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

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

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

 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.

 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).

 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.

 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.

 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’.

 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]

 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.

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

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

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

 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 ).

 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.

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

 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

 ((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)

 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)

 Mixed degree  Uniform substitution

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