I'm having some problems with the formalization in FOL. I'll start with an example:
'' Every state has a capital but city capital of two different states exists''
I'll give you my answer with how i interpreted the phrase:
$ ( \forall (x) . ( \exists (y) . state(x) \to ( city(y) \land capital (y,x))) \land \lnot (\exists (y). ( \exists (z). \exists (h) . ( city(y) \to (state (z) \land (capital(y,z)) \land (city (y) \to (state (h) \land capital (y,h))) $
I interpreted like: for everystate, a city exists and this city is a capital of the state. And, there aren's two sames city that are the capital for one country. My problems are: generally, how do i prove if my answer is correct? What laws should i use to prove it? Because my professor had a simpler and more elegant answer than me, but i thought mine, even if longer, thought i was correct, but i can't prove it. Obviously this question does apply in all my other exercises. I've done alot of pages of them, but i don't generally know how to check them. So: - How to i check that my phrase is correct? - Any tips and tricks?
Hint: Quantifiers are related to logical connectives as follows: $\forall x(x\in A\rightarrow P(x))$ and $\exists y(y\in B\wedge Q(y))$.