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R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy http://www.sfu.ca/llep/ Simon Fraser University

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What is truth said doughty Pilate. But snappy answer came there none and 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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Conversational understanding of truth will do for observation sentences. Theoretical sentences (causality, necessity, implication and so on) require something more.

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

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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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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 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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Inference relations are represented by relations between hypergraphs. α entails β iff the α -hypergraph, H α is in the relation, Bob Loblaw, to the β -hypergraph, H β. What the inference relation is is determined by how we characterize Bob Loblaw.

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

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Definition 2 Definition 1

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Definition 3 Definition 4

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We are now in a position to define Bob Loblaw. We consider four definitions.

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Definition one

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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) FDE A ^ B├ B A ├ A v B A ^ (B v C) ├ (A ^ B) v (A v C) ~~A ├ A A ├ ~~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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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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Definition two

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First-degree analytic entailment (FDAE): R FDAE : subsumption + prescriptive principle In the class of h-models, R FDAE corresponds to FDAE.

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

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First degree analytic entailment (FDAE) FDAE A ^ B├ B A ├ A v B A ^ B ├ A v B A ^ (B v C) ├ (A ^ B) v (A v C) ~~A ├ A A ├ ~~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 ^ A A ^ B ├ B ^ A ~~A ├ A A ├ ~~A A ^ (B v C) ├ (A ^ B) v (A v C) A ├ B ^ C / A ├ B A ├ B, C ├ D / A ^ B ├ C ^ D A ├ B, C ├ D / A v B ├ C v D A v (B ^ ~B) ├ A A ├ B, B ├ C / A ├ C f (A) / A ├ A A ├ 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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Definition three

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First-degree Parry entailment (FDPE)

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First degree Parry entailment (FDPE) FDPE A ^ B├ B A ├ A v B A ^ B ├ A v B A ├ A v ~A A ^ (B v C) ├ (A ^ B) v (A v C) ~~A ├ A A ├ ~~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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Definition four

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First-degree sub-entailment (FDSE)

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A ^ B├ B A ├ A v B A ^ (B v C) ├ (A ^ B) v (A v C) ~~A ├ A A ├ ~~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 B A ├ A v ~A

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

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