oposition
and an empirical generalization does not come in the _meaning_ of the
proposition; it comes in the nature of the _evidence_ for it. In the
empirical case, the evidence consists in the particular instances.
We believe that all men are mortal because we know that there are
innumerable instances of men dying, and no instances of their living
beyond a certain age. We do not believe it because we see a connexion
between the universal _man_ and the universal _mortal_. It is true that
if physiology can prove, assuming the general laws that govern living
bodies, that no living organism can last for ever, that gives a
connexion between _man_ and _mortality_ which would enable us to assert
our proposition without appealing to the special evidence of _men_
dying. But that only means that our generalization has been subsumed
under a wider generalization, for which the evidence is still of the
same kind, though more extensive. The progress of science is constantly
producing such subsumptions, and therefore giving a constantly wider
inductive basis for scientific generalizations. But although this gives
a greater _degree_ of certainty, it does not give a different _kind_:
the ultimate ground remains inductive, i.e. derived from instances, and
not an _a priori_ connexion of universals such as we have in logic and
arithmetic.

Two opposite points are to be observed concerning _a priori_ general
propositions. The first is that, if many particular instances are known,
our general proposition may be arrived at in the first instance by
induction, and the connexion of universals may be only subsequently
perceived. For example, it is known that if we draw perpendiculars
to the sides of a triangle from the opposite angles, all three
perpendiculars meet in a point. It would be quite possible to be first
led to this proposition by actually drawing perpendiculars in many
cases, and finding that they always met in a point; this experience
might lead us to look for the general proof and find it. Such cases are
common in the experience of every mathematician.

The other point is more interesting, and of more philosophical
importance. It is, that we may sometimes know a general proposition in
cases where we do not know a single instance of it. Take such a case as
the following: We know that any two numbers can be multiplied together,
and will give a third called their _product_. We know that all pairs
of integers the product of which is