User:Philogo/LogconSandbox

Truth functions and Interpretative function
Logical connective symbols can be defined by means of an interpretative function and a functionally complete set of truth-functions. Let I be an interpretative function, from sentences onto {true,false}, let Φ, Ψ be any two sentences and let the following the truth function fnand be defined as:-
 * fnand(T,T)=F; fnand(T,F)=fnand(F,T)=fnand(F,F)=T

Then, for convenience, we define fnot, for fand etc. 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 we define fnot, for fand, etc 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

etc.
 * I(~)=I=fnot
 * I(&)=I(^)=I(🇦🇩)=fand
 * I(v)=I= for
 * I(~Φ)=I(Φ=I(I(Φ)=fnot(I(Φ))
 * I(Φ🇦🇩Ψ) = I(🇦🇩)(I(Φ), I(Ψ))= fand(I(Φ), I(Ψ))

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 interpretion of all its non-logical symbols. refs