R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University.
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Transcript of R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University.
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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
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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.)
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Theory and Observation
Conversational understanding of truth will do for observation sentences.
Theoretical sentences (causality, necessity, implication and so on) require something more.
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Articulation
G. W. Leibniz: All truths are analytic. Contingent truths are infinitely so. Only God can articulate the analysis.
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Leibniz realized
Every wff of classical propositional logic has a finite analysis into articulated form:
Viz. its CNF (A conjunction of disjunctions of literals).
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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.
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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.
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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.
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Articular Models (a-models)
Each atom is assigned a hypergraph on the power set of the universe .
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A-models cont’d
Definition 2
Definition 1
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A-models cont’d
Definition 3
Definition 4
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Contradictions and Tautologies
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A-models cont’d
We are now in a position to define Bob Loblaw.
We consider four definitions.
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A STRANGELY FAMILIAR CASE
Definition one
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FDE (Anderson & Belnap)
α├ β iff DNF(α) ≤ CNF(β) Definition 5:
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Subsumption
In the class of a-models, the relation of subsumption corresponds to FDE.
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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.
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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 ;
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FIRST-DEGREE ANALYTIC ENTAILMENT
Definition two
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First-degree analytic entailment (FDAE): RFDAE: subsumption + prescriptive principle
In the class of h-models, RFDAE
corresponds to FDAE.
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Analytic Implication
Kit Fine: analytic implication Strict implication + prescriptive principle Arthur Prior
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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
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FIRST-DEGREE PARRY ENTAILMENT
Definition three
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Definition Three
First-degree Parry entailment (FDPE)
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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.
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SUB-ENTAILMENTDefinition four
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Definition Four
First-degree sub-entailment (FDSE)
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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
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Future Research
First-degree modal logics Higher-degree systems Other non-Boolean algebras