R. E. Jennings
[email protected]
Y. Chen
[email protected]
Laboratory for Logic and Experimental Philosophy
Simon Fraser University
 What
 The
is a proposition?
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.
R
is universal.
⧠
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).
is a simple hypergraph iff ∀E, E’∈ H,
E⊄E’.
H
 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
= <U, N’ > if N’ (x) is a simple hypergraph.
 PL closed under [RM].
L
• 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, 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 iff N(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├AVB
•
•
•
•
•
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).
(A is true) iff ∀ E ∈ HA, ∃ v ∈ E
such that ∃ v’ ∈ E: v’ = U – v. (N)
 Necessarily
 (N)
is closed under ⊦ and ˄.
 A⊦B
/ necessarily A→B is true.
Anderson & Belnap
 D1
D2 … Dn
 C1
C2 … C m

∀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