Show that the following formulas are consistent

Question

How do I prove if the following formulas are consistent?

∀x$\neg$S(x,x)

∃x P(x)

∀x∃y S(x,y)

∀x(P(x)$\to$∃y S(y,x))

I think I proved part of it...

There is at least one value for x such that P(x) is True

This means that for ∀x(P(x)$\to$∃y S(y,x)) ... the P(x) part is False because we do not know if it is True for all x.

I am not even sure if my approach is correct, can someone please explain to me how to prove if they are consistent? Unfortunately my teacher does not provide good resources for me to figure this out on my own, Thanks.

Answer 1

The easiest way to show that it is consistent is to explicitly construct a model for this theory. (Then we're done: recall that $T$ is consistent $\leftrightarrow$ $T$ has a model)

Due to axioms 1 and 3, we need to have at least 2 elements: the simplest model is then $A=\{0,1\}$, and make this into an L-Structure by $P_A=\{0\}$, $S_A=\{(0,1),(1,0)\}$. It's then pretty easy to check that this defines a model, and we are done: T is consistent. $\square$

Answer 2

This means that for $∀x(P(x)→ ∃y S(y,x))$ ... the $P(x)$ part is False because we do not know if it is True for all x.

No. This statement does not require $P(x)$ to be universally true, only that whenever it is true, then so too is $\exists y\;S(y,x)$.

This is called "restricted universal quantification".

$∀x(P(x)→ ∃y S(y,x))\;$ reads: "For every $x$ which satisfies $P(x)$, there is some $y$ which satisfies $S(y,x)$."