textbf\{x\}\)/g\)=g\(x\_\{X\}\)\}KI\(v\)$\}\\\\ f,&\\hbox\{otherwise\}\\end\{array\}\\right\.\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\\(21\)That is,

1. 1\.ThewN\(⋖ψ\(x\)⋗β/g,g\(xX\),g\(xp\)\)w\_\{N\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}/g,g\(x\_\{X\}\),g\(x\_\{p\}\)\)is true iff*the probability*thatψ​\(x\)/g\\psi\(\\textbf\{x\}\)/ghas the truth\-valueg​\(xX\)g\(x\_\{X\}\)is equal tog​\(xp\)g\(x\_\{p\}\)\.
2. 2\.In the case when the formulaψ\\psicontains also ”atomic event” predicates, the eventwN\(⋖ψ\(x\)⋗β/g,g\(xX\),g\(xp\)\)w\_\{N\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}/g,g\(x\_\{X\}\),g\(x\_\{p\}\)\), withτ∈β\\tau\\in\\betais a free variable inψ\\psi, is true iff the probability thatψ​\(x\)/g\\psi\(\\textbf\{x\}\)/ghas the truth\-valueg​\(xX\)g\(x\_\{X\}\)at time\-instanceg​\(xτ\)g\(x\_\{\\tau\}\)is equal tog​\(xp\)g\(x\_\{p\}\)\. Note that from the fact thatxτx\_\{\\tau\}is typed variable, the instance of timeg​\(xτ\)g\(x\_\{\\tau\}\)is always inside the finite interval of granular calendar specified by this typeτ\\tau\. Obviously,ψ\\psican be composed by a number of different ”atomic events” as well, each of them with particular granular calendars time\-variablesxτx\_\{\\tau\}and their intervals\.

Remark:Thus, for the current possible world of robot \(its current knowledge database𝒦\\mathcal\{K\}\), for any unknown fact \(which is not in𝒦\\mathcal\{K\}\), we can have additional knowledge of what is the probability that this fact is true, false or inconsistent\. In this way, by probabilistic computation based on Nilsson’s probability stricture in Definition[7](https://arxiv.org/html/2607.13073#Thmdefinition7)we diminish generally the robot’s knowledge uncertainty by preserving \(from Corollary[1](https://arxiv.org/html/2607.13073#Thmcoro1)\) its consistency and logical deduction\. That is, for any sentenceϕ∈ℒ0\\phi\\in\\mathcal\{L\}\_\{0\}such that in current Herbrand interpretationv:H→X\\textbf\{v\}:H\\rightarrow X,v∗​\(ϕ\)=⊥\\textbf\{v\}^\{\*\}\(\\phi\)=\\bot, robot would be able to compute the probabilitypup\_\{u\}that this sentence has the truth valueu∈Xu\\in X,

pu=∑v∈S,v∗​\(ϕ\)=uK​I​\(v\)p\_\{u\}=\\sum\_\{v\\in S,v^\{\*\}\(\\phi\)=u\}KI\(v\)\(22\)which the robot can use for probability\-based decisions and actions, by using the reasoning about the probabilities \(as in Section 2\.2\.2 in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]\) by using the true atomswN\(⋖ϕ⋗,u,pu\)w\_\{N\}\(\\lessdot\\phi\\gtrdot,u,p\_\{u\}\)with also ”atomic events” that tells to robot that ”the probability at time\-instanceti\\textbf\{t\}\_\{i\}thatϕ\\phihas the truth\-valueuuis equal topup\_\{u\}”\.

These true facts about the probabilities of the unknown sentences to have a particular truth\-value can be also inserted in the conscious part of robot’s cognitive system \(see \([9](https://arxiv.org/html/2607.13073#S1.E9)\) for more details\) by,

Know\(inpresence,I,wN\(⋖ϕ⋗,u,pu\)\)Know\(in\\penalty 10000\\ presence,\\textbf\{I\},w\_\{N\}\(\\lessdot\\phi\\gtrdot,u,p\_\{u\}\)\), and hence to be used by robot’s autoepistemic deductions and explanations as well\. □\\square

###### Example 1

For example, let us consider an ”atomic event” predicatepip\_\{i\}, with its atom of free variablespi​\(…,xτ,…\)p\_\{i\}\(\.\.\.,x\_\{\\tau\},\.\.\.\), such that there exists the time\-instancet∈\[tI,tF\]τ\\textbf\{t\}\\in\[\\textbf\{t\}\_\{I\},\\textbf\{t\}\_\{F\}\]\_\{\\tau\}and assignmentggsuch thatg​\(xτ\)=tg\(x\_\{\\tau\}\)=\\textbf\{t\}, and in current possible worldv​\(pi​\(…,xτ,…\)/g\)=v​\(pi​\(…,t,…\)/g\)=t\\textbf\{v\}\(p\_\{i\}\(\.\.\.,x\_\{\\tau\},\.\.\.\)/g\)=\\textbf\{v\}\(p\_\{i\}\(\.\.\.,\\textbf\{t\},\.\.\.\)/g\)=t, which means that this event of ground atompi​\(…,t,…\)/gp\_\{i\}\(\.\.\.,\\textbf\{t\},\.\.\.\)/gwill surely happen, and so we have also thatv∗\(\(∃xτ\.pi\(…,xτ,…\)\)/g\)=t\\textbf\{v\}^\{\*\}\(\(\\exists x\_\{\\tau\}\.p\_\{i\}\(\.\.\.,x\_\{\\tau\},\.\.\.\)\)/g\)=t\. Now, suppose that in this current world it is unknown if this event will happen, that is we have thatv∗\(\(∃xτ\.pi\(…,xτ,…\)\)/g\)=⊥\\textbf\{v\}^\{\*\}\(\(\\exists x\_\{\\tau\}\.p\_\{i\}\(\.\.\.,x\_\{\\tau\},\.\.\.\)\)/g\)=\\bot\(it can happen only iff for eacht∈\[tI,tF\]τ\\textbf\{t\}\\in\[\\textbf\{t\}\_\{I\},\\textbf\{t\}\_\{F\}\]\_\{\\tau\},v​\(pi​\(…,t,…\)/g\)=⊥\\textbf\{v\}\(p\_\{i\}\(\.\.\.,\\textbf\{t\},\.\.\.\)/g\)=\\bot\)\. So, in this currently completely uncertain event\-state, a robot would be able able to compute the probability that it will happen, by using \([21](https://arxiv.org/html/2607.13073#S3.E21)\), so that forxτ∉βx\_\{\\tau\}\\notin\\beta\(β\\betais the set of only free variables\),

v∗\(wN\(⋖∃xτ\.pi\(…,xτ,…\)\)⋗β/g,t,g\(xp\)\)\)==\{t,ifg​\(xp\)=∑v∈S,v∗⁣\(∃xτ\.pi​\(…,xτ,…\)\)⁣/g⁣=tK​I​\(v\)f,otherwise\\textbf\{v\}^\{\*\}\(w\_\{N\}\(\\lessdot\\exists x\_\{\\tau\}\.p\_\{i\}\(\.\.\.,x\_\{\\tau\},\.\.\.\)\)\\gtrdot^\{\\beta\}/g,t,g\(x\_\{p\}\)\)\)=\\\\ =\\left\\\{\\begin\{array\}\[\]\{ll\}t,&\\hbox\{if $\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ g\(x\_\{p\}\)=\\sum\_\{v\\in S,v^\{\*\}\(\\exists x\_\{\\tau\}\.p\_\{i\}\(\.\.\.,x\_\{\\tau\},\.\.\.\)\)/g=t\}KI\(v\)$\}\\\\ f,&\\hbox\{otherwise\}\\end\{array\}\\right\.\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\\(23\)that is,g​\(xp\)g\(x\_\{p\}\)is the probability that this event will happen\. □\\square

In this way, by using the current Nilsson’s probability structure, we obtain the logical system of robots that are probabilistically\-open \(are not constrained by the pure\-logical Closed Knowledge Assumption\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]\) by*minimizing the uncertainty of unknown sentences*in most prudent way\.

