I've seen statements of the type $\forall x \in X, P(x)$ and also $P(x) \forall x \in X$.
Is there any logical difference?
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10$\begingroup$ $P(x)\forall x\in X$ is an abuse of notation. You can certainly write it in language, "$P(x)$ for all $x\in X,$" but it is wrong to use $\forall$ as a replacement symbol for the words "for all." In language, putting a "for all" phrase after the statement can sometimes be ambiguous, too. However, assuming this is an abuse of notation, the second is just a wrong way of writing the first. $\endgroup$– Thomas AndrewsOct 1 at 20:23
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$\begingroup$ Putting the forall sign after what it quantities is simply wrong. $\endgroup$– Mariano Suárez-ÁlvarezOct 1 at 20:42
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2$\begingroup$ You will most often forgive this abuse of notation on blackboards, where space makes abbreviation more important, or because editing a sentence after you've started it will be more pain than it is worth. $\endgroup$– Thomas AndrewsOct 1 at 20:49
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$\begingroup$ Be very careful, even if you are writing the quantifiers in English, not to mix prefix and postfix notation, as the results can be ambiguous: "there exists an integer $y$ such that $x < y$ for all integers $x$" can be read in two different ways and is true under one reading and false under the other reading. $\endgroup$– Rob ArthanOct 3 at 20:10
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$\begingroup$ I'm not sure what you mean. The way I read that $\forall x \in \mathbb{Z}, \exists y \in \mathbb{Z}, x < y$ is true, but that is not what is said, what you are saying reads as $ \exists y \in \mathbb{Z}, \forall x \in \mathbb{Z}, x < y$, which is false. $\endgroup$– user129393192Oct 3 at 20:23
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The answer to your question depends on what you mean by "logical difference." If I observed a professor scribble something on the board akin to "$x$ is $P$, $\forall x \in \mathbb{R}$," then I would make no distinction between the meaning of such a statement and that of "$\forall x \in \mathbb{R}$, $x$ is $P$" because the scope of the quantifier is clear and I understand the professor is more interested in communicating a point than expressing a syntactically correct statement in the language of a formal system. With that being said, it is indeed true that $Px\forall x$ is not a well-formed formula in the language of first order logic (FOL).
Every formal system such as FOL contains a set of rules known as the syntax, or grammar, of the system. These rules define which strings of symbols are and are not well-formed formulas of the system's language. The syntax of FOL dictates that a quantifier, either $\forall$ or $\exists$, together with its accompanying variable $x$, always appear before the formula that is within its scope. Thus, the syntax of FOL allows for strings such as $\forall x Px$, but it does not produce strings such as $Px \forall x$.
In natural language, however, there is much more flexibility in how statements are worded. One can certainly say "for every $x$, $x$ is $P$" or "$x$ is $P$ for every $x$," and the meaning will be clear to the listener. But such levels of flexibility are typically not tolerated in formal systems because the idea is to keep the language precise, unambiguous, and efficient, leaving no room for confusion.
answered Oct 1 at 22:57