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 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, we define f_not, f_or f_and etc. 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 we define f_not, f_or f_and, etc 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($\neg$)=f_not
• I(&)=I(^)=I($\wedge$)=f_and
• I(v)=I($\lor$)= f_or
• I(~Φ)=I($\neg$Φ=I$\neg$(I(Φ)=f_not(I(Φ))
• I(Φ$\wedge$Ψ) = I($\wedge$)(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 interpretion of all its non-logical symbols. refs