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As far as I understand, Tarski's semantic conception of truth

(T) X is true if and only if p

(where p is a sentence of the object language whose truth value is in question, and X is the name of the sentence expressed in metalanguage to which the truth predicate applies) is an important contribution to mathematical logic.

Yet, it does not seem to me that this conception of truth is often used (if at all) in mathematical logic textbooks. Rather one usually assumes that a sentential variable p if true if v(p)=1 (where v is a truth assignment of the set of sentential variables to, say, {0,1}), before to extend the definition by recursion to all sentences of the language.

Why is Tarski's conception of truth simply ignored in mathematical logic textbooks?

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    Do you maybe mean that Tarski does not show up in introductory logic materials? There are certainly textbooks that mention him... (Note Tarski has written an intro to logic book.) – Joseph Weissman Dec 9 '16 at 13:56
  • Is not the derivation of a truth valuation from an interpretation a standard thing? Or is there more to "Tarski's semantic conception of truth" than can I have gleaned from a quick search? – Hurkyl Dec 9 '16 at 15:25
  • Extension by recursion is a direct consequence of the Tarski's conception of truth. And so is the notion of semantic validity which also dominates logic textbooks. If anything, Tarski's conception is used too much to the exclusion of others, see Why is Tarski's notion of logical validity preferred to deductive one? – Conifold Dec 9 '16 at 19:22
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It is not.

See any mathematical logic textbook; e.g. Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001), page 83.

To say that:

M⊨t=s[v] iff v(t)=v(s)

where t and s are terms, i.e."names" and v(t) and v(s) are the elements of the domain of the interpretation M (i.e. objects) that are the reference (assigned by the function v) of the said terms is nothing other than saying:

the sentence t=s holds (it is true) in M iff t is equal to s.

See also Tarski's Truth Definition and :

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