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A tautology is a statement which is true by dint only of the logical connectives contained therein. My question is about an open formula which contains an unquantified variable. For example:

  • P: ($x$ is a cat) or not ($x$ is a cat).

This appears to be a tautology, and is certainly true in any universe when we precede the statement by a "$\forall x$".

QUESION: Is P considered to be 1) a tautology and/or 2) an open formula?

ryang
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Trevor J Richards
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  • I don't know about terminology, but $P$ gives a way of assigning to each model $M$ of the relevant axioms a predicate $P_M : M \rightarrow \mathbf{Bool}$ such that for all $x \in M$, we have that $P_M(x)$ is true. So $P_M$ doesn't literally equal "true"; rather, it is the case that for all $x \in M$, the value $P_M(x)$ equals "true." Hope that helps. – goblin GONE Apr 21 '16 at 15:15
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    Good question. The answer, unfortunately, is "it depends." In some presentations of logic, a truth value in the structure $M$ is not assigned to such open statements. In other presentations, the open statement is assigned the same truth value in $M$ as the sentence obtained when all the free variables are universally quantified. – André Nicolas Apr 21 '16 at 15:18
  • When I took that class they told me that a tautology in predicate logic was a substitution instance of a tautology in propositional logic. So yes, your $P$ is a tautology; it's obtained by subtituting "$x$ is a cat" for $A$ in the tautology "$A$ or not $A$". – David C. Ullrich Apr 21 '16 at 15:20
  • @AndréNicolas Thanks very much. Now a stupid question: Which is the "right" way to do it? I think I have an aesthetic preference for the latter option you mentioned. Is there a down-side to assigning truth-values to open statements in this way? – Trevor J Richards Apr 21 '16 at 15:22
  • @AndréNicolas Sounds like your definition of "tautology" is different from what I was taught. Are you saying any valid first-order sentence is a tautology? Just curious.. – David C. Ullrich Apr 21 '16 at 15:22
  • There are various opinions here on the definition of "tautology" in predicate logic. For what it's worth I looked, and wikipedia agres with my definition "In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula". https://en.wikipedia.org/wiki/Tautology_(logic) – David C. Ullrich Apr 21 '16 at 15:28
  • @DavidC.Ullrich: When doing predicate calculus, I do not use the term "tautology" and I prefer to not assign truth values in an $L$-structure to formulas with free occurrences of variable symbols. However, that approach does have advantages when we define truth in $M$ via valuations. One problem with answering "logic" questions is that definitions that people use tend to vary fairly widely, albeit in inessential ways. – André Nicolas Apr 21 '16 at 15:30
  • @AndréNicolas Yes, that's a problem, heh. In fact the definition of "tautology" that I recall from years ago (Enderton) is the same as what Wikipedia currently says; that definition simply has nothing to do with structures for first-order logic. – David C. Ullrich Apr 21 '16 at 15:32
  • @TrevorRichards: I (slightly) prefer not to assign truth values to formulas with free occurrences of variables. – André Nicolas Apr 21 '16 at 15:33
  • @DavidC.Ullrich: Another complication is that nowadays "propositional logic" and the full predicate calculus are the only items discussed in a course in mathematical logic. However, there is a traditional branch of logic that sits uneasily between the two, what one could call monadic predicate calculus, with its own terminology. – André Nicolas Apr 21 '16 at 15:48
  • Very interesting discussion. Thanks to André Nicolas and David C. Ullrich . – Trevor J Richards Apr 21 '16 at 15:57

2 Answers2

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Of course this depends on the definition of "tautology" in predicate logic. The definition I learned years ago is the same as what Wikipedia says: "In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable)." By that definition yes, $P$ is a tautology, being a substitution instance of the propositional-logic tautology "A or not A".

(Regarding your doubt about open formulas, note it says "formula" above, not "sentence".)

We should probably note that this definition of "tautology" is purely syntactic, has nothing to do with assigning truth values in structures for first-order logic. Which it seems to me is as it "should" be; it "should" be purely syntactic. (Although all the other definitions we've seen in this thread involve the semantics of first-order logic I've never seen a definition of "tautology" anywhere else that was not syntactic.)

David C. Ullrich
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    I just want to emphasize that, in the context of formal logic, "tautology" definitely is a purely syntactic notion. Unfortunately, it seems common to introduce propositional logic, usually ones first introduction to formal logic, by identifying it with Boolean algebra (perhaps unintentionally). It (apparently) takes a while before people realize they are not the same thing. I strongly suspect many students would be surprised to be told that a truth table is not a proof. I'm not suggesting the commenters are making this conflation, but I am suggesting they are feeding in to it. – Derek Elkins left SE Apr 21 '16 at 23:54
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An open formula is not possible to evaluate at all. However one often define that $M\models \varphi(x)$ if $M\models \forall x\varphi(x)$. In this case if we translate $\forall x\varphi(x)$ to propositional logic, we only get a single propositional variable $P$ which certainly is not a tautology. So in general I would say No.

However a formula such as $\forall x(P(x)\rightarrow P(x))$ is equivalent to $\forall xP(x)\rightarrow \forall xP(x)$. If we translate $\forall xP(x)\rightarrow \forall xP(x)$ to propositional logic we get $R\rightarrow R$, which is a tautology. Thus we may possibly be able to say that $P(x)\to P(x)$ is a tautology, since if we add $\forall x$ it is equivalent to a formula which may be translated to a tautology. But if this is allowed, depends on how you define a tautology in predicate logic.

Ove Ahlman
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  • Everyone agrees that it depends on the definition of "tautology". I've actually seen a definition in several places. Every time I've seen a definition it's been the same as what I learned in a logic course years ago. Hence the question: Have you actually seen any of the various definitions you mention here? – David C. Ullrich Apr 21 '16 at 15:55