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SECTION IL., 1883. Timo SAN] Trans. Roy. Soc. CANADA. 
An Addition to the Logical Square of Opposition. 
By THE Rev. J. Chark Murray, LL.D. 
(Read May 23, 1883.) 
I do not think it necessary in this paper to enter into an explanation of the general 
logical doctrine of the opposition of propositions, as my intention is merely to indicate a 
useful addition to the diagram commonly used to illustrate that doctrine. That diagram 
is asquare, at the four angles of which are placed the symbols of the four kinds of proposi- 
tions, A, E, I, O, while the four sides of the square, along with its two diagonals, are 
supposed to illustrate the relation between the two opposite propositions of the several 
kinds of opposition. The relation between two opposites, however, is but very imperfectly 
exhibited by the simple square ; and it occurred to me, that a very slight addition to this 
diagram would render it a great deal more serviceable for its purpose. This addition, 
accordingly, I have been accustomed to use among my students for many years. 
The addition is based on the nature of the relation between opposites, which the 
square is designed to symbolize. Now, the relation of logical opposition is one that is 
expressed by the inferences which may be drawn from one opposite to another. If, 
for example, two propositions are so diametrically opposed, that one of them must be true 
and the other must be false, then that opposition is expressed by saying, that you can 
infer the falsity of the one from the truth of the other, and the truth of the one from 
the falsity of the other. In other cases, however, as is well known, the relation of 
two propositions is such, that you can infer only the falsity of one from the truth of the 
other, or only the truth of one from the falsity of the other, and so on. Now, the object 
of the proposed improvement on the logical diagram is to exhibit these varions relations to 
the eye. But here a difficulty arises, which has been very generally overlooked by 
logicians. The word some, which is the common expression of particularity in a 
proposition, is beset by an important ambiguity. It may mean either “some at least,” that 
is, “possibly all,” or “some at most,” that is, “some only, not all” Even such an eminent 
expositor of logical doctrine as Whately confounds these two meanings ; and his account of 
the opposition of propositions is consequently in part unintelligible. 
Now, the only way to avoid this ambiguity in an illustrative diagram is to adopt two 
squares, each intended to exhibit the relations of opposites under one interpretation of the 
word some. Then the relations are fully displayed by means of arrows between the symbols 
of the two propositions forming each pair of opposites. The arrows point in the direction in 
which inferences may be drawn. A transverse bar on the stem of an arrow symbolizes 
