The laws of thought are fundamental axiomatic rules upon which rational discourse itself is based. The rules have a long tradition in the history of philosophy. They are laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc.
The three classic laws of thought are attributed to Aristotle and were foundational in scholastic logic. They are:
1. The law of identity.
The law of identity states that an object is the same as itself: A ≡ A.
For the law of identity, Aristotle,[1] wrote:
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.) - Metaphysics, Book VII, Part 17
2. The law of non-contradiction
In logic, the law of non-contradiction ... states, in the words of Aristotle, that
"one cannot say of something that it is and that it is not in the same respect and at the same time". (2) [note Aristotle's use of indices: 'respect' & 'time']
see the Principle of contradiction
3. The law of the excluded middle
Aristotle wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the "facts" themselves:
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. … And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call 'man', and others were to call 'not-man'; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact. (Metaphysics 4.4, W.D. Ross (trans.), GBWW 8, 525–526). (3)
see the Law of excluded middle
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That everything be 'the same with itself and different from another' (law of identity) is the self-evident first principle upon which all symbolic communication systems (languages) are founded, for it governs the use of those symbols (names, words, pictograms, etc.) which denote the various individual concepts within a language, so as to eliminate ambiguity in the conveyance of those concepts between the users of the language. Such a principle (law) is necessary because symbolic designators have no inherent meaning of their own, but derive their meaning from the language users themselves, who associate each symbol with an individual concept in a manner that has been conventionally prescribed within their linguistic group. The degree to which this law must be obeyed depends upon the kind of language that one is utilizing. In a natural language there is considerable tolerance for violations since there are other means whereby one can determine which of a number of different concepts one is intended to call to mind by the use of a given symbol, such as the context in which the symbol is used. However, in the language of mathematics or formal logic, there is no such tolerance. If, for example, the symbol “+” were allowed to denote both the function of addition and some other mathematical function, then we would be unable to evaluate the truth value of a proposition such as, “2+2=4”, for the truth of such a proposition would be contingent upon which of the possible functions the symbol “+” was intended to denote. The same is true of symbols such as '2' and '4', for if these symbols did not denote conventionally prescribed quantities, then one could not attribute proper meaning to them, and the proposition would be rendered unintelligible.
The law of non-contradiction and the law of excluded middle are not separate laws per se, but correlates of the law of identity. That is to say, they are two interdependent and complementary principles that inhere naturally (implicitly) within the law of identity, as its essential nature. To understand how these supplementary laws relate to the law of identity, one must recognize the dichotomizing nature of the law of identity. By this I mean that whenever we 'identify' a thing as belonging to a certain class or instance of a class, we intellectually set that thing apart from all the other things in existence which are 'not' of that same class or instance of a class. In other words, the proposition, “A is A and A is not ~A” (law of identity) intellectually partitions a universe of discourse (the domain of all things)into exactly two subsets, A and ~A, and thus gives rise to a dichotomy. As with all dichotomies, A and ~A must then be 'mutually exclusive' and 'jointly exhaustive' with respect to that universe of discourse. In other words, 'no one thing can simultaneously be a member of both A and ~A' (law of non-contradiction), whilst 'every single thing must be a member of either A or ~A' (law of excluded middle).
What's more, since we cannot think without that we make use of some form of language (symbolic communication), for thinking entails the manipulation and amalgamation of simpler concepts in order to form more complex ones, and therefore, we must have a means of distinguishing these different concepts. It follows then that the first principle of language (law of identity) is also rightfully called the first principle of thought, and by extension, the first principle reason (rational thought).
Socrates, in a Platonic dialogue, described three principles derived from introspection. He asserted that these three axioms contradict each other.
[F]irst , that nothing can become greater or less, either in number or magnitude, while remaining equal to itself … Secondly, that without addition or subtraction there is no increase or diminution of anything, but only equality … Thirdly, that what was not before cannot be afterwards, without becoming and having become.
— Plato, Theaetetus, 155
The law of noncontradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini,[1] and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.[2]
The three classic laws of thought are attributed to Aristotle and were foundational in scholastic logic. They are:
The Persian philosopher, Ibn Sina (Avicenna), once wrote the following response to opponents of the law of noncontradiction:
"Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned." (Avicenna, Metaphysics)[3]
John Locke claimed that the principles of identity and contradiction were general ideas and only occurred to people after considerable abstract, philosophical thought. He characterized the principle of identity as "Whatsoever is, is." The principle of contradiction was stated as "It is impossible for the same thing to be and not to be." To Locke, these were not innate or a priori principles.
