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Have a hard time to conclude when a given predicate logic formula is invalid/valid. For example, this is one that I have spent a lot of time on

$$\vdash \exists x P(x) \wedge \exists x (P(x) \rightarrow Q(x)) \rightarrow \exists x Q(x)$$

To start with I have just looked at it but it makes no sense at all. So I tried to decompose it into words

If for some P and if some P result in some Q is true, then some Q is true.

But it does not really make it easier. How should I tackle a problem like this? If I conclude it is valid I "just" perform natural deduction, but if it is invalid (as I expect), how should I start constructing a counter example when I do not really understand the formula itself?

Natalie Clarius
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Salviati
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    If (there is a number that is Even and there is a number that (if it is Even, then it is a Mult of 2)), then there is a number that is a Mult of 2. – Mauro ALLEGRANZA Jan 03 '20 at 12:32
  • That makes much more sense! It sounds then like this expression is valid, right? – Salviati Jan 03 '20 at 12:44
  • No, a single example does not mean that the formula is always true. Try with $x > 0$ for $P$ and $x < 0$ for $Q$ and $\mathbb N$ as domain. – Mauro ALLEGRANZA Jan 03 '20 at 14:25

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