2 What quantifiers?

Propositional logic, as discussed in module 0023, is quite useful. However, it is not universally useful. Recall the conditional proposition example from that module:

If $x < y$, $x \le y$.

Although this is, intuitively, correct, it is not quite accepted as a proper proposition. The reason is that we don't know what is $x$ and what is $y$. You may say that $x$ and $y$ can be any numbers. But that is implicit! The question is, how can we say that properly?

Furthermore, there are other propositions that cannot be expressed without quantifiers. For example:

``I am the best.''

On the surface, this is sufficient. But that is only because we already understand the word ``best''. ``Best'' implies that ``no one is better''. Fine, we'll rewrite the proposition using the explicit meaning of ``best'':

``No one is better than I am.''

Here, we run into another problem. ``No one'' is not a particular subject, which is required in a proposition. Or, in logic term, we can say that the name ``no one'' is not bound to anyone explicitly. Now, we are really going for the obscure but necessary notation:

``It is not the case that (there exist person x such that x is better than I am).''

The parantheses are only there to help the grouping of phrases in this example. One may point out that x is, again, a name that is not bound. Well, x is bound by the phrase ``there exist person x''.

Is this all very confusing? It is probably so, if this is the first time you encounter this kind of discussion. We'll spend more time on examples first, then apply quantifiers to propositional logic.