What is to do if I have to give one correct Interpretation of a formula F?

Question

How do I solve a task such as: "Give one correct Interpretation for the following formula F".

$$F = \forall a \exists b~[P(a,b) \land P(f(a,b),a) \land P(f(a,b),b) ]$$

First, I define a modell $M=\langle \Bbb N,\leq_{\Bbb N} ,+_{\Bbb N} \rangle $. Then I choose my interpretations. $$\mathcal I(P)=\leq_{\Bbb N}, \mathcal I(f)=+_{\Bbb N} $$

Then I get: $$F = \forall a \exists b ~ [(a\leq b) \land ((a+b) \leq a) \land ((a+b)\leq b) ]$$

Now I have to look if this formula is true for all N. So can I say, for all a there is a b that this formula F is true.

So, I chosse $a=5$ in my first case and now there should be one value for b that F evaluates as correct.

$$F = \forall a \exists b [5\leq b \land (5+b) \leq 5 \land (5+b)\leq b ]$$ So for each a there is b=0 then that formula F evaluates to true. This means, the model M is correct

Answer 1

You chose $a=5$ and found $b=0$ which makes the formula true. Great, that's solid evidence.

But it's not a proof.

To prove that the model $M$ is a correct interpretation of the formula $F$, you must do the same with $a=4$, and $a=3$, with $a=42$, with $a=485430676$, and in fact you must do the same for each $a \in N$ --- that's what the quantifier $\forall$ instructs you to do.

Answer 2

Use $M= \langle \mathbb{N}, \ge, +\rangle$ and for any $a$ we can choose $b=0$ to have $a \ge 0$ (correct), $a+0 \ge a$ (correct) and $a+0 \ge 0$ (correct) , whatever $a$ is.

Answer 3

If you're just looking for one interpretation that satisfies your formula, you can just choose an interpretation where $P(x,y)$ is true for all $x$ and $y$.

You can then let $f$ mean whatever you want.