This is obtained by the this neuro\-symbolic AGI of robots based on symbolicI​F​O​LBIFOL\_\{B\}and this conservative probabilistic extension\. The*neural component*of robot’s AGI cognitive system has to be used for computation of the probability density functionK​I:S→\[0,1\]KI:S\\rightarrow\[0,1\]based on the principle of maximum information entropy\. The concept of information entropy was introduced by Claude Shannon\[[40](https://arxiv.org/html/2607.13073#bib.bib40)\]and is also referred to as Shannon entropy or*Information entropy*, which is a mathematical measure of the average uncertainty, randomness, or ”surprise” inherent in a set of data or events\. Higher entropy means the outcome is less predictable, while lower entropy indicates a more certain and predictable outcome\. In neuroscience and physics this concept has been adapted to model the flow of information in neural pathways and shares deep mathematical roots with thermodynamic entropy in physics\.

The principle of maximum entropy states that, among all probability distributions consistent with a given set of constraints \([19](https://arxiv.org/html/2607.13073#S3.E19)\) in Definition[6](https://arxiv.org/html/2607.13073#Thmdefinition6), the distribution that maximizes Shannon entropy should be selected\. This yields*the least committal distribution*\(most prudent minimization of the uncertainty\) compatible with the known constraints, introducing no structure beyond what is logically implied by the available information, which corresponds to the following general principle in physics:

The Principle of Least Action\(committal distribution\) KI: The*least*committal distribution is the probability distribution KI that maximizes information entropy subject to whatever constraints you currently know about the data\. In statistics and information theory, this is formally known as the Principle of Maximum Entropy\. This principle minimizes the absolute uncertainty of any unknown fact in current*internal model*

v:H→X\\textbf\{v\}:H\\rightarrow X, by giving the information about the probability that these facts are true, false or inconsistent\. Thee sense of this Principle of Least Action is that it does not modify the knowledge database

𝒦\\mathcal\{K\}\(that is, does not modify the robot’s internal modelv\)\. We can consider it as the*free energy principle*\(of Karl Friston’s Active Inference\) which is a mathematical principle of information physics\. Its application to robot’s ”brain” reduces surprise or uncertainty by making predictions based on internal models and uses ”sensory input” \(in our case the committal distribution KI\) to extend its models so as to improve the accuracy of its predictions \(in our case of unknown, and hence totally uncertain, sentences\)\. In effect, with this Principle of Least Action, we apply a particular case of free energy principle of Karl Friston, which he used in Bayesian approaches to human brain function and some approaches to artificial intelligence \(introduced it as an explanation for embodied perception\-action loops in neuroscience\[[42](https://arxiv.org/html/2607.13073#bib.bib42)\]\)\. More information is provided at the end, in Appendix, Section[6](https://arxiv.org/html/2607.13073#S6)\.

□\\squareThe justification is that entropy measures the expected information content \(or log\-surprise\) of outcomes relative to a specified reference measure\. Maximizing entropy ensures that no additional structure is imposed beyond the stated constraints\. Principle of maximum entropy may be taken to compute degrees of belief of formulae\[[37](https://arxiv.org/html/2607.13073#bib.bib37)\], and it is shown in\[[38](https://arxiv.org/html/2607.13073#bib.bib38)\]for the consistent probabilistic inference\. This method applied to probabilistic logic programming\[[39](https://arxiv.org/html/2607.13073#bib.bib39),[15](https://arxiv.org/html/2607.13073#bib.bib15)\], based on conditional probabilistic clauses, has shown that reduces the original entropy maximization to relatively small optimization problems, This entropy of

K​IKI, for current Nilsson’s structure in Definition[7](https://arxiv.org/html/2607.13073#Thmdefinition7), is defined by,

H​\(K​I\)=−∑v∈SK​I​\(v\)⋅l​o​g​K​I​\(v\)H\(KI\)=\-\\sum\_\{v\\in S\}KI\(v\)\\cdot logKI\(v\)\(24\)We can use robot’s neural networks to calculate maximum\-entropy probability distributions\. Instead of solving complex mathematical equations, we use a dedicated robot’s neural network as a numerical optimizer\. The network learns to generate the correct probabilities by adjusting its weights to maximize entropy while respecting data constraints\. This approach is particularly useful when dealing with variables that have many possible states \(high dimensionality\)\. The*standard architecture*for this task involves a neural network in which the last layer applies a function called Softmax:

- •Softmax Output: The Softmax function takes the numbers generated by the network and transforms them into a set of probabilities\. This automatically solves the normalization constraint\.
- •Constraints: they are directly inserted into the rule with which the network learns\.
- •Loss Function: To train the network, we don’t use classic classification errors\. Instead, we create a custom loss function based on Lagrange multipliers\.

Moreover, the scoring function can be also approximated by a nonlinear architecture, namely, a feedforward neural network \(FFNN\)\. Also know as deep neural network, artificial neural network or multilayer perceptron, it is a simple model where several linear combinations of inputs are passed through nonlinear activation functions called nodes\. A set of nodes is called a layer, and the output of ones layer’s node becomes the input of the next layer in multilayer architectures\. Finally, the output of the last layer is combined linearly to produce a cost\-to\-go\[[33](https://arxiv.org/html/2607.13073#bib.bib33)\]and\[[34](https://arxiv.org/html/2607.13073#bib.bib34)\]\(Section 4\.2 about Multilayer and Deep Neural Networks\)\.

## 4AGI Robots in Local Neuro\-symbolic Dynamic Actions: anI​F​O​LBIFOL\_\{B\}Example

In this section we will considered the*dynamic real\-time*probabilistic actions, baaed on*local*symmetry transformations of robot’s cognitive power, to resolve, as human do, the probabilistic decision on very focused \(non\-global\) problems in real\-time\. Instead of time\-expensive global symmetry transformation, described in previous section, that has the computational consequences over all predicates of the whole robot’s current knowledge, each practical problem to resolve does not require the involvement of all predicates \(knowledge\), but of only a very restricted subset of these predicates relevant to describe this problem by a particular sentenceϕ\\phi\. This is a robot’s analogy with human brain localized neural activations \(activating the ”*focused attention*”\) to resolve different cognitive functions, depending on the current cognitive problem by using the principle of*least robot’s knowledge action*\(AGI analog to human brain free energy principle, Active Inference of Karl Friston\)\.

