Volltext: Dictionary of philosophy and psychology including many of the principal conceptions of ethics, logics, aesthetics ... and giving a terminology in English, French, German and Italian, vol. 1 [a-laws] (1)

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 
(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 
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. 
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.) 


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