Principle of compositionality

Statement

Instead of using truth tables, logical connective symbols can be interpreted by means of an interpretation function and a functionally complete set of truth-functions (Gamut 1991), as detailed by the principle of compositionality of meaning. Let $I$ be an interpretation function, let $\Phi, \Psi$ be any two sentences and let the truth function f_nand be defined as:

• f_nand(T,T) = F; f_nand(T,F) = f_nand(F,T) = f_nand(F,F) = T

Then, for convenience, f_not, f_or f_and and so on are defined by means of f_nand:

• f_not(x) = f_nand(x,x)
• f_or(x,y) = f_nand(f_not(x), f_not(y))
• f_and(x,y) = f_not(f_nand(x,y))

or, alternatively f_not, f_or f_and and so on are defined directly:

• f_not(T) = F; f_not(F) = T;
• f_or(T,T) = f_or(T,F) = f_or(F,T) = T; f_or(F,F) = F
• f_and(T,T) = T; f_and(T,F) = f_and(F,T) = f_and(F,F) = F

Then I(~) = I() = f_not; I(&) = I() = f_and; I(v) = I() = f_or; I(~Φ) = I(Φ) = I()(I(Φ)) = f_not(I(Φ)); I(ΦΨ) = I()(I(Φ), I(Ψ)) = f_and(I(Φ), I(Ψ)) etc.

Thus if S is a sentence that is a string of symbols consisting of logical symbols v₁...v_n representing logical connectives, and non-logical symbols c₁...c_n, then if and only if $I(v_{1})...I(v_{n})$ have been provided interpreting v₁ to v_n by means of f_nand (or any other set of functional complete truth-functions) then the truth-value of $I(s)$ is determined entirely by the truth-values of c₁...c_n, i.e. of $I(c_{1})...I(c_{n})$. In other words, as expected and required, S is true or false only under an interpretation of all its non-logical symbols.