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Proposition and Necessity. R. E. Jennings [email protected] Y. Chen [email protected] 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

Propositionand Necessity

R. E. Jennings

[email protected]

Y. Chen

[email protected]

Laboratory for Logic and Experimental Philosophy

Simon Fraser University

slide2
What is a proposition?

The set of necessities at a point ⧠(x).

primordial necessity
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 = is a filter.

the leibnizian account
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
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
Necessities in CPN frame

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.

entering hypergraph
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
Locale frame
  • 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]
hypergraph semantics
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.
slide10
⧠(F)
  • F =
    • N(x) is a simple hypergraph.
    • ∀ A∈Φ, F ⊨ ⧠A iffN(x)R HA
  • [K], [RN], [RM(⊦)]
  • →?
necessarily a is true
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
Articular Models (a-models)

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

slide15
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
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
Problem of entailment

Anderson & Belnap

  • D1 D2 … Dn
  • C1 C2 … Cm
  • ∀1≤ i ≤ n, ∀1≤ j ≤ m, di∩ cj≠ Ø
a b con d
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
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)

slide20
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
Problem of degree

Mixed degree

Uniform substitution

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