The calculus of logic

On Syllogism.

The forms of categorical propositions already deduced are

whereof the two first give, by solution, 1 - = '(1 - ). All not-s are not-s, = '(1 - ), No s are s. To the above scheme, which is that of Aristotle, we might annex the four categorical propositions

the two first of which are similarly convertible into

If now the two premises of any syllogism are expressed by equations of the above forms, the elimination of the common symbol will lead us to an equation expressive of the conclusion.

the elimination of gives

the interpretation of which is

All s are s,

the form of the coefficient ' indicates that the predicate of the conclusion is limited by both the conditions which separately limit the predicates of the premises.

The elimination of gives

which is interpretable into Some s are s. It is always necessary that one term of the conclusion should be interpretable by means of the equations of the premises. In the above case both are so.

Instead of directly eliminating let either equation be transformed by solution as in (19). The first gives

being equivalent to + (1 - ), in which is arbitrary. Eliminating 1 - between this and the second equation of the system, we get

the interpretation of which is

No s are s.

Had we directly eliminated , we should have had

the reduced solution of which is

in which is an arbitrary elective symbol. This exactly agrees with the former result.

These examples may suffice to illustrate the employment of the method in particular instances. But its applicability to the demonstration of general theorems is here, as in other cases, a more important feature. I subjoin the results of a recent investigation of the Laws of Syllogism. While those results are characterized by great simplicity and bear, indeed, little trace of their mathematical origin, it would, I conceive, have been very difficult to arrive at them by the examination and comparison of particular cases.

Laws of Syllogism deduced from the Elective Calculus.

We shall take into account all propositions which can be made out of the classes , , , and referred to any of the forms embraced in the following system,

It is necessary to recapitulate that quantity (universal and particular) and quality (affirmative and negative) are understood to belong to the terms of propositions which is indeed the correct view.[8]

Thus, in the proposition All s are s, the subject All s is universal-affirmative, the predicate (some) s particular-affirmative.

In the proposition, Some s are s, both terms are particular-affirmative.

The proposition No s are s would in philosophical language be written in the form All s are not-s. The subject is universal-affirmative, the predicate particular-negative.

In the proposition Some s are not-s, the subject is particular-affirmative, the predicate particular-negative. In the proposition All not-s are s the subject is universal-negative, the predicate particular-affirmative, and so on.

In a pair of premises there are four terms, viz. two subjects and two predicates; two of these terms, viz. those involving the or not- may be called the middle terms, the two others the extremes, one of these involving X or not-, the other or not-.

The following are then the conditions and the rules of inference.

Case 1st. The middle terms of like quality.

Condition of Inference. One middle term universal.

Rule. Equate the extremes.

Case 2nd. The middle terms of opposite qualities.

1st. Condition of Inference. One extreme universal.

Rule. Change the quantity and quality of that extreme, and equate the result to the other extreme.

2nd. Condition of inference. Two universal middle terms.

Rule. Change the quantity and quality of either extreme, and equate the result to the other extreme.

I add a few examples,

This belongs to Case 1. All s is the universal middle term. The extremes equated give All s are s, the stronger term becoming the subject.

This belongs to Case 2, and satisfies the first condition. The middle term is particular-affirmative in the first premise, particular-negative in the second. Taking All s as the universal extreme, we have, on changing its quantity and quality, Some not-s, and this equated to the other extreme gives

All Xs are (some) not-s = No s are s.

If we take All s as the universal extreme we get

No Zs are Xs.

This also belongs to Case 2, and satisfies the first condition. The universal extreme All s becomes, some not-s, whence

Some Zs are not-Xs.

This belongs to Case 2, and satisfies the second condition. The extreme Some s becomes All not-s,

∴ All not-s are s.

The other extreme treated in the same way would give

All not-s are s,

which is an equivalent result.

If we confine ourselves to the Aristotelian premises A, E, I, O, the second condition of inference in Case 2 is not needed. The conclusion will not necessarily be confined to the Aristotelian system.

This belongs to Case 2, and satisfies the first condition. The result is

Some not-s are not-s.

These appear to me to be the ultimate laws of syllogistic inference. They apply to every case, and they completely abolish the distinction of figure, the necessity of conversion, the arbitrary and partial[9] rules of distribution, &c. If all logic were reducible to the syllogism these might claim to be regarded as the rules of logic. But logic, considered as the science of the relations of classes has been shewn to be of far greater extent. Syllogistic inference, in the elective system, corresponds to elimination. But this is not the highest in the order of its processes. All questions of elimination may in that system be regarded as subsidiary to the more general problem of the solution of elective equations. To this problem all questions of logic and of reasoning, without exception, may be referred. For the fuller illustrations of this principle I must however refer to the original work. The theory of hypothetical propositions, the analysis of the positive and negative elements, into which all propositions are ultimately resolvable, and other similar topics are also there discussed.

Undoubtedly the final aim of speculative logic is to assign the conditions which render reasoning possible, and the laws which determine its character and expression. The general axiom (A) and the laws (1), (2), (3), appear to convey the most definite solution that can at present be given to this question. When we pass to the consideration of hypothetical propositions, the same laws and the same general axiom which ought perhaps also to be regarded as a law, continue to prevail; the only difference being that the subjects of thought are no longer classes of objects, but cases of the coexistent truth or falsehood of propositions. Those relations which logicians designate by the terms conditional, disjunctive, &c., are referred by Kant to distinct conditions of thought. But it is a very remarkable fact, that the expressions of such relations can be deduced the one from the other by mere analytical process. From the equation = , which expresses the conditional proposition, "If the proposition is true the proposition is true," we can deduce

which expresses the disjunctive proposition, "Either and are together true, or is true and is false, or they are both false," and again the equation (1 - ) = 0, which expresses a relation of coexistence, viz. that the truth of and the falsehood of do not coexist. The distinction in the mental regard, which has the best title to be regarded as fundamental, is, I conceive, that of the affirmative and the negative. From this we deduce the direct and the inverse in operations, the true and the false in propositions, and the opposition of qualities in their terms.

The view which these enquiries present of the nature of language is a very interesting one. They exhibit it not as a mere collection of signs, but as a system of expression, the elements of which are subject to the laws of the thought which they represent. That those laws are as rigorously mathematical as are the laws which govern the purely quantitative conceptions of space and time, of number and magnitude, is a conclusion which I do not hesitate to submit to the exactest scrutiny.