Law of identity
- This article uses forms of logical notation. For a concise description of the symbols used in this notation, see List of logic symbols.
In logic, the law of identity is the first of the so-called three classic laws of thought. It states that an object is the same as itself: A → A (if you have A, then you have A); While this can also be listed as A ≡ A (A if-and-only-if A,) this is redundant.[1] Any reflexive relation upholds the law of identity. When discussing equality, the fact that "A is A" is a tautology.
History
The earliest use of the law appears to occur in Plato's dialogue Theaetetus (185a), where Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing.
Socrates: How about sounds and colours: in the first place you would admit that they both exist?
Theaetetus: Yes.
Socrates: And that either of them is different from the other, and the same with itself?
Theaetetus: Certainly.
Socrates: And that both are two and each of them one?
Theaetetus: Yes.
Parmenides the Eleatic (circa BCE. 490) formulated the principle Being is (eon emmenai) as the foundation of his philosophy.
Aristotle identifies the law in Book VII of the Metaphysics:
Now "why a thing is itself" is a meaningless inquiry (for—to give meaning to the question 'why'—the fact or the existence of the thing must already be evident—e.g., that the moon is eclipsed—but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical, unless one were to answer, 'because each thing is inseparable from itself, and its being one just meant this.' This, however, is common to all things and is a short and easy way with the question.)
Aristotle highlights "the fact that a thing is itself" because the objective of his inquiry at that point in the Metaphysics concerns "substance" and to provide answers to the question "what kind of thing substance should be said to be", given that "substance is a principle and a cause" of being. He further argues that while it is true that the question, "why a thing is itself" is meaningless, "the fact that a thing is itself" has meaning because we can then restate the why question to inquire "why something is predictable of something" given that each something is itself unique. For Aristotle, "substance is actuality" and it is the actual "thing that is itself" something that proceeds another such something in time.
Aristotle makes the claim that, "the fact that a thing is itself" allows for "a fixed constant nature of sensible things", and thus when confronted with the pronouncement that "this is bread" one can proceed to eat with confidence and not demand that for each and every such pronouncement evidence be provided to demonstrate that "this is not bread". Thus "the fact that a thing is itself" perhaps finds its greatest utility to man by providing "a fixed constant nature of sensible things" in an ever changing universe of being---Metaphysics, Book VII, Part 17
Thus in Aristotle we see the first logical presentation of the law of identity, "the fact that a thing is itself", to help answer the question "what kind of thing should be said to be". However, Aristotle never claimed that [A = A, 1 = 1, or A ≡ A], none of which correspond symbolically to "the fact that a thing is itself," for the simple reason that Aristotle never explicitly made the claim "thing is thing."
Both Thomas Aquinas (Met. IV., lect. 6) and Duns Scotus (Quaest. sup. Met. IV., Q. 3) follow Aristotle. Antonius Andreas, the Spanish disciple of Scotus (d. 1320) argues that the first place should belong to the law "Every Being is a Being" (Omne Ens est Ens, Qq. in Met. IV., Q. 4), but the late scholastic writer Francisco Suarez (Disp. Met. III., § 3) disagreed, also preferring to follow Aristotle.
Leibniz claimed that the law of Identity, which he expresses as 'Everything is what it is,' is the first primitive truth of reason which is affirmative, and the law of noncontradiction, is the first negative truth (Nouv. Ess. IV., 2, § i), arguing that "the statement that a thing is what it is, is prior to the statement that it is not another thing" (Nouv. Ess. IV.. 7, § 9). Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A."[2]
Locke (Essay Concerning Human Understanding IV. vii. iv. ("Of Maxims") says:
... whenever the mind with attention considers any proposition, so as to perceive the two ideas signified by the terms, and affirmed or denied one of the other to be the same or different; it is presently and infallibly certain of the truth of such a proposition; and this equally whether these propositions be in terms standing for more general ideas, or such as are less so: e.g. whether the general idea of Being be affirmed of itself, as in this proposition, "whatsoever is, is"; or a more particular idea be affirmed of itself, as "a man is a man"; or, "whatsoever is white is white" ...
J.S. Mill formulates the law as: "Whatever is true in one form of words, is true in every other form of words, which conveys the same meaning" (Exam. of Hamilton, p. 409). Shakespeare has Juliet Capulet state the same idea as "A rose by any other name would smell as sweet", in the 1597 play Romeo and Juliet.
