Page 17 - Judaic Logic

INTRODUCTION 11

A: All X are Y. E: No X is Y.
I: Some X are Y. O: Some X are not Y.
IO: Some X are Y and Some X are not Y.

“X” and “Y” (or any specific equivalents) are referred to as the terms, the former
being called the subject and the latter the predicate. The relational expressions “is (are)”
and “is (are) not” are known as copulae, the former having positive polarity and the latter
negative (note that the “not” is used here to negate the “is”, even though placed after it).
Expressions like “all”, “some” are called quantifiers: they serve to tell us the extension
(i.e. the number or proportion) of the subject which the predication (i.e. copula and
predicate) refers to. So much for the various features of individual propositions.
A and E are characterized as general (or universal) propositions, because they each
concern the whole of the subject, each and every instance of it which ever has appeared or
may ever appear. A may also be expressed in the form “Every X is Y”. It should be clear
that “No X is Y” means “Every X is-not Y”, the only difference between A and E being
the polarity of their copulae. I and O are called particular propositions; they each concern
at least part of the subject, and again differ only in their polarity; note well that such
propositions are ambiguous with regard to just how much of the subject they address.
Often, in practice, we fail to explicitly specify the quantity involved, taking for granted that
it is well understood (as in “swans are white”); in case of doubt, such a statement may be
dealt with as, minimally, a particular.
IO represents the conjunction of I and O, and may be classed as (extensionally)
contingent. Though here presented as a compound, IO is also a proposition in its own
right; it could equally be expressed in exclusive form, as “Only some X are Y” or “Only
some X are not Y” (different emphasis, same logical significance). What distinguishes IO
from its elements I and O, is that it is more definite about quantity than they are. It follows
from the various definitions, and it is important to note, that I can be interpreted to mean
“either A or IO” (that is, “either All X are Y or Only some X are Y”), and likewise O can
be read as “either E or IO” (that is, “either No X is Y or Only some X are not Y”).

The foregoing definitions and correlations, together with certain self-evident
principles, enable us to infer the following oppositions, as they are called. (Note that the
expression “opposition”, in the specialized sense used in logic, does not necessarily signify
conflict, but is intended in the sense of ‘face-off’.)
* A implies I; that is, the first cannot be true without the second being also true.
Remember that “all” is one of the possible outcomes of “at least some”, and therefore
conceptually presupposes it. Logic demands that we acknowledge the meaning and
implications of what we say (this principle is known as the Law of Identity). Likewise,
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E implies O; and of course both I and O are implicit in IO. But note that these relations are
not reversed: I does not imply A, nor IO; O does not imply E, nor IO.
* A and O are contradictory; that is, they cannot be both true and they cannot be
both false, one must be true and the other false. The general statement “All X are Y” tells
us that every single X is Y, and is therefore incompatible with any claim that “Some X are
not Y” which would mean that one or more X is not Y; for we must admit that nothing can
at once and in the same respect both have and not-have a given characteristic (this
principle is known as the Law of Non-contradiction). Also, since there is no alternative

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And indeed, 'call a spade a spade'.