Articular Models for Paraconsistent Systems The project so far

1. Articular Models for Paraconsistent SystemsThe project so far R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy http://www.sfu.ca/llep/ Simon Fraser University

2. Inarticulation 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. Theory and Observation • Conversational understanding of truth will do for observation sentences. • Theoretical sentences (causality, necessity, implication and so on) require something more.

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

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

6. 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.

7. 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.

8. Hypergraphic Representation • Inference relations are represented by relations between hypergraphs. • α entails βiff 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.

9. Articular Models (a-models) Each atom is assigned a hypergraph on the power set of the universe .

10. A-models cont’d Definition 1 Definition 2

11. A-models cont’d Definition 3 Definition 4

12. Contradictions and Tautologies

13. A-models cont’d • We are now in a position to define Bob Loblaw. • We consider four definitions.

14. A strangely familiar case Definition one

15. FDE (Anderson & Belnap) • α├βiff 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) • 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. 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

18. 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;

19. First-Degree Analytic Entailment Definition two

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

21. Analytic Implication • Kit Fine: analytic implication • Strict implication + prescriptive principle • Arthur Prior

22. First degree analytic entailment (FDAE) 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 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.

23. First-Degree Parry Entailment Definition three

24. Definition Three First-degree Parry entailment (FDPE)

25. First degree Parry entailment (FDPE) While the prescriptive principle in FDAE preserves vertices of hypergraphs that semantically represent wffs, that in FDPE preserves atoms of wffs. 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

26. Sub-entailment Definition four

27. Definition Four • First-degree sub-entailment (FDSE)

28. FDSE 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. Future Research • First-degree modal logics • Higher-degree systems • Other non-Boolean algebras