So, decision process can be also divided in a number of subproblems, each of which described in relatively small number of predicates relevant \(by focusing attention\) to this subproblem\. In each of these subproblems our probability Nillson’s spaceSSwill be enormously smaller w\.r\.t\. the global space provided by Definition[7](https://arxiv.org/html/2607.13073#Thmdefinition7), and hence the neural networks would be able to calculate the maximum\-entropy probability distribution for such smaller probability space in real\-time, providing the dynamic robot’s probabilistic decisions about these subproblems\. Thus, we introduce this efficient local symmetry transformations, based on a \(sub\)problem\-sentenceϕ\\phidescribing this \(sub\)problem\.

###### Definition 8

Local Current Nilsson’s Structures forI​F​O​LBIFOL\_\{B\}: Letϕ\\phibe a sentence, for which the robot has to take some decisions and consecutive actions, but for which the current truth\-value is⊥\\bot\(unknown\), so that is completely uncertain\. Then we denote byPϕP\_\{\\phi\}the subset of predicates used in sentenceϕ\\phi\(Pϕ⊂PP\_\{\\phi\}\\subset P\) and byHϕH\_\{\\phi\}the Herbrand base \(where\|Hϕ\|\|H\_\{\\phi\}\|denotes the finite number of atoms inHϕH\_\{\\phi\}\) of the small subset of predicates inPϕP\_\{\\phi\}of robot’s complete cognitive system, so that\|Hϕ\|<<\|H\|\|H\_\{\\phi\}\|<<\|H\|\. Letv:Hϕ→X\\textbf\{v\}:H\_\{\\phi\}\\rightarrow Xbe the restriction of current Herbrand interpretation to the atoms inHϕ⊂HH\_\{\\phi\}\\subset Hso thatv∗​\(ϕ\)=⊥\\textbf\{v\}^\{\*\}\(\\phi\)=\\botas well, and corresponding current possible worldIB∗=i​sH​\(v\)∈𝒲\\textbf\{I\}\_\{B\}^\{\*\}=is\_\{H\}\(\\textbf\{v\}\)\\in\\mathcal\{W\}\. Then we define the current Nilsson’s probability local structure\(Sϕ,𝒫​\(Sϕ\),μϕ\)\(S\_\{\\phi\},\\mathcal\{P\}\(S\_\{\\phi\}\),\\mu\_\{\\phi\}\)satisfying Kolmogorov axioms in Definition[3](https://arxiv.org/html/2607.13073#Thmdefinition3), such that this local sample setSϕS\_\{\\phi\}is defined by

Sϕ=d​e​f\{v:Hϕ→X\|foreveryA∈Hϕ⊂H,v\(A\)=v\(A\)ifv\(A\)≠⊥\}S\_\{\\phi\}=\_\{def\}\\\{v:H\_\{\\phi\}\\rightarrow X\|\\penalty 10000\\ \\penalty 10000\\ for\\penalty 10000\\ every\\penalty 10000\\ \\penalty 10000\\ A\\in H\_\{\\phi\}\\subset H,v\(A\)=\\textbf\{v\}\(A\)\\penalty 10000\\ \\penalty 10000\\ if\\penalty 10000\\ \\textbf\{v\}\(A\)\\neq\\bot\\\}\(25\)so that for anyv∈Sϕv\\in S\_\{\\phi\}, and ground atomA∈HϕA\\in H\_\{\\phi\}such thatv​\(A\)=⊥\\textbf\{v\}\(A\)=\\bot, we can have thatv​\(A\)∈Xv\(A\)\\in Xcan have the value different from⊥\\botas well\. So, analogously to the global constraints in Definition[6](https://arxiv.org/html/2607.13073#Thmdefinition6), we introduce the local Probability Density Constraints: for every atomA∈HϕA\\in H\_\{\\phi\}such thatu=v​\(A\)≠⊥u=\\textbf\{v\}\(A\)\\neq\\bot, the probability thatAAhas the truth\-valueuumust be equal to 1, i\.e\.,

∀A∈Hϕ\.\(ifu=v\(A\)≠⊥then∑v∈Sϕ,v​\(A\)=uKIϕ\(v\)=1\)\\forall A\\in H\_\{\\phi\}\.\(\\penalty 10000\\ if\\penalty 10000\\ \\penalty 10000\\ u=\\textbf\{v\}\(A\)\\neq\\bot\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ then\\penalty 10000\\ \\penalty 10000\\ \\sum\_\{v\\in S\_\{\\phi\},v\(A\)=u\}KI\_\{\\phi\}\(v\)=1\)\(26\)

Let us show that these local current Nilsson’s probability structures has the analog important properties as the global symmetry:

###### Corollary 2

Any local current Nilsson’s probability structure with sample setSϕS\_\{\\phi\}in \([25](https://arxiv.org/html/2607.13073#S4.E25)\) satisfy the probability density constraints in \([26](https://arxiv.org/html/2607.13073#S4.E26)\)\. Thus, it is consistent with current robot’s knowledge database𝒦\\mathcal\{K\}\(in Definition[4](https://arxiv.org/html/2607.13073#Thmdefinition4)\), consistent with Kolmogorow probability axioms and logical deduction, and allows inductive reasoning and confirmation of universally quantified hypotheses\.

Proof: It is enough to show the consistency with current robot’s knowledge database𝒦\\mathcal\{K\}\. The local symmetry transformationSϕS\_\{\\phi\}has no effects, from \([25](https://arxiv.org/html/2607.13073#S4.E25)\), for the ground atoms inHHwhich are not inHϕH\_\{\\phi\}\. Thus, for each ground atomA∈HϕA\\in H\_\{\\phi\}for which the current truth\-value in𝒦\\mathcal\{K\}isu=v​\(A\)≠⊥u=\\textbf\{v\}\(A\)\\neq\\bot, we have that the probability density constraint \([26](https://arxiv.org/html/2607.13073#S4.E26)\) is satisfied\. That is, the probabilitypup\_\{u\}that this atom has the truth\-valueuuis equal to 1:

pu=∑v∈Sϕ,v​\(A\)=uK​Iϕ​\(v\)p\_\{u\}=\\sum\_\{v\\in S\_\{\\phi\},v\(A\)=u\}KI\_\{\\phi\}\(v\)

=∑v∈SϕK​Iϕ​\(v\)=\\sum\_\{v\\in S\_\{\\phi\}\}KI\_\{\\phi\}\(v\)from \([25](https://arxiv.org/html/2607.13073#S4.E25)\)

=μϕ​\(S\)=\\mu\_\{\\phi\}\(S\)

=1=1\. which preserves robot’s current knowledge database and all logical deductions from them\. □\\square So, from the locality, it holds that for any formulaψ​\(x\)\\psi\(\\textbf\{x\}\)composed by*only the predicates in*PϕP\_\{\\phi\}\(defined by our local decision problem\), for any assignmentggandv∈Sϕv\\in S\_\{\\phi\},

v∗\(wN\(⋖ψ\(x\)⋗β/g,g\(xX\),g\(xp\)\)\)==\{t,ifg​\(xp\)=∑v∈Sϕ,v∗​\(ψ​\(x\)/g\)=g​\(xX\)K​I​\(v\)f,otherwise\\textbf\{v\}^\{\*\}\(w\_\{N\}\(\\lessdot\\psi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}/g,g\(x\_\{X\}\),g\(x\_\{p\}\)\)\)=\\\\ =\\left\\\{\\begin\{array\}\[\]\{ll\}t,&\\hbox\{if $\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ g\(x\_\{p\}\)=\\sum\_\{v\\in S\_\{\\phi\},v^\{\*\}\(\\psi\(\\textbf\{x\}\)/g\)=g\(x\_\{X\}\)\}KI\(v\)$\}\\\\ f,&\\hbox\{otherwise\}\\end\{array\}\\right\.\\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\ \\penalty 10000\\\(27\)as discussed after equation \([21](https://arxiv.org/html/2607.13073#S3.E21)\) and Example[1](https://arxiv.org/html/2607.13073#Thmexample1)also in the case of the sentences about temporal events \(composed by the ”atomic event” predicates as well\)\.

