The attitude towards the more general
and the more special in logic is connected with the usage of the
word “kind” which is liable to cause
confusion. We talk of kinds of numbers, kinds of
propositions, kinds of proofs; and, also, of kinds of apples,
kinds of paper, etc. In one sense what
defines the kind are properties, like sweetness, hardness,
etc. In the other the different kinds
are different grammatical structures. A treatise on
pomology may
be called incomplete if there exist kinds of
apples which it doesn't mention. Here we have
a standard of completeness in nature. Supposing on the
other hand there was a game resembling that of chess but simpler,
no pawns being used in it. Should we call this game
incomplete? Or should we call it a game “more
complete than chess” which in some way contained chess but
added new elements? The contempt for what seems the
less general case in logic springs from the idea that it is
incomplete.
It is in fact 30.
confusing to talk of
cardinal arithmetic as something special as opposed to something
more general. Cardinal arithmetic bears no mark of
incompleteness; nor does an arithmetic which is cardinal and
finite. (There are no subtle distinctions between
logical forms as there are between the tastes of different kinds of
apples).