If A is true, it is true ;
It cannot be that A is both true and not true ;
A is either true or not true ;
and these are expressions of the law of identity, the law of contradiction, and the law of excluded middle respectively.
It has been already shewn in section [108] that a similar result is obtainable if we write S for P in the following trio of equivalent propositions:
Every S is P ;
Nothing is both S and not P ;
Everything is P or not S.
These results indicate the close relations that exist between the three laws. But it is a mistake to suppose that we can regard one only of them as fundamental and the two others as deducible from this one. For the laws of thought stand at the foundation of all proof, and they must be postulated in order that the equivalences above assumed may themselves be shewn to be valid.
422. The Laws of Thought in relation to Immediate Inferences.—Granting that the laws of thought stand at the foundation of all proof, it is a further question what inferences, if any, can be shewn to be valid by their aid alone.
Hamilton claims that the law of identity is the principle of all logical affirmation, the law of contradiction of all logical negation, 465 and the law of excluded middle of all logical disjunction. By logical affirmation we may here understand affirmation which can be based on purely formal considerations without reference to the matter of thought, and we may interpret logical negation and logical disjunction similarly. The three laws of thought are accordingly held by Hamilton to justify what we have elsewhere called formal propositions, according as they are affirmative, negative, or disjunctive respectively. The division into affirmative, negative, and disjunctive is, however, of the nature of a cross division; and the question arises where we are to place formal hypotheticals such as the following:—If it is true that whatever is S is P, then it is true that whatever is not P is not S ; If it is true that all S is M and that all M is P, then it is true that all S is P. Apparently, since they are affirmative, they are to be brought under the law of identity; and inasmuch as the principle of any formal inference whatsoever may be expressed in a formal proposition similar in character to the above propositions, we find that Hamilton practically lays down the doctrine that in the three laws of thought (if not in the law of identity alone) we have a sufficient foundation upon which to base all logical inference.
This doctrine may, in the first place, be briefly considered with special reference to immediate inferences.
It may be granted that the process of obversion can be based exclusively on the laws of contradiction and excluded middle. From All S is P we pass to No S is not-P by the law of contradiction; and from No S is P we pass to All S is not-P by the law of excluded middle.
But it is a different matter when we pass to the consideration of the processes of conversion and contraposition; and it will be found that attempts to base these processes exclusively on the three laws of thought usually resolve themselves either into bare assertions or else into practical denials that conversion and contraposition are processes of inference at all.