Mediative Fuzzy Logic: From Type-1 Foundations to Type-2, Type-3 and Quantum Extensions

# Mediative Fuzzy Logic: From Type-1 Foundations to Type-2, Type-3 and Quantum Extensions
Source: [https://arxiv.org/html/2605.22900](https://arxiv.org/html/2605.22900)
###### Abstract

Mediative Fuzzy Logic was conceived as a practical scheme for reconciling hesitant or conflicting assessments in fuzzy control and decision\-making\. However, its logical and semantic foundations remain underdeveloped, especially beyond operational type\-1 settings\. This article develops a unified account of the type\-1 core together with interval type\-2, granular type\-3, and quantum extensions\. We characterize the mediative operator as a convex aggregation controlled by hesitation and contradiction, model mediative truth values as independent truth\-falsity pairs in a continuous bilattice\-like structure, and introduce a propositional system extending a standard t\-norm\-based fuzzy logic with a mediative connective\. We establish soundness, paraconsistency, and conservativity over the underlying fuzzy base for formulas without mediation, and formulate coherent semantic extensions to interval type\-2 truth values, granule\-indexed local evaluations, and effects and density operators on Hilbert spaces\. An autonomous\-braking sensor\-fusion example illustrates how the framework supports transparent, conservative, and safety\-first decisions under incomplete, heterogeneous, and mildly contradictory evidence\. Under suitable assumptions, the higher\-level formulations reduce to the type\-1 case, clarifying coherence across levels and reliably supporting future work in intelligent decision systems\.

###### keywords:

fuzzy logic , mediative reasoning , type\-2 fuzzy sets , type\-3 fuzzy sets , granular computing , quantum logic

††journal:Information Sciences\\affiliation

organization=Instituto Politécnico Nacional \- CITEDI,addressline=AV\. Instituto Politécnico Nacional 1310, city=Tijuana, postcode=22435, state=Baja California, country=México

## 1Introduction

Real\-world decision problems are typically based on information that is not only imprecise but also incomplete and, in many cases, genuinely contradictory\. Classical fuzzy logic is well adapted to graded truth\[[41](https://arxiv.org/html/2605.22900#bib.bib3),[42](https://arxiv.org/html/2605.22900#bib.bib4),[22](https://arxiv.org/html/2605.22900#bib.bib5)\], and its logical foundations are now well established\[[18](https://arxiv.org/html/2605.22900#bib.bib2)\]\. However, in the standard setting, falsity is not an independent degree: it is obtained by applying a fixed fuzzy negationNNto the truth degree, typically the standard negationN​\(μ\)=1−μN\(\\mu\)=1\-\\mu\. This assumption becomes too restrictive when different experts, sensors, or models provide conflicting assessments of the same proposition\.

To relax this constraint, intuitionistic fuzzy sets and related frameworks separate truth and falsity degrees and introduce an additional component of hesitation\[[3](https://arxiv.org/html/2605.22900#bib.bib8),[4](https://arxiv.org/html/2605.22900#bib.bib9),[6](https://arxiv.org/html/2605.22900#bib.bib11)\]\. Such approaches naturally represent incomplete information; yet, they are not primarily tailored to explicitly capture and reason with persistent contradictions across sources\. Persistent inconsistency is more naturally addressed within paraconsistent and bilattice\-based logics\[[10](https://arxiv.org/html/2605.22900#bib.bib16),[16](https://arxiv.org/html/2605.22900#bib.bib18),[2](https://arxiv.org/html/2605.22900#bib.bib20)\], which support controlled reasoning in the presence of conflicting evidence\. In parallel, intuitionistic fuzzy sets and their aggregation operators are widely used in information fusion as a principled framework for combining uncertain and potentially conflicting evidence from multiple sources\[[5](https://arxiv.org/html/2605.22900#bib.bib60),[40](https://arxiv.org/html/2605.22900#bib.bib61)\]\.

Mediative Fuzzy Logic \(MFL\) was first introduced in an operational form as a scheme that combines an agreement channel and a non\-agreement channel into a single mediative output\[[26](https://arxiv.org/html/2605.22900#bib.bib35),[27](https://arxiv.org/html/2605.22900#bib.bib36)\]\. In that formulation, each proposition is described by a membership function \(agreement\), a non\-membership function \(non\-agreement\), and the associated hesitation and contradiction fuzzy sets\. The mediative operator then aggregates the outputs of the two subsystems so that the resulting value jointly reflects an intuitionistic\-style hesitation margin and a contradiction index induced by the contradiction fuzzy setCC\. In applications to control and diagnosis, MFL has been shown to handle both hesitation and contradiction in a smooth and interpretable manner, with successful implementations in medical diagnosis and pandemic modeling\[[19](https://arxiv.org/html/2605.22900#bib.bib41),[38](https://arxiv.org/html/2605.22900#bib.bib43),[37](https://arxiv.org/html/2605.22900#bib.bib44)\]\. Nevertheless, the original formulation was not cast as a fully axiomatized logic with explicit algebraic semantics, and most existing works treat MFL primarily as a fuzzy inference scheme rather than as a proof\-theoretic logic\[[11](https://arxiv.org/html/2605.22900#bib.bib37),[24](https://arxiv.org/html/2605.22900#bib.bib39)\]\.

At a foundational level, our aim is to provide algebraic and logical foundations for Mediative Fuzzy Logic beyond the original operational control setting\. Starting from a type\-1 mediative operator with clear algebraic semantics, we define a propositional logic MFL\-T1 and develop its basic metatheory, taking as fuzzy base a standard t\-norm\-based fuzzy logic, such as Hájek’s Basic Logic \(BL\)\[[17](https://arxiv.org/html/2605.22900#bib.bib40)\]or Łukasiewicz logic\[[18](https://arxiv.org/html/2605.22900#bib.bib2)\]\. Our guiding question is whether mediative fuzzy semantics can reconcile paraconsistency and safety\-first aggregation while remaining compatible with standard fuzzy infrastructures\.

Building on this type\-1 core, we ask how mediative reasoning behaves at higher types\. Type\-2 and type\-3 Mediative Fuzzy Logic \(MFL\-T2 and MFL\-T3\) were introduced by Castillo and Melin in their proposals for mediative fuzzy control and type\-3 mediative systems\[[11](https://arxiv.org/html/2605.22900#bib.bib37),[24](https://arxiv.org/html/2605.22900#bib.bib39),[23](https://arxiv.org/html/2605.22900#bib.bib38)\]\. These contributions outline mediative architectures from type\-1 to type\-3 and report several control applications; however, they are formulated primarily at the level of system design and do not provide a granular semantics or a logical calculus in which higher\-type mediative truth values are treated as structured objects\.

In this paper, we develop a systematic mediative semantics for interval type\-2 truth values\(μ~p,ν~p\)\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\)within an MFL\-T2 framework\. In this setting, type\-2 hesitation and contradiction are made explicit, and a mediative evaluationM~p\\tilde\{M\}\_\{p\}is obtained via standard type\-reduction mechanisms used in type\-2 fuzzy decision models\[[21](https://arxiv.org/html/2605.22900#bib.bib53),[8](https://arxiv.org/html/2605.22900#bib.bib12),[9](https://arxiv.org/html/2605.22900#bib.bib13)\]\. In contrast to previous combinations of mediative fuzzy logic with type\-2 fuzzy controllers and higher\-type mediative systems\[[11](https://arxiv.org/html/2605.22900#bib.bib37),[23](https://arxiv.org/html/2605.22900#bib.bib38),[24](https://arxiv.org/html/2605.22900#bib.bib39)\], or with general type\-2 fuzzy logic systems and their decision models\[[20](https://arxiv.org/html/2605.22900#bib.bib56),[25](https://arxiv.org/html/2605.22900#bib.bib55)\], our construction treats\(μ~p,ν~p\)\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\)as mediative truth values in their own right and links their type\-2 footprints of uncertainty directly to the paraconsistent mediative operator\.

This semantic viewpoint also supports a granular interpretation at type\-3\. For MFL\-T3, we connect mediative semantics with granular computing and hierarchical reasoning\[[7](https://arxiv.org/html/2605.22900#bib.bib14),[31](https://arxiv.org/html/2605.22900#bib.bib15),[30](https://arxiv.org/html/2605.22900#bib.bib62)\]and introduce explicit aggregation operators over families of local mediative evaluations indexed by granules \(e\.g\., experts, sensors, or time slices\)\. Accordingly, MFL\-T2 captures second\-order uncertainty, whereas MFL\-T3 accounts for hierarchical evidence in multi\-source and multi\-level settings, and both remain conceptually coherent with the type\-1 core\. We complement these higher\-type semantics with an explicit logical treatment that keeps the type\-1 core intact while clarifying how type\-2 uncertainty and type\-3 heterogeneity propagate through connectives\.

Beyond higher\-type fuzzy sets, we also introduce Quantum Mediative Fuzzy Logic \(QMFL\), whose semantics is given in terms of quantum effects and density operators\. Our construction relates mediative truth degrees to effect algebras in the sense of Foulis and Bennett\[[15](https://arxiv.org/html/2605.22900#bib.bib21)\]and to fuzzy approaches to quantum logics\[[32](https://arxiv.org/html/2605.22900#bib.bib28),[33](https://arxiv.org/html/2605.22900#bib.bib29)\]\. The logical perspective is informed by many\-valued and modal quantum logics\[[14](https://arxiv.org/html/2605.22900#bib.bib30),[1](https://arxiv.org/html/2605.22900#bib.bib32)\]and by algebraic and lattice\-theoretic studies of effect algebras andLL\-valued quantum structures\[[28](https://arxiv.org/html/2605.22900#bib.bib33),[29](https://arxiv.org/html/2605.22900#bib.bib34),[13](https://arxiv.org/html/2605.22900#bib.bib24)\]\. We also build on recent work on axiomatic treatments of effect\-algebra\-based logics\[[39](https://arxiv.org/html/2605.22900#bib.bib26),[12](https://arxiv.org/html/2605.22900#bib.bib27)\]\.

The contributions of this paper can be summarized as follows\.

- •We axiomatize the mediative operator as a convex aggregation controlled by hesitation and contradiction parameters, and we establish basic properties such as boundedness between its two input degrees \(the outputs of the agreement and non\-agreement channels\) and reductions to type\-1 and intuitionistic\-fuzzy combinations\.
- •We introduce mediative truth values as pairs\(μ,ν\)∈\[0,1\]2\(\\mu,\\nu\)\\in\[0,1\]^\{2\}of truth and falsity degrees, from which hesitationπ\\piand contradictionζ\\zetaare derived, and we endow this space with suitable conjunction, disjunction, and negation operations, together with truth and information orders, so that\[0,1\]2\[0,1\]^\{2\}carries a continuous bilattice\-like structure\.
- •We define a propositional logic MFL\-T1 with standard fuzzy connectives plus a mediative connective, and we provide a Hilbert\-style axiom system that extends a chosen fuzzy base logic, such as Hájek’s Basic Logic \(BL\) or Łukasiewicz logic\.
- •We prove that MFL\-T1 is sound for its mediative semantics, paraconsistent in a precise sense, and a conservative extension of the underlying fuzzy logic on formulas without the mediative operator\.
- •We extend the mediative semantics to type\-2 fuzzy sets \(MFL\-T2\), modeling second\-order uncertainty about truth, falsity, hesitation, and contradiction, and we discuss how standard type\-reduction methods can be used to interpret mediative degrees under second\-order uncertainty\.
- •We propose a type\-3 granular extension \(MFL\-T3\) that organizes mediative truth into multi\-level granular structures indexed by arbitrary granules \(e\.g\., experts, sensors, or time slices\) within a unified framework\.
- •We introduce Quantum Mediative Fuzzy Logic \(QMFL\), whose semantics is formulated in terms of quantum effects and states on Hilbert spaces, and we illustrate a simple embedding of type\-1 mediative degrees into QMFL\.
- •We illustrate the safety\-first interpretation of mediative truth degrees through a detailed case study of sensor fusion for obstacle detection and braking decisions in autonomous driving\.
- •We clarify how standard type\-1 fuzzy and intuitionistic fuzzy semantics arise as special cases of MFL\-T1 by imposing suitable constraints on mediative truth values and parameters, under which the mediative operator reduces to the usualtt\-norm\-based combinations\.
- •We establish reduction results that connect the different levels of the framework: granular MFL\-T3 reduces to MFL\-T2 or MFL\-T1 when granules become homogeneous, and QMFL reduces to the classical mediative semantics when the relevant effects commute and the quantum states are diagonal in a common basis, thereby ensuring multi\-level coherence\.

Existing works on Mediative Fuzzy Logic\[[11](https://arxiv.org/html/2605.22900#bib.bib37)\]and its type\-3 extension\[[24](https://arxiv.org/html/2605.22900#bib.bib39)\]do not develop, to the best of our knowledge, the kind of granular semantics used here, where mediative truth values are treated as families of local evaluations indexed by arbitrary granules \(e\.g\., experts, sensors, or time slices\) and combined through explicit aggregation operators\. Accordingly, the definitions in Section 6 provide a first granular formalization of MFL\-T3 at the semantic level\. In addition, the quantum extension proposed here makes the connection with effect\-algebraic semantics explicit, aligning mediative reasoning with effect\-based viewpoints in quantum granular computing\[[36](https://arxiv.org/html/2605.22900#bib.bib52)\]\. To our knowledge, there is also no prior work that treats interval type\-2 pairs as mediative truth values in their own right\.

The paper is organized as follows\. Section[2](https://arxiv.org/html/2605.22900#S2)reviews the intuitionistic\-fuzzy and paraconsistent background\. Section[3](https://arxiv.org/html/2605.22900#S3)introduces the type\-1 mediative operator and its algebraic semantics\. Section[4](https://arxiv.org/html/2605.22900#S4)defines the propositional logic MFL\-T1 and establishes its basic metatheoretical properties\. Sections[5](https://arxiv.org/html/2605.22900#S5)–[7](https://arxiv.org/html/2605.22900#S7)develop MFL\-T2, MFL\-T3, and QMFL, respectively\. Section[8](https://arxiv.org/html/2605.22900#S8)presents the safety\-first sensor\-fusion case study, and Section[9](https://arxiv.org/html/2605.22900#S9)concludes\. For convenience, Appendix A collects the main notation and logical symbols used throughout the paper \(Table A\.1\)\.

## 2Background and motivation

In this section, we recall the intuitionistic\-fuzzy and paraconsistent perspectives that motivate Mediative Fuzzy Logic\. The goal is not to provide an exhaustive survey, but to highlight aspects directly relevant to the mediative semantics developed in the subsequent sections\.

In classical fuzzy logic, each propositionppis assigned a truth degreeμp∈\[0,1\]\\mu\_\{p\}\\in\[0,1\], and the degree of falsity is determined by a fixed negation, typically1−μp1\-\\mu\_\{p\}\[[42](https://arxiv.org/html/2605.22900#bib.bib4)\]\. In this setting, uncertainty is collapsed into a single dimension: onceμp\\mu\_\{p\}is fixed, there is no independent degree of falsity or inconsistency\. In contrast, intuitionistic fuzzy sets associate each element or propositionppwith a pair\(μp,νp\)\(\\mu\_\{p\},\\nu\_\{p\}\)of truth and falsity degrees, subject to the constraintμp\+νp≤1\\mu\_\{p\}\+\\nu\_\{p\}\\leq 1\[[3](https://arxiv.org/html/2605.22900#bib.bib8),[4](https://arxiv.org/html/2605.22900#bib.bib9),[6](https://arxiv.org/html/2605.22900#bib.bib11)\]\. The remaining amountπp=1−μp−νp\\pi\_\{p\}=1\-\\mu\_\{p\}\-\\nu\_\{p\}is interpreted as hesitation \(or indeterminacy\) and explicitly captures incomplete information\.

Paraconsistent logics, on the other hand, admit the possibility that bothppand¬p\\neg pmay hold to significant degrees without collapsing into triviality\[[35](https://arxiv.org/html/2605.22900#bib.bib57)\]\. Bilattice\-based logics, such as the Dunn–Belnap logic of first\-degree entailment, provide a well\-known example in which each proposition is evaluated by two coordinates capturing truth and knowledge \(or information\)\[[10](https://arxiv.org/html/2605.22900#bib.bib16),[16](https://arxiv.org/html/2605.22900#bib.bib18),[2](https://arxiv.org/html/2605.22900#bib.bib20)\]\. These frameworks support controlled reasoning in the presence of contradictions, but they are not usually formulated in terms of fuzzy membership and non\-membership functions\.

Mediative Fuzzy Logic \(MFL\) was originally proposed in an operational form to combine an agreement channel and a non\-agreement channel into a single mediative output\[[26](https://arxiv.org/html/2605.22900#bib.bib35)\]\. In that formulation, each proposition or element is described by an agreement membership function and a non\-agreement membership \(or non\-membership\) function, and the mediative operator aggregates the corresponding fuzzy\-system outputs\. In applications to control and diagnosis, MFL has been shown to handle both hesitation and contradiction in a smooth and interpretable manner, with successful implementations in medical diagnosis and pandemic modeling\[[19](https://arxiv.org/html/2605.22900#bib.bib41),[38](https://arxiv.org/html/2605.22900#bib.bib43),[37](https://arxiv.org/html/2605.22900#bib.bib44)\]\. Nevertheless, the original formulation was not cast as a fully axiomatized logic with clear algebraic semantics\. Most existing works treat MFL as a fuzzy inference scheme rather than as a logic in the proof\-theoretic sense\[[11](https://arxiv.org/html/2605.22900#bib.bib37),[24](https://arxiv.org/html/2605.22900#bib.bib39)\]\. In particular, when only hesitation is present, MFL reduces to an intuitionistic\-style combination, whereas in the presence of contradiction, the mediative output systematically takes an intermediate value between the agreement and non\-agreement channels\.

