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If you will eventually be teaching the basic methods of proof (conditional proof, proof by contradiction, etc.) in your course for math majors, you might consider starting with the truth tables for NOT, AND and OR and postpone the truth table for IMPLIES until they understand some of those basic methods of proof. Then they should be able to understand a ...

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Your self-answer is wrong, because a tautology must be true under every interpretation, but your example is certainly not so. There is no single word for what you want, but the standard terminology is that your formula is true under standard/intended interpretation. And you can note that it is conventional to say just "true" to mean that when ...

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In my opinion, the truth-table definition of material implication is disturbing because "if ..., then ..." is used in mathematics in two distinct ways (and no similar distinction exists for the other connectives like NOT, AND, and OR): "If $p$, then $q$" means that you can start out with $p$, make some deductions, and end up with $q$. As ...

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Expanding my comment into an answer: So you want the umbrella term for a predicate (propositional function), like the triangle or Cauchy–Schwarz inequality, that's always true, but not necessarily so regardless of interpretation. So, ‘validity’ and ‘tautology’ are ruled out. If the object is an equality instead of inequality, I'd have suggested ‘identity’. ...

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