Dr. Richard writes:
To say “Person A is x and then say Person A is not-x” is the violation of the Law of Identity. This is only one of several errors contained in the statement following the proposition A is both x and non-x. I will deal only with it.
“The three Aristotelian laws of logic are: the Law of Identity, the Law of Contradiction, the Law of Excluded Middle. The last two are merely corollaries or restatements of the first.
“The Law of Identity states that everything which is, is what it is— that a thing is itself—that a thing's identity is that which it is. In algebraic formulation, the Law of Identity is stated as: A is A.
“The Law of Identity was discussed in the first lecture as the basic principle of metaphysics. It is crucially important now to realize that the Law of Identity is also the basic principle of epistemology. It is the link between the two sciences, the bridge between existence and consciousness, between reality and knowledge.
“As a principle of metaphysics, the Law of Identity tells us that everything which is, is what it is. As a principle of epistemology, it tells us that contradictions cannot exist, that a thing cannot be A and not-A and, therefore, if we reach a contradiction in our thinking and draw two contradictory conclusions about the same thing in the same context, we have made an error. Our thinking is wrong. Our thinking does not correspond to reality, and we must check our conclusions, our reasons for them, our premises.
The Law of Contradiction and the Law of Excluded Middle are specifically epistemological restatements of the Law of Identity. They are epistemological rules to guide man's thinking.
“The Law of Contradiction states that nothing can be A and non-A at the same time and in the same respect. Thus, if the Law of Identity states that a table is a table, then the Law of Contradiction states that nothing can be a table and not a table at the same time and in the same respect. A tablecloth cannot be green and non-green at the same time and in the same respect. Contradictions cannot exist. At the same time and in the same respect, no attribute can be present and absent, no effort can succeed and fail to succeed, no action can be good and not good, no entity can be man and not man, no man can be alive and dead.
“It is the Law of Contradiction which teaches us that if all men are mortal, and Socrates is a man, it is rational to infer that Socrates is mortal and irrational to conclude that he is not—that if a theory conflicts with the facts of reality, the theory must be rejected; that a theory, to be valid, must integrate all and clash with none of the relevant evidence—that if a scientist's investigations lead him to the formulation of two mutually exclusive hypotheses, at least one of these hypotheses is wrong—that no proposition can be both true and false—that if a mother simultaneously demands of her children that they think independently and that they obey her on blind faith, she is not a practitioner of rationality, and neither is an employer who gives switching and irreconcilable orders, and neither is a moralist who preaches that self-abnegation leads to self- fulfillment, and neither is a political theorist who declares that men can achieve freedom only by establishing a slave society. Logic is the art of non-contradictory identification.
“The Law of Excluded Middle states that everything is either A or non-A at a given time and in a given respect. For instance, either this is a table, or it is not a table. Either this tablecloth is green, or it is not green. Observe, and this is very important, that I have not said: either this is a table, or it is a chair. I have said: either this is a table, or it is not a table. I have said: either this tablecloth is green, or it is not green.
“In accordance with the Law of Excluded Middle, I could also have said: either this tablecloth is red, or it is not red. This would be true in regard to this particular tablecloth. It is not red. But I could not say, as a universal principle: either this tablecloth is green, or it is red. This would be an obvious absurdity. A tablecloth may be green or red or blue or yellow or plaid or polka-dotted, etc. But whatever color a particular tablecloth happens to be, the universal alternative is only: that color or not that color, green or not green, red or not red, polka-dotted or not polka-dotted, etc.— this particular identity or not this particular identity, without specifying any other kind of particular identity. The Law of Excluded Middle states: everything is either A or non-A. It does not state: everything is either A or B.”