The following sections formalize this informal picture into a precise algebraic and logical framework and extend mediative reasoning to type\-2, type\-3, and quantum settings\.

## 3Type\-1 Mediative Fuzzy Logic: operator and algebraic semantics

We study the mediative operator at an abstract level, independently of any particular logical language\. Our starting point is a pair of numerical outputsa,b∈\[0,1\]a,b\\in\[0,1\], typically generated by an agreement channel and a non\-agreement channel, together with two parametersπ,ζ∈\[0,1\]\\pi,\\zeta\\in\[0,1\]representing hesitation and contradiction, respectively\.

###### Definition 3\.1\(Mediative operator\)\.

Fora,b∈\[0,1\]a,b\\in\[0,1\]and parametersπ,ζ∈\[0,1\]\\pi,\\zeta\\in\[0,1\], define

ℳ​\(a,b;π,ζ\):=\(1−π−ζ2\)​a\+\(π\+ζ2\)​b\.\\mathcal\{M\}\(a,b;\\pi,\\zeta\):=\\Bigl\(1\-\\pi\-\\frac\{\\zeta\}\{2\}\\Bigr\)a\+\\Bigl\(\\pi\+\\frac\{\\zeta\}\{2\}\\Bigr\)b\.

We impose the following axioms onℳ\\mathcal\{M\}, which constrain the admissible parameters\(π,ζ\)\(\\pi,\\zeta\):

- M1\(weight normalization\)\.Letw1=1−π−ζ/2w\_\{1\}=1\-\\pi\-\\zeta/2andw2=π\+ζ/2w\_\{2\}=\\pi\+\\zeta/2\. Then0≤w1,w2≤10\\leq w\_\{1\},w\_\{2\}\\leq 1andw1\+w2=1w\_\{1\}\+w\_\{2\}=1\.
- M2\(intuitionistic reduction\)\.Ifζ=0\\zeta=0, then ℳ​\(a,b;π,0\)=\(1−π\)​a\+π​b,\\mathcal\{M\}\(a,b;\\pi,0\)=\(1\-\\pi\)\\,a\+\\pi\\,b,\(1\)so that the mediative operator reduces to a convex combination controlled only by the hesitation parameterπ\\pi\.
- M3\(type\-1 reduction\)\.Ifπ=0\\pi=0andζ=0\\zeta=0, then ℳ​\(a,b;0,0\)=a,\\mathcal\{M\}\(a,b;0,0\)=a,\(2\)that is, the mediative operator reduces to the agreement channel’s output\.

The next results formalize basic properties ofℳ\\mathcal\{M\}\.

###### Theorem 3\.1\(Convexity and boundedness\)\.

Letw1=1−π−ζ/2w\_\{1\}=1\-\\pi\-\\zeta/2andw2=π\+ζ/2w\_\{2\}=\\pi\+\\zeta/2, and suppose thatw1,w2≥0w\_\{1\},w\_\{2\}\\geq 0andw1\+w2=1w\_\{1\}\+w\_\{2\}=1\(axiom M1\)\. Then, for alla,b∈\[0,1\]a,b\\in\[0,1\], we have

min⁡\(a,b\)≤ℳ​\(a,b;π,ζ\)≤max⁡\(a,b\)\.\\min\(a,b\)\\;\\leq\\;\\mathcal\{M\}\(a,b;\\pi,\\zeta\)\\;\\leq\\;\\max\(a,b\)\.\(3\)

###### Proof sketch\.

Under Axiom \(M1\), we havew1,w2≥0w\_\{1\},w\_\{2\}\\geq 0andw1\+w2=1w\_\{1\}\+w\_\{2\}=1, hence

ℳ​\(a,b;π,ζ\)=w1​a\+w2​b\\mathcal\{M\}\(a,b;\\pi,\\zeta\)=w\_\{1\}a\+w\_\{2\}bis a convex combination ofaaandbb\. Any convex combination of two real numbers lies between their minimum and maximum, which yields \([3](https://arxiv.org/html/2605.22900#S3.E3)\)\. ∎

By Axiom \(M1\), the admissible parameter pairs\(π,ζ\)\(\\pi,\\zeta\)are precisely those for which the induced weights satisfyw1,w2∈\[0,1\]w\_\{1\},w\_\{2\}\\in\[0,1\]andw1\+w2=1w\_\{1\}\+w\_\{2\}=1\(equivalently,π≥0\\pi\\geq 0,ζ≥0\\zeta\\geq 0, andπ\+ζ/2≤1\\pi\+\\zeta/2\\leq 1\)\. Hence, the mediative operator is a well\-defined convex aggregation of the two channelsaaandbband cannot extrapolate beyond the interval spanned by its inputs\. Moreover, in the induced type\-1 semantics we haveπ​\(μ,ν\)=max⁡\{0,1−μ−ν\}\\pi\(\\mu,\\nu\)=\\max\\\{0,1\-\\mu\-\\nu\\\}andζ​\(μ,ν\)=max⁡\{0,μ\+ν−1\}\\zeta\(\\mu,\\nu\)=\\max\\\{0,\\mu\+\\nu\-1\\\}for\(μ,ν\)∈\[0,1\]2\(\\mu,\\nu\)\\in\[0,1\]^\{2\}, soπ,ζ∈\[0,1\]\\pi,\\zeta\\in\[0,1\]and, in particular, they are nonnegative degrees\.

###### Theorem 3\.2\(Reductions\)\.

For alla,b∈\[0,1\]a,b\\in\[0,1\], the mediative operatorℳ\\mathcal\{M\}satisfies:

1. 1\.Ifζ=0\\zeta=0, then \([1](https://arxiv.org/html/2605.22900#S3.E1)\) holds\. In other words, the mediative operator reduces to the intuitionistic\-style convex combination controlled solely by the hesitation degreeπ\\pi\.
2. 2\.Ifπ=ζ=0\\pi=\\zeta=0, then \([2](https://arxiv.org/html/2605.22900#S3.E2)\) holds, recovering the underlying type\-1 fuzzy evaluation \(the agreement\-channel output\)\.

###### Proof\.

Both items follow directly by substituting the corresponding parameter values into the definition ofℳ\\mathcal\{M\}\. ∎

###### Theorem 3\.3\(Effect of contradiction\)\.

Leta,b∈\[0,1\]a,b\\in\[0,1\]and assumeπ=0\\pi=0and0<ζ<10<\\zeta<1\. Then

ℳ​\(a,b;0,ζ\)=\(1−ζ2\)​a\+\(ζ2\)​b\.\\mathcal\{M\}\(a,b;0,\\zeta\)=\\Bigl\(1\-\\frac\{\\zeta\}\{2\}\\Bigr\)a\+\\Bigl\(\\frac\{\\zeta\}\{2\}\\Bigr\)b\.
In particular, ifa\>ba\>b, then

a\>ℳ​\(a,b;0,ζ\)\>b\.a\>\\mathcal\{M\}\(a,b;0,\\zeta\)\>b\.By symmetry, ifa<ba<b, then

a<ℳ​\(a,b;0,ζ\)<b\.a<\\mathcal\{M\}\(a,b;0,\\zeta\)<b\.

###### Proof sketch\.

Forπ=0\\pi=0, the weights becomew1=1−ζ/2w\_\{1\}=1\-\\zeta/2andw2=ζ/2w\_\{2\}=\\zeta/2, both strictly between0and11when0<ζ<10<\\zeta<1\. Hence

ℳ​\(a,b;0,ζ\)=w1​a\+w2​b\\mathcal\{M\}\(a,b;0,\\zeta\)=w\_\{1\}a\+w\_\{2\}bis a strict convex combination ofaaandbb\. Ifa\>ba\>b, any strict convex combinationw1​a\+w2​bw\_\{1\}a\+w\_\{2\}bwithw1,w2∈\(0,1\)w\_\{1\},w\_\{2\}\\in\(0,1\)lies strictly betweenaaandbb, yieldinga\>ℳ​\(a,b;0,ζ\)\>ba\>\\mathcal\{M\}\(a,b;0,\\zeta\)\>b\. The casea<ba<bis analogous\. ∎

As ensured by Axiom \(M1\),ℳ​\(a,b;π,ζ\)\\mathcal\{M\}\(a,b;\\pi,\\zeta\)is a convex aggregation and cannot extrapolate beyond the interval spanned byaaandbb\. In the instantiated semantics whereπ=π​\(μ,ν\)\\pi=\\pi\(\\mu,\\nu\)andζ=ζ​\(μ,ν\)\\zeta=\\zeta\(\\mu,\\nu\)are obtained by a positive\-part construction, these quantities are nonnegative by definition\.

### 3\.1Mediative truth values and algebraic semantics

We now introduce an algebraic semantics for type\-1 MFL in terms of pairs of truth and falsity degrees\. These pairs constitute mediative truth values and provide the basis for defining hesitation and contradiction in a uniform way, consistent with the agreement/non\-agreement interpretation of MFL\.

LetV=\[0,1\]×\[0,1\]V=\[0,1\]\\times\[0,1\]\. A mediative truth value is any pair\(μ,ν\)∈V\(\\mu,\\nu\)\\in V, whereμ\\muis interpreted as a degree of truth \(or agreement\) andν\\nuas a degree of falsity \(or non\-agreement\)\. From each pair, we derive two secondary quantities, hesitation and contradiction\.

###### Definition 3\.2\(Hesitation and contradiction\)\.

For\(μ,ν\)∈V\(\\mu,\\nu\)\\in Vwe define

π​\(μ,ν\)=max⁡\(0,1−μ−ν\),ζ​\(μ,ν\)=max⁡\(0,μ\+ν−1\)\.\\pi\(\\mu,\\nu\)=\\max\(0,1\-\\mu\-\\nu\),\\qquad\\zeta\(\\mu,\\nu\)=\\max\(0,\\mu\+\\nu\-1\)\.Thusπ​\(μ,ν\)\>0\\pi\(\\mu,\\nu\)\>0precisely whenμ\+ν<1\\mu\+\\nu<1\(incomplete information\), whereasζ​\(μ,ν\)\>0\\zeta\(\\mu,\\nu\)\>0precisely whenμ\+ν\>1\\mu\+\\nu\>1\(overdetermined, possibly contradictory information\)\.

Note thatπ​\(μ,ν\)\\pi\(\\mu,\\nu\)andζ​\(μ,ν\)\\zeta\(\\mu,\\nu\)cannot be simultaneously positive\. This cleanly separates hesitation from contradiction and will be useful when interpreting the mediative operator in terms of agreement and non\-agreement channels\.

To endowVVwith logical operations, we fix a left\-continuoustt\-normTTand its dualtt\-conormSS, and define conjunction, disjunction, and negation coordinatewise on mediative truth values\.

###### Definition 3\.3\(Conjunction, disjunction and negation onVV\)\.

Fix a left\-continuoustt\-normTTand its dualtt\-conormSS\. For\(μ1,ν1\),\(μ2,ν2\)∈V=\[0,1\]2\(\\mu\_\{1\},\\nu\_\{1\}\),\(\\mu\_\{2\},\\nu\_\{2\}\)\\in V=\[0,1\]^\{2\}, define

\(μ1,ν1\)∧\(μ2,ν2\):=\(T​\(μ1,μ2\),S​\(ν1,ν2\)\),\(\\mu\_\{1\},\\nu\_\{1\}\)\\wedge\(\\mu\_\{2\},\\nu\_\{2\}\):=\\bigl\(T\(\\mu\_\{1\},\\mu\_\{2\}\),\\,S\(\\nu\_\{1\},\\nu\_\{2\}\)\\bigr\),\(μ1,ν1\)∨\(μ2,ν2\):=\(S​\(μ1,μ2\),T​\(ν1,ν2\)\),\(\\mu\_\{1\},\\nu\_\{1\}\)\\vee\(\\mu\_\{2\},\\nu\_\{2\}\):=\\bigl\(S\(\\mu\_\{1\},\\mu\_\{2\}\),\\,T\(\\nu\_\{1\},\\nu\_\{2\}\)\\bigr\),and

¬\(μ,ν\):=\(ν,μ\)\.\\neg\(\\mu,\\nu\):=\(\\nu,\\mu\)\.

Conjunction decreases the truth coordinate and increases the falsity coordinate, whereas disjunction increases truth and decreases falsity\. Negation swaps the truth and falsity coordinates, in accordance with the intuition that the falsity ofppcorresponds to the truth of¬p\\neg p\. In this way,\(V,∧,∨,¬\)\(V,\\wedge,\\vee,\\neg\)forms a bilattice\-like structure over pairs of truth and falsity degrees, on top of which the mediative evaluation is defined\.

###### Definition 3\.4\(Mediative evaluation\)\.

Let\(μ,ν\)∈V\(\\mu,\\nu\)\\in V\. The mediative evaluation of\(μ,ν\)\(\\mu,\\nu\)is the scalar

M​\(μ,ν\):=ℳ​\(a​\(μ,ν\),b​\(μ,ν\);π​\(μ,ν\),ζ​\(μ,ν\)\),M\(\\mu,\\nu\):=\\mathcal\{M\}\\\!\\bigl\(a\(\\mu,\\nu\),\\,b\(\\mu,\\nu\);\\,\\pi\(\\mu,\\nu\),\\,\\zeta\(\\mu,\\nu\)\\bigr\),where, in the basic case, we seta​\(μ,ν\)=μa\(\\mu,\\nu\)=\\muandb​\(μ,ν\)=1−νb\(\\mu,\\nu\)=1\-\\nu\.

Thus,M​\(μ,ν\)M\(\\mu,\\nu\)is the mediative score associated with a proposition whose degrees of agreement and non\-agreement areμ\\muandν\\nu, respectively\. It combines the agreement channela​\(μ,ν\)=μa\(\\mu,\\nu\)=\\muand the lack\-of\-disagreement channelb​\(μ,ν\)=1−νb\(\\mu,\\nu\)=1\-\\nu, with mixing weights given by the hesitationπ​\(μ,ν\)\\pi\(\\mu,\\nu\)and the contradictionζ​\(μ,ν\)\\zeta\(\\mu,\\nu\)extracted from\(μ,ν\)\(\\mu,\\nu\)\. The choicesa​\(μ,ν\)=μa\(\\mu,\\nu\)=\\muandb​\(μ,ν\)=1−νb\(\\mu,\\nu\)=1\-\\nureflect the idea that the mediative score should increase with agreement and and decrease with disagreement aboutpp\.

## 4Propositional Mediative Fuzzy Logic \(MFL\-T1\)

We define a propositional logic, MFL\-T1, whose semantics is based on mediative truth values\.

As usual, we use the definable constants⊤:=\(p→p\)\\top:=\(p\\to p\)and⊥⁣:=⁣¬⁣⊤\\bot:=\\neg\\top, for an arbitrary fixed atomic propositionpp\.

###### Definition 4\.1\(Language\)\.

The language of MFL\-T1 extends that of the chosen fuzzy base logic \(BL or Łukasiewicz logic\): it includes the connectives∧,∨,¬,→\\wedge,\\vee,\\neg,\\to, together with a unary mediative connectiveMed\\operatorname\{Med\}\. We writeMed⁡φ\\operatorname\{Med\}\\,\\varphifor its application to a formulaφ\\varphi\.

###### Definition 4\.2\(Implication onVV\)\.

Fix a left\-continuoustt\-normTTon\[0,1\]\[0,1\]and let⇒T\\Rightarrow\_\{T\}denote its residuum\. For\(μ1,ν1\),\(μ2,ν2\)∈V=\[0,1\]2\(\\mu\_\{1\},\\nu\_\{1\}\),\(\\mu\_\{2\},\\nu\_\{2\}\)\\in V=\[0,1\]^\{2\}, define

\(μ1,ν1\)→\(μ2,ν2\):=\(μ1⇒Tμ2,ν2⇒Tν1\)\.\(\\mu\_\{1\},\\nu\_\{1\}\)\\to\(\\mu\_\{2\},\\nu\_\{2\}\):=\\bigl\(\\mu\_\{1\}\\Rightarrow\_\{T\}\\mu\_\{2\},\\;\\nu\_\{2\}\\Rightarrow\_\{T\}\\nu\_\{1\}\\bigr\)\.

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

A mediative valuation on the language of MFL\-T1 is a functionvvthat assigns to each formulaφ\\varphia mediative truth valuev​\(φ\)∈Vv\(\\varphi\)\\in Vand satisfies, for all formulasφ,ψ\\varphi,\\psi:

- \(V1\)For each atomic propositionpp, the valuev​\(p\)v\(p\)is arbitrary inVV\.
- \(V2\)v​\(φ∧ψ\)=v​\(φ\)∧v​\(ψ\)v\(\\varphi\\wedge\\psi\)=v\(\\varphi\)\\wedge v\(\\psi\)\.
- \(V3\)v​\(φ∨ψ\)=v​\(φ\)∨v​\(ψ\)v\(\\varphi\\vee\\psi\)=v\(\\varphi\)\\vee v\(\\psi\)\.
- \(V4\)v​\(¬φ\)=¬v​\(φ\)v\(\\neg\\varphi\)=\\neg v\(\\varphi\)\.
- \(V5\)v​\(φ→ψ\)=v​\(φ\)→v​\(ψ\)v\(\\varphi\\to\\psi\)=v\(\\varphi\)\\to v\(\\psi\), where→\\toonVVis as in Definition[4\.2](https://arxiv.org/html/2605.22900#S4.Thmdefinition2)\.
- \(V6\)v​\(Med⁡\(φ\)\)=\(M​\(v​\(φ\)\),1−M​\(v​\(φ\)\)\)v\(\\operatorname\{Med\}\(\\varphi\)\)=\\bigl\(M\(v\(\\varphi\)\),\\,1\-M\(v\(\\varphi\)\)\\bigr\)\.

