# 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 fnand be defined as:

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

Then, for convenience, fnot, for fand and so on are defined by means of fnand:

• fnot(x)=fnand(x,x)
• for(x,y)= fnand(fnot(x), fnot(y))
• fand(x,y)=fnot(fnand(x,y))

or, alternatively fnot, for fand and so on are defined directly:

• fnot(T)=F; fnot(F)=T;
• for(T,T)=for(T,F)=for(F,T)=T;for(F,F)=F
• fand(T,T)=T; fand(T,F)=fand(F,T)=fand(F,F)=F

Then

• I(~)=I(¬)=fnot
• I(&)=I(^)=I(&)=fand
• I(v)=I= for
• I(~Φ)=IΦ)=I(¬)(I(Φ))=fnot(I(Φ))
• I(Φ&Ψ) = I(&)(I(Φ), I(Ψ))= fand(I(Φ), I(Ψ))

etc.

Thus if S is a sentence that is a string of symbols consisting of logical symbols v1...vn representing logical connectives, and non-logical symbols c1...cn, then if and only if I(v1)...I(vn) have been provided interpreting v1 to vn by means of fnand (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...cn, i.e. of I(c1)...I(cn). In other words, as expected and required, S is true or false only under an interpretation of all its non-logical symbols.