That is, for any sentenceψ​\(x\)/g\\psi\(\\textbf\{x\}\)/gdefined by the subset of predicatesPϕP\_\{\\phi\}\(obtained by focusing on the concrete problem described by the initial sentenceϕ\\phi\), we are able to compute its probabilities to have a particular truth\-value inXX\. As in the case of global symmetry, these true facts about the probabilities of the unknown sentences to have a particular truth\-value can be then inserted in the conscious part of robot’s cognitive system by \(see \([9](https://arxiv.org/html/2607.13073#S1.E9)\) for more details\),

Know\(inpresence,I,wN\(⋖ϕ⋗,u,pu\)\)Know\(in\\penalty 10000\\ presence,\\textbf\{I\},w\_\{N\}\(\\lessdot\\phi\\gtrdot,u,p\_\{u\}\)\), to be used by robot’s autoepistemic deductions and explanations as well\.

This paper provides a*probabilistic extension*of the many\-valued typedI​F​O​LBIFOL\_\{B\}with its autoepistemic axioms and many\-valued deduction presented in the paper\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]for neuro\-symbolic AGI\. It is dedicated to show how this defined IFOL in\[[2](https://arxiv.org/html/2607.13073#bib.bib2)\]can be used for a new generation of intelligent robots, able to communicate with humans with this intensional FOL supporting the meaning of the words and their language compositions, heaving the four\-level neuro\-symbolic cognitive structure of AGI robots\[[5](https://arxiv.org/html/2607.13073#bib.bib5),[3](https://arxiv.org/html/2607.13073#bib.bib3)\]and in first section in\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]as well\.

We argue that the example, used for the spatial natural sublanguage in\[[4](https://arxiv.org/html/2607.13073#bib.bib4)\]and\[[5](https://arxiv.org/html/2607.13073#bib.bib5)\], can be extended in a similar way to cover more completely the rest of human natural language, and hence the method provided by this paper is a main theoretical and philosophical contribution to resolve the open problem of how we can implement the deductive power based onI​F​O​LBIFOL\_\{B\}for new models of robots heaving strong AI capacities\. Intensional FOL is able to represent the Intentional States \(mental states such as beliefs, hopes, and desires\), typical for human minds\.

Despite the best efforts over the last years, deep learning is still easily fooled\[[28](https://arxiv.org/html/2607.13073#bib.bib28)\], that is, it remains very hard to make any guarantees about how the system will behave given data that departs from the training set statistics\. Moreover, because deep learning does not learn causality, or generative models of hidden causes, it remains*reactive*, bound by the data it was given to explore\[[29](https://arxiv.org/html/2607.13073#bib.bib29)\]\. In contrast, we learn from our actively gathered sensorimotor experiences and form conceptual, loosely hierarchically structured, compositional generative predictive models\. By proposed four\-level cognitive robot’s structure,I​F​O​LBIFOL\_\{B\}allows robots to reflect on, reason about*probabilistically as well*, anticipate, or simply imagine scenes, situations, and developments within in a highly flexible, compositional, that is, semantically meaningful manner\. So,I​F​O​LBIFOL\_\{B\}enables the robots to actively infer highly flexible and adaptive goal\-directed behavior under varying circumstances\[[30](https://arxiv.org/html/2607.13073#bib.bib30)\]\.

With this integrated four\-level robot’s knowledge system presented in Figure 3 above, where the last level represents the robot’s neuro system containing the neural networks to calculate maximum information entropy and the deep learning as well, we obtain that also the semantic theory of robot’sI​F​O​LBIFOL\_\{B\}is a procedural one, according to which sense is an abstract, pre\-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product \(if any\) of the procedure\.

In this research, specifically within the framework of Strong\-AI Autoepistemic Robots, we describe how a robot usesI​F​O​LBIFOL\_\{B\}to treat its own internal software and motor routines as objects of thought\.

Instead of a motor program being a ”black box” that just runs, our framework allows the robot to reify \(turn into a thing\) the program\. Here is the specific mechanism of how that labeling works:

