Law of thought

This article is about the axiomatic rules. For George Boole's book on logic An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, see The Laws of Thought.

The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc.

According to the 1999 Cambridge Dictionary of Philosophy, laws of thought are laws by which or in accordance with which valid thought proceeds, or that justify valid inference, or to which all valid deduction is reducible. Laws of thought are rules that apply without exception to any subject matter of thought, etc.; sometimes they are said to be the object of logic. The term, rarely used in exactly the same sense by different authors, has long been associated with three equally ambiguous expressions: the law of identity (ID), the law of contradiction (or non-contradiction; NC), and the law of excluded middle (EM).

Sometimes, these three expressions are taken as propositions of formal ontology having the widest possible subject matter, propositions that apply to entities per se: (ID), everything is (i.e., is identical to) itself; (NC) no thing having a given quality also has the negative of that quality (e.g., no even number is non-even); (EM) every thing either has a given quality or has the negative of that quality (e.g., every number is either even or non-even). Equally common in older works is use of these expressions for principles of metalogic about propositions: (ID) every proposition implies itself; (NC) no proposition is both true and false; (EM) every proposition is either true or false.

Beginning in the middle to late 1800s, these expressions have been used to denote propositions of Boolean Algebra about classes: (ID) every class includes itself; (NC) every class is such that its intersection ("product") with its own complement is the null class; (EM) every class is such that its union ("sum") with its own complement is the universal class. More recently, the last two of the three expressions have been used in connection with the classical propositional logic and with the so-called protothetic or quantified propositional logic; in both cases the law of non-contradiction involves the negation of the conjunction ("and") of something with its own negation and the law of excluded middle involves the disjunction ("or") of something with its own negation. In the case of propositional logic the "something" is a schematic letter serving as a place-holder, whereas in the case of protothetic logic the "something" is a genuine variable. The expressions "law of non-contradiction" and "law of excluded middle" are also used for semantic principles of model theory concerning sentences and interpretations: (NC) under no interpretation is a given sentence both true and false, (EM) under any interpretation, a given sentence is either true or false.

The expressions mentioned above all have been used in many other ways. Many other propositions have also been mentioned as laws of thought, including the dictum de omni et nullo attributed to Aristotle, the substitutivity of identicals (or equals) attributed to Euclid, the so-called identity of indiscernibles attributed to Gottfried Wilhelm Leibniz, and other "logical truths".

The expression "laws of thought" gained added prominence through its use by Boole (1815–64) to denote theorems of his "algebra of logic"; in fact, he named his second logic book An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (1854). Modern logicians, in almost unanimous disagreement with Boole, take this expression to be a misnomer; none of the above propositions classed under "laws of thought" are explicitly about thought per se, a mental phenomenon studied by psychology, nor do they involve explicit reference to a thinker or knower as would be the case in pragmatics or in epistemology. The distinction between psychology (as a study of mental phenomena) and logic (as a study of valid inference) is widely accepted.

The three classical laws

The classical "Three Laws of Thought" are the three fundamental linguistic principles without which there could be no intelligible communication. Their formal expression is generally attributed to Aristotle, and they were foundational in later scholastic logic. The law of identity is the most fundamental of the three, the law of noncontradiction and the law of excluded middle being corollaries.

The law of identity

The law of identity states "that every thing is the same with itself and different from another": A is A and not ~A.

Regarding this law, Aristotle wrote:

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.)

— Aristotle, Metaphysics, Book IV, Part 4 (translated by W.D. Ross)

More than two millennia later, George Boole alluded to the very same principle as did Aristotle when Boole made the following observation with respect to the nature of language and those principles that must inhere naturally within them:

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.

— George Boole, An Investigation of the Laws of Thought

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". As an illustration of this law, he wrote:

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.

— Aristotle, Metaphysics, Book IV, Part 4 (translated by W.D. Ross)

The law of excluded middle

Regarding the law of excluded middle, Aristotle wrote:

But on the other hand there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate. This is clear, in the first place, if we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false

— Aristotle, Metaphysics, Book IV, Part 7 (translated by W.D. Ross)

Rationale

That everything be "the same with itself and different from another" (law of identity) is one of the "self-evident logical principles" 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—such as the context in which the symbol is used—by which one can determine which of a number of different concepts one is intended to call to mind by the use of a given symbol. 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", since 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". 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 exactly separate laws; rather, they are 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. 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 both "jointly exhaustive" (with respect to that universe of discourse) and "mutually exclusive". 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).

