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I have seen that $\forall x\in X [P(x)]$ is shorthand notation for $\forall x [x\in X\rightarrow P(x)]$. Is this formally correct? If so, could $\forall x\in X [P]$ be represented in this notation where $x$ is free in $P$? Also, I was wondering if there was a similar representation for $\exists x\in X[P(x)]$. Any help would be greatly appreciated.

W. G.
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1 Answers1

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I think what you note to be shorthand, is especially used in mathematical proofs. In formal logic, your shorthand would certainly be understood, but is not encouraged, particularly when students are first encountering formal logic.

Your second statement is correct and more formal (logically speaking): $$\forall x\big(x\in X \to P(x)\big)$$

But in situations, for example, when you want to discuss or present properties of real numbers, you can pre-define your "Domain of Discourse to be the set of real numbers" or the "universe of real numbers", after which you can simply use, e.g., in your example, $$\forall x (P(x))$$ And that is fully appropriate.

That said, you'll find many formulas in math that include as a predicate: $x \in \mathbb R,$ or $\mathbb R(x)$ meaning $x \in \mathbb R$. You'll also find many formulas in which you have a "Let ... " sentence to specify the domain of discourse first, but also using a less formal style.

For example, Wikipedia presents the "formal" epsilon-delta definition of a limit as follow:

Let $ f$ be a real-valued function defined on a subset $D$ of the real numbers. Let $c$ be a limit point of $D$ and let $L$ be a real number.

$$\lim _{x\to c}f(x)=L\iff (\forall \varepsilon >0,\,\exists \ \delta >0,\,\forall x\in D,\,0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon )$$

This is very succinct, and typical of definitions in mathematics. And is formal.

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amWhy
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  • I like your symbol in the last example. By the way, if it's allowable to choose, then $\forall x\in D,,(0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon)$ vs. $\forall x\in D,,0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon$ which is better? That is to say, should we set $\forall$ prior to $\rightarrow$? But I'm not sure if it is possible to define our own predicate logic formula construction rule so that $\forall$ has a higher priority then $\rightarrow$. – Eric Jul 24 '17 at 15:27
  • If you're talking about parentheses, I personally would use use the parentheses like you did in in your former rather than as in the example in Wikipedia. I.e. I would write $$ \lim _{x\to c}f(x)=L\iff \big( \forall \varepsilon >0,,\exists \ \delta >0,,\forall x\in D,,(0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon )\big)$$ – amWhy Jul 24 '17 at 15:41