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# Proposition and Necessity - PowerPoint PPT Presentation

Proposition and 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 ). Primordial necessity.

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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).

### 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.

### 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).

### 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.

• ### 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.

### 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’.

### 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]

### 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.

### ⧠(F)

• F = <U, N >

• N(x) is a simple hypergraph.

• ∀ A∈Φ, F ⊨ ⧠A iffN(x)R HA

• [K], [RN], [RM(⊦)]

• →?

### Necessarily (A is true)

A is necessarily true;

(Necessarily A) is true. ⊨⧠A

HA→B is interpreted as H¬A˅B.

### Articular Models (a-models)

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

### 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).

### 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.

### Problem of entailment

Anderson & Belnap

• D1 D2 … Dn

• C1 C2 … Cm

• ∀1≤ i ≤ n, ∀1≤ j ≤ m, di∩ cj≠ Ø

### 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

### 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)

• 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)

### Problem of degree

Mixed degree

Uniform substitution