**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 *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, *f*_{not}, *f*_{or} *f*_{and} and so on are defined 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 *f*_{not}, *f*_{or} *f*_{and} and so on are defined 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*(¬)=*f*_{not}*I*(&)=*I*(^)=*I*(&)=*f*_{and}*I*(*v*)=*I*=*f*_{or}*I*(~*Φ*)=*I*(¬*Φ*)=*I*(¬)(*I*(*Φ*))=*f*_{not}(*I*(*Φ*))*I*(*Φ*&*Ψ*) =*I*(&)(*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 *v*_{1}...*v*_{n} representing logical connectives, and non-logical symbols *c*_{1}...*c*_{n}, then if and only if *I*(*v*_{1})...*I*(*v*_{n}) have been provided interpreting *v*_{1} 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 *c*_{1}...*c*_{n}, i.e. of *I*(*c*_{1})...*I*(*c*_{n}). 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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