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

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

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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 = is a filter.

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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).

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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.

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M = 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.

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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 hypergraph iff ∀E, E’∈ H, E⊄E’.

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

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We use hypergraphs instead of sets to represent wffs. Classically, inference relations are represented by subset relations between sets. α entails β iff the α -hypergraph, H α is in the relation R to the β -hypergraph, H β. H α RH β. : ∀ E ∈ H β, ∃ E’ ∈ H α : E’ ⊆ E.

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F = N (x) is a simple hypergraph. ∀ A ∈Φ, F ⊨ ⧠ A iff N (x)R H A [K], [RN], [RM( ⊦ )] →?

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

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

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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 ).

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

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Anderson & Belnap D 1 D 2 … D n C 1 C 2 … C m ∀1≤ i ≤ n, ∀1≤ j ≤ m, d i ∩ cj ≠ Ø

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C 1 C 2 … C n C1 C2 … Cm ∀1≤ i ≤ n, ∃1≤ j ≤ m, cj ⊆ d i ∀1≤ i ≤ n, ∃1≤ j ≤ m, cj ⊢ d i

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((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)

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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)

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Mixed degree Uniform substitution

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