### 4\.1Axiomatic system

We build MFL\-T1 on top of a fuzzy base logic such as Hájek’s Basic Logic \(BL\) or Łukasiewicz logic\[[18](https://arxiv.org/html/2605.22900#bib.bib2)\], which provides the axioms and rules for the connectives∧,∨,→,¬\\wedge,\\vee,\\to,\\neg\. We then add axiom schemata governing the mediative connective\.

#### 4\.1\.1Axiom schemata forMed\\operatorname\{Med\}

We use the abbreviationφ↔ψ:=\(φ→ψ\)∧\(ψ→φ\)\\varphi\\leftrightarrow\\psi:=\(\\varphi\\to\\psi\)\\wedge\(\\psi\\to\\varphi\)\.

1. \(Med1\)\(φ→ψ\)→\(Med⁡\(φ\)→Med⁡\(ψ\)\)\.\(\\varphi\\to\\psi\)\\to\(\\operatorname\{Med\}\(\\varphi\)\\to\\operatorname\{Med\}\(\\psi\)\)\.
2. \(Med2\)Med⁡\(⊤\)↔⊤andMed⁡\(⊥\)↔⊥\.\\operatorname\{Med\}\(\\top\)\\leftrightarrow\\top\\ \\ \\text\{and\}\\ \\ \\operatorname\{Med\}\(\\bot\)\\leftrightarrow\\bot\.
3. \(Med3\)\(φ↔ψ\)→\(Med\(φ\)↔Med\(ψ\)\)\.\(\\varphi\\leftrightarrow\\psi\)\\to\(\\operatorname\{Med\}\(\\varphi\)\\leftrightarrow\\operatorname\{Med\}\(\\psi\)\)\.

### 4\.2Metatheoretical properties of MFL\-T1

We record several basic metatheoretical properties of MFL\-T1\. The proofs rely on standard techniques from fuzzy logic and algebraic logic and are omitted\. For the notation used below, see Appendix A and Table[4](https://arxiv.org/html/2605.22900#S10.T4)\.

##### Soundness

MFL\-T1 is sound with respect to the mediative semantics: for every setΓ\\Gammaof formulas and every formulaφ\\varphi, ifΓ⊢mφ\\Gamma\\vdash\_\{m\}\\varphi, thenΓ⊧mφ\\Gamma\\models\_\{m\}\\varphi\. Equivalently, wheneverΓ⊢mφ\\Gamma\\vdash\_\{m\}\\varphiandvvis a mediative valuation such thatM​\(v​\(ψ\)\)=1M\(v\(\\psi\)\)=1for allψ∈Γ\\psi\\in\\Gamma, we also haveM​\(v​\(φ\)\)=1M\(v\(\\varphi\)\)=1\. In particular, if⊢mφ\\vdash\_\{m\}\\varphi, then⊧mφ\\models\_\{m\}\\varphi, i\.e\.,M​\(v​\(φ\)\)=1M\(v\(\\varphi\)\)=1for every mediative valuationvv\.

##### Paraconsistency

MFL\-T1 is paraconsistent: there exist formulasφ\\varphiand mediative valuationsvvsuch that bothM​\(v​\(φ\)\)M\(v\(\\varphi\)\)andM​\(v​\(¬φ\)\)M\(v\(\\neg\\varphi\)\)can take high values simultaneously, while the explosion principle

\(φ∧¬φ\)→ψ\(\\varphi\\wedge\\neg\\varphi\)\\to\\psiis not derivable in MFL\-T1 for arbitraryψ\\psi\.

##### Reduction to the base fuzzy logic

Letφ\\varphibe a formula that does not contain the mediative connectiveMed\\operatorname\{Med\}\. Thenφ\\varphiis derivable in MFL\-T1 if and only if it is derivable in the underlying fuzzy base logic \(BL or Łukasiewicz logic\)\. Thus, MFL\-T1 is a conservative extension of the chosen base logic\.

##### Reduction to intuitionistic fuzzy and type\-1 cases

If mediative valuations are restricted so thatμ\+ν≤1\\mu\+\\nu\\leq 1for all atomic propositions, the mediative evaluation reduces to an intuitionistic\-style combination controlled only by hesitation\. In the further special case whereν=1−μ\\nu=1\-\\mufor all atoms, the mediative evaluation coincides with the underlying type\-1 fuzzy evaluation, and MFL\-T1 reduces to the base fuzzy logic at the semantic level\.

## 5Type\-2 Mediative Fuzzy Logic \(MFL\-T2\)

This section develops a semantic treatment of Type\-2 Mediative Fuzzy Logic \(MFL\-T2\) along the lines proposed in\[[11](https://arxiv.org/html/2605.22900#bib.bib37)\]\. The key point is that the primary degrees of truth and falsity are themselves uncertain; we model this second\-order variability—due to noise, calibration drift, or changing environmental conditions—by interval type\-2 fuzzy sets\. For simplicity, we restrict attention to the interval case, which already captures the main mechanisms we need\.

###### Definition 5\.1\(Footprint of uncertainty \(FOU\)\[[25](https://arxiv.org/html/2605.22900#bib.bib55)\]\)\.

LetA~\\tilde\{A\}be an interval type\-2 fuzzy set on\[0,1\]\[0,1\]with lower and upper membership functionsAL,AU:\[0,1\]→\[0,1\]A^\{L\},A^\{U\}:\[0,1\]\\to\[0,1\]\. The footprint of uncertainty \(FOU\) ofA~\\tilde\{A\}is the set

FOU​\(A~\)=\{\(x,u\)∈\[0,1\]×\[0,1\]:AL​\(x\)≤u≤AU​\(x\)\}\.\\mathrm\{FOU\}\(\\tilde\{A\}\)=\\bigl\\\{\\,\(x,u\)\\in\[0,1\]\\times\[0,1\]:A^\{L\}\(x\)\\leq u\\leq A^\{U\}\(x\)\\,\\bigr\\\}\.In the present setting,FOU​\(A~\)\\mathrm\{FOU\}\(\\tilde\{A\}\)collects all admissible pairs\(x,u\)\(x,u\)wherex∈\[0,1\]x\\in\[0,1\]is a candidate primary degree andu∈\[0,1\]u\\in\[0,1\]is an admissible membership level consistent with the lower and upper membership functions\.

At the type\-1 level, each propositionppis assigned a mediative truth value\(μp,νp\)\(\\mu\_\{p\},\\nu\_\{p\}\)\. At the type\-2 level, these scalar degrees are replaced by interval type\-2 fuzzy sets, so that the truth value ofppbecomes a pair\(μ~p,ν~p\)\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\), equivalently the pair of footprints of uncertaintyFOU​\(μ~p\)\\mathrm\{FOU\}\(\\tilde\{\\mu\}\_\{p\}\)andFOU​\(ν~p\)\\mathrm\{FOU\}\(\\tilde\{\\nu\}\_\{p\}\)\.

###### Definition 5\.2\(Type\-2 mediative truth values\)\.

A type\-2 mediative truth value for an atomic propositionppis a pair

v\(2\)​\(p\)=\(μ~p,ν~p\),v^\{\(2\)\}\(p\)=\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\),whereμ~p\\tilde\{\\mu\}\_\{p\}andν~p\\tilde\{\\nu\}\_\{p\}are interval type\-2 fuzzy sets on\[0,1\]\[0,1\]with lower and upper membership functionsμpL,μpU\\mu\_\{p\}^\{L\},\\mu\_\{p\}^\{U\}andνpL,νpU\\nu\_\{p\}^\{L\},\\nu\_\{p\}^\{U\}, respectively\. Here, the superscriptsL,UL,Urefer to lower/upper membership functions, whereas the underline/overline notationμ¯p,μ¯p\\underline\{\\mu\}\_\{p\},\\overline\{\\mu\}\_\{p\}andν¯p,ν¯p\\underline\{\\nu\}\_\{p\},\\overline\{\\nu\}\_\{p\}is reserved for the projected scalar bounds induced by these memberships \(Definition[5\.3](https://arxiv.org/html/2605.22900#S5.Thmdefinition3)\)\.

Equivalently,v\(2\)​\(p\)v^\{\(2\)\}\(p\)can be represented by the pair of footprints of uncertaintyFOU​\(μ~p\)\\mathrm\{FOU\}\(\\tilde\{\\mu\}\_\{p\}\)andFOU​\(ν~p\)\\mathrm\{FOU\}\(\\tilde\{\\nu\}\_\{p\}\)\. By “atomic proposition” we mean a propositional variable \(a formula with no connectives\); this valuation is extended inductively to compound formulas by the semantic clauses for the connectives\.

###### Definition 5\.3\(Projected interval bounds\)\.

LetA~\\tilde\{A\}be an interval type\-2 fuzzy set on\[0,1\]\[0,1\]with lower and upper membership functionsALA^\{L\}andAUA^\{U\}\. AssumeAU≢0A^\{U\}\\not\\equiv 0\. We define the*outer*\(conservative\) projection ofA~\\tilde\{A\}by projecting the support ofAUA^\{U\}:

ProjU​\(A~\)=\[a¯U,a¯U\],a¯U=inf\{x∈\[0,1\]:AU​\(x\)\>0\},a¯U=sup\{x∈\[0,1\]:AU​\(x\)\>0\}\.\\mathrm\{Proj\}^\{U\}\(\\tilde\{A\}\)=\[\\underline\{a\}^\{U\},\\overline\{a\}^\{U\}\],\\qquad\\underline\{a\}^\{U\}=\\inf\\\{x\\in\[0,1\]:A^\{U\}\(x\)\>0\\\},\\quad\\overline\{a\}^\{U\}=\\sup\\\{x\\in\[0,1\]:A^\{U\}\(x\)\>0\\\}\.IfAL≢0A^\{L\}\\not\\equiv 0, we also define the*inner*\(guaranteed\) projection by projecting the support ofALA^\{L\}:

ProjL​\(A~\)=\[a¯L,a¯L\],a¯L=inf\{x∈\[0,1\]:AL​\(x\)\>0\},a¯L=sup\{x∈\[0,1\]:AL​\(x\)\>0\}\.\\mathrm\{Proj\}^\{L\}\(\\tilde\{A\}\)=\[\\underline\{a\}^\{L\},\\overline\{a\}^\{L\}\],\\qquad\\underline\{a\}^\{L\}=\\inf\\\{x\\in\[0,1\]:A^\{L\}\(x\)\>0\\\},\\quad\\overline\{a\}^\{L\}=\\sup\\\{x\\in\[0,1\]:A^\{L\}\(x\)\>0\\\}\.More generally, one may replace these support\-based projections byα\\alpha\-cut projections, using\{x:AU​\(x\)≥α\}\\\{x:A^\{U\}\(x\)\\geq\\alpha\\\}and\{x:AL​\(x\)≥α\}\\\{x:A^\{L\}\(x\)\\geq\\alpha\\\}for a prescribedα∈\(0,1\]\\alpha\\in\(0,1\]\. Once an interval has been selected \(outer, inner, orα\\alpha\-cut based\), subsequent steps treat it as the admissible range of the primary degree and propagate it through the same interval computations used later \(forπ\\pi,ζ\\zeta, and the mediative evaluation\)\.

###### Definition 5\.4\(Interval bounds for type\-2 connectives\)\.

Letφ,ψ\\varphi,\\psibe formulas, and assume each is associated with projected scalar bounds

μφ∈\[μ¯φ,μ¯φ\],νφ∈\[ν¯φ,ν¯φ\],μψ∈\[μ¯ψ,μ¯ψ\],νψ∈\[ν¯ψ,ν¯ψ\],\\mu\_\{\\varphi\}\\in\[\\underline\{\\mu\}\_\{\\varphi\},\\overline\{\\mu\}\_\{\\varphi\}\],\\quad\\nu\_\{\\varphi\}\\in\[\\underline\{\\nu\}\_\{\\varphi\},\\overline\{\\nu\}\_\{\\varphi\}\],\\qquad\\mu\_\{\\psi\}\\in\[\\underline\{\\mu\}\_\{\\psi\},\\overline\{\\mu\}\_\{\\psi\}\],\\quad\\nu\_\{\\psi\}\\in\[\\underline\{\\nu\}\_\{\\psi\},\\overline\{\\nu\}\_\{\\psi\}\],obtained, for instance, from Definition[5\.3](https://arxiv.org/html/2605.22900#S5.Thmdefinition3)\. LetTTbe thett\-norm andSSthe associatedtt\-conorm used in Definition[3\.3](https://arxiv.org/html/2605.22900#S3.Thmdefinition3)\. Let⇒T\\Rightarrow\_\{T\}denote the residuum ofTTas in Definition[4\.2](https://arxiv.org/html/2605.22900#S4.Thmdefinition2)\.

SinceTTandSSare nondecreasing in each argument, conservative bounds for∧\\wedgeand∨\\veeare obtained by endpoint evaluation\. For¬\\neg, we use the bilattice negation \(swap of truth and falsity bounds\)\. For→\\to, recall that the residuum⇒T\\Rightarrow\_\{T\}is antitone in its first argument and monotone in its second; conservative bounds again follow from endpoint evaluation using the worst\-case combinations of premise and conclusion bounds\.

μ¯φ∧ψ\\displaystyle\\underline\{\\mu\}\_\{\\varphi\\wedge\\psi\}=T​\(μ¯φ,μ¯ψ\),\\displaystyle=T\(\\underline\{\\mu\}\_\{\\varphi\},\\underline\{\\mu\}\_\{\\psi\}\),μ¯φ∧ψ\\displaystyle\\overline\{\\mu\}\_\{\\varphi\\wedge\\psi\}=T​\(μ¯φ,μ¯ψ\),\\displaystyle=T\(\\overline\{\\mu\}\_\{\\varphi\},\\overline\{\\mu\}\_\{\\psi\}\),\(4\)ν¯φ∧ψ\\displaystyle\\underline\{\\nu\}\_\{\\varphi\\wedge\\psi\}=S​\(ν¯φ,ν¯ψ\),\\displaystyle=S\(\\underline\{\\nu\}\_\{\\varphi\},\\underline\{\\nu\}\_\{\\psi\}\),ν¯φ∧ψ\\displaystyle\\overline\{\\nu\}\_\{\\varphi\\wedge\\psi\}=S​\(ν¯φ,ν¯ψ\),\\displaystyle=S\(\\overline\{\\nu\}\_\{\\varphi\},\\overline\{\\nu\}\_\{\\psi\}\),μ¯φ∨ψ\\displaystyle\\underline\{\\mu\}\_\{\\varphi\\vee\\psi\}=S​\(μ¯φ,μ¯ψ\),\\displaystyle=S\(\\underline\{\\mu\}\_\{\\varphi\},\\underline\{\\mu\}\_\{\\psi\}\),μ¯φ∨ψ\\displaystyle\\overline\{\\mu\}\_\{\\varphi\\vee\\psi\}=S​\(μ¯φ,μ¯ψ\),\\displaystyle=S\(\\overline\{\\mu\}\_\{\\varphi\},\\overline\{\\mu\}\_\{\\psi\}\),ν¯φ∨ψ\\displaystyle\\underline\{\\nu\}\_\{\\varphi\\vee\\psi\}=T​\(ν¯φ,ν¯ψ\),\\displaystyle=T\(\\underline\{\\nu\}\_\{\\varphi\},\\underline\{\\nu\}\_\{\\psi\}\),ν¯φ∨ψ\\displaystyle\\overline\{\\nu\}\_\{\\varphi\\vee\\psi\}=T​\(ν¯φ,ν¯ψ\),\\displaystyle=T\(\\overline\{\\nu\}\_\{\\varphi\},\\overline\{\\nu\}\_\{\\psi\}\),μ¯¬φ\\displaystyle\\underline\{\\mu\}\_\{\\neg\\varphi\}=ν¯φ,μ¯¬φ=ν¯φ,\\displaystyle=\\underline\{\\nu\}\_\{\\varphi\},\\quad\\overline\{\\mu\}\_\{\\neg\\varphi\}=\\overline\{\\nu\}\_\{\\varphi\},ν¯¬φ\\displaystyle\\underline\{\\nu\}\_\{\\neg\\varphi\}=μ¯φ,ν¯¬φ=μ¯φ\.\\displaystyle=\\underline\{\\mu\}\_\{\\varphi\},\\quad\\overline\{\\nu\}\_\{\\neg\\varphi\}=\\overline\{\\mu\}\_\{\\varphi\}\.
For implication, we use the residuum⇒T\\Rightarrow\_\{T\}of the chosentt\-normTT\. Since⇒T\\Rightarrow\_\{T\}is antitone in its antecedent and monotone in its consequent, the lower bound is attained at\(first=¯,second=¯\)\(\\text\{first\}=\\overline\{\\phantom\{\\mu\}\},\\,\\text\{second\}=\\underline\{\\phantom\{\\mu\}\}\)and the upper bound at\(first=¯,second=¯\)\(\\text\{first\}=\\underline\{\\phantom\{\\mu\}\},\\,\\text\{second\}=\\overline\{\\phantom\{\\mu\}\}\):

μ¯φ→ψ\\displaystyle\\underline\{\\mu\}\_\{\\varphi\\to\\psi\}=μ¯φ⇒Tμ¯ψ,\\displaystyle=\\overline\{\\mu\}\_\{\\varphi\}\\Rightarrow\_\{T\}\\underline\{\\mu\}\_\{\\psi\},μ¯φ→ψ\\displaystyle\\overline\{\\mu\}\_\{\\varphi\\to\\psi\}=μ¯φ⇒Tμ¯ψ,\\displaystyle=\\underline\{\\mu\}\_\{\\varphi\}\\Rightarrow\_\{T\}\\overline\{\\mu\}\_\{\\psi\},\(5\)ν¯φ→ψ\\displaystyle\\underline\{\\nu\}\_\{\\varphi\\to\\psi\}=ν¯ψ⇒Tν¯φ,\\displaystyle=\\overline\{\\nu\}\_\{\\psi\}\\Rightarrow\_\{T\}\\underline\{\\nu\}\_\{\\varphi\},ν¯φ→ψ\\displaystyle\\overline\{\\nu\}\_\{\\varphi\\to\\psi\}=ν¯ψ⇒Tν¯φ\.\\displaystyle=\\underline\{\\nu\}\_\{\\psi\}\\Rightarrow\_\{T\}\\overline\{\\nu\}\_\{\\varphi\}\.
In what follows, we propagate interval bounds through compound formulas using the endpoint rules in \([4](https://arxiv.org/html/2605.22900#S5.E4)\) and \([5](https://arxiv.org/html/2605.22900#S5.E5)\)\.

