Truth Function - Principle of Compositionality

Principle of Compositionality

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 Φ, Ψ 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(^)=I(&)=f_and
• I(v)=I= f_or
• I(~Φ)=I(¬Φ)=I(¬)(I(Φ))=f_not(I(Φ))
• I(Φ&Ψ) = 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(v₁)...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(c₁)...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.