Furthermore, we cannot think conceptually without making use of some form of language (symbolic communication), for thinking conceptually entails the manipulation and amalgamation of simpler concepts in order to form more complex concepts. Therefore we must have a means of distinguishing these different concepts, namely symbols or signs. 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 of reason (rational thought).

Plato

In one of Plato's Socratic dialogues, Socrates described three principles derived from introspection:

[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

Indian logic

The law of non-contradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini, and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.

Avicenna's commentary

The Persian philosopher Ibn Sina (Avicenna) once wrote the following response to opponents of the law of non-contradiction:

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

Locke

John Locke claimed that the principles of identity and contradiction (i.e. the law of identity and the law of non-contradiction) were general ideas and only occurred to people after considerable abstract, philosophical thought. He characterized the principle of identity as "Whatsoever is, is." He stated the principle of contradiction as "It is impossible for the same thing to be and not to be." To Locke, these were not innate or a priori principles.{{

Leibniz

Gottfried Leibniz formulated two additional principles, either or both of which may sometimes be counted as a law of thought:

• principle of sufficient reason
• identity of indiscernibles

In Leibniz's thought, as well as 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 19th centuries, although they were subject to greater debate in the 19th century. As turned out to be the case with the law of continuity, these two laws 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).

Schopenhauer

Four laws

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:

1. A subject is equal to the sum of its predicates, or a = a.
2. No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.
3. Of every two contradictorily opposite predicates one must belong to every subject.
4. Truth is the reference of a judgment to something outside it as its sufficient reason or ground.

Also:

The laws of thought can be most intelligibly expressed thus:

1. Everything that is, exists.
2. Nothing can simultaneously be and not be.
3. Each and every thing either is or is not.
4. Of everything that is, it can be found why it is.

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.

— 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:

1. A is A.
2. A is not not-A.
3. A is either A or not-A.
4. If A then B (A implies B).

Two laws

Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. In the ninth chapter of the second volume of The World as Will and Representation, he wrote:

It seems to me that the doctrine of the laws of thought could be simplified if we were to set up only two, the law of excluded middle and that of sufficient reason. The former thus: "Every predicate can be either confirmed or denied of every subject." Here it is already contained in the "either, or" that both cannot occur simultaneously, and consequently just what is expressed by the laws of identity and contradiction. Thus these would be added as corollaries of that principle which really says that every two concept-spheres must be thought either as united or as separated, but never as both at once; and therefore, even although words are joined together which express the latter, these words assert a process of thought which cannot be carried out. The consciousness of this infeasibility is the feeling of contradiction. The second law of thought, the principle of sufficient reason, would affirm that the above attributing or refuting 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. So far as a judgement satisfies the first law of thought, it is thinkable; so far as it satisfies the second, it is true, or at least in the case in which the ground of a judgement is only another judgement it is logically or formally true.

Boole

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 algebra, 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.

Welton

In the 19th century, the Aristotelian laws of thoughts, as well as 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.

Contemporary developments

Some (such as dialetheists) argue that the law of non-contradiction is denied by paraconsistent logic, however, "negation" in paraconsistent logic is not really negation in the formal sense; it is merely a subcontrary-forming operator.

The law of the excluded middle is not part of the execution of intuitionistic logic, but neither is it negated. Intuitionistic logic merely forbids the use of the operation as part of what it defines as a "constructive proof", which is not the same as demonstrating it invalid (this is comparable to the use of a particular building style in which screws are forbidden and only nails are allowed; it does not necessarily disprove or even question the existence or usefulness of screws, but merely demonstrates what can be built without them).

References

• Aristotle, "The Categories", Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Aristotle, "On Interpretation", Harold P. Cooke (trans.), pp. 111–179 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Aristotle, "Prior Analytics", Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Boole, George, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.
• Russell, Bertrand, The Problems of Philosophy, Williams and Norgate, London, 1912.
• Arthur Schopenhauer, The World as Will and Representation, Volume 2, Dover Publications, New York, 1966, ISBN 0-486-21762-0

External links

• James Danaher, "The Laws of Thought", The Philosopher, Volume LXXXXII No. 1
• Peter Suber, "Non-Contradiction and Excluded Middle", Earlham College