###### Definition 5\.5\(Type\-2 hesitation and contradiction \(interval bounds\)\)\.

Letμ∈\[μ¯,μ¯\]\\mu\\in\[\\underline\{\\mu\},\\overline\{\\mu\}\]andν∈\[ν¯,ν¯\]\\nu\\in\[\\underline\{\\nu\},\\overline\{\\nu\}\]be the projected interval bounds induced byμ~\\tilde\{\\mu\}andν~\\tilde\{\\nu\}\(Definition[5\.3](https://arxiv.org/html/2605.22900#S5.Thmdefinition3)\)\. Define

H​\(μ,ν\):=max⁡\{0,1−μ−ν\},C​\(μ,ν\):=max⁡\{0,μ\+ν−1\}\.H\(\\mu,\\nu\):=\\max\\\{0,\\,1\-\\mu\-\\nu\\\},\\qquad C\(\\mu,\\nu\):=\\max\\\{0,\\,\\mu\+\\nu\-1\\\}\.Since1−μ−ν1\-\\mu\-\\nuis nonincreasing in each argument andμ\+ν−1\\mu\+\\nu\-1is nondecreasing in each argument, and sincemax⁡\{0,⋅\}\\max\\\{0,\\cdot\\\}preserves monotonicity, conservative interval envelopes are obtained by endpoint evaluation:

HL=max⁡\{0,1−μ¯−ν¯\},HU=max⁡\{0,1−μ¯−ν¯\},H\_\{L\}=\\max\\\{0,\\,1\-\\overline\{\\mu\}\-\\overline\{\\nu\}\\\},\\qquad H\_\{U\}=\\max\\\{0,\\,1\-\\underline\{\\mu\}\-\\underline\{\\nu\}\\\},CL=max⁡\{0,μ¯\+ν¯−1\},CU=max⁡\{0,μ¯\+ν¯−1\}\.C\_\{L\}=\\max\\\{0,\\,\\underline\{\\mu\}\+\\underline\{\\nu\}\-1\\\},\\qquad C\_\{U\}=\\max\\\{0,\\,\\overline\{\\mu\}\+\\overline\{\\nu\}\-1\\\}\.In the type\-1 semantics,H​\(μ,ν\)H\(\\mu,\\nu\)andC​\(μ,ν\)C\(\\mu,\\nu\)coincide with the hesitationπ\\piand contradictionζ\\zeta, respectively\.

###### Definition 5\.6\(Type\-2 mediative evaluation: type\-reduced and envelope modes\)\.

Letv\(2\)​\(p\)=\(μ~p,ν~p\)v^\{\(2\)\}\(p\)=\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\)be an interval type\-2 mediative assignment for an atomic propositionpp\. We consider two complementary ways of assigning a mediative degree\.

*\(i\) Type\-reduced \(crisp\) mode\.*First type\-reduceμ~p\\tilde\{\\mu\}\_\{p\}andν~p\\tilde\{\\nu\}\_\{p\}to a crisp pair\(μ¯p,ν¯p\)\(\\bar\{\\mu\}\_\{p\},\\bar\{\\nu\}\_\{p\}\)\(for instance, by centroid type\-reduction via the Karnik–Mendel procedures\), and then set

M¯p:=M​\(μ¯p,ν¯p\)\.\\bar\{M\}\_\{p\}:=M\(\\bar\{\\mu\}\_\{p\},\\bar\{\\nu\}\_\{p\}\)\.
*\(ii\) Envelope \(interval\) mode\.*Alternatively, extract projected boundsμp∈\[μ¯p,μ¯p\]\\mu\_\{p\}\\in\[\\underline\{\\mu\}\_\{p\},\\overline\{\\mu\}\_\{p\}\]andνp∈\[ν¯p,ν¯p\]\\nu\_\{p\}\\in\[\\underline\{\\nu\}\_\{p\},\\overline\{\\nu\}\_\{p\}\]from the footprints ofμ~p\\tilde\{\\mu\}\_\{p\}andν~p\\tilde\{\\nu\}\_\{p\}\(Definition[5\.3](https://arxiv.org/html/2605.22900#S5.Thmdefinition3)\), and define the envelope of admissible mediative scores by

\[ML​\(p\),MU​\(p\)\]:=\[minμ∈\[μ¯p,μ¯p\]ν∈\[ν¯p,ν¯p\]⁡M​\(μ,ν\),maxμ∈\[μ¯p,μ¯p\]ν∈\[ν¯p,ν¯p\]⁡M​\(μ,ν\)\]\.\[M\_\{L\}\(p\),M\_\{U\}\(p\)\]:=\\Bigl\[\\min\_\{\\mu\\in\[\\underline\{\\mu\}\_\{p\},\\overline\{\\mu\}\_\{p\}\]\\atop\\nu\\in\[\\underline\{\\nu\}\_\{p\},\\overline\{\\nu\}\_\{p\}\]\}M\(\\mu,\\nu\),\\;\\max\_\{\\mu\\in\[\\underline\{\\mu\}\_\{p\},\\overline\{\\mu\}\_\{p\}\]\\atop\\nu\\in\[\\underline\{\\nu\}\_\{p\},\\overline\{\\nu\}\_\{p\}\]\}M\(\\mu,\\nu\)\\Bigr\]\.BecauseM​\(μ,ν\)M\(\\mu,\\nu\)is continuous and piecewise affine \(with regime boundaryμ\+ν=1\\mu\+\\nu=1\), its extrema over a rectangle are attained on the boundary\. In practice, it is enough to evaluateMMat the four corners of\[μ¯,μ¯\]×\[ν¯,ν¯\]\[\\underline\{\\mu\},\\overline\{\\mu\}\]\\times\[\\underline\{\\nu\},\\overline\{\\nu\}\]and, whenever the rectangle intersects the lineμ\+ν=1\\mu\+\\nu=1, also at the intersection points with the rectangle edges\.

##### Extension from atoms to formulas

In mode \(i\), define a type\-1 valuationv¯​\(p\)=\(μ¯p,ν¯p\)\\bar\{v\}\(p\)=\(\\bar\{\\mu\}\_\{p\},\\bar\{\\nu\}\_\{p\}\)and extend it inductively to all formulas using the same connective clauses as in MFL\-T1\. In mode \(ii\), extend the projected bounds inductively to all formulas using Definition[5\.4](https://arxiv.org/html/2605.22900#S5.Thmdefinition4), and compute the corresponding envelope\[ML​\(φ\),MU​\(φ\)\]\[M\_\{L\}\(\\varphi\),M\_\{U\}\(\\varphi\)\]by the same optimization as above\.

###### Proposition 5\.1\(Reduction to the type\-1 case\)\.

If for every atomic propositionppthe interval type\-2 setsμ~p\\tilde\{\\mu\}\_\{p\}andν~p\\tilde\{\\nu\}\_\{p\}degenerate to crisp degreesμp\\mu\_\{p\}andνp\\nu\_\{p\}, then in both modes \(i\) and \(ii\) the induced evaluation of formulas in MFL\-T2 coincides with the type\-1 semantics of MFL\-T1\.

###### Proof\.

By structural induction on formulas\. In the base case, type\-reduction returns\(μ¯p,ν¯p\)=\(μp,νp\)\(\\bar\{\\mu\}\_\{p\},\\bar\{\\nu\}\_\{p\}\)=\(\\mu\_\{p\},\\nu\_\{p\}\)and the projected intervals collapse to singletons\. The inductive step follows because the connective clauses are the same as in MFL\-T1 \(mode \(i\)\) or are their monotone interval extensions \(mode \(ii\)\), which also collapse to singletons\. Hence, both modes coincide with the type\-1 evaluation\. ∎

##### Conservative decision rules from envelopes

The envelope mode supports conservative decision policies: for a decision thresholdτ\\tau, one may requireML​\(φ\)≥τM\_\{L\}\(\\varphi\)\\geq\\tauto assertφ\\varphidecisively and useMU​\(φ\)≥τM\_\{U\}\(\\varphi\)\\geq\\tauto flagφ\\varphias potentially true and request additional evidence\.

##### Axioms and proof system

The axiomatic system introduced in Section[4](https://arxiv.org/html/2605.22900#S4)is kept unchanged\. In MFL\-T2, we only enrich the semantic domain by allowing interval type\-2 uncertainty in the truth and falsity degrees assigned to atomic propositions; syntactic derivability is defined exactly as in type\-1 logic\.

## 6Type\-3 Granular Mediative Fuzzy Logic \(MFL\-T3\)

Type\-3 Granular Mediative Fuzzy Logic \(MFL\-T3\), originally proposed in\[[11](https://arxiv.org/html/2605.22900#bib.bib37)\], extends mediative reasoning to higher\-order uncertainty by combining it with type\-3 fuzzy systems\. The goal is to represent knowledge from multiple experts and to capture variability arising from heterogeneous evidence streams \(e\.g\., distributed sensing\)\. Here we provide a granular, multi\-level formalization in which mediative truth is organized into families indexed by*granules*\(e\.g\., information sources, evidence modalities, time slices, sensor channels, or expert groups\)\[[30](https://arxiv.org/html/2605.22900#bib.bib62),[34](https://arxiv.org/html/2605.22900#bib.bib63)\]\.

In MFL\-T3, a truth value is not treated as a single mediative pair \(type\-1\) nor as a single footprint of uncertainty \(type\-2\), but as a structured family of local evaluations indexed by granules\. This representation captures heterogeneity across sources and contexts and supports principled aggregation of potentially conflicting evidence within the mediative semantics\.

LetGGbe a finite nonempty set of granules \(e\.g\., expert, sensor, and time triples\)\. For each granuleg∈Gg\\in Gwe consider a*local*mediative valuationvgv\_\{g\}assigning to each atomic propositionppeither a type\-1 or a type\-2 mediative truth value:

- •in the type\-1 case, vg​\(p\)=\(μp,g,νp,g\)∈\[0,1\]2;v\_\{g\}\(p\)=\(\\mu\_\{p,g\},\\nu\_\{p,g\}\)\\in\[0,1\]^\{2\};
- •in the type\-2 case, vg​\(p\)=\(μ~p,g,ν~p,g\),v\_\{g\}\(p\)=\(\\tilde\{\\mu\}\_\{p,g\},\\tilde\{\\nu\}\_\{p,g\}\),whereμ~p,g\\tilde\{\\mu\}\_\{p,g\}andν~p,g\\tilde\{\\nu\}\_\{p,g\}are interval type\-2 fuzzy sets on\[0,1\]\[0,1\]with footprints of uncertaintyFOU​\(μ~p,g\)\\mathrm\{FOU\}\(\\tilde\{\\mu\}\_\{p,g\}\)andFOU​\(ν~p,g\)\\mathrm\{FOU\}\(\\tilde\{\\nu\}\_\{p,g\}\), as in Section[5](https://arxiv.org/html/2605.22900#S5)\.

A type\-3 mediative truth assignment forppis the indexed family

v\(3\)​\(p\):=\(vg​\(p\)\)g∈G\.v^\{\(3\)\}\(p\):=\(v\_\{g\}\(p\)\)\_\{g\\in G\}\.More generally, a type\-3 valuation for formulas is

v\(3\)​\(φ\):=\(vg​\(φ\)\)g∈G,v^\{\(3\)\}\(\\varphi\):=\(v\_\{g\}\(\\varphi\)\)\_\{g\\in G\},where eachvgv\_\{g\}extends to compound formulas by the same compositional clauses as in MFL\-T1 and MFL\-T2\. In particular, all connectives \(conjunction, disjunction, negation, implication, and the mediative connective\) are evaluated*locally*at each granule, yielding a family of local truth values prior to any cross\-granule aggregation\.

Here “type\-3” is used in the sense of a higher\-level granular family of mediative evaluations, i\.e\., a granule\-indexed valuation, without committing to any particular formalization of type\-3 fuzzy sets beyond what is needed for the present semantics\.

###### Definition 6\.1\(Local and granular mediative evaluations\)\.

Fix a finite set of granulesGG\. For each granuleg∈Gg\\in Gand formulaφ\\varphi, define a*local*scalar mediative degreeMg​\(φ\)M\_\{g\}\(\\varphi\)as follows:

- •*Type\-1 local evaluation\.*Ifvg​\(φ\)=\(μφ,g,νφ,g\)v\_\{g\}\(\\varphi\)=\(\\mu\_\{\\varphi,g\},\\nu\_\{\\varphi,g\}\), set Mg​\(φ\):=M​\(μφ,g,νφ,g\)\.M\_\{g\}\(\\varphi\):=M\(\\mu\_\{\\varphi,g\},\\nu\_\{\\varphi,g\}\)\.
- •*Type\-2 local evaluation \(via type\-reduction\)\.*Ifvg​\(φ\)=\(μ~φ,g,ν~φ,g\)v\_\{g\}\(\\varphi\)=\(\\tilde\{\\mu\}\_\{\\varphi,g\},\\tilde\{\\nu\}\_\{\\varphi,g\}\), obtain type\-reduced intervals TR​\(μ~φ,g\)=\[μφ,gl,μφ,gr\],TR​\(ν~φ,g\)=\[νφ,gl,νφ,gr\],\\mathrm\{TR\}\(\\tilde\{\\mu\}\_\{\\varphi,g\}\)=\[\\mu\_\{\\varphi,g\}^\{l\},\\mu\_\{\\varphi,g\}^\{r\}\],\\qquad\\mathrm\{TR\}\(\\tilde\{\\nu\}\_\{\\varphi,g\}\)=\[\\nu\_\{\\varphi,g\}^\{l\},\\nu\_\{\\varphi,g\}^\{r\}\],via a centroid type\-reduction method \(e\.g\., Karnik–Mendel procedures\[[21](https://arxiv.org/html/2605.22900#bib.bib53),[20](https://arxiv.org/html/2605.22900#bib.bib56),[25](https://arxiv.org/html/2605.22900#bib.bib55)\]\)\. Choose a crisp representative of each interval; for definiteness, we use the midpoint μ¯φ,g:=μφ,gl\+μφ,gr2,ν¯φ,g:=νφ,gl\+νφ,gr2,\\bar\{\\mu\}\_\{\\varphi,g\}:=\\frac\{\\mu\_\{\\varphi,g\}^\{l\}\+\\mu\_\{\\varphi,g\}^\{r\}\}\{2\},\\qquad\\bar\{\\nu\}\_\{\\varphi,g\}:=\\frac\{\\nu\_\{\\varphi,g\}^\{l\}\+\\nu\_\{\\varphi,g\}^\{r\}\}\{2\},and set Mg​\(φ\):=M​\(μ¯φ,g,ν¯φ,g\)\.M\_\{g\}\(\\varphi\):=M\(\\bar\{\\mu\}\_\{\\varphi,g\},\\bar\{\\nu\}\_\{\\varphi,g\}\)\.

If one prefers to preserve uncertainty at the local level, one may instead propagate bounds and report an intervalMg​\(φ\)∈\[Mg,L​\(φ\),Mg,U​\(φ\)\]M\_\{g\}\(\\varphi\)\\in\[M\_\{g,L\}\(\\varphi\),M\_\{g,U\}\(\\varphi\)\], as in Definition[5\.6](https://arxiv.org/html/2605.22900#S5.Thmdefinition6)\.

