Law of thought
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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 and logic. They are laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc.
The Three Classical Laws 
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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 Non-contradiction and Law of Excluded Middle being merely corollaries.
The British philosopher Bertrand Russell, though held to be one of the 20th century's premier logicians, failed to fully recognize the true nature of the three classical laws of thought, as is evidenced by his statement: “The name 'laws of thought' is also misleading, for what is important is not the fact that we think in accordance with these laws, but the fact that things behave in accordance with them; in other words, the fact that when we think in accordance with them we think truly.”  Like so many others, Russell employed the laws without truly understanding that they are linguistic principles, and that they do not concern the behaviour of things, but what may be rightfully predicated of things. For even if things did not behave accordingly, we would still require the laws of thought in order that our thoughts or discourse be intelligible.
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.
For the Law of Identity, 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." (Metaphysics 4.4, W.D. Ross (trans.))
More than two millennia later, George Boole made the following observation with respect to the nature of language and those principles that must inhere naturally within them. In so doing, he alludes to the very same principle as did Aristotle:
“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.” (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".  (note Aristotle's use of indices:'respect' and 'time')
"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.))
The Law of Excluded Middle 
Aristotle wrote of the Law of Excluded Middle:
"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; ..." (Metaphysics 4.7, W.D. Ross (trans.))
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That everything be 'the same with itself and different from another' (law of identity) is one of the self-evident first 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 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 is meant 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, we cannot think without making 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).
[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.
Indian logic 
The law of noncontradiction 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 Commentaries 
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).
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:
- A subject is equal to the sum of its predicates, or a = a.
- No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.
- Of every two contradictorily opposite predicates one must belong to every subject.
- Truth is the reference of a judgment to something outside it as its sufficient reason or ground.
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's four laws can be schematically presented in the following manner:
- A is A.
- A is not not-A.
- A is either A or not-A.
- 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, 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.
Contemporary developments 
Some (such as dialetheists) argue 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 only nails are allowed, and screws are forbidden- it does not necessarily disprove or even question the existence or usefulness of screws, but merely demonstrates what can be built without them).
As such, while some doubt these principles, to this day they have yet to be successfully challenged or had any coherent alternatives presented.
- Problems of Philosophy, Chapter VII
- “These three laws are samples of self-evident logical principles, but are not really more fundamental or more self-evident than various other similar principles: for instance. the one we considered just now, which states that what follows from a true premiss is true.” (Bertrand Russell, Problems of Philosophy, Chapter VII)
- Frits Staal (1988), Universals: Studies in Indian Logic and Linguistics, Chicago, pp. 109–28 (cf. Bull, Malcolm (1999), Seeing Things Hidden, Verso, p. 53, ISBN 1-85984-263-1)
- Dasgupta, Surendranath (1991), A History of Indian Philosophy, Motilal Banarsidass, p. 110, ISBN 81-208-0415-5
- Avicenna, Metaphysics, I; commenting on Aristotle, Topics I.11.105a4–5
- 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