Are predicates like $P(x)$ functions of $x$?

Question

Is it a function which can take values 'true' and 'false' depending on $x$? Then is $P(x)\to Q(x)$ equivalent to writing $P(x)=Q(x)$ for all $x$? Am I understanding this right?

EDIT- So I was supposed to prove that James is mortal using logic given that:

1. James is a man
2. All men are mortal

I did this:

$P(x)$: x is a man

$Q(x)$: x is mortal

$P(James)$

$\forall x(P(x)\rightarrow Q(x))$

$P(James)\rightarrow Q(James)$

$Q(James)$

Is this proof correct? In my proof, there are two lines which just say $P(James)$ and $Q(James)$. I took $P(James)$ as simply a shorthand for 'James is a man', and similarly for $Q(James)$. I don't see how the functional interpretation of $P(x)$ is equivalent to this shorthand interpretation of $P(x)$

Answer 1

I don't know that I perfectly understand your question. If this doesn't answer it, please let me know and I'll update my answer.

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Yes, you can look at predicates as being functions from your domain of discourse to truth values. In general $P(x)$ might be something like $x>5$. This is not true or false, but becomes true or false once we fix a value of $x$. However, $\exists x. P(x)$ is no longer a "function" -- it is true, since $6$ (among other things) will witness the existence of such an $x$.

However, $P(x) \to Q(x)$ is not saying that $P(x) = Q(x)$. Instead it says that $\lnot P(x) \lor Q(x)$. Said another way, it says $\{ x ~|~ P(x)\} \subseteq \{ x~|~ Q(x)\}$.

Remember, though, $P(x) \to Q(x)$ still depends on $x$. Saying that $x>5 \to x>7$ is false when $x=6$ but it's true when $x=8$. However, once we quantify, and write $\forall x. (x> 5 \to x>7)$ then we get something which is either true or false (in this instance, it is false).

To get $P(x) = Q(x)$ you want $P(x) \leftrightarrow Q(x)$, which is an abbreviation for $(P(x) \to Q(x)) \land (Q(x) \to P(x))$. This says, following the above, that $\{ x ~|~ P(x)\} = \{ x~|~ Q(x)\}$.

Edit: In your example, we might take $P(x)$ to be "$x$ is a man" and $Q(x)$ to be "$x$ is mortal".

Then if we assume $P(\text{James})$ and $\forall x. P(x) \to Q(x)$, what can we say?

Well, since we know for ALL $x$, $P(x) \to Q(x)$, we know that, in particular for James, $P(\text {James}) \to Q(\text{James})$. Further, since we know $P(\text{James})$, we can conclude $Q(\text{James})$. That is, "James is mortal".

There is a lot of mathematics which is cantered around formalizing this argument. It is called "proof theory", and might be interesting for you to read about. At a high level, though, what I have just said is how it works.

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I hope this helps ^_^

Answer 2

Is it a function which can take values 'true' and 'false' depending on $x$? Then is $P(x)\to Q(x)$ equivalent to writing $P(x)=Q(x)$ for all $x$? Am I understanding this right?

Yes and no.

Yes, $P(x)\to Q(x)$ is a function that maps a term, $x$, to truth values.

However, no.   In classical predicate logic, $P(x)\to Q(x)$ evaluates as true whenever $Q(x)$ evaluates as true or $P(x)$ evaluates as false.   So the predicates need not be equal.   Indeed if you consider Truth to be greater than Falsity, the comparison is more $P(x)\leqslant Q(x)$ ; $P(x)$ implies $Q(x)$ exactly when $Q(x)$ is at least as true as $P(x)$.