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We call a statement truth functional if its truth value depends on truth value of its parts. Like A⊃B can be true or false, depending on truth values of A and B.

But, it's not the case with contradictions and tautologies. Like Av~A will always be true, no matter truth value of A.

From this I conclude that tautologies and contradictions are not truth functional. Am I correct? Or do I miss something?

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    No. We call an expression truth functional if its truth value can be determined from truth values of its parts alone (without extra information about their structure and "meaning"). It no more has to "depend" on them than the value of x-x in arithmetic has to "depend" on the value of x. If it can be determined even without knowing x so much the better, f(x)=0 is still a function of x.
    – Conifold
    Sep 16, 2021 at 6:40
  • See Truth function. Sep 16, 2021 at 7:06
  • @Conifold OK, got it
    – user161005
    Sep 16, 2021 at 8:28
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    Usually the logical connectives (⊃,~,V, etc) are said to truth-functional, not the statements that contain them.
    – Eliran
    Sep 16, 2021 at 15:09

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Short answer : a constant function is still a function. As long as the output exists and is unique for a given input, its a function.


Consider this mathematical function : f(x)=10 . It maps every real number to number 10. That does not prevent it from being a function.

Same thing for g [(x,y)] = 1000 . No matter what couple of numbers is taken as input, he output is 1000. It maps every point in the XY plane to number 1000.

If you consider a well formed formula as a function, it takes a truth value ( or an n--tuple of truth values) as input and gives back a truth value as output.

For example the expression " (P v ~ P)" is a function from the set : T, F to the set : T, F . For every input , it gives back as output the value : T.

The formula : " If (P-->Q) and P then Q" is a function from the set of couples : TT, TF, FT, FF to the set : T, F. For every input it gives back T as output.

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