there.
I was taking a look at the solution to a First Order Logic (FOL) problem I had that asked if a statement was FOL Valid/Consequent. I was not able to produce a counter-example world, and when I saw the solution, I realized that my instructor had replaced one of the predicates that took a single argument, with a predicate that took two arguments!
Is this even possible!? And if so, what governs the rules behind it?
The statement is as follows...
∃x(P(x) ^ Q(x))
∃x(P(x) ^ R(x))
Therefore... ∃x∃y(P(x) ^ P(y) ^ x≠y)
The provided counterexample is as follows.
Let... P(x) mean Small(x)
Let... Q(x) mean Cube(x)
Let... R(x) mean Left(x,a)
These being the case, and these being plugged into the original FO statements, the premises would be true and the conclusion would be false.
∃x(Small(x) ^ Cube(x))
∃x(Small(x) ^ LeftOf(x,a))
Therefore... ∃x∃y(Small(x) ^ Small(y) ^ x≠y)
I appreciate any insight that anyone may give regarding this question.
Thank you ^^~