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  1. In this question it was said in a comment that it is not good style to use $\in$ in the quantificational part of a predicate logic formula.
    Is this so? I have actually seen this quite often in definitions (like $\forall x \in \mathbb{R}:\ ...$) and find this by far more convenient than definining additonal predicates for sets that already have an unambiguous set symbol (like $\mathbb{R}$) and then spelling out the set membership by an additional predicate $P(x) \land \ ...$ (for $\exists x$) or $P(x) \to \ ...$ (for $\forall x$) respectively.
    In some statements (usually translations of natural language sentences into predicate logic such as "All bachelors are not married") this predicate version instead of $\in$ notaton might make sense, but is it in general not good style to write $\exists x \in P : R(x)$ but only $\exists x : P(x) \land R(X)$?
  2. Similarly, is it permissible to use the $\subseteq$ in the quantificational part of a statement?
    E.g. if I wanted to say "for all subsets $B$ of this particular set $A$", with the $\in$-relation, I'd have to make the rather complicated assertion $\forall B \in \mathbb{P}(A)$ by saying "for all sets that are an element of $A$'s power set".
    Could I just use $\forall B \subseteq A :\ ...$ instead? It looks weird to me, but I see no reason not to use it if $\in$ is allowed as well.
    In case neither $\in$ or $\subseteq$ is allowed, how would I formalize this statement - would I need to define a predicate "is a subset of $A$"?
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Technically, it's not appropriate, in the sense that the syntax of first-order logic doesn't allow for it.

That said, it is indeed efficient. More than that, it can be viewed as an abbreviation for perfectly correct syntax: as you observe, we can just replace "$\forall x\in A(...)$" with "$\forall x(x\in A\rightarrow . . . )$," and replace "$\exists x\in A(...)$" with "$\exists x(x\in A\wedge ...)$". (A similar trick works for "$\subseteq$".)

Whether this matters comes down to what context you are in. There are certain contexts where it's important to pay attention to the precise form of a first-order sentence, and there abbreviations can mislead you as to the true complexity of the formulas involved. However, for the vast majority of purposes there is no problem.

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    $\begingroup$ It should also be remarked that in many contexts we really don't care whether our use of quantifiers stick to the syntax of first-order logic. Often we write formulas with quantifiers in them merely as a way to be precise about how each variable is quantified and which of them may depend on each other how, and in such cases even thinking about the the bounded quantifiers as abbreviations is likely to be counterproductive. $\endgroup$ – Henning Makholm Sep 9 '16 at 14:56
  • $\begingroup$ Thank you, this seems reasonable. So there is no difference in the possibility of use between $\in$ and $\subseteq$ either, so I could write both $\forall B : B \subseteq A \to \ ...$ and $\forall B \subseteq A : \ ... $ (as long as context permits) the same way I have both notations available for $\in$? To clarify, I am aware that $\in$ is of course not the same as $\subseteq$, I am just wondering whether they are to the same degree usueable like this in predicate logic formulae, because I can't recall having seen the subset relation there very often. $\endgroup$ – lemontree Sep 9 '16 at 17:36
  • $\begingroup$ By the way, is this abbreviation the same as the use of the alternative style $(\forall x : P(x)) Q(x)$ in place of $\forall x P(x) \to Q(x)$, or is this a different issue? As far as I see it, it should be the same as what you described, just with predicates instead of $\in$/$\subseteq$, so the choice of style should be similar here? $\endgroup$ – lemontree Sep 9 '16 at 17:51
  • $\begingroup$ @lemontree Yes, that's basically right. $\endgroup$ – Noah Schweber Sep 9 '16 at 17:57
  • $\begingroup$ @NoahSchweber Just been looking at logic recently. Is it not more so the case that $\in$ has no place unless you're doing something with sets? $\in$ is a predicate of two set variables. Then once you're dealing with sets you can define notation $\forall x \in A: P(x)\; : \iff \forall x: x\in A \implies P(x)$. $\endgroup$ – snulty Nov 7 '16 at 12:04
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In pretty much all of everyday mathematical writing, not only is it good notation to use restricted quantifiers; it is actually desirable to do so consistently, for clarity and to avoid mistakes in writing. For example, $x$ is often used to denote a real number; but often enough, our mathematical statement applies only for positive or non-negative $x$, or perhaps $x$ represents a set, a vector, a function, or some other type of mathematical object. This restricted quantification also reflects how we naturally think. For example, consider $$(\forall\varepsilon)(\forall x)(\exists\delta)[ \delta\in\Bbb R_+^n\;\&\;((\varepsilon>0\; \&\; x\in\Bbb R^n)\implies X(x,\varepsilon,\delta))],$$where $X(x,\varepsilon,\delta)$ is some mathematical statement involving (unquantified) $x$, $\varepsilon$, and $\delta$. It starts off by introducing things which could be almost any imaginable sort of mathematical object. Only after this hyper-abstract introduction are we allowed to know in which sets these objects respectively reside. How much more natural it is to write $$(\forall\,\varepsilon>0)(\forall\,x\in\Bbb R^n)(\exists\,\delta\in\Bbb R_+^n)[X(x,\epsilon,\delta)].$$

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  • $\begingroup$ The example seems very reasonable, thanks. $\endgroup$ – lemontree Sep 9 '16 at 17:52

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