Compactness of a supervaluational language

Extract

Compactness of supervaluational semantics has long been an open problem. Recently, Peter Woodruff [2] showed that the full quantificational language is not compact. In the present note, I will show that the quantifier-free fragment of the language is compact. Since this result can be easily extended to the monadic quantificational language, and since Woodruff's result only requires the presence of one binary predicate, the two results combined give a complete solution of the problem.

My language L contains infinitely many individual constants, infinitely many n-ary predicates (for every n > 0), the usual connectives, the symbols E! and =, and parentheses. (Atomic) sentences are defined as usual, and A^b/a is the result of uniformly substituting b for a in the sentence A.

An interpretation of L is an ordered pair I = 〈D, φ〉, where D is a set (possibly empty) and φ is a unary function, total from the set of n-ary predicates into the power set of Dⁿ, and partial from the set of constants into D.

A classical valuation for an interpretation I = 〈D, φ〉 is a total unary function V from the set of sentences into {T, F} such that

(a) if φ is defined for all of a₁, …, a_m, then V(Pa₁ … a_n) = T iff 〈φ(a₁), …, φ(a_n)〉 ∈ φ(P);

(b) if φ(a) and φ(b) are both defined, then V(a = b) = T iff φ(a) = φ(b;

(c) if exactly one of φ(a) and φ(b) is defined, then V(a = b) = F;

(d) V(a = a) = T;

(e) if V(a = b) = T and A is atomic, then V(A) = V(A^b/a);

(f)V(E!a) = T iff φ(a) is defined;

(g) V(~ A) = T iff V(A) = F, and similarly for the other connectives.