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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


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 = <U, R> 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 = <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
    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 = <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
    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
    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 .



    Hypergraph operations
    Hypergraph operations


    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)


    • 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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