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

Aristotle

Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

 and were foundational in scholastic logic. They are:

• law of identity
  Law of identity
  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 ; While this can also be listed as A ≡ A this is redundant Any reflexive relation upholds the law of identity...
  
  
• law of noncontradiction
  Law of noncontradiction
  In classical logic, the law of non-contradiction is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g...
  
  
• law of excluded middle
  Law of excluded middle
  In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
  
  

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

Law of identity

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 ; While this can also be listed as A ≡ A this is redundant Any reflexive relation upholds the law of identity...

, Aristotle,[1] wrote:
2. The law of non-contradiction


In logic, the law of non-contradiction ... states, in the words of Aristotle, that

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:

see the Law of excluded middle

Law of excluded middle

In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....

Rationale

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).

Plato

Socrates

Socrates

Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...

, in a Platonic dialogue, described three principle

Principle

A principle is a law or rule that has to be, or usually is to be followed, or can be desirably followed, or is an inevitable consequence of something, such as the laws observed in nature or the way that a system is constructed...

s derived from introspection

Introspection

Introspection is the self-observation and reporting of conscious inner thoughts, desires and sensations. It is a conscious and purposive process relying on thinking, reasoning, and examining one's own thoughts, feelings, and, in more spiritual cases, one's soul...

. He asserted that these three axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s contradict each other.

Indian logic

The law of noncontradiction

Law of noncontradiction

In classical logic, the law of non-contradiction is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g...

 is found in ancient Indian logic

Indian logic

The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...

 as a meta-rule in the Shrauta Sutras

Kalpa (Vedanga)

Kalpa is one of the six disciplines of Vedanga, treating ritual.Tradition does not single out any special work in this branch of the Vedanga; but sacrificial practice gave rise to a large number of systematic sutras for the several classes of priests...

, the grammar of Pāṇini, and the Brahma Sutras

Brahma Sutras

The Brahma sūtras , also known as Vedānta Sūtras , are one of the three canonical texts of the Vedānta school of Hindu philosophy. A thorough study of Vedānta requires a close examination of these three texts, known in Sanskrit as the Prasthanatrayi, or the three starting points...

attributed to Vyasa

Vyasa

Vyasa is a central and revered figure in most Hindu traditions. He is also sometimes called Veda Vyasa , or Krishna Dvaipayana...

. It was later elaborated on by medieval commentators such as Madhvacharya

Madhvacharya

Madhvācārya was the chief proponent of Tattvavāda "Philosophy of Reality", popularly known as the Dvaita school of Hindu philosophy. It is one of the three most influential Vedānta philosophies. Madhvācārya was one of the important philosophers during the Bhakti movement. He was a pioneer in...

.

Aristotle

The three classic laws of thought are attributed to Aristotle

Aristotle

Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

 and were foundational in scholastic logic. They are:

• law of identity
  Law of identity
  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 ; While this can also be listed as A ≡ A this is redundant Any reflexive relation upholds the law of identity...
  
  
• law of noncontradiction
  Law of noncontradiction
  In classical logic, the law of non-contradiction is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g...
  
  
• law of excluded middle
  Law of excluded middle
  In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
  
  

Avicenna and Persian Logic

The Persian philosopher, Ibn Sina

Avicenna

Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā , commonly known as Ibn Sīnā or by his Latinized name Avicenna, was a Persian polymath, who wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived...

 (Avicenna), once wrote the following response to opponents of the law of noncontradiction:

Locke

John Locke

John Locke

John Locke FRS , widely known as the Father of Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social...

 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

A priori and a posteriori (philosophy)

The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...

principles.

Leibniz

Gottfried Leibniz

Gottfried Leibniz

Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

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

• principle of sufficient reason
  Principle of sufficient reason
  The principle of sufficient reason states that anything that happens does so for a reason: no state of affairs can obtain, and no statement can be true unless there is sufficient reason why it should not be otherwise...
  
  
• identity of indiscernibles
  Identity of indiscernibles
  The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa...
  
  

In Leibniz's thought and generally in the approach of rationalism

Rationalism

In epistemology and in its modern sense, rationalism is "any view appealing to reason as a source of knowledge or justification" . In more technical terms, it is a method or a theory "in which the criterion of the truth is not sensory but intellectual and deductive"...

, the latter two principles are regarded as clear and incontestable axioms. They were widely recognized in Europe

Europe

Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...

an 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

Law of Continuity

The Law of Continuity is a heuristic principle introduced by Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite"...

), they involve matters which, in contemporary terms, are subject to much debate and analysis (respectively on determinism

Determinism

Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...

 and extensionality

Extensionality

In logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties...

). Leibniz's principles were particularly influential in German thought. In France the Port-Royal Logic

Port-Royal Logic

Port-Royal Logic, or Logique de Port-Royal, is the common name of La logique, ou l'art de penser, an important textbook on logic first published anonymously in 1662 by Antoine Arnauld and Pierre Nicole, two prominent members of the Jansenist movement, centered around Port-Royal. Blaise Pascal...

was less swayed by them. Hegel quarrelled with the identity of indiscernibles in his Science of Logic

Science of Logic

Hegel's work The Science of Logic outlined his vision of logic, which is an ontology that incorporates the traditional Aristotelian syllogism as a sub-component rather than a basis...

(1812–1816).

Four Laws

Arthur Schopenhauer

Arthur Schopenhauer

Arthur Schopenhauer was a German philosopher known for his pessimism and philosophical clarity. At age 25, he published his doctoral dissertation, On the Fourfold Root of the Principle of Sufficient Reason, which examined the four separate manifestations of reason in the phenomenal...

 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

On the Fourfold Root of the Principle of Sufficient Reason

On the Fourfold Root of the Principle of Sufficient Reason was originally published as a doctoral dissertation in 1813. The German philosopher Arthur Schopenhauer revised this important work and re-published it in 1847....

, §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:
To show that they are the foundation of reason, he gave the following explanation:
Schopenhauer's four laws can be schematically presented in the following manner:

1. A is A.
2. A is not both B and not-B.
3. A is either B or not-B.
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. "It seems to me," he wrote in the second volume of The World as Will and Representation

The World as Will and Representation

The World as Will and Representation is the central work of the German philosopher Arthur Schopenhauer. The first edition was published in December 1818, and the second expanded edition in 1844. In 1948, an abridged version was edited by Thomas Mann....

, 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:
Law 2 is as follows:
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 … ."

Boole

The title of George Boole

George Boole

George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...

'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

Boolean logic

Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...

 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

Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....

.

Welton

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:

Contemporary developments

The law of non-contradiction is denied by paraconsistent logic

Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

. The law of the excluded middle is denied by intuitionistic logic

Intuitionistic logic

Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...

. Thus, two of Aristotle's laws of thought, which were once considered indubitable, are today openly doubted.

External links

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