---
tag: PT0G50689S
kind: statement
role: proposition
name: Principle of compositionality
sources:
- wikipedia: en:604707@r1358780673
locator: Truth function, section Principle of compositionality
note: adopted
mention: wikipedia://en/604707#NAPE
sections: [Statement]
inventory: [5PBQ9RW337]
---
## Statement
Instead of using [[truth tables|05RSK4Q657]], 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 $\Phi, \Psi$ 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*() = *f*and; *I*(*v*) = *I*() = *f*or; *I*(~Φ) = *I*(Φ) = *I*()(*I*(Φ)) = *f*not(*I*(Φ)); *I*(ΦΨ) = *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.