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I am in trouble , I searched a lot about how to solve this kind of questions but I did not get any answers.

I understand how can I know when the sentences is valid and Unsatisfiable in propositional logic, but in FOL I can't.

Can someone help me how can I solve this kind of questions? Because it comes always in the past exams in AI.

Let us have the following:

1- ¬(Speed(Processor(MyPC))=Speed(Processor(MyPC))) 
2- ¬ ∃ x ( Speed(x) = Speed(MyPC) ) 
3- Speed(MyPC)=MyPC 
4- ∀ y ∃ x (Speed(x) ∧ Speed(y))= Speed(MyPC)

Note :

    MyPC: 
    a constant symbols that represents my PC.

    Speed(x): 
    a function symbol that refers to speed of x

The answer for 1 is Syntactically wrong
2 is Unsatisfiable
3 is Neither valid nor unsatisfiable
4 is Syntactically wrong

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  • 1
    $\begingroup$ Welcome to Computer Science! The title you have chosen is not well suited to representing your question. Please take some time to improve it; we have collected some advice here. Thank you! $\endgroup$ – Raphael Jan 1 '17 at 20:04
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1 uses a function symbol Processor which is not in the logic signature. Hence, it's outside the syntax.

4 tries to equate a proposition with Speed(MyPC). You can't equate propositions in FOL. Outside the syntax.

3 is not universally true for every function Speed and every argument MyPC. For instance interpret MyPC as the natural $0$, and Speed as the successor function to obtain $\mathsf{succ}(0)=0$ which is false. Hence the formula is not valid. However, it is satisfiable: now interpret MyPC as zero and Speed as the identity function to obtain $\mathsf{id}(0)=0$ which is true. The formula has a model and a countermodel: satisfiable and invalid.

2 ∃ x ( Speed(x) = Speed(MyPC) ) is always true, since we can take x to be MyPC. This holds no matter how we interpret Speed and MyPC. Hence is it valid. (Alternatively, we can even show it is a theorem of FOL using a proof system, and then rely on the correctness theorem to claim it is valid.) Since formula 2 is the negation of a valid formula, it is unsatisfiable.

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