ARTICULAR MODELS FOR PARACONSISTENT SYSTEMS

THE PROJECT SO FAR

R. E. JenningsY. Chen

Laboratory for Logic and Experimental Philosophyhttp://www.sfu.ca/llep/

Simon Fraser University

Inarticulation

What is truthsaid doughty Pilate.But snappy answer came there noneand he made good his escape.Francis Bacon: Truth is noble.Immanuel Jenkins: Whoop-te-doo!*

(*Quoted in Tessa-Lou Thomas. Immanuel Jenkins: the myth and the man.)

Theory and Observation

Conversational understanding of truth will do for observation sentences.

Theoretical sentences (causality, necessity, implication and so on) require something more.

Articulation

G. W. Leibniz: All truths are analytic. Contingent truths are infinitely so. Only God can articulate the analysis.

Leibniz realized

Every wff of classical propositional logic has a finite analysis into articulated form:

Viz. its CNF (A conjunction of disjunctions of literals).

Protecting the analysis

Classical Semantic representation of CNF’s:

the intersection of a set of unions of truth-sets of literals. (Propositions are single sets.)

Taking intersections of unions masks the articulation.

Instead, we suggest, make use of it. An analysed proposition is a set of sets of

sets.

Hypergraphs

Hypergraphs provide a natural way of thinking about Normal Forms.

We use hypergraphs instead of sets to represent wffs.

Classically, inference relations are represented by subset relations between sets.

Hypergraphic Representation

Inference relations are represented by relations between hypergraphs. α entails β iff the α-hypergraph, Hα is in the

relation, Bob Loblaw, to the β-hypergraph, Hβ .

What the inference relation is is determined by how we characterize Bob Loblaw.

Articular Models (a-models)

Each atom is assigned a hypergraph on the power set of the universe .

A-models cont’d

Definition 2

Definition 1

A-models cont’d

Definition 3

Definition 4

Contradictions and Tautologies

A-models cont’d

We are now in a position to define Bob Loblaw.

We consider four definitions.

A STRANGELY FAMILIAR CASE

Definition one

FDE (Anderson & Belnap)

α├ β iff DNF(α) ≤ CNF(β) Definition 5:

Subsumption

In the class of a-models, the relation of subsumption corresponds to FDE.

First-degree entailment (FDE)

A ^ B├ B A ├ A v BA ^ (B v C) ├ (A ^ B) v (A v C)~~A ├ AA ├ ~~A~(A ^ B) ├ ~A v ~B~(A v B) ├ ~A ^ ~B[Mon] Σ ├ A / Σ, Δ ├ A[Ref] A Σ / Σ ├ A[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B

A. R. Anderson & N. Belnap, Tautological entailments, 1962.

FDE is determined by a subsumption in the class of a-models.

FD entailment preserves the cardinality of a set of contradictions.

Two approaches from FDE to E

A&B ((A→A)→B)→B; (A→B)→((B→C)→(A→C)); (A→(A→B))→(A→B); (A→B) ∧ (A→C) ├

A→B∧C; (A→C) ∧ (B→C) ├ AVB→C; (A→~A)→~A; (A→~B)→(B→~A); NA ∧NB→N(A∧B).

NA=def (A→A)→A

R&C (A→B) ∧ (A→C) ├ A→B∧C; (A→C) ∧ (B→C) ├ AVB→C; A→C ├ A∧B→C ; (A→B)├ AVC→ BVC; A→ B∧C ├ A→C ;

FIRST-DEGREE ANALYTIC ENTAILMENT

Definition two

First-degree analytic entailment (FDAE): RFDAE: subsumption + prescriptive principle

In the class of h-models, RFDAE

corresponds to FDAE.

Analytic Implication

Kit Fine: analytic implication Strict implication + prescriptive principle Arthur Prior

First degree analytic entailment (FDAE)

A ^ B├ BA ├ A v BA ^ B ├ A v BA ^ (B v C) ├ (A ^ B) v (A v C)~~A ├ AA ├ ~~A~(A ^ B) ├ ~A v ~B~(A v B) ├ ~A ^ ~B[Mon] Σ ├ A / Σ, Δ ├ A[Ref] A Σ / Σ ├ A[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B

FDAE preserves classical contingency and colourability.

First-Degree fragment of Parry’s original system

A ├ A ^ AA ^ B ├ B ^ A~~A ├ AA ├ ~~AA ^ (B v C) ├ (A ^ B) v (A v C)A ├ B ^ C / A ├ BA ├ B, C ├ D / A ^ B ├ C ^ D

A ├ B, C ├ D / A v B ├ C v D

A v (B ^ ~B) ├ AA ├ B, B ├ C / A ├ Cf (A) / A ├ AA ├ B, B ├ A / f (A) ├ f (B), f (B) ├ f(A)A, B ├ A ^ B~ A ├ A, A ├ B / ~ B ├ B

FIRST-DEGREE PARRY ENTAILMENT

Definition three

Definition Three

First-degree Parry entailment (FDPE)

First degree Parry entailment (FDPE)

A ^ B├ BA ├ A v BA ^ B ├ A v BA ├ A v ~AA ^ (B v C) ├ (A ^ B) v (A v C)~~A ├ AA ├ ~~A~(A ^ B) ├ ~A v ~B~(A v B) ├ ~A ^ ~B[Mon] Σ ├ A / Σ, Δ ├ A[Ref] A Σ / Σ ├ A[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B

While the prescriptive principle in FDAE preserves vertices of hypergraphs that semantically represent wffs, that in FDPE preserves atoms of wffs.

SUB-ENTAILMENTDefinition four

Definition Four

First-degree sub-entailment (FDSE)

FDSE

A ^ B├ BA ├ A v BA ^ (B v C) ├ (A ^ B) v (A v C)~~A ├ AA ├ ~~A~(A ^ B) ├ ~A v ~B~(A v B) ├ ~A ^ ~B[Mon] Σ ├ A / Σ, Δ ├ A[Ref] A Σ / Σ ├ A[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B

Comparing with FDAE and FDPE:

A ^ B ├ A v BA ├ A v ~A

Future Research

First-degree modal logics Higher-degree systems Other non-Boolean algebras