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I learned in discrete math class that a sentence like $x+2=4$ is not a proposition, because the truth value depends on the value of $x$. It is a propositional function. However, my question is, are propositional functions which are always true or always false, themselves propositions? So, would $x=x$ and also $x \neq x$ be propositions? Note, I am not talking about quantifiers, I mean just $x=x$ by itself, not $(\forall x) x =x $.

user107952
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    $\forall x(x=x)$ is a proposition, as is $\exists x(x=x)$; with $x$ a dummy variable, $x=x$ is a unary [predicate](https://en.wikipedia.org/wiki/Predicate(mathematical_logic))_, as is something less trivial such as $x+2=4$. – J.G. Feb 01 '23 at 22:19
  • $x+2=4 \iff x \in {2}$. Similarly $x=x \iff x \in \Omega$ and $x \not=x \iff x \in \emptyset$. – Henry Feb 01 '23 at 22:22
  • In this answer, I explained that while $x=x$ is logically valid, it is technically nevertheless an open formula (propositional function) rather than a sentence/proposition. – ryang Feb 02 '23 at 03:28

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