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I read in some sources that valid formulas are tautologies (valid under every evaluation). In the others, I read that these are formulas that have conclusions true when premises are true. Are these just equivalent definitions because ⊨ P → Q is equivalent to P ⊨ Q?

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Yes. These definitions are equivalent, noting that not all valid formula have the form $P\to Q$, though any formula $Q$ is equivalent to $\top\to Q$.

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  • $\begingroup$ wait, do you mean that we want the ⊨ P → Q form because every valid formula is equivalent to ⊤→Q? $\endgroup$ – Val Sep 25 '12 at 15:19
  • $\begingroup$ If you want the two statements to be equivalent, then I need to be able to talk about statements that are not of the form $P\to Q$. Fortunately, any formula (not just any valid formula) $Q$ is equivalent to $\top\to Q$. $\endgroup$ – Dave Clarke Sep 25 '12 at 19:55
  • $\begingroup$ wait, wait. I wanted to know why there are two definitions of valid formula. True in every interpretation/valuation, the always valid tautology is designated ⊨ Q. Now you say that any formula Q can be denoted ⊨ Q. Something is wrong then. BTW, T in your formula, does it stay for Theory or True? $\endgroup$ – Val Sep 25 '12 at 20:35
  • $\begingroup$ $\top$ means true. It is possible to have multiple definitions of a concept, as long as they are equivalent. Different definitions may be better suited for different purposes. I never said that any formula $Q$ can be denoted $\models Q$. I merely gave you a tautology, namely $Q\leftrightarrow (\top \to Q)$. $\endgroup$ – Dave Clarke Sep 25 '12 at 22:07

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