African Spir proclaims the law of identity as the fundamental law of knowledge, which is opposed to the changing appearance of the empirical reality.[3]
A Linguistic Principle
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That everything is necessarily "the same with itself and different from another" is the self-evident[citation needed] first principle of language, for it governs the designation or "identification" of individual concepts within any symbolic language, so as to avoid any ambiguity in the communicating of concepts between the users of that language. Such a principle is necessary because a "symbolic designator" (name, word, sign, etc.) has no inherent meaning of its own, but derives its meaning from the language user who correlates the given designator with a conventionally prescribed concept that has been previously learned. To put it another way, the principle (law) states that although it is permissible to call the same concept by many different names, words, signs, etc., a fact that makes it possible for there to be different languages, it is not permissible, within any single linguistic group, to call different concepts by the same designator, else the users of the language will not know which of the possible concepts they are intended to call to mind when they encounter that designator.
Exceptions are only permissible where the users are able to readily discern which of the different concepts they are intended to call to mind by the context in which the designator is used. Since our ability to generate valid conclusions from premises is dependent upon our having a clear understanding of the concepts expressed in those premises, it follows that any ambiguity in the symbolic denotation of those concepts will hamper our ability to reason soundly. It is for this reason that the law of identity is considered the self-evident first principle of thought (reason).
In the introduction to his treatise An Investigation of the Laws of Thought, George Boole wrote: “There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed.” While his remarks pertain here to the development of a scientific language of formal reasoning or Logic, they apply equally well to natural languages, for all languages are essentially systems of symbolic communication. Linguistic terms, like all symbols, are devoid of any inherent meaning, and so must derive their meaning from the users of the language, who attribute meaning to them in a manner that is conventionally prescribed within their particular linguistic group. Consequently, the same “indispensable conditions” apply to the employment of terms in a natural language as to the symbols employed in Boole's formal logic.
In his Metaphysics (Book IV), Aristotle gives the following explication of that linguistic principle which later came to be known as the Law of Identity: “First then this at least is obviously true, that the word 'be' or 'not be' has a definite meaning, so that not everything will be “so and not so”. Again, if “man” has one meaning, let this be “two-footed animal”; by having one meaning I understand this: if “man” means “X”, then if A is a man, “X” will be what “being a man” means for him. (It makes no difference even if one were to say a word has several meanings, if only they are limited in number; for to each definition there might be assigned a different word. For instance, we might say that “man” has not one meaning but several, one of which would have one definition, viz. “two-footed animal”, while there might be also several other definitions if only they were limited in number; for a peculiar name might be assigned to each of the definitions. If, however, they were not limited but one were to say that the word has an infinite number of meanings, obviously reasoning would be impossible; for not to have one meaning is to have no meaning, and if words have no meaning our reasoning with one another, and indeed with ourselves, has been annihilated; for it is impossible to think of anything if we do not think of one thing; but if this is possible, one name might be assigned to this thing.) Let it be assumed then, as was said at the beginning, that the name has a meaning and has one meaning; it is impossible, then, that “being a man” should mean precisely “not being a man”, if “man” not only signifies something about one subject but also has one significance (for we do not identify “having one significance” with “signifying something about one subject” since on that assumption even “musical” and “white” and “man” would have had one significance, so that all things would have been one; for they would all have had the same significance).”
Trivia
In 2002 Jonathon Keats held a petition drive to pass "A = A" as statutory law in Berkeley, California. Specifically, the proposed law stated that, "every entity shall be identical to itself". Any entity caught being unidentical to itself was to be subject to a fine of up to one tenth of a cent. The law did not pass.[4]
See also
Allusions
People
- Thomas Aquinas
- Aristotle
- René Descartes
- Keith Donnellan
- Thomas Hobbes
- David Kaplan
- Saul Kripke
- Gottfried Wilhelm von Leibniz
- John Locke
- Plato
- W.V. Quine
- Hilary Putnam
- Baruch Spinoza
- John Searle
- African Spir
References
- ^ this is because one form of the biconditional is the conjunction of two conditionals. So in this form, A ≡ A can be reduced to (A → A) * (A → A) which is redundant.
- ^ La philosophie éternelle ou traditionnelle, la métaphysique, la logique, la raison et l'intelligence
- ^ Forschung nach der Gewissheit in der Erkenntniss der Wirklichkeit, Leipzig, J.G. Findel, 1869 and Denken und Wirklichkeit: Versuch einer Erneuerung der kritischen Philosophie, Leipzig, J. G. Findel, 1873.
- ^ San Francisco Chronicle
- ^ [1]
- ^ [2]