Logic

Tautology is popularly defined two main ways: 1) An argument that derives its conclusion from one of its premises, or 2) logical statements that are necessarily true, as in (A∨~A). How are these two definitions reconciled? The second definition is only a statement; it has no premises or conclusions.

Comments

You've definitely put your finger on a problem. I'd say that for most purposes the two definitions aren't reconcilable because they belong to different discourses or contexts. The first usage is more colloquial and rhetorical. The second is a technical definition. The term "valid" is used in similarly different ways. In common discourse, one can make a "valid point"; but in technical terms only arguments or inferences, not points, can be valid. There is, however, at least one way to make the two definitions consistent: assume in the first that the premise from which the conclusion is derived is the same as the conclusion. From two premises A and B, the conclusion A follows. This, of course, becomes a variant of begging the question. Conversely, I suppose, one could argue that a tautology follows from itself, making the second definition applicable to an argument. (Note that the way you've phrased the first definition is a bit odd, since all good arguments derive their conclusions from their premises, sometimes from just one of their premises.)

There is a connection between (1) and (2) in that there is a connection between an argument's being a good one and some statement's being logically true. It can be stated somewhat generally like this: The argument whose premises are X1, X2, ..., Xn, and whose conclusion is Y is logically correct (or valid) if and only if the statement "If X1 and X2 and ... and Xn, then Y" is logically true (or a tautology). For instance, the argument whose two premises are "Either A or B" and "not-B" and whose conclusion is "A" is logically correct. And this amounts to saying that the statement "If it is both the case that A or B and the case that not-B, then A" is logically true.

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