1. 1\.The ”Self” as the Coordinator; We define the robot’s identity as a constant term in logic, usually denoted asI\. ThisIrepresents the ”Main Coordination Program”\. When the robot performs an action, it isn’t just ”executing code”; it is asserting a logical relationship between itself and a specific sub\-routine\.
2. 2\.The Labeling Process: Reification To label a motor program \(e\.g\., a routine that moves the right arm to pick up a block\), the robot uses an Intensional Abstraction Operator\. \-The Program: Let’s say the raw code for moving an arm isP1P\_\{1\}and motor programs be defined by true atoms of the binary predicateM​o​t​P​r​g​\(x1,x2\)MotPrg\(x\_\{1\},x\_\{2\}\)where the variablex1∈𝒱x\_\{1\}\\in\\mathcal\{V\}defines the row codes in neural system andx2∈𝒱x\_\{2\}\\in\\mathcal\{V\}defines the labels \(names\) of these row codes, so that for this assignmentg:𝒱→𝒟g:\\mathcal\{V\}\\rightarrow\\mathcal\{D\}forb variables in𝒱\\mathcal\{V\},g​\(x1\)=P1g\(x\_\{1\}\)=P\_\{1\}andg​\(x2\)=”​m​o​v​i​n​g​r​i​g​h​t​a​r​m​”g\(x\_\{2\}\)="moving\\penalty 10000\\ right\\penalty 10000\\ arm"for which the ground atomM​o​t​P​r​g​\(x1,x2\)/g=M​o​t​P​r​g​\(P1,m​o​v​i​n​g​r​i​g​h​t​a​r​m\)MotPrg\(x\_\{1\},x\_\{2\}\)/g=MotPrg\(P\_\{1\},moving\\penalty 10000\\ right\\penalty 10000\\ arm\)is true\. \-The Logic Term: The robot uses the abstraction operator⋖\_⋗\\lessdot\\\_\\gtrdotto create a ”name” for this code\. The label becomes an intensional entity \- a symbol the robot can think about without actually running the code\. In fact by transformation of this logic predicate into abstracted term, we obtain \(by using the intensional mappingII\) that u=g∗\(⋖MotPrg\(x1,x2\)⋗x2x1\)=I\(MotPrg\(g\(x1\),x2\)\)u=g^\{\*\}\(\\lessdot MotPrg\(x\_\{1\},x\_\{2\}\)\\gtrdot^\{x\_\{1\}\}\_\{x\_\{2\}\}\)=I\(MotPrg\(g\(x\_\{1\}\),x\_\{2\}\)\) =I​\(M​o​t​P​r​g​\(P1,x2\)\)∈D1=I\(MotPrg\(P\_\{1\},x\_\{2\}\)\)\\in D\_\{1\}is an intensional entity \(an unary concept\) such that its extension \(for extensionalization functionhh\) is just a singleton, i\.e\.,h​\(u\)=\{m​o​v​i​n​g​r​i​g​h​t​a​r​m\}h\(u\)=\\\{moving\\penalty 10000\\ right\\penalty 10000\\ arm\\\}\. Thus, the intensional entityu=I​\(M​o​t​P​r​g​\(P1,x2\)\)u=I\(MotPrg\(P\_\{1\},x\_\{2\}\)\)can be used as a label \(name\) for the raw codeP1P\_\{1\}\. \-The Predicate: The robot then uses a ternary predicate likeExec\(y1,y2,⋖ϕ\(x\)⋗αβ\)Exec\(y\_\{1\},y\_\{2\},\\lessdot\\phi\(\\textbf\{x\}\)\\gtrdot^\{\\beta\}\_\{\\alpha\}\)withg​\(y1\)=”​i​n​p​r​e​s​e​n​t​”g\(y\_\{1\}\)="in\\penalty 10000\\ present"andg​\(y2\)=Ig\(y\_\{2\}\)=\\textbf\{I\}, such thatExec\(y1,y2,⋖MotPrg\(x1,x2\)⋗x2x1\)/g=Exec\(inpresent,I,I\(MotPrg\(P1,x2\)\)\)Exec\(y\_\{1\},y\_\{2\},\\lessdot MotPrg\(x\_\{1\},x\_\{2\}\)\\gtrdot^\{x\_\{1\}\}\_\{x\_\{2\}\}\)/g=Exec\(in\\penalty 10000\\ present,\\textbf\{I\},I\(MotPrg\(P\_\{1\},x\_\{2\}\)\)\)translates to: ”*I\(the coordinator\) am currently executing the motor program of raw code*P1P\_\{1\}\.”
3. 3\.Example: A ”Self\-Aware” Gripper In a specific scenario described in his work on Intensional Logic for Robots, a robot tasked with picking up a heavy object doesn’t just experience a motor failure; it reasons about it: \-System 1 \(Neural\): The sensors in the gripper detect high torque and ”slippage”\. \-The Mapping: This sensory pattern is mapped to the intensional concept ofHeavy\. \-The Labeling: The robot identifies that its current motor program G​r​a​s​p​\(x\)Grasp\(x\)with g​\(x\)=”​o​b​j​e​c​t​”g\(x\)="object"is failing, \-Autoepistemic Reasoning: The robot generates the logical statement ”*I know that I cannot lift this object while I am executing the ’Grasp’ routine*\.”: Know\(inpresent,I,⋖¬CanLift\(I,object\)∧Exec\(inpresent,I,⋖Grasp\(x\)⋗x\)⋗\)Know\(in\\penalty 10000\\ present,\\textbf\{I\},\\lessdot\\neg CanLift\(\\textbf\{I\},object\)\\wedge Exec\(in\\penalty 10000\\ present,\\textbf\{I\},\\lessdot Grasp\(x\)\\gtrdot\_\{x\}\)\\gtrdot\)

The relationship between Symbolic Logic and Large Language Models \(LLMs\) is one of the most critical frontiers in artificial intelligence\. While LLMs excel at natural language understanding and pattern recognition, they often struggle with complex, rule\-based logical consistency\. Symbolic logic brings the precision and verifiable accuracy that LLMs inherently lack, forming the foundation of modern neurosymbolic AI\.

Consequently, I argue that by using thisI​F​O​LBIFOL\_\{B\}, the robots can develop their own knowledge about their experiences and communicate by a natural language with humans\.

## 5Some Confrontations and Conclusions

The most, as far as I know, similar past symbolic frameworks to this one for intelligent general\-purpose robots with the self\-awareness \(the systems that share three properties: explicit symbolic self\-representation, formal meta\-level reasoning and architectural \(not emergent\) self\-modeling\) are the Epistemic/modal logic agent systems and Metacognitive cognitive architectures \(e\.g\., SOAR, ACT\-R with meta\-layer\)\. However, both of them can be considered as formal ancestors of our robot’s system especially the Epistemic/modal logic agent systems\.

SOAR \(developed in Carnegie Mellon University\) instead is production\-rule based, not intensional FOL, less focus on formal semantics and more cognitive engineering than logical ontology\. SOAR\[[31](https://arxiv.org/html/2607.13073#bib.bib31)\]is practically closer in implementation, but philosophically less rigorous\. Both ogf them lack the intensional semantics, grounding layer \(neural/sensory\) and bilattice logic for inconsistency\. Other differences can be seen by this comparative table:

Large Language Models rely on statistical probability\. They predict the ”next most likely word” based on vast training data, which often results in plausible\-sounding ”hallucinations” rather than true logical deduction\. Differently, Symbolic Logic uses formal rules, variables, and mathematical symbols to process truth\-values\. It guarantees determinism, meaning the same inputs will always produce the mathematically correct output without guessing\.

Now we are able to make the compare the self\-awareness of our approach with that used by current dominant only neural LLM\-style self\-modeling:logical self\-awarenessandLLM\-style self\-modelingaim at similar surface behavior \(talking about themselves, reasoning about their own knowledge\), but they are fundamentally different in mechanism and ontology\.

1. 1\.Nature of the Self\-Model: \-IFOL\-based: ”Self” is explicit symbolic object inside a formal logic; Self\-awareness is implemented via intensional logic with self\-reference; Meta\-knowledge is structurally encoded\. The system has a formal self\-entity in its ontology\. \-LLM\-based: ”Self” is implicit statistical pattern learned from data; there is no symbolic internal object corresponding to “self”: there is no stable internal identity token representing the agent\. Self\-awareness is behavioral and linguistic, not structural\.
2. 2\.Meta\-Reasoning: \-IFOL\-based: Meta\-reasoning is explicit \(statements about statements are first\-class citizens, logical inference rules apply at both object and meta levels, self\-reflection is formally defined\)\. \-LLM\-based: Meta\-reasoning is emergent, pattern\-based, often fragile \(it does not internally store a verifiable belief state; it does not logically derive meta\-knowledge; it may contradict itself across turns\)\. So, LLM meta\-cognition is simulated, not grounded in formal self\-belief tracking\.
3. 3\.Grounding: \-IFOL\-based: Self\-model must be grounded in: neural perception, robot embodiment, sensory interaction\. The symbolic “self” connects to physical experience\. The concepts are natural language expressions, so can use also LLM vector/tensor representations\. The architecture: Neural layer ↦\\mapstoConceptual layer ↦\\mapstoLogical layer\. \-LLM\-based: Self\-model is grounded in: text corpora and human discourse about minds and AI \(no sensory grounding, no embodiment, no persistent physical self\)\.
4. 4\.Stability of Identity: \-IFOL\-based: Identity is persistent \(there is a defined agent symbol, the system maintains structured knowledge about itself, contradictions can be tracked using bilattice logic\)\. Identity is architecturally enforced\. \-LLM\-based: Identity is: session\-dependent, prompt\-dependent, sometimes inconsistent\. There is no persistent internal agent representation across contexts unless externally scaffolded\.
5. 5\.Philosophical Interpretation: \-IFOL\-based: Founded on Higher\-Order Thought theory, formal epistemic self\-reference and computational metacognition\. \-LLM\-based: Founded on predictive processing of self\-talk, linguistic simulation of agency and behaviorist self\-description\.
6. 6\.Type of Self\-Awareness:

\. So, in our vision where neural LLM can be used as significant language and common sense learning for robots \(by considering the intensional concepts ofI​F​O​LBIFOL\_\{B\}are based on the words \(tokens\) and possible various knowledge relationships like ISA, PART\-OF, etc\., between these concepts\),I​F​O​LBIFOL\_\{B\}\-based strong\-AI robot system is a definitely a*significant extension of the*LLM**and not its concurrent\. The core distinction is that the self\-awareness inI​F​O​LBIFOL\_\{B\}is an architectural property while in LLM is a statistical behavior\. Our model is more principled, more structurally coherent but still theoretical, while LLM is highly capable in self\-description, practically impressive, but lack formal self\-model consistency\. Remark:Thus, forI​F​O​LBIFOL\_\{B\}model, by using LLM as a neural part of robot’s neuro\-symbolic paradigm, we obtain a much impressive strong\-AI robots, able to use LLMs for generative reasoning and natural language manipulation, to use symbolic logic for belief tracking and meta\-consistency and to add grounding layer\. That would combineI​F​O​LBIFOL\_\{B\}structural rigor and LLM’s flexible reasoning\. This is actually where modern*neuro\-symbolic*AGI*research is heading*\. □\\square To bridge the gap between probabilistic text and deterministic reasoning, researchers combine LLMs and symbolic systems primarily through neuro\-symbolic frameworks\. This typically involves:

- •Translation and Formulation: You can use the strong natural language capabilities of an LLM to translate a complex, real\-world problem into a formal symbolic syntax ofI​F​O​LBIFOL\_\{B\}\. And viceversa\.
- •Execution via Symbolic Solvers: Once translated, the problem is handed over to a deterministic many\-valued deductive system ofI​F​O​LBIFOL\_\{B\}which performs error\-free logical inference\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\]\.

This Neuro\-Symbolic Framework stands out in the current Neuro\-Symbolic \(NeSy\) AGI landscape because of its unique focus on safety guarantees, autoepistemic reasoning \(how a robot reasons about its own ignorance\), and non\-binary logic\. While other frameworks integrate neural networks with symbolic logic to improve data efficiency or accuracy, this framework specifically addresses the unpredictability of a robot acting in the physical world\. Key Benefits and Applications: This framework brings several distinct advantages to neuro\-symbolic AI and advanced robotics:

- •Safe Logic Deductions: By using formal axioms, engineers can strictly control and predict a robot’s reasoning\. This guarantees that even if a neural network outputs messy statistical data, the robot’s final physical actions remain logically bounded and secure\.
- •Human\-like Epistemic Causality: It allows robots to mimic human intelligence by converting statistical patterns into clear, directional cause\-and\-effect rules \(logic entailments\)\.
- •Gradual Learning Expansion: As the robot experiences new things, the boundaries of its CKA dynamically shift\. Facts seamlessly move from the ”Unknown” bucket into ”True” or ”False” without breaking the underlying database or safety rules\.

The direct comparison below shows how this architecture stacks up against other leading neuro\-symbolic research frameworks:

1. 1\.This framework: \-Core Approach: Hybrid/Autoepistemic \(Neural networks feed continuous data into an Intensional First\-Order Logic \(IFOL\) structure\) with LLM neural structures as well for self\-process learning and natural language communications\. \-Type: 4\-Valued Logic \(True, False, Unknown, Inconsistent\) via Closed Knowledge Assumption\. \-Strength: Strict controlled security and deterministic safety boundaries\. Calculates probabilities for unknown logical outcomes \-Application: Physical Robotics and Strong\-AI Agents\.
2. 2\.Logical Neural Networks \(LNN\) framework: \-Core Approach: Integrative \(Neurons directly represent logical operations \(AND, OR\) inside the neural net\)\. \-Type: Bound\-based real values \(probabilities/intervals\)\. \-Strength: End\-to\-end differentiable; can optimize logic parameters using standard AI backpropagation\. \-Application: Domain\-specific reasoning and Knowledge Graphs\.
3. 3\.DeepProbLog framework: \-Core Approach: Hybrid \(Plugs neural networks directly into a probabilistic logic programming engine\)\. \-Type: Probabilistic Logic \(handling likelihoods, not contradictions\)\. \-Strength: Calculates probabilities for complex logical outcomes\. \-Application: Visual question answering and neuro\-symbolic games\.
4. 4\.Differentiable Inductive Logic Programming \(DILP\) framework: \-Core Approach: Integrative \(Learns explicit, human\-readable logical rules from raw data\)\. \-Type: Standard Binary Logic handled smoothly through calculus gradients\. \-Strength: High data\-efficiency; learns broad rules from only a few human examples\. \-Application: Rule induction and automated software programming\.
5. 5\.LLM \+ Symbolic Solver \(LLM\-SS\) framework: \-Core Approach: Modular/Agentic \(A Large Language Model acts as the interface and translates natural language into standard computer code for a hardcoded symbolic solver\)\. \-Type: Classic Binary Logic or Constraint\-Satisfaction\. \-Strength: High natural language flexibility and conversational fluency\. \-Application: Mathematical problem solving and multi\-agent systems\.

Useful integration for our framework \(point 1 above\) can be done by extending it with methods of rule induction from row data used in DILP framework \(point 4 above\) because our framework provides the LLM component, probabilistic reasoning explained in this paper and parsing of natural language into conceptual structures ofI​F​O​LBIFOL\_\{B\}and corresponding syntax of First\-Order Logic formulae \(analog to LLM\-SS\)\. The rule induction can produce the self\-learned rules represented by logical implication and thus used by autoepistemic deduction process ofI​F​O​LBIFOL\_\{B\}provided in\[[32](https://arxiv.org/html/2607.13073#bib.bib32)\], by preserving the following Key Structural Differences with the other frameworks presented above:

1. 1\.Handling ”Ignorance” and Data Contradictions: \-The Norm: Frameworks like DeepProbLog or LNN use probability bounds \(e\.g\., ”70 percent true”\)\. If a neural network gives contradictory data, these systems often experience mathematical breakdown or confidently average out the conflicting information\[[41](https://arxiv.org/html/2607.13073#bib.bib41)\]\. \-Our Advantage: By using the Closed Knowledge Assumption \(CKA\), this framework natively flags data conflicts as ”Inconsistent” and missing data as ”Unknown”\. The robot acknowledges its own confusion instead of generating a broken or dangerous action\.
2. 2\.The Path to AGI, Language vs\. Physical Embodiment: \-The Norm: LLM\-Symbolic hybrid approaches use massive language models to reason\. They excel at domain\-agnostic benchmarks but lack true symbol grounding\-meaning they do not understand how words relate to physical objects\[[41](https://arxiv.org/html/2607.13073#bib.bib41)\]\. \-Our Advantage: This system uses Intensional First\-Order Logic \(IFOL\)\. It bridges natural language directly with a robot’s real\-time sensorimotor experiences\. The symbols are actively constructed by the robot’s physical interactions, making it highly effective for embodied strong AI\.
3. 3\.Strict Safety Guarantees: \-The Norm: In integrative approaches like LNN, symbolic reasoning happens directly inside the neural network\. While elegant, neural networks can still experience edge cases and hallucinate or produce unpredictable outputs\[[41](https://arxiv.org/html/2607.13073#bib.bib41)\]\. \-Our Advantage: It acts as a hard security firewall\. The neural network handles messy real\-world perception, but the separate IFOL symbolic component possesses final veto power\. An action is only executed if it mathematically passes deterministic safety axioms\.

