LAWS OF THOUGHT
all of that which is other than a—that is,
if everything must be one or the other (a or b)
and if nothing can be both. These two pro¬
perties constitute the definition of a pair of con¬
tradictories (whether terms or propositions),
namely, they are mutually exclusive, and they
are together exhaustive ; expressed in the lan¬
guage of ‘ exact logic/ these properties are
(writing x for the negative of x and + for or) :
(i) XX < o,
what is at once x and x
does not exist, or, in the
language of propositions,
the conjoint occurrence
of x and x does not take
place.
(2) 00 < x + x,
everything is either x
or x, or, in the language
of propositions, what can
occur is either x or x, or,
reality entails x or x—
there is no tertium quid.
Together these properties constitute the
requirements of contradiction or of exact
negation ; it is a very inelegant piece of
nomenclature (besides that it leads to actual
confusion) to refer to (1) alone as the ‘prin¬
ciple of contradiction/ Better names for
them are (1) exclusion and (2) exhaustion (in
place of excluded middle). In the common
phraseology we are obliged to commit the
absurdity of saying that two terms or propo¬
sitions may satisfy the ‘ principle of contra¬
diction ’ and still not be contradictory (since
they may lack the quality of being exhaustive).
The mere fact that (1) has been called the
principle of contradiction has given it a pre¬
tended superiority over the other which it by
no means deserves ; they are of equal impor¬
tance in the conducting of reasoning processes.
In fact, for every formal argument which rests
upon (1) there is a corresponding argument
which rests upon (2) : thus in the case of
the fundamental law of Transposition (q. v.),
which affirms the identity of these two
propositions, (m) the student who is not*
a citizen is not a voter ; (n) every student is
either a citizen or not a voter ; that (m) follows
from (n) depends upon one of these prin¬
ciples, and that (n) follows from (m) depends
upon the other. These two names, exhaus¬
tion and exclusion, have the great advantage
that they permit the formation of adjectives ;
thus we may say that the test for the contra¬
dictoriness of two terms or propositions which
are not on their face the negatives one of
another is that they should be (1) mutually
exclusive and (2) together exhaustive.
It may be noticed that if two terms are
exhaustive but not exclusive, their negatives
are exclusive but not exhaustive. Thus
within the field of number, ‘ prime ’ and
‘ even ’ are exclusive (no number can be both)
but not exhaustive (except in the limiting
case of two, some numbers can be neither),
while ‘ not even ’ and ‘ not prime ’ are ex¬
haustive and not exclusive.
In the case of propositions, ‘ contrary ’ and
‘ subcontrary ’ are badly chosen names for the
Opposition (q. v.) of A and E, 0 and I, re¬
spectively, of the traditional logical scheme ;
they do not carry their meaning on their face,
and hence are unnecessarily difficult for the
learner to bear in mind. A and E should be
said to be mutually exclusive (but not ex¬
haustive), 0 and I to be conjointly exhaustive
(but not exclusive). This relation of qualities
is then seen to be a particular case merely of
the above-stated general rule.
Again, ‘no a is 6’ and ‘all a is b ’ are
exclusive but not exhaustive, while ‘ some a
is b ’ and ‘ some a is not b ’ are exhaustive but
not exclusive (provided in both cases that a
exists).
Laws of thought is not a good name for
these two characteristics; they should rather
be called the laws (if laws at all) of negation.
Properly speaking, the laws of thought are all
the rules of logic ; of these laws there is one
which is of far more fundamental importance
than those usually referred to under the
name, namely, the law that if a is & and b is
c, it can be concluded that a is c. This is the
great law of thought, and everything else is of
minor importance in comparison with it. It
is singular that it is not usually enumerated
under the name. Another law of thought of
equal consequence with those usually so called
is, according to Sigwart, the law that the
double negative is equivalent to an affirmative,
x = x, or
(3) * < x, I (4) x < x.
But these are not fundamental, for from the
principles of
Exclusion, I Exhaustion,
(r) xx < o, I (2) 00 < x + x,
it follows
by (2) that I by (1) that
x < x, I x < X.
(C.L.E.)
Literature : for the history of these princi¬
ples see Ueberweg, Syst. d. Logik, §§ 75-80 ;
Prantb, Gesch. d. Logik (see ‘ principium ’ in
the indices to the four volumes). There are
additional notes in an appendix to Hamilton,
Lects. on Logic. (c.s.p.)
END OF VOL. I.