Lastly, every logical term is postulated as having existence in thought, or existence in external reality, according to the universe in question. It is only as representing existence that a term is used in logic. Thus, when from every x is v, and every r is z, we infer every x is z, our premises are supposed to imply the assertions that Y, z exist.
We now come to the copula or relation by which terms arc con nected, and the proposition or jet/gaunt, which asserts or denies the connection. And here the logicians admit only one copula into the form of the proposition : that is, identiication, affirmed by is, denied by is not. If we say A goes before n, they call this proposition a material connection between A and n. They hold that the form of the above which is essential to its consideration in pure logic is " A is a thing which goes before n." And they further maintain that there are three fundamental forms of thought, which they call the principles of identity, difference, and excluded middle. Of identity, A is A; of difference, A is not not-A (or n, other than A); of excluded middle (or of nothing between), a is either A or not-A. That every thought of our minds must be fashioned so as not to contradict either of these principles is most evident ; as evident as the things themselves ; as sure as eggs are eggs. But the logician assumes a right to say that only these propositions are of the pure form of thought, true by the form : and that all others are true by the matter and not by the form.
Became all assertion identifies something with something, and all denial separates something from something, he have it down that Identity and difference constitute the only forme of comparison of a.s, or the whole form. For ourselves, we believe that there is matter in the proposition " A is A," one specific relation, identity : we do not lee why, except by arbitrary separation founded upon great importance, the process of identification and the relation of identity are declared to be of so different a nature from all other processes and relations as to make the difference of form and matter. To us the pure form of the proposition, divested of all matter, is the 'assertion or denial of the following :—x stands in relation L to Y. The proposition "x is x" is brought under the general form by two introductions of matter : L is identity, and x and v are the same symbols.
What is the test of distinction between a consequence which is true by its form, and one which is only true by its matter ? When the assertion is true by its form, it can only be refused admission by impossibility in its matter ; and when it can only be refused admission by impossibility in its matter, it is true by its form ; and by impossi bility we mean incompatibility with the conditions of the universe understood. If the universe be the whole sphere of possible thought,
the impossible is that which contradicts itself, and nothing else ; a five-footed quadruped, for example. But if it be the terrestrial sphere of actual existence, the impossible is only the non-existent on earth ; a rational quadruped, for example. If the universe be limited to the idea of animal, a stone is an impossibility. For instance, " A is A," and " A is not not-A," are true by form, in our view ; how can they be denied or rather, when can we doubt of admitting them ? for assertion must be opposed to non-assertion, not to denial. Only when A contra dicts itself. A horse which is not a horse, and a cow which is not a cow, are they the same things, or different ? and must they be either one or the other I We leave the logicians to settle it ; fur ourselves we cannot tell : we do not see how to affirm, and we do not see how to deny. But we do see that " A is A " can only be questioned, whatever the result may be, when the matter of A is impossible; and we recognise it as therefore true in form.
Again, let x be an existing animal ; it follows that the tail of x is the tail of an animal. Is this consequence formal or material ? Formal, because this is true whatever a tail may be, so long as there is a tail; and it cannot be refused assertion except when x has no tail. A guinea pig, for instance, puts this proposition out of the pale of assertion, and equally out of that of denial ; the tail of a non-tailed' animal is beyond us. There are those who will assert that the tail of a von-tailed animal is no tail ; but we are satisfied that the tail of a non-tailed ani mal is just as much a tail as no-tall ; and therein lies the contradiction and the impossibility. If we were clear that the tail of a non-tailed animal is no tail, we should know how to think of it ; for we can think of every subject which comes under a predicate. This very guinea pig is here produced by us because it has been brought forward by an acute thinker to prove that the consequence above given is material, and not formal. It has been made to prove that `x is an animal, there fore the tail of x is the tail of an animal' is a special inference gained from our material knowledge of the thing thought about, and not a general inference necessitated by the universal laws of thinking. We answer that though the lack of tail in a guinea-pig is not a law of logic, but a material incident of the object, yet we know the consequence to be necessitated by the laws of thinking, because we must go to im possible matter, we must make the tail of x a non-existence, before we can refuse to assert it.