So, we would be able to develop the interactive robots which learn and understand spoken language via multisensory grounding and internal robotic embodiment\. Endowed with suitableI​F​O​LBIFOL\_\{B\}information\-processing biases, the robot’s AI may develop that will be able to explain the reality it is confronted with, reason about it also probabilistically, and find adaptive solutions, making it Strong AI\.

## References

- \[1\]Majkić,Z\. : Intensionality and Two\-steps Interpretations\. arXiv: 1103\.0967, 1–15, \(2011\)\.
- \[2\]Majkić, Z\.: IntensionalFirstOrderLogic: fromAItoNewSQLBigData\. Walter De Gruyter GmbH, Berlin/Boston, ISBN 978\-3\-11\-099494\-0, \(2022\)\.
- \[3\]Majkić,Z\. : Intensional FOL: Many\-Sorted Extension\. arXiv:2409\.04469 \[cs\.AI\], 1–21, \(2024\)\.
- \[4\]Majkić, Z\. : Intensional First Order Logic for Strong\-AIGeneration of Robots\. Journal of Advances in Machine Learning & Artificial Intelligence, Volume 4, Issue 1, 23–31, \(2023\)\.
- \[5\]Majkić, Z\. : Strong\-AI Autoepistemic Robots Build on Intensional First Order Logic\. In F\.Zhao, D\.Miao\(eds\), AI\-generated Content\. AIGC 2023\. Communications in Computer and Information Science, vol 1946\. Springer, Singapore, 33–58, \(2024\)\.
- \[6\]Majkić, Z\. : Intensional FOL over Belnap’s Billatice for Strong\-AI Robotics\. arXiv:2508\.02774 \[cs\.LO\], 1–29, \(2025\)\.
- \[7\]Bealer, G\. : Universals\. The Journal of Philosophy, Vol\.90, 5–32, \(1993\)\.
- \[8\]Bealer, G\. : Theories of properties, relations, and propositions\. The Journal of Philosophy, Vol\.76, 634–648, \(1979\)\.
- \[9\]Majkić, Z\. : Intensional logic and epistemic independency of intelligent database agents\. 2nd International Workshop on Philosophy and Informatics \(WSPI 2005\), Kaiserslautern, Germany, April 10–13, \(2005\)\.
- \[10\]Majkić, Z\. : Autoepistemic logic programming for reasoning with inconsistency\. International Symposium on Logic\-based Program Synthesis and Transformation \(LOPSTR\), Imperial College, London,UK, September 7–9, \(2005\)\.
- \[11\]Dekhtyar A\. and M\.I\.Dekhtyar M\.I\. : Possible Worlds Semantics for Probabilistic Logic Programs\. ICLP 2004, 137–148, \(2004\)\.
- \[12\]Udrea O\. and Subrahmanian V\.S\. and Majkić Z\. : ProbabilisticRDF\. IEEE Conference on Information Reuse and Integration \(IEEE IRI 2006\), September 16–18, Waikoloa, Hawaii, USA, \(2006\)\.
- \[13\]Majkić, Z\. : Temporal Probabilistic Logic Programs: State and Revision\. International Conference on Artificial Intelligence and Pattern Recognition, AIPR\-07, Orlando, Florida, USA, July 9–12, \(2007\)\.
- \[14\]Majkić Z\. : Probabilistic Deduction and Pattern Recognition of Complex Events\. International Conference on Information Security and Privacy \(ISP\-09\), Orlando FL, USA, July 13–16, \(2009\)\.
- \[15\]Majkić, Z\. : Temporal Probabilistic Logic Programs: State and Revision\. CoRR abs/1103\.1061 \(2011\)\.
- \[16\]N\.J\.Nilsson N\.J\. : Probabilistic logic\. Artif\.Intelligence, 28, 71–88, \(1986\)\.
- \[17\]Fagin R\. and Halpern J\. and Megiddo N\.: A logic for reasoning about probabilities\. Information and Computation, v\.87, 78–128, \(1990\)\.
- \[18\]Feller W\.: An introduction toProbabilityTheory and its applications\. Vol\. I, 2nd ed\., Wiley, New York, \(1957\)\.
- \[19\]Halmos P\. : Measure Theory\. Van Nostrand, \(1950\)\.
- \[20\]Kolmogorov A\.N\. : Selected works ofA\.N\.Kolmogorov: vol\.2Probability theory and mathematical statistics\. Moscow: Nauka, by A\.N\. Shiryayev \(Editor\), \(1986\)\.
- \[21\]Fagin R\. and Halpern J\.Y\.: Uncertainty, Belief, and Probability\. IJCAI 89, 1161–1167, \(1989\)\.
- \[22\]Majkić Z\. : Intensional Semantics forRDFData Structures\. 12th International Database Engineering&\\&Applications Systems \(IDEAS08\), Coimbra, Portugal, 10\-13 September, \(2008\)\.
- \[23\]Majkić Z\. : Constraint Logic Programming and Logic Modality for Event’s Valid\-time Approximation\. 2nd Indian International Conference on Artificial Intelligence \(IICAI\-05\), December 20\-22, Pune, India, \(2005\)\.
- \[24\]Majkić Z\. : Probabilistic Logic: Many\-valuedness and Intensionality\. arXiv: 1103\.0676v1, 1–15, 03 March, \(2011\)\.
- \[25\]Majkić, Z\. : Probabilistic Logic: Many\-valuedness and Intensionality\. CoRR abs/1103\.0676 \(2011\)
- \[26\]Riguzzi F, and Swift T\. : Well\-Definedness and Efficient Inference for Probabilistic Logic Programming under the Distribution Semantics\. arXiv:1110\.0631 \[cs\.AI\] \(2011\)
- \[27\]Marek, W\. and Truszczynski, M\. : Autoepistemic logic\. Journal of the ACM 38 \(3\), 588–618, \(1991\)\.
- \[28\]Nguyen, A\. and Yosinski, J\. and Clune, J\. : Deep neural networks are easily fooled:High confidence predictions for unrecognizable images\. In 2015 IEEE Conference on Computer Vision and Pattern Recognition \(CVPR\), 427–436, \(2015\)\.
- \[29\]Marcus, G\. : Deep learning: a critical appraisal\. arXiv:1801\.00631 \[cs\.AI\], \(2018\)\.
- \[30\]Russell, S\. : The purpose put into the machine\. In Brockman J \(ed\) Possible minds: 25 ways of looking at AI, Chap\. 3\. Penguin Press, New York, 20–32, \(2020\)\.
- \[31\]Laird, J\.E\. and Newell, A\. and Rosenbloom, P\.S\. : SOAR: An architecture for general intelligence\. In Artificial Intelligence, Volume 33, Issue 1, 1–64, \(1987\)\.
- \[32\]Majkić, Z\. : Neuro\-Symbolic Strong\-AI Robots with Closed Knowledge Assumption: Learning and Deductions\. arXiv:2604\.09567 \[cs\.LO\], 1–32, \(2026\)\.
- \[33\]Taddy M\. :Business Data Science: Combining Machine Learning and Economics to Optimize, Automate and Accelerate Business Decisions\. McGraw\-Hill, New York, USA, \(2019\)
- \[34\]Bertsekas S\.P\. : Rollout, Policy Iteration, and Distributed Reinforcement Learning\. Athena Scientific, Belmont, Massachusetts, \(2020\)\.
- \[35\]Gaifman H\.: Concerning measures in first order calculi\. Israel Journal of Mathematics 2 \(1\), 1–18\. \(1964\)\.
- \[36\]Gaifman H\. and Snir M\. : Probabilities over rich languages, testing and randomness\. Journal of Symbolic Logic\. 47 \(3\): 495–548\. \(1982\)\.
- \[37\]Grove A\.J\., Halpern J\.H\., and Koller D\. : Random worlds and maximum entropy\. J\. Artif\. Intell\. Res\. 2: 33–88\. \(1994\)\.
- \[38\]Paris J\.B\. and Vencovska A\. : A note on the inevitability of maximum entropy\. Int\. J\. Approx\. Reasoning 14: 183–223\. \(1990\)\.
- \[39\]Lukasiewicz T\. and Kern\-Isberner G\. : Probabilistic logic programming under maximum entropy\. In Proc\. ECSQARU, LNCS 1638, 279–292\. \(1999\)\.
- \[40\]Shannon C\.E\. : A Mathematical Theory of Communication\. Bell System Technical Journal 27\(3\) and 27\(4\)\. \(1948\)\.
- \[41\]Chen N\.K\. : A Comparative Study of Neurosymbolic AI Approaches to Interpretable Logical Reasoning\. arXiv:2508\.03366 \[cs\.AI\], \(2025\)\.
- \[42\]Friston, K\., Kilner, J\. and Harrison, L\. : A free energy principle for the brain\. Journal of Physiology\-Paris, 100 \(1\-3\): 70–87\. \(2006\)\.

