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.

Read more about this topic:  Truth Function

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... The principle of compositionality has been the subject of intense debate ... Indeed, there is no general agreement as to how the principle is to be interpreted, although there have been several attempts to provide formal definitions of it ... Szabó, 2012) Scholars are also divided as to whether the principle should be regarded as a factual claim, open to empirical testing a analytic truth, obvious from the nature ...

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