In this paper we shall use a logic with truth values ranging over all the real numbers x such that 0 ≦ x ≦ 1.1 will be “complete truth” and 0 will be “complete falsity.” The primitive sentential connectives are ‘⊃’ and ‘∼’; other connectives are ‘∨’ and ‘·’. Assume that ‘p’ and ‘q’ are sentential variables, whose truth values are respectively x and y. Then

1.1. ‘p ⊃ q’ has the value min(1 − x + y, 1),

1.2. ‘∼p’ has the value 1 − x,

1.3. ‘p∨q’ has the value max(x, y), and

1.4. ‘p·q’ has the value min (x, y).

‘∨’ and ‘·’ can be defined as follows:

It is the purpose of this paper to prove a theorem which will be stated in the next section. The following symbolism and convention will be used throughout the paper:

S is a logical formula.

ν (S) is the value of S.

‘p’, ‘pi1, ’p2, …, ‘q’, are sentential variables.

ν(p) = x and ν(x1) = x1, etc.

ν(S) = σ and ν(S1) = σ1, etc.

If S contains the sentential variables ‘p1’, ‘p2’, …, then we write for S, S(p1, P2, …). Also ν{S(p1, p2, …)) = σ(x1, x2, …).

A logical formula is defined in the usual manner. 1. A sentential variable is a logical formula; 2. if S is a logical formula then ·S is a logical formula; and 3. if S and S′ are logical formulae then (S ⊃ S′) is a logical formula.