Why do we even need first order logic?

Question

I don't understand why propositional logic isn't enough. Can't any first order statement be encoded in the form of a propositional logic? What does first order logic do for us that we cannot do in propositional logic?

Answer 1

Propositional logic cannot account for, amongst other things, the validity of such arguments as

Socrates is a man.
All men are mortal.
$\therefore$ Socrates is mortal.

In propositional logic, we cannot do any better than to translate this argument as (e.g.)

$S$
$M$
$\therefore R$

which is plainly invalid.

In first-order logic, we can formulate this intuitively valid argument in such a way that it does turn out valid, thanks to the universal quantifier:

$Ms$
$\forall x(Mx \to Rx)$
$\therefore Rs$

To generalise, first-order logic allows us to get at the internal structure of certain propositions in a way that is not possible with mere propositional logic. The possession or non-possession of important logical properties turns on the precise nature of these internal structures. So it is important that we have adequate tools at hand to analyse them. First-order logic gives us many of these tools.