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Articular Models for Paraconsistent Systems The project so far. R. E. Jennings Y. Chen. Laboratory for Logic and Experimental Philosophy http://www.sfu.ca/llep/ Simon Fraser University. Inarticulation. What is truth said doughty Pilate. But snappy answer came there none - PowerPoint PPT Presentation
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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 ^ DA ├ B, C ├ D / A v B ├ C v DA 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)
FDSEA ^ 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