A*granular aggregation operator*forφ\\varphiis a mapping

Aφ:\[0,1\]G→\[0,1\],\(Mg​\(φ\)\)g∈G⟼MG​\(φ\),A\_\{\\varphi\}:\[0,1\]^\{G\}\\to\[0,1\],\\qquad\(M\_\{g\}\(\\varphi\)\)\_\{g\\in G\}\\longmapsto M\_\{G\}\(\\varphi\),and the resulting group\-level \(global\) mediative degree is

MG​\(φ\):=Aφ​\(\(Mg​\(φ\)\)g∈G\)\.M\_\{G\}\(\\varphi\)\\;:=\\;A\_\{\\varphi\}\\bigl\(\(M\_\{g\}\(\\varphi\)\)\_\{g\\in G\}\\bigr\)\.Typical choices forAφA\_\{\\varphi\}include \(i\) weighted averages with weights reflecting the reliability or relevance of each granule; \(ii\) ordered weighted averaging \(OWA\) operators; and \(iii\) hierarchical combinations respecting predefined groupings \(e\.g\., expert groups, sensor classes, or time windows\)\.

The granular setting becomes meaningful precisely when the family\(Mg​\(φ\)\)g∈G\(M\_\{g\}\(\\varphi\)\)\_\{g\\in G\}is heterogeneous, since thenAφA\_\{\\varphi\}can encode domain policies \(e\.g\., robustness to outliers or priority of trusted sources\) that are not captured by a single unindexed mediative evaluation\.

###### Theorem 6\.1\(Consistency under homogeneous granules and reduction to lower types\)\.

Assume that for each formulaφ\\varphi, the aggregation operatorAφA\_\{\\varphi\}is idempotent, i\.e\.,

Aφ​\(\(c\)g∈G\)=cfor every constant family​\(c\)g∈G\.A\_\{\\varphi\}\\bigl\(\(c\)\_\{g\\in G\}\\bigr\)=c\\quad\\text\{for every constant family \}\(c\)\_\{g\\in G\}\.If all granules assign the same local mediative degreeMg​\(φ\)=cM\_\{g\}\(\\varphi\)=ctoφ\\varphi, then the global mediative degree satisfiesMG​\(φ\)=cM\_\{G\}\(\\varphi\)=c\.

Moreover, fix an atomic propositionppand assume that the local truth values forppare identical across granules, i\.e\.,vg​\(p\)=vg′​\(p\)v\_\{g\}\(p\)=v\_\{g^\{\\prime\}\}\(p\)for allg,g′∈Gg,g^\{\\prime\}\\in G\. Then, for every formulaφ\\varphiin the sublanguage generated by\{p\}\\\{p\\\}, the type\-3 assignmentv\(3\)​\(φ\)v^\{\(3\)\}\(\\varphi\)reduces, up to this common value, to a lower\-type assignment: definev\(2\)​\(p\):=vg​\(p\)v^\{\(2\)\}\(p\):=v\_\{g\}\(p\)\(for anyg∈Gg\\in G\) when the common value is type\-2, and identifyv\(1\)​\(p\):=vg​\(p\)v^\{\(1\)\}\(p\):=v\_\{g\}\(p\)when the common value is type\-1\. In this homogeneous case, MFL\-T3 reduces to MFL\-T2 \(or further to MFL\-T1\)\.

###### Proof\.

IfMg​\(φ\)=cM\_\{g\}\(\\varphi\)=cfor allg∈Gg\\in G, then\(Mg​\(φ\)\)g∈G=\(c\)g∈G\(M\_\{g\}\(\\varphi\)\)\_\{g\\in G\}=\(c\)\_\{g\\in G\}and, by idempotence,MG​\(φ\)=Aφ​\(\(c\)g∈G\)=cM\_\{G\}\(\\varphi\)=A\_\{\\varphi\}\(\(c\)\_\{g\\in G\}\)=c\.

For the reduction claim, assume thatvg​\(p\)=vg′​\(p\)v\_\{g\}\(p\)=v\_\{g^\{\\prime\}\}\(p\)for allg,g′∈Gg,g^\{\\prime\}\\in G\. Since each local valuationvgv\_\{g\}extends to compound formulas by the same compositional clauses as in MFL\-T1/MFL\-T2, it follows by structural induction thatvg​\(φ\)=vg′​\(φ\)v\_\{g\}\(\\varphi\)=v\_\{g^\{\\prime\}\}\(\\varphi\)for every formulaφ\\varphibuilt from the single atompp\. HenceMg​\(φ\)M\_\{g\}\(\\varphi\)is constant ingg, and the first part yieldsMG​\(φ\)=Mg​\(φ\)M\_\{G\}\(\\varphi\)=M\_\{g\}\(\\varphi\)\.

Identifying the common local value as a lower\-type assignment \(type\-2 or type\-1, depending on the case\), therefore, yields a semantics that is indistinguishable from the type\-3 one in homogeneous situations\. ∎

MFL\-T3 is particularly suitable in domains where evidence is both heterogeneous and dynamically evolving, such as longitudinal medical diagnosis involving multiple specialists, smart\-grid monitoring with diverse sensing infrastructures, or ensembles of machine\-learning models deployed alongside human oversight\.

##### Axioms and proof system

The axiomatic system introduced in Section[4](https://arxiv.org/html/2605.22900#S4)is kept unchanged\. In MFL\-T3 we enrich only the semantic domain by indexing valuations over a granular setGGand by adding an explicit granular aggregation stage; syntactic derivability is defined exactly as in the type\-1 logic\.

## 7Quantum Mediative Fuzzy Logic \(QMFL\)

Quantum Mediative Fuzzy Logic \(QMFL\) extends mediative truth values to a quantum setting in which evidence is represented by quantum states and effects on a Hilbert space\. Algebraically, the framework fits naturally within effect\-algebraic semantics in quantum theory\[[15](https://arxiv.org/html/2605.22900#bib.bib21)\]\. From a granular viewpoint, QMFL can also be seen as an instance of effect\-based granular computing: quantum effects serve as granules, and their combinations implement mediative aggregation\.

Letℋ\\mathcal\{H\}be a finite\-dimensional Hilbert space\. The central semantic objects are quantum effects, following the effect\-based granulation viewpoint\[[36](https://arxiv.org/html/2605.22900#bib.bib52)\]\.

A \(quantum\)*effect*onℋ\\mathcal\{H\}is a self\-adjoint operatorEEsuch that

0⪯E⪯I,0\\preceq E\\preceq I,\(6\)where⪯\\preceqdenotes the Löwner \(positive semidefinite\) order\. A*quantum state*is a density operatorρ\\rhoonℋ\\mathcal\{H\}, i\.e\.,ρ⪰0\\rho\\succeq 0andTr⁡\(ρ\)=1\\operatorname\{Tr\}\(\\rho\)=1\. The Born expectationTr⁡\(ρ​E\)\\operatorname\{Tr\}\(\\rho E\)is interpreted as a graded degree associated with the effectEE\.

###### Definition 7\.1\(Quantum mediative structure\)\.

For each propositionpp, consider two effectsEp\+E\_\{p\}^\{\+\}andEp−E\_\{p\}^\{\-\}onℋ\\mathcal\{H\}, representing a*positive channel*\(evidence in favour ofpp\) and a*negative channel*\(evidence in favour of¬p\\neg p\), respectively\. For a quantum stateρ\\rho, define

μp​\(ρ\):=Tr⁡\(ρ​Ep\+\),νp​\(ρ\):=Tr⁡\(ρ​Ep−\),\\mu\_\{p\}\(\\rho\):=\\operatorname\{Tr\}\\\!\\bigl\(\\rho E\_\{p\}^\{\+\}\\bigr\),\\qquad\\nu\_\{p\}\(\\rho\):=\\operatorname\{Tr\}\\\!\\bigl\(\\rho E\_\{p\}^\{\-\}\\bigr\),and set the associated hesitation and contradiction degrees as

πp​\(ρ\):=max⁡\{0,1−μp​\(ρ\)−νp​\(ρ\)\},ζp​\(ρ\):=max⁡\{0,μp​\(ρ\)\+νp​\(ρ\)−1\}\.\\pi\_\{p\}\(\\rho\):=\\max\\\{0,\\,1\-\\mu\_\{p\}\(\\rho\)\-\\nu\_\{p\}\(\\rho\)\\\},\\qquad\\zeta\_\{p\}\(\\rho\):=\\max\\\{0,\\,\\mu\_\{p\}\(\\rho\)\+\\nu\_\{p\}\(\\rho\)\-1\\\}\.The pair\(μp​\(ρ\),νp​\(ρ\)\)\(\\mu\_\{p\}\(\\rho\),\\nu\_\{p\}\(\\rho\)\)is the \(type\-1\) mediative truth value ofppin the quantum stateρ\\rho\.

The effectsEp\+E\_\{p\}^\{\+\}andEp−E\_\{p\}^\{\-\}are not required to be complementary \(e\.g\.,Ep−≠I−Ep\+E\_\{p\}^\{\-\}\\neq I\-E\_\{p\}^\{\+\}\)\. They encode two independent channels of graded evidence; accordingly, semantic contradiction is admissible\.

###### Definition 7\.2\(Quantum mediative effect and degree\)\.

Define the mediative weights

w1,p​\(ρ\):=1−πp​\(ρ\)−ζp​\(ρ\)2,w2,p​\(ρ\):=πp​\(ρ\)\+ζp​\(ρ\)2,w\_\{1,p\}\(\\rho\):=1\-\\pi\_\{p\}\(\\rho\)\-\\frac\{\\zeta\_\{p\}\(\\rho\)\}\{2\},\\qquad w\_\{2,p\}\(\\rho\):=\\pi\_\{p\}\(\\rho\)\+\\frac\{\\zeta\_\{p\}\(\\rho\)\}\{2\},\(7\)wherew1,p​\(ρ\),w2,p​\(ρ\)∈\[0,1\]w\_\{1,p\}\(\\rho\),w\_\{2,p\}\(\\rho\)\\in\[0,1\]andw1,p​\(ρ\)\+w2,p​\(ρ\)=1w\_\{1,p\}\(\\rho\)\+w\_\{2,p\}\(\\rho\)=1\.

The*quantum mediative effect*associated withppin the stateρ\\rhois

Mp​\(ρ\):=w1,p​\(ρ\)​Ep\+\+w2,p​\(ρ\)​\(I−Ep−\)\.M\_\{p\}\(\\rho\):=w\_\{1,p\}\(\\rho\)\\,E\_\{p\}^\{\+\}\+w\_\{2,p\}\(\\rho\)\\,\\bigl\(I\-E\_\{p\}^\{\-\}\\bigr\)\.\(8\)
The corresponding*quantum mediative degree*ofppin the stateρ\\rhois

Mq​\(p,ρ\):=Tr⁡\(ρ​Mp​\(ρ\)\)\.M\_\{q\}\(p,\\rho\):=\\operatorname\{Tr\}\\\!\\bigl\(\\rho\\,M\_\{p\}\(\\rho\)\\bigr\)\.\(9\)

The operatorMp​\(ρ\)M\_\{p\}\(\\rho\)depends onρ\\rhothrough the weightsw1,p​\(ρ\)w\_\{1,p\}\(\\rho\)andw2,p​\(ρ\)w\_\{2,p\}\(\\rho\)\. Operationally, this can be read as an adaptive, evidence\-conditioned effect constructed from the current informational stateρ\\rho; the scalarMq​\(p,ρ\)M\_\{q\}\(p,\\rho\)in \([9](https://arxiv.org/html/2605.22900#S7.E9)\) is the corresponding Born expectation and yields a graded mediative score in\[0,1\]\[0,1\]\.

###### Theorem 7\.1\(Effect\-algebra compatibility\)\.

For every propositionppand every quantum stateρ\\rho, the operatorMp​\(ρ\)M\_\{p\}\(\\rho\)is an effect, i\.e\., it satisfies \([6](https://arxiv.org/html/2605.22900#S7.E6)\) withE=Mp​\(ρ\)E=M\_\{p\}\(\\rho\)\.

###### Proof\.

SinceEp\+E\_\{p\}^\{\+\}andEp−E\_\{p\}^\{\-\}are effects, we have0⪯Ep\+⪯I0\\preceq E\_\{p\}^\{\+\}\\preceq Iand0⪯Ep−⪯I0\\preceq E\_\{p\}^\{\-\}\\preceq I\. Hence0⪯I−Ep−⪯I0\\preceq I\-E\_\{p\}^\{\-\}\\preceq I, so bothEp\+E\_\{p\}^\{\+\}andI−Ep−I\-E\_\{p\}^\{\-\}are effects\. Moreover,w1,p​\(ρ\),w2,p​\(ρ\)∈\[0,1\]w\_\{1,p\}\(\\rho\),w\_\{2,p\}\(\\rho\)\\in\[0,1\]andw1,p​\(ρ\)\+w2,p​\(ρ\)=1w\_\{1,p\}\(\\rho\)\+w\_\{2,p\}\(\\rho\)=1, soMp​\(ρ\)M\_\{p\}\(\\rho\)is a convex combination of two effects \(cf\. \([8](https://arxiv.org/html/2605.22900#S7.E8)\)\)\. Therefore0⪯Mp​\(ρ\)⪯I0\\preceq M\_\{p\}\(\\rho\)\\preceq I\. ∎

###### Theorem 7\.2\(Consistency with the classical mediative evaluation\)\.

For every propositionppand every quantum stateρ\\rho,

Mq​\(p,ρ\)=M​\(μp​\(ρ\),νp​\(ρ\)\),M\_\{q\}\(p,\\rho\)\\;=\\;M\\\!\\bigl\(\\mu\_\{p\}\(\\rho\),\\nu\_\{p\}\(\\rho\)\\bigr\),whereM​\(μ,ν\)M\(\\mu,\\nu\)denotes the type\-1 mediative evaluation defined by the mediative operator\.

###### Proof\.

By Definition[7\.2](https://arxiv.org/html/2605.22900#S7.Thmdefinition2),

Mq​\(p,ρ\)=Tr⁡\(ρ​Mp​\(ρ\)\)=Tr⁡\(ρ​\[w1,p​\(ρ\)​Ep\+\+w2,p​\(ρ\)​\(I−Ep−\)\]\)\.M\_\{q\}\(p,\\rho\)=\\operatorname\{Tr\}\\\!\\bigl\(\\rho\\,M\_\{p\}\(\\rho\)\\bigr\)=\\operatorname\{Tr\}\\\!\\Bigl\(\\rho\\bigl\[w\_\{1,p\}\(\\rho\)E\_\{p\}^\{\+\}\+w\_\{2,p\}\(\\rho\)\(I\-E\_\{p\}^\{\-\}\)\\bigr\]\\Bigr\)\.By linearity of the trace andTr⁡\(ρ\)=1\\operatorname\{Tr\}\(\\rho\)=1,

Mq​\(p,ρ\)=w1,p​\(ρ\)​Tr⁡\(ρ​Ep\+\)\+w2,p​\(ρ\)​\(1−Tr⁡\(ρ​Ep−\)\)=w1,p​\(ρ\)​μp​\(ρ\)\+w2,p​\(ρ\)​\(1−νp​\(ρ\)\)\.M\_\{q\}\(p,\\rho\)=w\_\{1,p\}\(\\rho\)\\operatorname\{Tr\}\\\!\\bigl\(\\rho E\_\{p\}^\{\+\}\\bigr\)\+w\_\{2,p\}\(\\rho\)\\bigl\(1\-\\operatorname\{Tr\}\(\\rho E\_\{p\}^\{\-\}\)\\bigr\)=w\_\{1,p\}\(\\rho\)\\,\\mu\_\{p\}\(\\rho\)\+w\_\{2,p\}\(\\rho\)\\,\\bigl\(1\-\\nu\_\{p\}\(\\rho\)\\bigr\)\.By definition of the type\-1 mediative evaluation, the right\-hand side is exactlyM​\(μ,ν\)M\(\\mu,\\nu\)evaluated atμ=μp​\(ρ\)\\mu=\\mu\_\{p\}\(\\rho\)andν=νp​\(ρ\)\\nu=\\nu\_\{p\}\(\\rho\)\. ∎

##### Classical reduction and absence of coherences

Ifρ\\rho,Ep\+E\_\{p\}^\{\+\}andEp−E\_\{p\}^\{\-\}commute \(hence are simultaneously diagonalizable\), thenμp​\(ρ\)\\mu\_\{p\}\(\\rho\)andνp​\(ρ\)\\nu\_\{p\}\(\\rho\)depend only on the diagonal entries in a common eigenbasis\. In this sense, QMFL reduces operationally to the classical mediative semantics when quantum coherences play no operational role\.

##### Minimal qubit\-level instantiation

A simple instantiation consistent with Definitions[7\.1](https://arxiv.org/html/2605.22900#S7.Thmdefinition1)and[7\.2](https://arxiv.org/html/2605.22900#S7.Thmdefinition2)can be given onℋ=ℂ2\\mathcal\{H\}=\\mathbb\{C\}^\{2\}by fixingρ=\|0⟩​⟨0\|\\rho=\|0\\rangle\\langle 0\|and choosing diagonal effects

Ep\+=\(μ000\),Ep−=\(ν000\),μ,ν∈\[0,1\]\.E\_\{p\}^\{\+\}=\\begin\{pmatrix\}\\mu&0\\\\ 0&0\\end\{pmatrix\},\\qquad E\_\{p\}^\{\-\}=\\begin\{pmatrix\}\\nu&0\\\\ 0&0\\end\{pmatrix\},\\qquad\\mu,\\nu\\in\[0,1\]\.Thenμp​\(ρ\)=μ\\mu\_\{p\}\(\\rho\)=\\muandνp​\(ρ\)=ν\\nu\_\{p\}\(\\rho\)=\\nu, and Theorem[7\.2](https://arxiv.org/html/2605.22900#S7.Thmtheorem2)yields

Mq​\(p,ρ\)=M​\(μ,ν\)\.M\_\{q\}\(p,\\rho\)=M\(\\mu,\\nu\)\.A concrete numerical instantiation for the obstacle detection case study is given in Section[8](https://arxiv.org/html/2605.22900#S8)\.

