The Laws of Logic

In dealing with perception-derived material, the guidance logic provides flows from a single imperative: Be consistent. Because contradictions do not exist in reality, a mental process that involves or implies a contradiction has departed from reality and is invalid; a conceptual product that contradicts any fact is false.

The fundamental laws of logic are:

1. The Law of Identity—Two existents, X and Y, are identical if they share the same characteristics.
2. The Law of Non-Contradiction—Either X or non-X is true, and there is no other possibility.
3. The Law of Excluded Middle—An existent X either has or does not have a characteristic C. Conversely, a characteristic C is either possessed by an existent X or it is not possessed by X.
4. The Law of Quantitative Comparison—Given two existents, X and Y, both having a characteristic, C, either X has C more than Y, less than Y, or is indifferent to Y. When X and Y are closely similar, it is possible, by using expert judgment, to estimate the dominance of one over another numerically and derive priority from such comparisons.

Later Aristotelians recognized that both the Law of Non-Contradiction and the Law of Excluded Middle stem from the axiom of identity: “A is A.” A thing is what it is. There are various ways to formulate these three laws plus one, but the following formulations capture the fundamental metaphysical issue of what it means to be:

1. The Law of Identity: To be is to have a specific identity. (A is A.)
2. The Law of Non-Contradiction: To have a given identity is to have no other. (A isn’t non-A.)
3. The Law of the Excluded Middle: Not to have a given identity is to have some other. (What isn’t A is non-A.)
4. The Law of Quantitative Comparison: To possess a characteristic is to possess it to some extent—either more or less than something else or equally so. (If A and B both have C, either A has more C, less C, or the same amount of C as B.)

For ease of memory and economy, the laws can also be stated more succinctly, though these are consequences of the more precise formulations:

1. Identity: Everything is something.
2. Non-Contradiction: A thing can’t be everything.
3. Excluded Middle: A thing can’t be nothing.
4. Quantitative Comparison: Things that share a characteristic must do so to some extent.

The Laws of Non-Contradiction and Excluded Middle are essentially reformulations of the Law of Identity designed to guide cognition. To think that a thing is both A and non-A in the same respect is to implicitly hold that the thing is everything in that respect. However, to be everything in a given respect is to be nothing in particular in that respect—i.e., to lack identity. For instance, consider a ball that simultaneously is and isn’t all red. Such a ball has no specific identity with respect to color. Similarly, if the ball is simultaneously here and not here (or neither here nor not here), it has no identity regarding location.

Knowledge is the awareness of the identity of things. Logic requires us to use our conceptual faculty in ways that recognize that things are what they are, rather than being contradictory or lacking identity. Thus, Ayn Rand defined logic as “the art of non-contradictory identification.”