Charles S. Peirce
Chap. X. The Copula and Simple Syllogism
MS 232 (Robin 383): Writings 3, 95-98
We have seen that all thought is in signs or at least is equivalent to what would be the signification of a sign. Now in order that an object may fulfill the function of a sign it is essential that it should be thought to be such this thought being itself a sign. There must be a sign which signifies that one thing is the sign of another. A sign which does this is called a proposition the corresponding thought a judgment. In the proposition then there is reference to two signs one of which is represented as standing for whatever the other stands for. To give a language the possibility of expressing propositions it is necessary that there should be some symbol which shall mean that a word placed in a certain relation to itsay, for example, following it in orderdenotes whatever is denoted by another word which is placed in some other relation to itsay, for example, preceding it. As this is necessary in every language so it is necessary in thought which is equivalent to a language. This symbol which is the soul of a proposition is called the copula. Let us illustrate. Take the proposition 'man is mortal'. The word man by itself stands for some one of those creatures but if nothing is added to it it is left indeterminate what one it is that it stands for. The word mortal by itself stands for something which dies while it remains indeterminate what thing of that sort it stands for. Now if we say 'man is mortal' we imply that no matter what member of the genus homo the word man stands for it is an individual which the word mortal also is proper to denotethat is the pole of the indication of the proposition. Every proposition of whatever kind may be expressed in the general form 'A is B'. The 'A' here is termed in logic the subject of the proposition the 'B' the predicate. If the verb is not the substantive verb then the common form in English 'is loving' in place of 'loves' suggests the manner in which the proposition may be thrown into this form. 'Every woman loves her child'. Here 'woman' takes the place of 'A' and 'the lover of her own child' takes the place of 'B'. If the proposition is a negative one as 'no woman hates her child' then 'woman' takes the place of 'A' and 'non-hater of her child' takes the place of 'B'. If the proposition is limited as to time then the limitation attaches either to the subject or to the predicate according to the nature of the limitation. Thus if we say 'every man has been born and will die' the subject is 'man' the predicate is 'that which has been born and which will die'. If we say 'At the time of Alfred few priests in England could read the Psalter' the subject is 'one of a certain majority of priests in England at the time of Alfred' the predicate is 'a person unable to read the Psalter'. The copula thus is not to be understood as implying either past present or future but as meaning simply that which is denoted by the subject is also denoted by the predicate. Some logicians have held that hypothetical propositions such as 'if it lightens it also thunders' are not capable of being reduced to the standard form we have given which those logicians term the form of categorical propositions but they conceive that hypothetical propositions have a distinct species. It may certainly be admitted that hypotheticals involve a conception not generally contained in either proposition that of the dependence of one thing upon another but there is nothing to prevent a categorical proposition from containing the same idea and it is certain that the whole meaning of hypothetical may be expressed in a categorical proposition. Thus 'if it lightens it will thunder on the same day'; this is the same as to say 'every day in which there is lightning there is thunder' and in general to say that 'if M happens then N happens' is the same as to say that 'whatever exists only if M happens exists only if N happens' or to use a form of expression which is apparently less intricate because it is less fully analyzed the above proposition is equivalent to saying that 'every state of things in which M happens is a state of things in which N happens'.
The copula expresses a certain relation between the two terms which form the subject and predicate of the proposition. We have defined this relation in terms of the properties of signs but for the purposes of formal logic it is more useful to define it in reference to its formal properties. Of these there are three. The 1st is that anything is in this relation to itselfthus 'man is man' &c. For any term whatever is proper to denote whatever is denoted by that. This I shall term the equiparant character of the copula. The 2nd formal property of the copula is that if any term A is in this relation to a second
term B which is itself in the same relation to a third term C then the first term A is in this relation to the 3rd term C. If A is B and B is C then A is C. For if C denotes whatever is denoted by B and B denotes whatever is denoted by A then C denotes whatever is denoted by A. This is termed the transitive character of the copula. The 3rd formal property of the copula is that if two terms stand reciprocally in this relation to each other then there is no distinction between the things which they signify. If A is B and B is A then there is no further distinction to be drawn between A and B. Of these three characters of the copula the second or transitive character is the most interesting from the point of view of formal logic. It follows immediately from this that it is good reasoning to conclude from the premises A is B and B is C the conclusion that A is C. This sort of reasoning is termed the simple syllogism and inasmuch as the transitive character of the copula is the only one by virtue of which one proposition depends upon others it follows that all reasoning must depend upon this principle and therefore that all reasoning can be reduced to the form of a syllogism however important the differences between one kind of reasoning and another may be and whatever other principles some inferences may involve. The relation expressed by the copula is by no means the only transitive relation. Examples of others are 'being greater than' or 'less than' and it is clear that the syllogism is equally valid for any kind of transitive relation whatever so that it is the same sort of reasoning to say A is greater than B, B is greater than C therefore A is greater than C as it is to say A is B, B is C therefore A is C. This was first pointed out by Mr. De Morgan.