##### Finite\-shot estimation and safety margins

In an actual quantum implementation,Mq​\(p,ρ\)=Tr⁡\(ρ​Mp​\(ρ\)\)M\_\{q\}\(p,\\rho\)=\\operatorname\{Tr\}\(\\rho\\,M\_\{p\}\(\\rho\)\)must be estimated from a finite number of measurement shots\. LetX1,…,XN∈\[0,1\]X\_\{1\},\\dots,X\_\{N\}\\in\[0,1\]denote the observed outcomes of an estimation procedure forTr⁡\(ρ​Mp​\(ρ\)\)\\operatorname\{Tr\}\(\\rho\\,M\_\{p\}\(\\rho\)\), and define

M^q:=1N​∑k=1NXk\.\\widehat\{M\}\_\{q\}:=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}X\_\{k\}\.Standard concentration bounds \(e\.g\., Hoeffding’s inequality\) imply thatM^q\\widehat\{M\}\_\{q\}concentrates aroundMq​\(p,ρ\)M\_\{q\}\(p,\\rho\)asNNincreases\. Consequently, in safety\-critical settings, decision rules based onM^q\\widehat\{M\}\_\{q\}should incorporate an explicit uncertainty margin so that finite\-shot fluctuations cannot systematically overturn conservative actions in the presence of strong \(possibly conflicting\) evidence\.

## 8Illustrative example: safety\-first mediative sensor fusion in obstacle detection

We present a case study illustrating the safety\-first interpretation of mediative truth degrees in a domain beyond the original control applications of MFL\. We consider a simple obstacle\-detection scenario in autonomous driving\. This example shows that, in a conservative regime with low higher\-order uncertainty, all mediative variants yield the same control decision, while still allowing a structured path to richer behaviours when uncertainty, granularity, or quantum effects matter\. Throughout, we use a strict safety\-first design: decision thresholds and aggregation weights are chosen so that strong evidence of danger—even if conflicting—favours braking or deceleration over proceeding as if no obstacle were present\.

Letppbe the proposition

p:“There is a dangerous obstacle within 20 meters in front of the vehicle\.”p:\\text\{\`\`There is a dangerous obstacle within 20 meters in front of the vehicle\.''\}Assume that two independent perception channels are available: a radar/LiDAR channel, which is robust under bad weather but has limited resolution, and a camera\-based detector, which is more precise under good visibility but sensitive to glare and occlusion\. Each channel provides an assessment ofppas a mediative truth value\(μ,ν\)\(\\mu,\\nu\), whereμ\\muis the degree of support forppandν\\nuis the degree of support for¬p\\neg p\.

### 8\.1Safety\-first aggregation and decision thresholds

Heterogeneous, uncertain,possibly conflictingevidenceCameraRadar / LiDARWeathersensorHumanexperts↑\\uparrowsafe↑\\uparrowdanger↑\\uparrowsafe↑\\uparrowdanger↑\\uparrowsafe↑\\uparrowdanger↑\\uparrowsafe↑\\uparrowdangerMediative encoding ofevidence intomediative truth–falsitypairs\(μ,ν\)\(\\mu,\\nu\)Legend \(for propositionpp\):↑\\uparrowdanger = degreeμ\\mu\(support forpp\),↑\\uparrowsafe = degreeν\\nu\(support for¬p\\neg p\); equivalentlyb=1−νb=1\-\\numeasureslack of support for safety\.Type\-1 Mediative Fuzzy Logic \(MFL\-T1\)Mediative operatorM​\(μ,ν\)M\(\\mu,\\nu\)Type\-2 Mediative Fuzzy Logic \(MFL\-T2\)Footprints of uncertainty forμ\\muandν\\nuType\-3 granular MFL \(MFL\-T3\)Granular familyv\(3\)​\(p\)v^\{\(3\)\}\(p\)over expert–sensor–time granulesQuantum Mediative Fuzzy Logic \(QMFL\)Hilbert\-space implementation ofmediative operatorMMwith quantum interferenceMFL variantsAlternative MFL\-based reasoning schemes\(one active at a time\)EmergencybrakeCautiousslow\-downProceedOutput degreeM​\(μ,ν\)M\(\\mu,\\nu\)orMqM\_\{q\}compared with safety\-biasedthresholds to triggerBrake / Slow\-down / Go

Figure 1:Instantiation of the mediative pipeline in the autonomous braking case study\.Figure[1](https://arxiv.org/html/2605.22900#S8.F1)summarizes how the proposed mediative pipeline is instantiated in this case study\. The left panel shows heterogeneous sources of evidence and their mediative encoding into truth–falsity pairs\(μ,ν\)\(\\mu,\\nu\)\. The central panel highlights the four alternative Mediative Fuzzy Logic variants \(MFL\-T1, MFL\-T2, MFL\-T3, and QMFL\), of which only one is active at a time\. The right panel depicts the safety\-first decision module, where the mediative degreeM​\(μ,ν\)M\(\\mu,\\nu\)orMqM\_\{q\}is compared against safety\-biased thresholds to trigger Emergency brake, Cautious slow\-down, or Proceed\.

Although the pipeline in Fig\.[1](https://arxiv.org/html/2605.22900#S8.F1)includes four potential evidence sources \(camera, radar/LiDAR, weather sensor, and human experts\), in the example below we use only two channels \(camera and radar/LiDAR\) to keep the notation and plots readable\. The mediative encoding and aggregation extend directly to additional sources using the same scheme\.

To maintain a strict safety\-first approach, we specify \(i\) how the two channels are aggregated and \(ii\) how the resulting mediative value maps to control actions\. We use a weighted aggregation of radar and camera assessments:

\(μ,ν\)=\(α​μradar\+\(1−α\)​μcam,α​νradar\+\(1−α\)​νcam\),\(\\mu,\\nu\)=\\bigl\(\\,\\alpha\\mu\_\{\\mathrm\{radar\}\}\+\(1\-\\alpha\)\\mu\_\{\\mathrm\{cam\}\},\\;\\alpha\\nu\_\{\\mathrm\{radar\}\}\+\(1\-\\alpha\)\\nu\_\{\\mathrm\{cam\}\}\\,\\bigr\),\(10\)whereα∈\[0,1\]\\alpha\\in\[0,1\]reflects the relative trust in the radar channel under the current context \(weather, visibility, road type, etc\.\)\. In poor visibility conditions, we takeα\\alphacloser to11, favoring radar; in more balanced conditions, we may takeα=0\.5\\alpha=0\.5\.

On top of the mediative evaluationM​\(μ,ν\)M\(\\mu,\\nu\), we adopt a simple decision policy based on a braking thresholdTbrake=0\.7T\_\{\\mathrm\{brake\}\}=0\.7:

- •ifM​\(μ,ν\)≥0\.7M\(\\mu,\\nu\)\\geq 0\.7: the vehicle should brake decisively \(emergency or strong braking\),
- •if0\.5≤M​\(μ,ν\)<0\.70\.5\\leq M\(\\mu,\\nu\)<0\.7: the vehicle should decelerate and seek additional sensing,
- •ifM​\(μ,ν\)<0\.5M\(\\mu,\\nu\)<0\.5: the vehicle may continue cautiously while monitoring for changes\.

The numerical examples below are chosen so that even mild contradictions are resolved in favour of safety when the radar channel strongly suggests the presence of an obstacle\.

##### Three evidence configurations

We consider three representative configurations\. In Cases 1 and 3, we assume poor visibility and setα=0\.7\\alpha=0\.7to prioritize the radar/LiDAR channel\. In Case 2, we model a highly ambiguous situation \(e\.g\., night glare\) and setα=0\.5\\alpha=0\.5to reflect a deliberately balanced fusion between radar/LiDAR and camera\. Table[1](https://arxiv.org/html/2605.22900#S8.T1)reports the radar/LiDAR and camera assessments, the aggregated mediative pair\(μ,ν\)\(\\mu,\\nu\)obtained from \([10](https://arxiv.org/html/2605.22900#S8.E10)\), the resulting hesitationπ\\pi, contradictionζ\\zeta, the mediative degreeM​\(μ,ν\)M\(\\mu,\\nu\), and the corresponding safety\-first control action\.

Table 1:Three illustrative evidence configurations and their aggregated mediative scores \(Cases 1 and 3 useα=0\.7\\alpha=0\.7, Case 2 usesα=0\.5\\alpha=0\.5\)\.Under the safety\-first thresholds adopted above, configurations 1 and 3 trigger decisive braking \(M​\(μ,ν\)≥0\.7M\(\\mu,\\nu\)\\geq 0\.7\), while configuration 2 triggers cautious slow\-down \(0\.5≤M​\(μ,ν\)<0\.70\.5\\leq M\(\\mu,\\nu\)<0\.7\)\. Note that configuration 3 exhibits explicit contradiction \(ζ\(3\)\>0\\zeta^\{\(3\)\}\>0\), modelling the case where one channel provides strong counter\-evidence while the other still suggests danger\.

### 8\.2Case 1: incomplete but non\-contradictory information

Suppose that under light fog, the radar detects a strong reflection consistent with an obstacle, while the camera has low confidence because the image is partially blurred\. We model this as

\(μradar,νradar\)=\(0\.8,0\.1\),\(μcam,νcam\)=\(0\.4,0\.2\)\.\(\\mu\_\{\\mathrm\{radar\}\},\\nu\_\{\\mathrm\{radar\}\}\)=\(0\.8,0\.1\),\\qquad\(\\mu\_\{\\mathrm\{cam\}\},\\nu\_\{\\mathrm\{cam\}\}\)=\(0\.4,0\.2\)\.Since fog favors the radar channel, we chooseα=0\.7\\alpha=0\.7\. The aggregated mediative value forppbecomes

\(μ,ν\)=\(0\.7⋅0\.8\+0\.3⋅0\.4,0\.7⋅0\.1\+0\.3⋅0\.2\)=\(0\.68,0\.13\)\.\(\\mu,\\nu\)=\\bigl\(0\.7\\cdot 0\.8\+0\.3\\cdot 0\.4,\\;0\.7\\cdot 0\.1\+0\.3\\cdot 0\.2\\bigr\)=\(0\.68,0\.13\)\.Hereμ\+ν=0\.81<1\\mu\+\\nu=0\.81<1, so the situation is one of incomplete information with hesitation

π​\(μ,ν\)=1−μ−ν=0\.19,ζ​\(μ,ν\)=0\.\\pi\(\\mu,\\nu\)=1\-\\mu\-\\nu=0\.19,\\qquad\\zeta\(\\mu,\\nu\)=0\.Using the mediative evaluation

M​\(μ,ν\)=ℳ​\(a,b;π,ζ\),a=μ,b=1−ν,M\(\\mu,\\nu\)=\\mathcal\{M\}\(a,b;\\pi,\\zeta\),\\qquad a=\\mu,\\;b=1\-\\nu,and recalling that whenζ=0\\zeta=0the operator reduces to the intuitionistic combination, we obtain

a=0\.68,b=1−0\.13=0\.87,a=0\.68,\\qquad b=1\-0\.13=0\.87,M​\(μ,ν\)=\(1−π\)​a\+π​b≈0\.81⋅0\.68\+0\.19⋅0\.87≈0\.716\.M\(\\mu,\\nu\)=\(1\-\\pi\)a\+\\pi b\\approx 0\.81\\cdot 0\.68\+0\.19\\cdot 0\.87\\approx 0\.716\.Thus, despite the hesitation, the mediative truth degree ofppis above the braking thresholdTbrake=0\.7T\_\{\\mathrm\{brake\}\}=0\.7\. This aligns with a safety\-first design: under poor visibility, strong radar evidence for an obstacle is enough to trigger decisive braking, even if the camera is uncertain\.

### 8\.3Case 2: strong but symmetric conflict

Consider now a situation at night with glare: the radar still reports a strong reflection consistent with an obstacle, but the camera confidently “sees” an empty road ahead\. We model this as

\(μradar,νradar\)=\(0\.9,0\.1\),\(μcam,νcam\)=\(0\.1,0\.9\)\.\(\\mu\_\{\\mathrm\{radar\}\},\\nu\_\{\\mathrm\{radar\}\}\)=\(0\.9,0\.1\),\\qquad\(\\mu\_\{\\mathrm\{cam\}\},\\nu\_\{\\mathrm\{cam\}\}\)=\(0\.1,0\.9\)\.To reflect an unbiased fusion in this particularly ambiguous situation, we takeα=0\.5\\alpha=0\.5, obtaining

\(μ,ν\)=\(0\.5⋅0\.9\+0\.5⋅0\.1,0\.5⋅0\.1\+0\.5⋅0\.9\)=\(0\.5,0\.5\)\.\(\\mu,\\nu\)=\\bigl\(0\.5\\cdot 0\.9\+0\.5\\cdot 0\.1,\\;0\.5\\cdot 0\.1\+0\.5\\cdot 0\.9\\bigr\)=\(0\.5,0\.5\)\.In this caseμ\+ν=1\\mu\+\\nu=1, so both hesitation and contradiction vanish:

π​\(μ,ν\)=max⁡\(0,1−μ−ν\)=0,ζ​\(μ,ν\)=max⁡\(0,μ\+ν−1\)=0\.\\pi\(\\mu,\\nu\)=\\max\(0,1\-\\mu\-\\nu\)=0,\\qquad\\zeta\(\\mu,\\nu\)=\\max\(0,\\mu\+\\nu\-1\)=0\.Semantically, the available evidence is perfectly balanced betweenppand¬p\\neg p\. The mediative evaluation is then

a=μ=0\.5,b=1−ν=0\.5,a=\\mu=0\.5,\\qquad b=1\-\\nu=0\.5,M​\(μ,ν\)=ℳ​\(a,b;0,0\)=a=0\.5\.M\(\\mu,\\nu\)=\\mathcal\{M\}\(a,b;0,0\)=a=0\.5\.The resulting mediative truth degree is exactly intermediate\. According to the policy above, the vehicle should decelerate and seek additional sensing \(for example, by re\-scanning, adjusting exposure, or relying more on other sensors\), but an immediate emergency stop is not required\. This configuration shows how MFL can represent a genuinely undecided state without forcing full braking or full trust in one channel\.

### 8\.4Case 3: explicit overdetermined contradiction with safety\-first resolution

To obtain a genuinely contradictory configuration, suppose instead that the vehicle receives conservative estimates in which each channel strongly defends its own assessment\. For instance, the radar is almost certain that an obstacle is present, while the camera almost certainly indicates no obstacle due to a misleading reflection or occlusion:

\(μradar,νradar\)=\(0\.95,0\.05\),\(μcam,νcam\)=\(0\.2,0\.9\)\.\(\\mu\_\{\\mathrm\{radar\}\},\\nu\_\{\\mathrm\{radar\}\}\)=\(0\.95,0\.05\),\\qquad\(\\mu\_\{\\mathrm\{cam\}\},\\nu\_\{\\mathrm\{cam\}\}\)=\(0\.2,0\.9\)\.In this configuration, we again takeα=0\.7\\alpha=0\.7to reflect a design choice that, under severe conflict, favours the safer channel \(radar\) without completely discarding the camera\. The aggregated value is

\(μ,ν\)=\(0\.7⋅0\.95\+0\.3⋅0\.2,0\.7⋅0\.05\+0\.3⋅0\.9\)=\(0\.725,0\.305\)\.\(\\mu,\\nu\)=\\bigl\(0\.7\\cdot 0\.95\+0\.3\\cdot 0\.2,\\;0\.7\\cdot 0\.05\+0\.3\\cdot 0\.9\\bigr\)=\(0\.725,0\.305\)\.Nowμ\+ν≈1\.03\>1\\mu\+\\nu\\approx 1\.03\>1, so we have overdetermined, potentially contradictory information, with

π​\(μ,ν\)=0,ζ​\(μ,ν\)=μ\+ν−1≈0\.03\.\\pi\(\\mu,\\nu\)=0,\\qquad\\zeta\(\\mu,\\nu\)=\\mu\+\\nu\-1\\approx 0\.03\.The mediative evaluation becomes

a=μ=0\.725,b=1−ν=0\.695,a=\\mu=0\.725,\\qquad b=1\-\\nu=0\.695,w1=1−π−ζ/2≈0\.985,w2=π\+ζ/2≈0\.015,w\_\{1\}=1\-\\pi\-\\zeta/2\\approx 0\.985,\\qquad w\_\{2\}=\\pi\+\\zeta/2\\approx 0\.015,M​\(μ,ν\)=w1​a\+w2​b≈0\.985⋅0\.725\+0\.015⋅0\.695≈0\.724\.M\(\\mu,\\nu\)=w\_\{1\}a\+w\_\{2\}b\\approx 0\.985\\cdot 0\.725\+0\.015\\cdot 0\.695\\approx 0\.724\.Here, the mediative truth degree is slightly above0\.720\.72, comfortably above the braking thresholdTbrakeT\_\{\\mathrm\{brake\}\}\. Even with explicit contradiction \(ζ\>0\\zeta\>0\), the system resolves the conflict in favor of safety because the more trusted radar channel dominates\. This shows that even mild contradictions should pushM​\(μ,ν\)M\(\\mu,\\nu\)into a braking regime when at least one reliable sensor strongly supports a hazard\.

