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Proposition and Necessity

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Propositionand Necessity

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 = <U, R> 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.

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.

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’.

- 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]

- 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.

- F = <U, N >
- N(x) is a simple hypergraph.
- ∀ A∈Φ, F ⊨ ⧠A iffN(x)R HA

- [K], [RN], [RM(⊦)]
- →?

A is necessarily true;

(Necessarily A) is true. ⊨⧠A

HA→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 ∈ HA, ∃ v∈ E such that ∃ v’∈ E: v’ = U – v. (N)

(N) is closed under ⊦ and ˄.

A⊦B / necessarily A→B is true.

Anderson & Belnap

- D1 D2 … Dn
- C1 C2 … Cm
- ∀1≤ i ≤ n, ∀1≤ j ≤ m, di∩ cj≠ Ø

C1 C2 … Cn

C1 C2 … Cm

∀1≤ i ≤ n, ∃1≤ j ≤ m, cj⊆ di

∀1≤ i ≤ n, ∃1≤ j ≤ m, cj⊢ di

((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