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Proposition and Necessity

This study investigates the nature of propositions and their necessities through a hypergraph framework. It begins by discussing the definition of a proposition and the assignment of primordial necessity to each point in a binary relational frame. The analysis includes CPN frames, hypergraph semantics, and the interpretation of entailment in relation to hypergraph structures. Additionally, the paper examines the problem of degree in logical entailments, providing a comprehensive approach to necessity in logic using hypergraphs.

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Proposition and Necessity

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  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

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