##### QMFL instantiation for the three evidence configurations

To connect QMFL with Table[1](https://arxiv.org/html/2605.22900#S8.T1), we fixℋ=ℂ2\\mathcal\{H\}=\\mathbb\{C\}^\{2\}andρ=\|0⟩​⟨0\|\\rho=\|0\\rangle\\langle 0\|\. For each configurationii, we encode the aggregated classical degrees\(μ\(i\),ν\(i\)\)\(\\mu^\{\(i\)\},\\nu^\{\(i\)\}\)as diagonal effects

Ep,i\+=\(μ\(i\)000\),Ep,i−=\(ν\(i\)000\)\.E\_\{p,i\}^\{\+\}=\\begin\{pmatrix\}\\mu^\{\(i\)\}&0\\\\ 0&0\\end\{pmatrix\},\\qquad E\_\{p,i\}^\{\-\}=\\begin\{pmatrix\}\\nu^\{\(i\)\}&0\\\\ 0&0\\end\{pmatrix\}\.\(11\)
LetMp,i​\(ρ\)M\_\{p,i\}\(\\rho\)be constructed from\(Ep,i\+,Ep,i−\)\(E\_\{p,i\}^\{\+\},E\_\{p,i\}^\{\-\}\)via Definition[7\.2](https://arxiv.org/html/2605.22900#S7.Thmdefinition2), and define the corresponding QMFL score asMq\(i\)​\(p\):=Tr⁡\(ρ​Mp,i​\(ρ\)\)M\_\{q\}^\{\(i\)\}\(p\):=\\operatorname\{Tr\}\\\!\\bigl\(\\rho\\,M\_\{p,i\}\(\\rho\)\\bigr\)\. Sinceμp​\(ρ\)=μ\(i\)\\mu\_\{p\}\(\\rho\)=\\mu^\{\(i\)\}andνp​\(ρ\)=ν\(i\)\\nu\_\{p\}\(\\rho\)=\\nu^\{\(i\)\}under this encoding, Theorem[7\.2](https://arxiv.org/html/2605.22900#S7.Thmtheorem2)yieldsMq\(i\)​\(p\)=M​\(μ\(i\),ν\(i\)\)M\_\{q\}^\{\(i\)\}\(p\)=M\(\\mu^\{\(i\)\},\\nu^\{\(i\)\}\)\. Therefore, in this conservative regime \(low higher\-order uncertainty and a minimal diagonal encoding\), QMFL reproduces the same numerical mediative scores as MFL\-T1 on the aggregated inputs, while the operational difference is thatMq\(i\)​\(p\)M\_\{q\}^\{\(i\)\}\(p\)would be estimated from finite\-shot measurements in a concrete quantum implementation\.

### 8\.5Comparison across MFL variants

The obstacle\-detection example can also be interpreted within MFL\-T2, MFL\-T3, and QMFL\. Under natural modelling assumptions \(interval type\-2 footprints centred around the type\-1 values, homogeneous granules for the two sensors, and diagonal quantum effects in a classical limit\), these variants induce the same scalar mediative degrees as type\-1 MFL in Cases 1–3 and therefore the same control actions\. This coincidence is a direct consequence of how the higher\-type semantics are instantiated in this simple scenario\.

For MFL\-T2, we use interval type\-2 footprints whose type\-reduction returns exactly the same crisp degrees\(μp,νp\)\(\\mu\_\{p\},\\nu\_\{p\}\)as in the type\-1 case, so the mediative operator sees the same input pair\. For MFL\-T3, all granules associated with a given proposition share the same local mediative truth value, and the granular aggregation operator is idempotent; hence, the aggregated value coincides with that common local one\. For QMFL, we restrict attention to diagonal effects and density matrices whose Born expectations reproduce the same degrees of truth and falsity as in the fuzzy semantics\. In other words, the case study is evaluated in a conservative fragment in which second\-order uncertainty, granular heterogeneity and quantum coherences are weak or absent, so the higher\-type and quantum variants collapse to the same scalar decision values as type\-1 MFL\. In more complex scenarios, where footprints of uncertainty \(FOUs\), granular structure, or non\-commuting effects play a substantive role, the four frameworks may diverge both semantically and numerically\. The corresponding scalar mediative evaluations for Cases 1–3, based on the aggregated inputs in Table[1](https://arxiv.org/html/2605.22900#S8.T1), are collected in Table[2](https://arxiv.org/html/2605.22900#S8.T2)\.

Table 2:Scalar mediative evaluations across MFL variants for the obstacle detection example\.Beyond the numerical analysis above, the obstacle\-detection scenario admits complementary readings in terms of MFL\-T2, MFL\-T3, and QMFL\. These perspectives highlight how second\-order uncertainty, granular structure, and quantum effects can be incorporated into mediative reasoning while preserving the safety\-first philosophy\.

### 8\.6Type\-2, type\-3 and quantum mediative perspectives on the example

The previous example was presented at the type\-1 level, with crisp mediative truth values\(μ,ν\)\(\\mu,\\nu\)for each fused assessment of the obstacle propositionpp\. We now outline how the same scenario can be treated in MFL\-T2, MFL\-T3, and QMFL, and which design considerations are specific to each extension\.

#### 8\.6\.1Type\-2 mediative extension \(MFL\-T2\)

In MFL\-T2, the mediative truth value ofppis no longer a single pair\(μ,ν\)\(\\mu,\\nu\)but a pair of type\-2 fuzzy sets\(μ~p,ν~p\)\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\)on\[0,1\]\[0,1\]\. Intuitively, this represents second\-order uncertainty about the degrees of agreement and non\-agreement, for instance, due to sensor noise, calibration errors, or changing environmental conditions\.

Under natural modelling assumptions \(interval type\-2 footprints centred around the type\-1 values\), the type\-2 extension yields the same scalar mediative degrees as type\-1 MFL in Cases 1–3 and therefore the same control actions\. Concretely, after aggregating the radar and camera outputs at the primary level, we wrap the crisp pair into an interval type\-2 pair\(μ~p,ν~p\)\(\\tilde\{\\mu\}\_\{p\},\\tilde\{\\nu\}\_\{p\}\), from which type\-2 hesitation and contradiction degrees are computed and fed to a type\-2 mediative operator\. The resulting mediative evaluationM~p\\tilde\{M\}\_\{p\}is itself a type\-2 fuzzy set \(or, after type\-reduction, an interval of plausible mediative values\)\.

A safety\-first design must then specify how to interpret this interval\. One conservative policy is to trigger decisive braking whenever the*lower*endpoint of the type\-reduced mediative interval exceeds the thresholdTbrakeT\_\{\\mathrm\{brake\}\}, and to trigger at least cautious deceleration whenever the*upper*endpoint exceedsTbrakeT\_\{\\mathrm\{brake\}\}\. In this way, MFL\-T2 captures uncertainty about the mediative degree itself and allows the control strategy to distinguish between clearly safe, borderline and clearly dangerous configurations at a second\-order level\.

#### 8\.6\.2Granular view in MFL\-T3

In MFL\-T3, mediative truth is organized in a granular way over a family of indices\. We write

g:=\(sensor,time window,context\),g:=\(\\text\{sensor\},\\ \\text\{time window\},\\ \\text\{context\}\),\(12\)and letGGdenote the set of admissible granules \(i\.e\., all indices of the form \([12](https://arxiv.org/html/2605.22900#S8.E12)\)\), so that each granuleg∈Gg\\in Gcarries its own mediative truth valuevg​\(p\)=\(μg,νg\)v\_\{g\}\(p\)=\(\\mu\_\{g\},\\nu\_\{g\}\)and mediative evaluationMgM\_\{g\}\. In the obstacle\-detection example, we may take granules of the form

g=\(radar or camera,current frame or recent window,driving context\),g=\(\\text\{radar or camera\},\\;\\text\{current frame or recent window\},\\;\\text\{driving context\}\),so that short\-term history \(several consecutive frames\), multiple sensors \(radar, cameras, possibly LiDAR\) and context \(weather, road type\) are all represented at the granular level\.

The MFL\-T3 semantics then aggregates the family\{vg​\(p\):g∈G\}\\\{v\_\{g\}\(p\):g\\in G\\\}through explicit granular aggregation operators as defined in Section[6](https://arxiv.org/html/2605.22900#S6)\. For obstacle detection, a safety\-first design requires these operators to satisfy two conditions: \(i\) monotonicity with respect to each granule \(stronger local evidence for an obstacle cannot decrease the global mediative degree\), and \(ii\) a dominance property ensuring that if any sufficiently trusted granule strongly supports the presence of a dangerous obstacle, then the aggregated mediative evaluation crosses the braking threshold\. At the same time, the granular structure allows temporal smoothing and cross\-checking between sensors: for instance, a single spurious camera frame can be downweighted if both radar and neighbouring frames disagree with it\.

#### 8\.6\.3Quantum mediative perspective \(QMFL\)

In QMFL, the semantics of the obstacle propositionppis specified by a quantum stateρ\\rhotogether with two effectsEp\+E\_\{p\}^\{\+\}andEp−E\_\{p\}^\{\-\}on a Hilbert spaceℋ\\mathcal\{H\}, encoding, respectively, support forppand support for¬p\\neg p\(cf\. Section[7](https://arxiv.org/html/2605.22900#S7)\)\. The mediative degrees are then obtained as Born expectations

μp​\(ρ\)=Tr​\(ρ​Ep\+\),νp​\(ρ\)=Tr​\(ρ​Ep−\),\\mu\_\{p\}\(\\rho\)=\\mathrm\{Tr\}\(\\rho E\_\{p\}^\{\+\}\),\\qquad\\nu\_\{p\}\(\\rho\)=\\mathrm\{Tr\}\(\\rho E\_\{p\}^\{\-\}\),from which hesitation and contradiction are derived, and the quantum mediative degreeMq​\(p,ρ\)M\_\{q\}\(p,\\rho\)is computed via the quantum mediative effectMp​\(ρ\)M\_\{p\}\(\\rho\)defined in Definition[7\.2](https://arxiv.org/html/2605.22900#S7.Thmdefinition2)\.

For the obstacle\-detection task, a quantum instantiation may arise in two ways\. In a*simulation*setting, the pair\(μ,ν\)\(\\mu,\\nu\)obtained from classical MFL fusion is encoded into diagonal density operators and diagonal effects, so thatMq​\(p,ρ\)M\_\{q\}\(p,\\rho\)reproduces the classical mediative value while enabling quantum\-inspired processing \(e\.g\., superpositions of hypotheses or quantum walks on state graphs\)\. In a more ambitious*hardware*setting, part of the perception or decision pipeline is implemented on a quantum device, andρ\\rhocaptures both sensor information and intrinsic quantum noise\.

From a design perspective, the safety\-first requirement translates into constraints on the encoding map from sensor evidence to\(ρ,Ep\+,Ep−\)\(\\rho,E\_\{p\}^\{\+\},E\_\{p\}^\{\-\}\)and on the decision rule based onMq​\(p,ρ\)M\_\{q\}\(p,\\rho\)\. In particular, finite\-shot estimation and device noise should not systematically push the estimated quantum mediative degree below the braking threshold in situations where classical evidence already mandates braking\. This typically leads to conservative margins in the threshold and to robust encoding schemes forρ\\rhoand the associated effects\.

### 8\.7MFL\-T2, MFL\-T3 and QMFL outcomes for the three cases

The three numerical cases in Sections[8\.2](https://arxiv.org/html/2605.22900#S8.SS2)to[8\.4](https://arxiv.org/html/2605.22900#S8.SS4)were analyzed at the type\-1 level, producing crisp mediative valuesM​\(μ,ν\)M\(\\mu,\\nu\)and associated control decisions\. We now describe how the same three scenarios can be handled in MFL\-T2, MFL\-T3, and QMFL, and summarize the corresponding outcomes\.

#### Type\-2 mediative degrees for the three cases \(MFL\-T2\)

In MFL\-T2, a mediative truth value is a pair\(μ~,ν~\)\(\\tilde\{\\mu\},\\tilde\{\\nu\}\)of type\-2 fuzzy sets on\[0,1\]\[0,1\]\. For the present example, we adopt a simple interval type\-2 representation by wrapping each crisp pair\(μ,ν\)\(\\mu,\\nu\)in a small rectangle

μ∈\[μ−,μ\+\],ν∈\[ν−,ν\+\],\\mu\\in\[\\mu^\{\-\},\\mu^\{\+\}\],\\qquad\\nu\\in\[\\nu^\{\-\},\\nu^\{\+\}\],with case\-dependent half\-widths chosen to reflect the expected sensor variability\. For each case, we then consider a pessimistic slice\(μ−,ν\+\)\(\\mu^\{\-\},\\nu^\{\+\}\)and an optimistic slice\(μ\+,ν−\)\(\\mu^\{\+\},\\nu^\{\-\}\), yielding diagonal\-corner mediative bounds

M−:=M​\(μ−,ν\+\),M\+:=M​\(μ\+,ν−\)\.M^\{\-\}:=M\(\\mu^\{\-\},\\nu^\{\+\}\),\\qquad M^\{\+\}:=M\(\\mu^\{\+\},\\nu^\{\-\}\)\.This provides an illustrative interval\[M−,M\+\]\[M^\{\-\},M^\{\+\}\]of plausible mediative values forpp\.

Table 3:Illustrative interval type\-2 mediative degrees for the three cases\.In Case 1, the interval\[ML,MU\]\[M\_\{L\},M\_\{U\}\]lies mostly above the braking thresholdTbrake=0\.7T\_\{\\mathrm\{brake\}\}=0\.7, so a safety\-first policy still recommends decisive braking despite second\-order uncertainty\. In Case 2, the entire interval lies well below0\.70\.7, confirming that emergency braking is not justified and that controlled deceleration is appropriate\. In Case 3, the whole interval is above0\.70\.7, so the safety\-first resolution of a strong conflict remains braking even under type\-2 uncertainty\. The precise widths of the intervals are only illustrative; what matters is that MFL\-T2 can express a band of mediative values and that the decision rule can be formulated in terms ofMLM\_\{L\}andMUM\_\{U\}\(e\.g\., braking wheneverML≥TbrakeM\_\{L\}\\geq T\_\{\\mathrm\{brake\}\}\)\.

#### Granular outcomes in MFL\-T3

In MFL\-T3, mediative truth is assigned at the level of granulesg∈Gg\\in G, where a typical granule has the form \([12](https://arxiv.org/html/2605.22900#S8.E12)\)\.

For the obstacle\-detection example, a minimal granular description for each case is obtained by taking two granules

gradar=\(radar,current window,context\),gcam=\(camera,current window,context\),g\_\{\\mathrm\{radar\}\}=\(\\text\{radar\},\\ \\text\{current window\},\\ \\text\{context\}\),\\qquad g\_\{\\mathrm\{cam\}\}=\(\\text\{camera\},\\ \\text\{current window\},\\ \\text\{context\}\),with mediative valuesvgradar​\(p\)v\_\{g\_\{\\mathrm\{radar\}\}\}\(p\)andvgcam​\(p\)v\_\{g\_\{\\mathrm\{cam\}\}\}\(p\)matching the radar and camera assessments used in Cases 1–3\. A type\-3 granular aggregation operator then combines\{vg​\(p\):g∈G\}\\\{v\_\{g\}\(p\):g\\in G\\\}into a global mediative valueMG​\(p\)M\_\{G\}\(p\)\.

If we choose a top\-level granular aggregator that, in the two\-granule case, reduces exactly to the type\-1 fusion scheme used in Sections[8\.2](https://arxiv.org/html/2605.22900#S8.SS2)–[8\.4](https://arxiv.org/html/2605.22900#S8.SS4)\(e\.g\., a weighted combination with the same parameterα\\alpha\), then the global MFL\-T3 outcomes coincide numerically with the type\-1 mediative values:

MG\(1\)​\(p\)≈0\.716,MG\(2\)​\(p\)=0\.5,MG\(3\)​\(p\)≈0\.724M\_\{G\}^\{\(1\)\}\(p\)\\approx 0\.716,\\quad M\_\{G\}^\{\(2\)\}\(p\)=0\.5,\\quad M\_\{G\}^\{\(3\)\}\(p\)\\approx 0\.724
for Cases 1, 2 and 3 respectively\. The advantage of the type\-3 setting is not a different scalar output in this minimal configuration, but rather the ability to systematically incorporate additional granules: multiple frames, multiple cameras, different radar modes, or contexts\. Safety\-first design then constrains the granular aggregation operator so that strong evidence for an obstacle in any sufficiently trusted granule forcesMG​\(p\)M\_\{G\}\(p\)aboveTbrakeT\_\{\\mathrm\{brake\}\}, while still allowing temporal smoothing and cross\-checking between sources\.

