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I have just learnt the identity ¬(∀xP(x)) ≡ ∃x¬P(x). I have noticed that in the place of P(x), there could be a proposition which has seemingly nothing to do with x, and this equivalence would still hold- for example, in the place of P(x) there could be the proposition "The sky is blue". This has led me to ask 2 questions:

  1. Is what I have noticed correct for this identity and for the related identity ¬(∃xP(x)) ≡ ∀x¬P(x)?
  2. Is a proposition which follows a quantifier in the same statement always regarded as a function of the variable referenced in the quantifier , even though this proposition may not use this variable amongst its own propositional variables? e.g., is "The sky is blue" still regarded as a function of x if it used above?

Thanks

Princess Mia
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    Think about the function $f(x)=3$. It is a function of $x$, albeit constant. – Jason May 24 '23 at 07:06
  • In general, $\forall x A \equiv A$ if $x$ is not free in formula $A$. – Mauro ALLEGRANZA May 24 '23 at 07:23
  • The so-called "propositional function" $P(x)$ is a function of variable $x$. If we add a quantifier in front, like e.g. $\forall x P(x)$ the result is a sentence, i.e. an expression whose truth value does not depend on the value assigned to variable $x$. "Everything is Mortal", like "The sky is blue" is not a function of $x$. – Mauro ALLEGRANZA May 24 '23 at 07:24
  • If the universe is nonempty, then $∀x,Q$ and $∃x,Q$ and $Q$ are equivalent to one another. This is why $( ¬∀x,A ≡ ∃x,¬A ).\quad$ 2. In that link, in case you are referring to my previous answer, no I did not claim what you said. $\quad$ 3. That question in the title (and in the body) doesn't actually reflect what you're trying to ask!
    • – ryang May 24 '23 at 07:34
  • @ryang thank you. Just to help me out, what title would you suggest for this post/ how to ask this question? Additionally, would you be able to elaborate on how a nonempty universe would affect this equivalency? – Princess Mia May 24 '23 at 07:35
  • @mauroALLEGRANZA as jason commented, what would be the reasoning behind deeming "everything is mortal" as not a function of x, if it could be a function of it and not make reference to it, like the example he posted? – Princess Mia May 24 '23 at 07:38
  • Not clear: "it is a Man" is the natural language equivalent of "x is Mortal": in order to verify its truth value we have to decide what is the reference of pronoun "it" (of variable "x"). "Everything is Mortal" is either true or false, fullstop. – Mauro ALLEGRANZA May 24 '23 at 07:41
  • @ryang sorry for the misinterpretation; this part has been deleted. – Princess Mia May 24 '23 at 07:49
  • @God I mean, clearly, in $∀x∀y(Px→Qy),: Px$ trivially isn't a function of $y;$ so, I suspect that your intended question is less trivial than what you actually wrote in the title and in point #2. $\quad$ P.S. Thanks for that edit. – ryang May 24 '23 at 10:29
  • @ryang when you state "If the universe is nonempty, then ∀xQ and ∃xQ and Q are equivalent to one another", just to clarify, Q doesn't depend on x in any way here right? If it did, it seems it would hold unless it were exactly in the form I have cited with the negations – Princess Mia May 24 '23 at 18:07
  • @God Don't understand your final sentence; yes $x$ is not free in $Q$ (which I use to denote a sentence not a predicate). – ryang May 24 '23 at 18:11
  • @ryang could P(x) represent a sentence here? My course note say ¬(∀x∃yP(x, y)) ≡ ∃x¬(∃yP(x, y)) ≡ ∃x∀y¬P(x, y), and justify it using ¬(∀xP(x)) ≡ ∃x¬P(x). What would be the rationale behind deeming these sentences functions of x, when I have seen a distinction between propositional functions of x and sentences? – Princess Mia May 24 '23 at 20:17
  • @God You seem to be painting some contrast, but I don't see any. Anyhow, $P(x)$ contains a free variable so isn't a sentence. – ryang May 25 '23 at 03:48