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  "body": "\n## Statement\n\nInstead 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*\u003csub\u003enand\u003c/sub\u003e be defined as:\n\n- *f*\u003csub\u003enand\u003c/sub\u003e(T,T) = F; *f*\u003csub\u003enand\u003c/sub\u003e(T,F) = *f*\u003csub\u003enand\u003c/sub\u003e(F,T) = *f*\u003csub\u003enand\u003c/sub\u003e(F,F) = T\n\nThen, for convenience, *f*\u003csub\u003enot\u003c/sub\u003e, *f*\u003csub\u003eor\u003c/sub\u003e *f*\u003csub\u003eand\u003c/sub\u003e and so on are defined by means of *f*\u003csub\u003enand\u003c/sub\u003e:\n\n- *f*\u003csub\u003enot\u003c/sub\u003e(*x*) = *f*\u003csub\u003enand\u003c/sub\u003e(*x*,*x*)\n- *f*\u003csub\u003eor\u003c/sub\u003e(*x*,*y*) = *f*\u003csub\u003enand\u003c/sub\u003e(*f*\u003csub\u003enot\u003c/sub\u003e(*x*), *f*\u003csub\u003enot\u003c/sub\u003e(*y*))\n- *f*\u003csub\u003eand\u003c/sub\u003e(*x*,*y*) = *f*\u003csub\u003enot\u003c/sub\u003e(*f*\u003csub\u003enand\u003c/sub\u003e(*x*,*y*))\n\nor, alternatively *f*\u003csub\u003enot\u003c/sub\u003e, *f*\u003csub\u003eor\u003c/sub\u003e *f*\u003csub\u003eand\u003c/sub\u003e and so on are defined directly:\n\n- *f*\u003csub\u003enot\u003c/sub\u003e(T) = F; *f*\u003csub\u003enot\u003c/sub\u003e(F) = T;\n- *f*\u003csub\u003eor\u003c/sub\u003e(T,T) = *f*\u003csub\u003eor\u003c/sub\u003e(T,F) = *f*\u003csub\u003eor\u003c/sub\u003e(F,T) = T; *f*\u003csub\u003eor\u003c/sub\u003e(F,F) = F\n- *f*\u003csub\u003eand\u003c/sub\u003e(T,T) = T; *f*\u003csub\u003eand\u003c/sub\u003e(T,F) = *f*\u003csub\u003eand\u003c/sub\u003e(F,T) = *f*\u003csub\u003eand\u003c/sub\u003e(F,F) = F\n\nThen *I*(~) = *I*() = *f*\u003csub\u003enot\u003c/sub\u003e; *I*(\u0026) = *I*() = *f*\u003csub\u003eand\u003c/sub\u003e; *I*(*v*) = *I*() = *f*\u003csub\u003eor\u003c/sub\u003e; *I*(~Φ) = *I*(Φ) = *I*()(*I*(Φ)) = *f*\u003csub\u003enot\u003c/sub\u003e(*I*(Φ)); *I*(ΦΨ) = *I*()(*I*(Φ), *I*(Ψ)) = *f*\u003csub\u003eand\u003c/sub\u003e(*I*(Φ), *I*(Ψ)) etc.\n\nThus if *S* is a sentence that is a string of symbols consisting of logical symbols *v*\u003csub\u003e1\u003c/sub\u003e...*v*\u003csub\u003e*n*\u003c/sub\u003e representing logical connectives, and non-logical symbols *c*\u003csub\u003e1\u003c/sub\u003e...*c*\u003csub\u003e*n*\u003c/sub\u003e, then if and only if $I(v_{1})...I(v_{n})$ have been provided interpreting *v*\u003csub\u003e1\u003c/sub\u003e to *v*\u003csub\u003e*n*\u003c/sub\u003e by means of *f*\u003csub\u003enand\u003c/sub\u003e (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*\u003csub\u003e1\u003c/sub\u003e...*c*\u003csub\u003e*n*\u003c/sub\u003e, 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.\n",
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