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12 JUDAIC LOGIC



to either being or not-being (this principle is known as the Law of the Excluded
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Middle ), we are forced to assert one or the other of our two sentences in any given case.
Similarly, E and I are contradictory.
* A, E and IO are all contrary to each other, mutually exclusive; that is, only one of
them may be true, and the other two must then be false. Furthermore, they are taken
together exhaustive; that is, one of them must be true, since there are no available forms
besides them. It follows that the contradictory of the conjunction of I and O is simply a
disjunctive statement of the form “either A or E”; for IO signifies a denial of all
universality, whether that of A or that of E.
Lastly, what is the relation between I and O as such? They are obviously
compatible, since they combine together within IO; that is, they may both be true at once.
But they cannot both be false at once, for then their contradictories A and E would both be
true, which is impossible. Their special opposition is therefore given a distinct name; they
are said to be subcontrary. Note that the concept of subcontrariety applies to a pair of
propositions, while the larger concept of exhaustiveness (above defined) applies to any
number of propositions.

These concepts of opposition are applicable to other forms, besides those above, note. Also,
there are other, related such concepts worth mentioning. Two propositions are mutual implicants, if
the truth of either implies the truth of the other and the falsehood of either implies the falsehood of
the other. If, however, the implication is only one-way, they are said to be subalternatives, and the
one which implies but is not implied is called the subalternant, while the one which is implied but
does not imply is called the subaltern. Two forms are said to be incompatible, if they are contrary
or contradictory; in all other cases, they are said to be compatible. The latter class includes forms
which are unconnected, or neutrally related, meaning that they are related neither by mutual
implication or subalternation, nor by contrariety or contradiction, nor by subcontrariety.
It should be noted, too, that the same concepts of opposition can be applied to terms, as
well as to propositions. Two terms, say X and Y, are mutual implicants, if all X are Y and all Y are
X (in such case, X and Y are equivalent classes or coextensive). X subalternates Y, if all X are Y but
not all Y are Y, in which case, Y is called a genus or overclass of X, and X is called a species or
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subclass of Y; mutatis mutandis for the reverse case, of X subalternated by Y . X and Y are
contradictory, if no X is Y, and no nonX is nonY. They are contrary, if no X is Y, but some nonX
are nonY. They are subcontrary, if some X are Y, though no nonX is nonY. And, finally, they are
unconnected, if all the categorical propositions relating them or their negations are contingent (note,
however, that they may still have conditional connections in such case).

There are of course many other forms, besides those listed above. Propositions may
also be singular (these involve an indicated instance of the subject “This X” or a proper
name; symbols R and G are used for the positive and negative variants, respectively). All
propositions other than singular are called plural; this class includes not only A, E, I, O,
but also majoritive or minoritive forms (those are introduced by the quantifiers “most” and
“few”, respectively) and with indeed any number or proportion we please (“lots of”, “a


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The three laws: Identity, Non-contradiction, and Exclusion of a Middle, are known as the
Laws of Thought. They were first formulated by Aristotle, who identified them as the foundations of
all logic.
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Note well here that the subclass implies the overclass, and not vice-versa; so that the
subclass is the subalternant and the overclass is the subaltern. The concept of subalternation is not
to be confused, as is easily done, with the concept of subordination. While the species implies the
genus (because whatever falls under the species is subsumed under the genus), the genus is said
to include the species (because species is narrower than genus). To indicate the inferiority, in the
latter sense, of the species to the genus, one may say that it is subordinate. Thus, to repeat, a
subclass is not subaltern to an overclass, but subordinate to it.
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