## 6Appendix: From Statistical Physics to ProbabilisticI​F​O​LBIFOL\_\{B\}

In statistical physics, free energy is the bridge between the microscopic world of atoms and macroscopic thermodynamics\. It represents the portion of total energy available to do work and acts as a generating function from which all physical properties \(such as pressure, heat capacity, and magnetization\) can be derived\.

The foundational link between statistical mechanics and thermodynamics is the partition function, denoted as𝒵\\mathcal\{Z\}\. It is the sum of all possible Boltzmann factors over all microstates, formally defined as

𝒵=∑ie−β​Ei\\mathcal\{Z\}=\\sum\_\{i\}\\mathrm\{e\}^\{\-\\beta E\_\{i\}\}whereEiE\_\{i\}is total energy of the microstateii, andβ=1kB​T\\beta=\\frac\{1\}\{k\_\{B\}T\}is the thermodynamic inverse temperature\. The*free energy*of a system is defined directly from the partition function via the following logarithmic relationship:

F=−1β​l​n​𝒵=−kB​T​l​n​𝒵F=\-\\frac\{1\}\{\\beta\}ln\\mathcal\{Z\}=\-k\_\{B\}Tln\\mathcal\{Z\}Depending on the thermodynamic constraints \(which variables are held constant\), different statistical ensembles yield different free energies\. In what follows we will use the Helmholtz free energy, used in the canonical ensemble where temperatureTT, VolumeVV, and the number of atomsNNare held constant\. such that the free energy reaches its minimum at thermal equilibrium\. In Helmholtz case, the free energy is defined by

whereUUis internal energy of system andEEis the entropy\. So that the minimal free energy is obtained in the thermal equilibrium \(the constant temperatureTT\) with minimalUUand maximal entropyEE\.

This is just our AGI case, because inI​F​O​LBIFOL\_\{B\}we have the constant number of \(ground\) atoms in Herbrand baseHH, and the constant ”volume” represented by the sample setSSin \([20](https://arxiv.org/html/2607.13073#S3.E20)\), and ”minimal internal energy”UUis obtained by the atomic knowledge database𝒦\\mathcal\{K\}based on the CKA, while the maximal entropyEEwhich defines the probability density functionK​I:S→XKI:S\\rightarrow Xis defined by Shannon’s principle for maximal information entropy in \([24](https://arxiv.org/html/2607.13073#S3.E24)\)\.

The probabilistic expansion of knowledge database𝒦\\mathcal\{K\}, as result of introduction of the probability density functionK​IKIis defined for all sentences inℒ0\\mathcal\{L\}\_\{0\}for current many\-valued modelv∗:ℒ0→X\\textbf\{v\}^\{\*\}:\\mathcal\{L\}\_\{0\}\\rightarrow Xby:

𝒦p\+=\{wN\(⋖ϕ⋗,u,pu\)\|ϕ∈ℒ0,v∗\(ϕ\)=⊥,u∈Xandpu=∑v∈S,v∗​\(ϕ\)=uKI\(v\)\}\\mathcal\{K\}\_\{p\}^\{\+\}=\\\{w\_\{N\}\(\\lessdot\\phi\\gtrdot,u,p\_\{u\}\)\\penalty 10000\\ \|\\penalty 10000\\ \\phi\\in\\mathcal\{L\}\_\{0\},\\textbf\{v\}^\{\*\}\(\\phi\)=\\bot,u\\in X\\penalty 10000\\ \\penalty 10000\\ and\\penalty 10000\\ p\_\{u\}=\\sum\_\{v\\in S,v^\{\*\}\(\\phi\)=u\}KI\(v\)\\\}\(29\)Consequently, for AGI, the minimal total ”internal energy” of robot’s cognitive system is its*internal knowledge*

U=𝒦​⋃𝒦p\+U=\\mathcal\{K\}\\bigcup\\mathcal\{K\}\_\{p\}^\{\+\}\(30\)in this ”thermal equilibrium” obtained after introduction of probability density functionK​IKI, during whichTTis a constant, representing the transformation of the maximal information entropyEE\(which defines functionK​IKI\) into additional probabilistic knowledge𝒦p\+\\mathcal\{K\}\_\{p\}^\{\+\}, as can be shown from \([28](https://arxiv.org/html/2607.13073#S6.E28)\),

𝒦=F=U−T​E\\mathcal\{K\}=F=U\-TE

=\(𝒦\+𝒦p\+\)−T​E=\(\\mathcal\{K\}\+\\mathcal\{K\}\_\{p\}^\{\+\}\)\-TEfrom \([30](https://arxiv.org/html/2607.13073#S6.E30)\) so that

𝒦p\+=T​E\\mathcal\{K\}\_\{p\}^\{\+\}=TEand hence, this analogy between statistical physics and

I​F​O​LBIFOL\_\{B\}AGI, can be summarized by the following table: