Mediative Fuzzy Logic: Type‑3 Model

• The Type‑3 Model is a granular extension of mediative fuzzy logic that assigns multiple local truth values to reconcile hesitant and conflicting evidence.
• It computes local mediative scores using a type‑1 basis and aggregates them through monotone, safety‑biased operators to guide decision-making.
• Its hierarchical design guarantees reduction to lower type models, making it ideal for multi‑source applications in safety‑critical systems like autonomous driving.

The Type‑3 Model in Mediative Fuzzy Logic (MFL‑T3) is the granular semantic extension of a mediative fuzzy framework designed to reconcile hesitant and conflicting evidence while preserving a safety‑first, non‑explosive behavior. In this construction, a formula is not assigned a single global truth value. Instead, it is evaluated locally across a finite family of granules—such as sensors, experts, time windows, or environmental contexts—and those local mediative evaluations are then aggregated into a global mediative degree. The logical language remains that of the type‑1 core, but the semantics are enriched from single truth–falsity pairs to granule‑indexed families of such evaluations, with optional interval type‑2 uncertainty at each granule (Ross, 21 May 2026).

1. Position of the Type‑3 model in the mediative hierarchy

Mediative Fuzzy Logic assigns to each proposition an independent degree of truth or agreement, denoted $\mu$, and an independent degree of falsity or non‑agreement, denoted $\nu$. From these, it derives hesitation $\pi$ and contradiction $\zeta$, and uses them to control a mediative operator that combines a positive and a negative channel in a convex way. The hierarchy developed in the framework consists of four levels: Type‑1 (MFL‑T1), where truth values are simple pairs $(\mu,\nu)\in[0,1]^2$; Type‑2 (MFL‑T2), where $\mu$ and $\nu$ are interval type‑2 fuzzy sets; Type‑3 (MFL‑T3), where truth is granular and indexed by local contexts; and a Quantum extension in which mediative scores are realized as Born expectations of mediative effects (Ross, 21 May 2026).

Within that hierarchy, the type‑3 model is defined semantically rather than syntactically. It does not introduce a new logical language or a new proof system. Instead, it reinterprets formula evaluation so that each proposition receives many local mediative truth values, one for each granule, prior to any aggregation. In the formulation of the paper, this is what “type‑3” means: a higher‑level structure of mediative evaluations $G\to V$ (or $G\to$ type‑2 values), rather than a single pair or a single interval (Ross, 21 May 2026).

A granule is an index representing a local context. The examples explicitly listed include a particular sensor, a particular time window, an expert or expert group, a specific environmental context such as weather regime or road type, and combinations such as sensor–time–context triplets. This gives MFL‑T3 an explicit granular computing interpretation rather than treating higher type merely as additional uncertainty over a single membership grade.

2. Type‑1 basis of the Type‑3 semantics

The type‑3 model is built on the type‑1 truth space

$V=[0,1]\times[0,1],$

where each mediative truth value is a pair $\nu$0, with $\nu$1 the degree of truth or agreement and $\nu$2 the degree of falsity or non‑agreement. From such a pair the framework derives

$\nu$3

for hesitation or incompleteness, and

$\nu$4

for contradiction or over‑determination. The paper states that $\nu$5 iff $\nu$6, $\nu$7 iff $\nu$8, and that $\nu$9 and $\pi$0 cannot both be positive at once (Ross, 21 May 2026).

This truth space is described as a continuous bilattice‑like structure with bilattice‑style connectives. Given a left‑continuous $\pi$1‑norm $\pi$2 and its dual $\pi$3‑conorm $\pi$4,

$\pi$5

$\pi$6

$\pi$7

and implication is residuated coordinate‑wise: $\pi$8

The mediative operator itself takes channel values $\pi$9, hesitation $\zeta$0, and contradiction $\zeta$1, and returns the convex aggregate

$\zeta$2

Its weights

$\zeta$3

satisfy $\zeta$4 and $\zeta$5, and the convexity theorem yields

$\zeta$6

In type‑1 MFL, the scalar mediative score attached to $\zeta$7 is obtained by choosing

$\zeta$8

and defining

$\zeta$9

The unary mediative connective is then interpreted by

$(\mu,\nu)\in[0,1]^2$0

These type‑1 constructions are the semantic ground floor on which the type‑3 model is erected (Ross, 21 May 2026).

3. Granule‑indexed semantics of MFL‑T3

The defining move of MFL‑T3 is to replace a single global valuation with a finite family of local valuations indexed by a nonempty finite set of granules $(\mu,\nu)\in[0,1]^2$1. For each $(\mu,\nu)\in[0,1]^2$2, one has a local mediative valuation $(\mu,\nu)\in[0,1]^2$3. For an atomic proposition $(\mu,\nu)\in[0,1]^2$4, the local truth at granule $(\mu,\nu)\in[0,1]^2$5 may be either type‑1,

$(\mu,\nu)\in[0,1]^2$6

or type‑2,

$(\mu,\nu)\in[0,1]^2$7

where the second form consists of interval type‑2 fuzzy sets with footprints of uncertainty and associated type‑reduction machinery (Ross, 21 May 2026).

The full type‑3 truth assignment is then the family

$(\mu,\nu)\in[0,1]^2$8

For a compound formula $(\mu,\nu)\in[0,1]^2$9, each $\mu$0 is extended locally using the same connectives as in MFL‑T1 or MFL‑T2, so that

$\mu$1

Accordingly, before aggregation, every formula has a vector of local mediative truth values indexed by granules.

At each granule, a local scalar mediative degree is computed. In the type‑1 local case,

$\mu$2

In the type‑2 local case, the paper specifies a type‑reduction step: $\mu$3 followed, for example, by midpoint selection,

$\mu$4

and then

$\mu$5

The paper also notes that one may instead retain interval uncertainty by tracking an interval $\mu$6.

This semantics makes the type‑3 model explicitly localist: each granule yields its own mediative score, and only afterward is a global degree formed.

4. Granular aggregation and reduction to lower types

The passage from local scores to a global value is governed by a granular aggregation operator

$\mu$7

which acts as

$\mu$8

Typical choices explicitly listed are simple weighted averages, OWA operators, and hierarchical aggregations, such as first per sensor type, then across sensor groups, and then across experts (Ross, 21 May 2026).

The semantic content of a type‑3 evaluation therefore has two layers: first, the family of local mediative values

$\mu$9

and second, the global scalar mediative degree $\nu$0 obtained by aggregating the local scalar scores.

A central structural property of the framework is its exact reduction to lower types under homogeneity conditions. If $\nu$1 and the aggregator is idempotent,

$\nu$2

then

$\nu$3

so MFL‑T3 reduces exactly to MFL‑T1, or to MFL‑T2 if the single local valuation is type‑2. More generally, the paper states a reduction theorem under homogeneous granules: if $\nu$4 is idempotent,

$\nu$5

and all granules assign the same local mediative degree $\nu$6, then

$\nu$7

If, more strongly, all local truth values coincide on atoms, then by induction on formula structure all $\nu$8 and all $\nu$9 coincide, and the global score equals the common type‑1 or type‑2 value. The hierarchy is summarized semantically as

$G\to V$0

with reduction maps obtained by collapsing footprints of uncertainty and then collapsing granules (Ross, 21 May 2026).

This coherence result is one of the paper’s principal claims about MFL‑T3. It ensures that granularity enriches the semantics without introducing artificial distortion when all local contexts agree.

5. Logical status and semantic properties of the Type‑3 model

The proof system of MFL‑T3 is unchanged from the type‑1 calculus. The paper explicitly states that all axioms remain those of MFL‑T1 and that type‑3 is purely semantical: valuations become granule‑indexed families, together with an extra aggregation layer $G\to V$1 acting on local mediative scores (Ross, 21 May 2026).

This arrangement imports the type‑1 metatheoretic results into the granular setting in a qualified way. For MFL‑T1, the paper establishes soundness for the mediative semantics, paraconsistency, and conservativity over the underlying fuzzy base for formulas without the mediative connective. It also states that high values for both $G\to V$2 and $G\to V$3 can coexist and that no explosion of the form $G\to V$4 holds in general (Ross, 21 May 2026).

For MFL‑T3, the paper emphasizes the following properties. First, idempotence and consistency of the aggregator guarantee that homogeneous granules reproduce the lower‑type semantics. Second, the reduction theorem shows that type‑3 is a conservative generalization in the semantic sense. Third, paraconsistency inheritance is intended: since each $G\to V$5 is itself a mediative valuation in the type‑1 sense, each local component is paraconsistent, and aggregation can be chosen so that global behavior does not enforce classical explosion. The paper is careful to note that it does not provide a separate formal paraconsistency theorem for type‑3. Fourth, although not stated as a theorem, it is presented as natural and intended to require $G\to V$6 to be monotone in each coordinate, possibly with further structural constraints respecting groupings of granules.

The absence of a separate completeness theorem is also explicitly noted. Type‑2, type‑3, and quantum mediative fuzzy logics are treated as coherent semantic extensions compatible with the type‑1 calculus rather than as independently axiomatized systems.

6. Operational reading, comparison, and applications

The paper’s autonomous‑braking example is developed at type‑1 level but is presented as the natural template for a type‑3 reading. In the type‑1 version, the proposition is “dangerous obstacle within 20m,” radar/LiDAR and camera evidence are pre‑fused into a single pair $G\to V$7 by a weighted average, and the resulting mediative score is compared with thresholds:

• $G\to V$8: emergency brake,
• $G\to V$9: cautious slow‑down,
• $G\to$0: proceed, still cautious.

In the contradictory but safety‑first case identified as Case 3, radar is almost sure of an obstacle and camera is almost sure of no obstacle, the fused value is approximately $G\to$1, so $G\to$2 and $G\to$3, yielding

$G\to$4

hence decisive braking (Ross, 21 May 2026).

The type‑3 interpretation restructures this situation. Rather than fusing radar and camera immediately into one pair, one takes granules such as

$G\to$5

assigns a local mediative truth value $G\to$6 to each granule, computes the local scores $G\to$7, and then aggregates them: $G\to$8 The paper gives the simpler instantiation

$G\to$9

with

$V=[0,1]\times[0,1],$0

and then

$V=[0,1]\times[0,1],$1

This construction supports several features stated explicitly in the paper: adding more granules such as further sensors or time history, assigning different weights or nonlinear aggregation policies per granule, and enforcing safety‑first rules such as forcing $V=[0,1]\times[0,1],$2 high if any highly trusted granule has $V=[0,1]\times[0,1],$3 above $V=[0,1]\times[0,1],$4. The high‑level pipeline is given as: encode local evidence as pairs $V=[0,1]\times[0,1],$5, compute local mediative scores, choose a granular aggregator that is monotone, idempotent, and safety‑biased, and decide brake, slow‑down, or proceed by comparing $V=[0,1]\times[0,1],$6 against thresholds (Ross, 21 May 2026).

In its comparison with standard fuzzy frameworks, the paper characterizes the novelty of the type‑3 model in four ways: mediative semantics comes first and granularity second; type‑3 values are explicit granule‑indexed mediative families

$V=[0,1]\times[0,1],$7

reduction theorems make the hierarchy coherent; and the entire construction preserves a safety‑first interpretation in multi‑source settings rather than merely introducing higher‑order uncertainty. This suggests a specialized role for MFL‑T3 in intelligent decision systems that must fuse incomplete, heterogeneous, and mildly contradictory evidence while retaining transparent aggregation rules.

The application areas stated in the paper are intelligent decision systems in safety‑critical domains, including autonomous driving, medical diagnosis, and smart grids and distributed monitoring. The practical characteristics singled out are scalability through hierarchical granules and inexpensive aggregators, interpretability of local mediative scores $V=[0,1]\times[0,1],$8, explainability through explicit hesitation and contradiction at each granule, and compatibility with existing fuzzy and type‑2 controllers used as local engines (Ross, 21 May 2026).