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Consider the following two statements:

Statement 1. There exists a $x\in S$ such that $p(x)$ holds, which can be written by notation as $\exists x\in S: p(x).$

Statement 2. For any $x\in S$, $p(x)$ holds, which can be written by notation as $\forall x\in S: p(x).$

But Zorich , in his book, interpretes the first one as $$\exists x\in S: p(x)=\exists x( x\in S \color{red}{\wedge} p(x)),$$ and the second one as $$\forall x\in S: p(x)=\forall x( x\in S \color{red}{\to} p(x)).$$

Why one is "AND" and another is "IMPLY"?

The cited webpage has not answer my question，because it just tell that all things are arranged like this， but do not give the reason why to make such arrangment.

WuKong
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1 Answers1

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I think it is better to view the question in the universe of discourse, so in the first statement $\exists x\in U:p(x)$, in the case of this element it has two properties, one it belongs to the universe of discourse AND it satisfies the conditions of $p(x)$.

For an analogy, say the universe of discourse would be planets in the Milky Way and $p(x)$ is that the planet has radius bigger than (insert some number). Here x just stands for a planet, but x could be any planet but putting it a slightly modified syntax of your first statement.

$$\exists x\in U:p(x) = \exists(x\in U \land p(x))$$

Now we see clearer that we need the two properties to hold. For the case of the latter statement, if we stick with the analogy of the former example, $$\forall x\in U: p(x) = \forall x(x\in U \rightarrow p(x))$$ This may make it clearer that the right hand of the equal sign start by saying out of ALL planets that would be all planets in the entirety of space, if that planet belongs to the Milky Way then it also has a radius bigger than (insert some number)

So to summarise, in the confusing can stem when you don't think in terms of your universe of discourse. Since all planets of the Milky Way have a radius bigger than (insert some number) then simply if a planet resides there it is satisfied but not all planets with a radius bigger than (insert some number) reside in the milky way.

I have recently self learned some logic from Daniel J. Velleman's book "How to prove it", I hope this crude analogy helps your understanding as I am not capable of giving you a rigorous answer.

Gabríel Snær Völuson
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