Charles S. Peirce
Chap. VIII. Of the Copula
MS 229 (Robin 386, 387): Writings 3, 90-92
Every idea occupies time. In the time in which one is thought another may be thought. In this case the latter is thought as a special case of the former. This is called the subsumption of the latter under the former.
Every judgment expresses a relation of ideas, and consequently involves a comparison of them, and a thinking of them together. Every thinking of ideas together is a process of subsumption. Consequently every proposition or expression of a judgment, may be put in the form "A is a case under B." Thus, when we say "every man is an animal," we say that man is included under animal as a special case of it. I shall use the symbol < to declare that the object of the term written before it is included under the object of the term written after it. Thus "man < mortal" will mean "man is mortal," and "-Ö < Ö " will mean that the negative of a square root is itself a square root of the same quantity. The symbol < or "is", its equivalent in words, is termed the copula by logicians. The term which precedes the copula is called the subject of the proposition, & that which follows it the predicate. The latter is said to be predicated of the former.
The student of logic needs to make himself expert in putting propositions into the canonical form a < b. The following are a few typical examples:
The soul is not mortal" = "Soul < immortal."
Fishes swim" = "Fish < thing that swims."
Cats kill mice" = "Cat < killer of mice."
Every man loves himself" = "Man < self-lover."
If it rains it is cloudy" = "What exists only if it
rains < what exists only if it is cloudy."
Some men are black" = "Whatever exists <
what exists only in states of things in which black men exist."
All the properties of the copula may be summed up in three propositions. They are these.
1st Anything is itself, or a < a. This is called the principle of identity.
2nd If a is b then whatever is a is b. This is called the dictum de omni. This is as much as to say that we can reason thus:
a < b
x < a
Ergo x < b
This form of argument is called Barbara.
Expertness in reducing arguments to this form is indispensible to the logician. Examples for practice are given in the appendix.
These two properties belong to various other relations besides that expressed by the copula. The only thing which distinguishes this from those is 3rd. If a and b have the same predicates (in true propositions) then there is no difference between a and b, so far as the objects they name are concerned. I shall term this the principle of the singleness of the same.
If the first two properties belong to any relation that is if anything to which the relation is applicable at all is in that relation with itself, and if what is in that relation to something else which is in that relelation to a third is itself in the same relation to the third, I term it a relation of containing.
An example of such a relation is being as small as. For everything is as small as itself, and if a is as small as b and b as small as c then a is as small as c.
Another such relation is the converse of that expressed by the copula or that which b has to a if a < b.
Another is the following of one proposition from another. For it is universally true that "If A then A" and also that we can reason
If A then B
If X then A
Ergo If X then B.
Now we may, if we choose, express any such relation by a sign similar to the copula,say by the sign < with accents as <', <'', etc.and then if we will only make the third property of the copula hold good by neglecting all differences between objects except such as subsist between a and b if a <' x is true while b <' x is not true, then we shall have a doctrine concerning these relations which will necessarily run precisely parallel with the logical doctrine concerning the copula.
Logic may be considered as the science of identity. If we let a <' b mean that a is as small as b, and neglect all differences between objects except such as consist in one being as small as something which the other is not as small as, we shall have a parallel science of equality, which is mathematics or the logic of quantity. If we let a <'' b mean that all b is a, and neglect all differences between terms except so far as there is something of which one can be predicated of which the other cannot be predicated, we shall have a science of the identity of qualities, which is only logic in another aspect. If we let a <"' b denote that b is a consequence of a, and neglect all differences between statements except so far as they lead to different consequences, we have the logic of conditionals.
It is plain that if there be two sets of objects which correspond in any way each to each singly, then for every relation among the objects of the first set there must be a corresponding relation among objects of the second set. And for every proposition concerning objects of the first set expressed with any quasi-copula <' there must be a corresponding proposition concerning objects of the second set either with the same or with some other quasi-copula <".
The further consideration of this subject must be postponed until after we have considered relations in general.
This is the place to mention a certain term which would never be suggested to us except by the study of the relations of terms. It is called Ens and I denote it by the symbol 1. It is defined by the proposition that anything whatever is Ens, or
x < 1 whatever thing x may name.