#### Quantum mediative degrees for the three cases \(QMFL\)

In QMFL, a propositionppis evaluated in a stateρ\\rhovia two effectsEp\+E\_\{p\}^\{\+\}andEp−E\_\{p\}^\{\-\}as in Section[7](https://arxiv.org/html/2605.22900#S7)\. A simple embedding of the type\-1 mediative values for the three cases into QMFL is obtained by consideringℋ=ℂ2\\mathcal\{H\}=\\mathbb\{C\}^\{2\}, fixingρ=\|0⟩​⟨0\|\\rho=\|0\\rangle\\langle 0\|, and encoding each aggregated crisp pair\(μ\(i\),ν\(i\)\)\(\\mu^\{\(i\)\},\\nu^\{\(i\)\}\)as diagonal effects defined in \([11](https://arxiv.org/html/2605.22900#S8.E11)\)\. LetMp,i​\(ρ\)M\_\{p,i\}\(\\rho\)be constructed from\(Ep,i\+,Ep,i−\)\(E\_\{p,i\}^\{\+\},E\_\{p,i\}^\{\-\}\)via Definition[7\.2](https://arxiv.org/html/2605.22900#S7.Thmdefinition2), and defineMq\(i\)​\(p\):=Tr​\(ρ​Mp,i​\(ρ\)\)M\_\{q\}^\{\(i\)\}\(p\):=\\mathrm\{Tr\}\\\!\\bigl\(\\rho\\,M\_\{p,i\}\(\\rho\)\\bigr\)\. Thenμp​\(ρ\)=μ\(i\)\\mu\_\{p\}\(\\rho\)=\\mu^\{\(i\)\}andνp​\(ρ\)=ν\(i\)\\nu\_\{p\}\(\\rho\)=\\nu^\{\(i\)\}; thus, Theorem[7\.2](https://arxiv.org/html/2605.22900#S7.Thmtheorem2)yieldsMq\(i\)​\(p\)=M​\(μ\(i\),ν\(i\)\)M\_\{q\}^\{\(i\)\}\(p\)=M\(\\mu^\{\(i\)\},\\nu^\{\(i\)\}\)\. Hence, at the level of ideal expectation values, QMFL reproduces the same mediative scores obtained classically:

Mq\(1\)​\(p\)≈0\.716,Mq\(2\)​\(p\)=0\.5,Mq\(3\)​\(p\)≈0\.724\.M\_\{q\}^\{\(1\)\}\(p\)\\approx 0\.716,\\qquad M\_\{q\}^\{\(2\)\}\(p\)=0\.5,\\qquad M\_\{q\}^\{\(3\)\}\(p\)\\approx 0\.724\.In a realistic quantum implementation, these expectation values would be estimated from a finite number of shots, introducing statistical fluctuations aroundMq\(i\)​\(p\)M\_\{q\}^\{\(i\)\}\(p\)\. A safety\-first design must therefore ensure that the encoding of sensor evidence and the chosen braking threshold are robust with respect to such fluctuations\. For example, one may introduce a safety margin by triggering braking whenever the estimatedMq\(i\)​\(p\)M\_\{q\}^\{\(i\)\}\(p\)exceedsTbrakeT\_\{\\mathrm\{brake\}\}by more than a small tolerance, so that quantum statistical noise cannot systematically suppress a braking action that would be required by the classical mediative evidence\.

## 9Conclusions

We have presented a unified account of Mediative Fuzzy Logic, from its type\-1 foundations to type\-2, type\-3, and quantum extensions\. At the core is a mediative operator that combines positive and negative channels under explicit control by hesitation and contradiction\. This provides a single semantic mechanism for reconciling incomplete and genuinely conflicting evidence while remaining compatible with standard fuzzy infrastructures\.

At the type\-1 level, mediative truth values are formalized as truth–falsity pairs equipped with bilattice\-like operations and a scalar mediative evaluation\. A propositional system \(MFL\-T1\) with a mediative connective is introduced\. The resulting logic is algebraically and axiomatically well behaved, supports paraconsistent reasoning, and reduces to standard fuzzy and intuitionistic\-fuzzy logics in the expected special cases\. These properties establish MFL\-T1 as a principled foundation for safety\-first aggregation in settings where contradiction should not entail triviality\.

We then extended the framework in three directions\. The type\-2 extension \(MFL\-T2\) captures second\-order uncertainty about truth, falsity, hesitation, and contradiction via interval type\-2 truth values, supporting both type\-reduced \(crisp\) and envelope \(interval\) interpretations\. The type\-3 extension \(MFL\-T3\) organizes mediative truth in a granular, multi\-level form, enabling explicit cross\-granule aggregation policies that reflect domain requirements such as robustness to outliers or priority of trusted sources\. Finally, the quantum extension \(QMFL\) embeds mediative reasoning in the effect\-algebraic setting of quantum theory and connects the classical mediative score to a Born expectation of an adaptive mediative effect, thereby opening a path to quantum implementations and to effect\-based quantum granular computing architectures\.

The autonomous\-driving sensor\-fusion example demonstrates how mediative truth degrees can be interpreted conservatively in a safety\-first regime and linked to concrete decision policies\. It further illustrates that the higher\-type and quantum variants reduce to type\-1 behaviour under appropriate modelling assumptions, while retaining a clear route to richer behaviours when uncertainty, granular heterogeneity, or non\-commuting effects become operationally significant\.

Future research directions include completeness and representation theorems for the extended logics, development of proof calculi tailored to mediative reasoning, and further applications in control, diagnosis, decision support, and quantum technologies\.

## 10Notation and Symbol Table

For convenience, Table[4](https://arxiv.org/html/2605.22900#S10.T4)collects the main syntactic, semantic, and operational symbols used throughout the paper\. We also recall the distinction between syntactic derivability, semantic entailment, and \(mediative\) validity in the present setting\.

The sequentΓ⊢mφ\\Gamma\\vdash\_\{m\}\\varphidenotes syntactic derivability in the propositional system MFL\-T1: the formulaφ\\varphican be obtained from the set of premisesΓ\\Gammaby a finite sequence of applications of the axioms and inference rules of the logic\. This is a proof\-theoretic notion and does not involve semantic evaluation\.

Semantic entailment is writtenΓ⊧mφ\\Gamma\\models\_\{m\}\\varphi\. It means that for every mediative valuationvv, if all premisesψ∈Γ\\psi\\in\\Gammaattain maximal mediative degree \(i\.e\.,M​\(v​\(ψ\)\)=1M\(v\(\\psi\)\)=1\), then so doesφ\\varphi\(i\.e\.,M​\(v​\(φ\)\)=1M\(v\(\\varphi\)\)=1\)\. This notion depends on the mediative semantics through the mapφ↦v​\(φ\)∈V\\varphi\\mapsto v\(\\varphi\)\\in Vand the induced scalar evaluationMM\.

Finally,⊧mφ\\models\_\{m\}\\varphidenotes mediative validity:φ\\varphiis valid ifM​\(v​\(φ\)\)=1M\(v\(\\varphi\)\)=1for every mediative valuationvv, equivalently∅⊧mφ\\emptyset\\models\_\{m\}\\varphi\. With these notions in place, soundness states that every formula derivable in MFL\-T1 is semantically entailed under the mediative semantics\.

Table 4:Logical, semantic, and operational symbols used across MFL\-T1, MFL\-T2, MFL\-T3, and QMFL\.SymbolMeaningIllustrative exampleφ,ψ\\varphi,\\psiPropositional formulasφ:=“obstacle detected”\\varphi:=\\text\{\`\`obstacle detected''\}p,qp,qAtomic propositionsp:=“radar detects obstacle”p:=\\text\{\`\`radar detects obstacle''\}⊤\\topTruth constant \(tautology\)⊤:=\(p→p\)\\top:=\(p\\to p\)⊥\\botFalsity constant⊥⁣:=⁣¬⁣⊤\\bot:=\\neg\\top¬\\negNegation¬φ\\neg\\varphi∧\\wedgeConjunctionφ∧ψ\\varphi\\wedge\\psi∨\\veeDisjunctionφ∨ψ\\varphi\\vee\\psi→\\toImplication \(residuum\-based\)φ→ψ\\varphi\\to\\psi↔\\leftrightarrowBi\-implication\(φ→ψ\)∧\(ψ→φ\)\(\\varphi\\to\\psi\)\\wedge\(\\psi\\to\\varphi\)Med⁡\(φ\)\\operatorname\{Med\}\(\\varphi\)Mediative connective applied toφ\\varphi“mediated evaluation ofφ\\varphi”vvMediative valuationv​\(φ\)=\(μφ,νφ\)v\(\\varphi\)=\(\\mu\_\{\\varphi\},\\nu\_\{\\varphi\}\)\(μφ,νφ\)\(\\mu\_\{\\varphi\},\\nu\_\{\\varphi\}\)Truth and falsity degrees \(type\-1\)\(0\.5,0\.0\)\(0\.5,0\.0\)π,ζ\\pi,\\zetaHesitation and contradiction degreesπ=max⁡\(0,1−μ−ν\),ζ=max⁡\(0,μ\+ν−1\)\\pi=\\max\(0,1\-\\mu\-\\nu\),\\;\\zeta=\\max\(0,\\mu\+\\nu\-1\)M​\(μ,ν\)M\(\\mu,\\nu\)Scalar mediative evaluationM​\(0\.5,0\.0\)=0\.75M\(0\.5,0\.0\)=0\.75Γ\\GammaSet of formulas \(theory\)Γ=\{φ,φ→ψ\}\\Gamma=\\\{\\varphi,\\varphi\\to\\psi\\\}⊢m\\vdash\_\{m\}Formal derivability in MFL\-T1 \(proof\-theoretic\)Γ⊢mφ\\Gamma\\vdash\_\{m\}\\varphi⊧m\\models\_\{m\}Semantic entailment under mediative semanticsM​\(v​\(ψ\)\)=1​∀ψ∈Γ⇒M​\(v​\(φ\)\)=1M\(v\(\\psi\)\)=1\\ \\forall\\psi\\in\\Gamma\\Rightarrow M\(v\(\\varphi\)\)=1⊧mφ\\models\_\{m\}\\varphiMediative validityM​\(v​\(φ\)\)=1M\(v\(\\varphi\)\)=1for all mediative valuationsvvType\-2 Mediative Fuzzy Logic \(MFL\-T2\)μ~,ν~\\tilde\{\\mu\},\\tilde\{\\nu\}Interval type\-2 truth and falsity fuzzy setse\.g\.,ProjU​\(μ~\)=\[0\.6,0\.8\]\\mathrm\{Proj\}^\{U\}\(\\tilde\{\\mu\}\)=\[0\.6,0\.8\]μ¯,μ¯\\underline\{\\mu\},\\overline\{\\mu\}Lower and upper bounds of truthμ∈\[μ¯,μ¯\]\\mu\\in\[\\underline\{\\mu\},\\overline\{\\mu\}\]ν¯,ν¯\\underline\{\\nu\},\\overline\{\\nu\}Lower and upper bounds of falsityν∈\[ν¯,ν¯\]\\nu\\in\[\\underline\{\\nu\},\\overline\{\\nu\}\]TTChosentt\-norme\.g\.,T​\(a,b\)=max⁡\(0,a\+b−1\)T\(a,b\)=\\max\(0,a\+b\-1\)\(Łukasiewicz\)SStt\-conorm dual toTTe\.g\.,S​\(a,b\)=min⁡\(1,a\+b\)S\(a,b\)=\\min\(1,a\+b\)\(Łukasiewicz\)⇒T\\Rightarrow\_\{T\}Residuum induced byTTa⇒Tb:=sup\{c∈\[0,1\]∣T​\(a,c\)≤b\}a\\Rightarrow\_\{T\}b:=\\sup\\\{c\\in\[0,1\]\\mid T\(a,c\)\\leq b\\\}H​\(μ,ν\)H\(\\mu,\\nu\)Hesitation functionH=max⁡\(0,1−μ−ν\)H=\\max\(0,1\-\\mu\-\\nu\)C​\(μ,ν\)C\(\\mu,\\nu\)Contradiction functionC=max⁡\(0,μ\+ν−1\)C=\\max\(0,\\mu\+\\nu\-1\)HL,HUH\_\{L\},H\_\{U\}Lower/upper hesitation boundsHL=max⁡\(0,1−μ¯−ν¯\),HU=max⁡\(0,1−μ¯−ν¯\)H\_\{L\}=\\max\(0,1\-\\overline\{\\mu\}\-\\overline\{\\nu\}\),\\ H\_\{U\}=\\max\(0,1\-\\underline\{\\mu\}\-\\underline\{\\nu\}\)CL,CUC\_\{L\},C\_\{U\}Lower/upper contradiction boundsCL=max⁡\(0,μ¯\+ν¯−1\),CU=max⁡\(0,μ¯\+ν¯−1\)C\_\{L\}=\\max\(0,\\underline\{\\mu\}\+\\underline\{\\nu\}\-1\),\\ C\_\{U\}=\\max\(0,\\overline\{\\mu\}\+\\overline\{\\nu\}\-1\)M¯p\\bar\{M\}\_\{p\},\[ML​\(p\),MU​\(p\)\]\[M\_\{L\}\(p\),M\_\{U\}\(p\)\]Type\-reduced / envelope mediative degreesM¯p:=M​\(μ¯p,ν¯p\)\\bar\{M\}\_\{p\}:=M\(\\bar\{\\mu\}\_\{p\},\\bar\{\\nu\}\_\{p\}\),\[ML,MU\]\[M\_\{L\},M\_\{U\}\]Type\-3 Granular Mediative Fuzzy Logic \(MFL\-T3\)ggGranule index \(source/context\)g=radar,camerag=\\text\{radar\},\\text\{camera\}GGSet of granulesG=\{gradar,gcam\}G=\\\{g\_\{\\mathrm\{radar\}\},g\_\{\\mathrm\{cam\}\}\\\}vg​\(φ\)v\_\{g\}\(\\varphi\)Local mediative valuation at granuleggvg​\(φ\)=\(μφ,g,νφ,g\)v\_\{g\}\(\\varphi\)=\(\\mu\_\{\\varphi,g\},\\nu\_\{\\varphi,g\}\)Mg​\(φ\)M\_\{g\}\(\\varphi\)Local scalar mediative degreeMg​\(φ\):=M​\(μφ,g,νφ,g\)M\_\{g\}\(\\varphi\):=M\(\\mu\_\{\\varphi,g\},\\nu\_\{\\varphi,g\}\)AφA\_\{\\varphi\}Granular aggregation operatorMG​\(φ\):=Aφ​\(\(Mg​\(φ\)\)g∈G\)M\_\{G\}\(\\varphi\):=A\_\{\\varphi\}\(\(M\_\{g\}\(\\varphi\)\)\_\{g\\in G\}\)MG​\(φ\)M\_\{G\}\(\\varphi\)Global mediative degree after aggregationMG​\(φ\)∈\[0,1\]M\_\{G\}\(\\varphi\)\\in\[0,1\]Quantum Mediative Fuzzy Logic \(QMFL\)ρ\\rhoQuantum state \(density operator\)ρ⪰0,Tr​\(ρ\)=1\\rho\\succeq 0,\\ \\mathrm\{Tr\}\(\\rho\)=1EEQuantum effect0⪯E⪯I0\\preceq E\\preceq IEp\+,Ep−E\_\{p\}^\{\+\},E\_\{p\}^\{\-\}Positive/negative evidence effects forppEp\+≠I−Ep−E\_\{p\}^\{\+\}\\neq I\-E\_\{p\}^\{\-\}allowedμp​\(ρ\),νp​\(ρ\)\\mu\_\{p\}\(\\rho\),\\nu\_\{p\}\(\\rho\)Born degrees \(truth/falsity channels\)μp​\(ρ\)=Tr​\(ρ​Ep\+\),νp​\(ρ\)=Tr​\(ρ​Ep−\)\\mu\_\{p\}\(\\rho\)=\\mathrm\{Tr\}\(\\rho E\_\{p\}^\{\+\}\),\\ \\nu\_\{p\}\(\\rho\)=\\mathrm\{Tr\}\(\\rho E\_\{p\}^\{\-\}\)w1,p​\(ρ\),w2,p​\(ρ\)w\_\{1,p\}\(\\rho\),w\_\{2,p\}\(\\rho\)Mediative weightsw1,p=1−πp−ζp/2,w2,p=πp\+ζp/2w\_\{1,p\}=1\-\\pi\_\{p\}\-\\zeta\_\{p\}/2,\\ w\_\{2,p\}=\\pi\_\{p\}\+\\zeta\_\{p\}/2πp​\(ρ\)\\pi\_\{p\}\(\\rho\)Quantum hesitation degreeπp=max⁡\(0,1−μp−νp\)\\pi\_\{p\}=\\max\(0,1\-\\mu\_\{p\}\-\\nu\_\{p\}\)ζp​\(ρ\)\\zeta\_\{p\}\(\\rho\)Quantum contradiction degreeζp=max⁡\(0,μp\+νp−1\)\\zeta\_\{p\}=\\max\(0,\\mu\_\{p\}\+\\nu\_\{p\}\-1\)Mp​\(ρ\)M\_\{p\}\(\\rho\)Quantum mediative effect forppMp​\(ρ\)=w1,p​Ep\+\+w2,p​\(I−Ep−\)M\_\{p\}\(\\rho\)=w\_\{1,p\}E\_\{p\}^\{\+\}\+w\_\{2,p\}\(I\-E\_\{p\}^\{\-\}\)Mq​\(p,ρ\)M\_\{q\}\(p,\\rho\)Quantum mediative degree \(Born expectation\)Mq​\(p,ρ\)=Tr​\(ρ​Mp​\(ρ\)\)M\_\{q\}\(p,\\rho\)=\\mathrm\{Tr\}\(\\rho\\,M\_\{p\}\(\\rho\)\)
## Declaration of competing interest

The author declares no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper\.

## Data availability

No datasets were generated or analyzed during the current study\. All theoretical results and examples are fully described in the article\.

## Declaration of generative AI and AI\-assisted technologies in the writing process

The author declares that no generative AI or AI\-assisted technologies were used to write the manuscript or to generate figures or analytical results\. Only standard text editing tools were employed\.

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