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Is "$(\exists z)$ the sky is blue" a proposition? I am unsure as this sentence certainly states something which could be either true or false; however the $(\exists z)$ seems meaningless here- I am unsure if the fact that $z$ is unused after it has been quantified is relevant here.

Thanks

Angelo
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Princess Mia
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2 Answers2

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It is a proposition. Here the predicate $P(z)$ would be the statement "the sky is blue." Although this has nothing to do with $z$, it is still a valid predicate in $z$ (analogous to a constant map from one set to another).

Does there exist some object $z$ such that the sky is blue? Why yes, no matter what object $z$ you pick, the sky is still blue. So in fact, we have the more general proposition: "$(\forall z)\text{ the sky is blue.}$"

Of course, this is assuming the sky really is always blue...

JMM
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    A tautology is a propositional form that is always valid, whereas a validity is just that. There’s nothing tautological about “the sky is blue” even if it is always true. It is a validity if we take it as an axiom, but it’s not an entailment of FOL in general. – PW_246 May 30 '23 at 04:23
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    The proposition "for some object, the sky is blue" is not a tautology, which is a very strong claim, nor even a validity, merely true in the standard interpretation (our world). – ryang May 30 '23 at 04:46
  • "the predicate P(z) would be the statement "the sky is blue" "... There is no $z$ in the statement "the sky is blue". It is simply a statement $P$. The OP's question is quite confusing, mixing natural language and predicate-logic: in natural language it is quite unnatural to say "there is something such that the sky is blue", while in predicate logic there is nothing wrong in writing $\exists x \varphi$ also in case that $x$ is not free in $\varphi$. – Mauro ALLEGRANZA May 30 '23 at 06:55
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You’re referring to vacuous quantification. While it is kind of odd, vacuous quantification is innocuous/redundant at the level of syntax, and it simplifies things at the level of semantics.

That is, $P$, $\exists x P$, and $\forall x P$ are all equivalent for a formula $P$ with $x$ not free. Vacuous quantification then allows us to more easily define what constitutes a well-formed formula (wff); namely, if $P$ is a formula and $x$ a variable, then $\forall x P$ and $\exists x P$ are formulas.

PW_246
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