Gottfried Leibniz formulated two additional principles, either or both of which may sometimes be counted as a law of thought:
In Leibniz's thought and generally in the approach of rationalism, the latter two principles are regarded as clear and incontestable axioms. They were widely recognized in European thought of the 17th, 18th, and (while subject to greater debate) nineteenth century. As turned out to be the case with another such (the so-called law of continuity), they involve matters which, in contemporary terms, are subject to much debate and analysis (respectively on determinism and extensionality). Leibniz's principles were particularly influential in German thought. In France the Port-Royal Logic was less swayed by them. Hegel quarrelled with the identity of indiscernibles in his Science of Logic (1812–1816).
Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason. He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33:
Also:
The laws of thought can be most intelligibly expressed thus:
There would then have to be added only the fact that once for all in logic the question is about what is thought and hence about concepts and not about real things.
- Everything that is, exists.
- Nothing can simultaneously be and not be.
- Each and every thing either is or is not.
- Of everything that is, it can be found why it is.
— Schopenhauer, Manuscript Remains, Vol. 4, "Pandectae II," §163
To show that they are the foundation of reason, he gave the following explanation:
Through a reflection, which I might call a self-examination of the faculty of reason, we know that these judgments are the expression of the conditions of all thought and therefore have these as their ground. Thus by making vain attempts to think in opposition to these laws, the faculty of reason recognizes them as the conditions of the possibility of all thought. We then find that it is just as impossible to think in opposition to them as it is to move our limbs in a direction contrary to their joints. If the subject could know itself, we should know those laws immediately, and not first through experiments on objects, that is, representations (mental images).
— Schopenhauer,On the Fourfold Root of the Principle of Sufficient Reason, §33.
Schopenhauer's four laws can be schematically presented in the following manner:
Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. "It seems to me," he wrote in the second volume of The World as Will and Representation, Chapter 9, "that the doctrine of the laws of thought could be simplified by our setting up only two of them, namely, the law of the excluded middle, and that of sufficient reason or ground." Here is Law 1:
The first law thus: “Any predicate can be either attributed or denied of every subject.” Here already in the “either, or” is the fact that both cannot occur simultaneously, and consequently the very thing expressed by the laws of identity and of contradiction. Therefore these laws would be added as corollaries of that principle, which really states that any two concept-spheres are to be thought as either united or separated, but never as both simultaneously; consequently, that where words are joined together which express the latter, such words state a process of thought that is not feasible. The awareness of this want of feasibility is the feeling of contradiction.
Law 2 is as follows:
The second law of thought, the principle of sufficient reason, would state that the above attribution or denial must be determined by something different from the judgment itself, which may be a (pure or empirical) perception, or merely another judgment. This other and different thing is then called the ground or reason of the judgment.
He further asserted that "Insofar as a judgment satisfies the first law of thought, it is thinkable; insofar as it satisfies the second, it is true … ."
The title of George Boole's 1854 treatise on logic, An Investigation on the Laws of Thought, indicates an alternate path. The laws are now incorporated into his boolean logic in which the classic Aristotelian laws come down to saying there are two and only two truth values. The Leibnizian principles are ignored, at the algebraic level, absent second-order logic.
In the 19th century the Aristotelian, and sometimes the Leibnizian, laws of thought were standard material in logic textbooks, and J. Welton described them in this way:
The Laws of Thought, Regulative Principles of Thought, or Postulates of Knowledge, are those fundamental, necessary, formal and a priori mental laws in agreement with which all valid thought must be carried on. They are a priori, that is, they result directly from the processes of reason exercised upon the facts of the real world. They are formal; for as the necessary laws of all thinking, they cannot, at the same time, ascertain the definite properties of any particular class of things, for it is optional whether we think of that class of things or not. They are necessary, for no one ever does, or can, conceive them reversed, or really violate them, because no one ever accepts a contradiction which presents itself to his mind as such.
— Welton, A Manual of Logic, 1891, Vol. I, p. 30.
The law of non-contradiction is denied by paraconsistent logic. The law of the excluded middle is denied by intuitionistic logic. Thus, two of Aristotle's laws of thought, which were once considered indubitable, are today openly doubted.