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R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University.

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Presentation on theme: "R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University."— Presentation transcript:

1 R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University

2 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.)

3  Conversational understanding of truth will do for observation sentences.  Theoretical sentences (causality, necessity, implication and so on) require something more.

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

5  Every wff of classical propositional logic has a finite analysis into articulated form:  Viz. its CNF (A conjunction of disjunctions of literals).

6  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.

7  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.

8  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.

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

10 Definition 2 Definition 1

11 Definition 3 Definition 4

12

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

14 Definition one

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

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

17 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,  FDE is determined by a subsumption in the class of a- models.  FD entailment preserves the cardinality of a set of contradictions.

18 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 ;

19 Definition two

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

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

22 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

23 Definition three

24 First-degree Parry entailment (FDPE)

25 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.

26 Definition four

27  First-degree sub-entailment (FDSE)

28 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

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


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