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There seems to be two forms of the conditional statement in predicate logic.

$\forall x\,(P(x)\Rightarrow Q(x))$

versus

$(\forall x\in S)\Rightarrow Q(x)$

$S=\{x:P(x)\}$

Are these equivalent? I'm a bit confused in the second form because It looks like $Q(x)$ is an open sentence however I know that this is just the universal statement:

$(\forall x\in S)Q(x)$

albeit written more explicitly. Are these two statements the same? Is the scope different? If the scope is restricted, or if they are not the same, what is their difference? To me the first says "for every $x$, if $x$ is $P(x)$ then $x$ is a $Q(x)$" and the second says "for every $x$ that is a $P(x)$, aforementioned $x$ is a $Q(x)$". Is more presupposed in this case?

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    I've removed [tag:analysis] tag. I don't really see relevance of this tag for this question.2012-08-02

2 Answers 2

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Your second formula "$(\forall x \in S)\Rightarrow Q(x)$" is not well-formed at all. What comes to the left of $\Rightarrow$ must be a complete formula, and "$\forall x\in S$" is not a complete formula -- it's a dangling quantifier with no body formula.

It is true, however, that $(\forall x \in S) Q(x)$ (without the spurious $\Rightarrow$) is the same as $(\forall x)(x\in S \Rightarrow Q(x))$ -- usually the former is considered a mere abbreviation of the latter.

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    fair enough, I just thought I would leave it open a little while longer in case others were interested in adding insight. but the connection I was speaking of was in reference to $\forall x\,(P(x)\Rightarrow Q(x))$. in your answer the connection you mention is between $(\forall x \in S) Q(x)$ and $(\forall x)(x\in S \Rightarrow Q(x))$. which doesn't make explicit the connection between the element test $x\in S$ and $P(x)$. Thats all.2012-07-31
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You can easily prove that $S=\{x:P(x)\}\rightarrow (\forall x(P(x)\rightarrow Q(x))\leftrightarrow (\forall x\in S)Q(x))$ where $(S=\{x:P(x)\})\equiv \forall x (x\in S \leftrightarrow P(x))$and $(\forall x\in S)Q(x)\equiv \forall x(x\in S \rightarrow Q(x))$ See my formal proof at http://www.dcproof.com/skyfire.htm