Proposition and Necessity

1. Propositionand Necessity R. E. Jennings jennings@sfu.ca Y. Chen nek@sfu.ca Laboratory for Logic and Experimental Philosophy Simon Fraser University

2. What is a proposition? The set of necessities at a point ⧠(x).

3. Primordial necessity Every point x in U is assigned a primordial necessity R(x) = { y | Rxy } . The set of necessities at a point ⧠(x)in a model of a binary relational frame F = <U, R> is a filter.

4. The `leibnizian’ account R is universal; The primordial necessity for every point is identical, which is U. Only the universally true is necessary (and what is necessary is universally true, and in fact, universally necessary).

5. CPN frame • A common primordial necessity • (x)(y)(z)(Rxz→Ryz) (CPN) • [K], [RM], [RN], [5], ⧠(⧠p→p), ⧠(p→ ⧠ ◊p). • R is serial and symmetric. • R satisfies CPN. • R is universal.

6. Necessities in CPN frame M = <F, V> M ⊨ ⧠A iff ℙ ⊆ ∥A∥M The set of necessities in a model, ⧠(M) is a filter on P (U), i.e. a hypergraph on U.

7. Entering hypergraph A hypergraph H is a pair H = (X, E) where X is a set of elements, called vertices, and E is a non-empty set of subsets of X called (hyper)edges. Therefore, E ⊆ P (X). H is a simple hypergraphiff∀E, E’∈ H, E⊄E’.

8. Locale frame • Weakening neighbourhood truth condition • F = <U, N > • N(x) is a set of propositions. • ∀ A∈Φ, F ⊨ ⧠A iff∃a∈ N(x): a⊆ ∥A∥F • L = <U, N’ > if N’ (x) is a simple hypergraph. • PL closed under [RM]. • N’ (x)≠∅ [RN] • N’ (x) is a singleton [K]

9. Hypergraph semantics • We use hypergraphs instead of sets to represent wffs. • Classically, inference relations are represented by subset relations between sets. • α entails βiff the α-hypergraph, Hα is in the relation R to the β-hypergraph, Hβ . • HαRHβ . : ∀ E ∈ Hβ , ∃ E’∈ Hα : E’ ⊆ E.

10. ⧠(F) • F = <U, N > • N(x) is a simple hypergraph. • ∀ A∈Φ, F ⊨ ⧠A iffN(x)R HA • [K], [RN], [RM(⊦)] • →?

11. Necessarily (A is true) A is necessarily true; (Necessarily A) is true. ⊨⧠A HA→B is interpreted as H¬A˅B.

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

13. Preliminary definitions

14. Hypergraph operations

15. FDE • First degree fragment of E • A ∧ B ├ A • A ├ A V B • A ┤├~~A • ~(A ∧ B) ┤├ ~A V ~B • ~(A V B) ┤├ ~A ∧ ~B • A V (B ∧ C)├ (A V C) ∧ (B V C) • A ∧(B V C)├ (A ∧ C) V(B ∧ C).

16. FDE with necessity Necessarily (A is true) iff∀ E ∈ HA, ∃ v∈ E such that ∃ v’∈ E: v’ = U – v. (N) (N) is closed under ⊦ and ˄. A⊦B / necessarily A→B is true.

17. Problem of entailment Anderson & Belnap • D1 D2 … Dn • C1 C2 … Cm • ∀1≤ i ≤ n, ∀1≤ j ≤ m, di∩ cj≠ Ø

18. A & B Con’d C1 C2 … Cn C1 C2 … Cm ∀1≤ i ≤ n, ∃1≤ j ≤ m, cj⊆ di ∀1≤ i ≤ n, ∃1≤ j ≤ m, cj⊢ di

19. Higher degree entailment ((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) ├ (A V B → C) (A → ~ A)├ ~ A (A → B)├(~ B → ~ A)

20. Higher degree E • ((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) → (A V B → C) • (A → ~ A) → ~ A • (A → B) → (~ B → ~ A)

21. Problem of degree Mixed degree Uniform substitution