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Truth functions and Interpretative function
[edit]Logical connective symbols can be defined by means of an interpretative function and a functionally complete set of truth-functions. [1] 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
- I(~)=I()=fnot
- I(&)=I(^)=I()=fand
- I(v)=I()= for
- I(~Φ)=I(Φ=I(I(Φ)=fnot(I(Φ))
- I(ΦΨ) = 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 interpretion of all its non-logical symbols. refs
- ^ Gamut, L.T.F (1991). "2". Logic, languagage and Meaning,. Vol. 1: Introduction to Logic. University of Chicago Press. pp. 54..64. ISBN 0-226-28285-3.
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