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Axioms of rational discourse
axiomatic rules due to various logicians and philosophers
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. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic. 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 as such: (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 the 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, ¬(A∧¬A), and the law of excluded middle involves the disjunction ("or") of something with its own negation, A∨¬A. 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 law of non contradiction (alternately the 'law of contradiction'): 'Nothing can both be and not be.' 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. >

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

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.

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 judgment 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 judgment is only another judgment it is logically or formally 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 an algebraic representation of his "laws of the mind", honed over the years into modern Boolean algebra.

Armed with his "system" he derives the "principle of [non]contradiction" starting with his law of identity: x2 = x. He subtracts x from both sides (his axiom 2), yielding x2 − x = 0. He then factors out the x: x(x − 1) = 0. For example, if x = "men" then 1 − x represents NOT men. So we have an example of the "Law of Contradiction":
"Hence: x(1 − x) will represent the class whose members are at once "men," and" not men," and the equation [x(1 − x)=0] thus express the principle, that a class whose members are at the same time men and not men does not exist. In other words, that it is impossible for the same individual to be at the same time a man and not a man. this is identically that "principle of contradiction" which Aristotle has described as the fundamental axiom of all philosophy. what has been commonly regarded as the fundamental axiom of metaphysics is but the consequence of a law of thought, mathematical in its form." (with more explanation about this "dichotomy" comes about cf Boole 1854:49ff)

This notion is found throughout Boole's "Laws of Thought" e. g. 1854:28, where the symbol "1" (the integer 1) is used to represent "Universe" and "0" to represent "Nothing", and in far more detail later (pages 42ff):
" Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse." In his chapter "The Predicate Calculus" Kleene observes that the specification of the "domain" of discourse is "not a trivial assumption, since it is not always clearly satisfied in ordinary discourse in mathematics likewise, logic can become pretty slippery when no D [domain] has been specified explicitly or implicitly, or the specification of a D [domain] is too vague (Kleene 1967:84).

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.

The sequel to Bertrand Russell's 1903 "The Principles of Mathematics" became the three volume work named Principia Mathematica (hereafter PM), written jointly with Alfred North Whitehead. Immediately after he and Whitehead published PM he wrote his 1912 "The Problems of Philosophy". His "Problems" reflects "the central ideas of Russell's logic".

At about the same time (1912) that Russell and Whitehead were finishing the last volume of their Principia Mathematica, and the publishing of Russell's "The Problems of Philosophy" at least two logicians (Louis Couturat, Christine Ladd Franklin) were asserting that two "laws" (principles) of contradiction" and "excluded middle" are necessary to specify "contradictories"; Ladd Franklin renamed these the principles of exclusion and exhaustion. The following appears as a footnote on page 23 of Couturat 1914:
"As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse)." In other words, the creation of "contradictories" represents a dichotomy, i. e. the "splitting" of a universe of discourse into two classes (collections) that have the following two properties: they are (i) mutually exclusive and (ii) (collectively) exhaustive. In other words, no one thing (drawn from the universe of discourse) can simultaneously be a member of both classes (law of non contradiction), but [and] every single thing (in the universe of discourse) must be a member of one class or the other (law of excluded middle).

Then there is the matter of "independence" of the axioms. In his commentary before Post 1921, van Heijenoort states that Paul Bernays solved the matter in 1918 (but published in 1926) – the formula ❋1.5 Associative Principle: p ⋁ (q ⋁ r) ⊃ q ⋁ (p ⋁ r) can be proved with the other four. As to what system of "primitive propositions" is the minimum, van Heijenoort states that the matter was "investigated by Zylinski (1925), Post himself (1941), and Wernick (1942)" but van Heijenoort does not answer the question.

Kleene (1967:33) observes that "logic" can be "founded" in two ways, first as a "model theory", or second by a formal "proof" or "axiomatic theory"; "the two formulations, that of model theory and that of proof theory, give equivalent results"(Kleene 1967:33). This foundational choice, and their equivalence also applies to predicate logic (Kleene 1967:318). In his introduction to Post 1921, van Heijenoort observes that both the "truth table and the axiomatic approaches are clearly presented". This matter of a proof of consistency both ways (by a model theory, by axiomatic proof theory) comes up in the more congenial version of Post's consistency proof that can be found in Nagel and Newman 1958 in their chapter V "An Example of a Successful Absolute Proof of Consistency". In the main body of the text they use a model to achieve their consistency proof (they also state that the system is complete but do not offer a proof) (Nagel & Newman 1958:45–56). But their text promises the reader a proof that is axiomatic rather than relying on a model, and in the Appendix they deliver this proof based on the notions of a division of formulas into two classes K1 and K2 that are mutually exclusive and exhaustive (Nagel & Newman 1958:109–113).

The (restricted) "first order predicate calculus" is the "system of logic" that adds to the propositional logic (cf Post, above) the notion of "subject predicate" i. e. the subject x is drawn from a domain (universe) of discourse and the predicate is a logical function f(x): x as subject and f(x) as predicate (Kleene 1967:74). Although Gödel's proof involves the same notion of "completeness" as does the proof of Post, Gödel's proof is far more difficult; what follows is a discussion of the axiom set.

Kurt Gödel in his 1930 doctoral dissertation "The completeness of the axioms of the functional calculus of logic" proved that in this "calculus" (i. e. restricted predicate logic with or without equality) that every valid formula is "either refutable or satisfiable" or what amounts to the same thing: every valid formula is provable and therefore the logic is complete. Here is Gödel's definition of whether or not the "restricted functional calculus" is "complete":
" whether it actually suffices for the derivation of every logico mathematical proposition, or where, perhaps, it is conceivable that there are true propositions (which may be provable by means of other principles) that cannot be derived in the system under consideration."

To supplement the four (down from five; see Post) axioms of the propositional calculus, Gödel 1930 adds the dictum de omni as the first of two additional axioms. Both this "dictum" and the second axiom, he claims in a footnote, derive from Principia Mathematica. Indeed, PM includes both as
❋10.1 ⊦ ∀xf(x) ⊃ f(y) ["I. e. what is true in all cases is true in any one case" ("Aristotle's dictum", rewritten in more modern symbols)]
❋10.2 ⊦∀x(p ⋁ f(x)) ⊃ (p ⋁ ∀xf(x)) [rewritten in more modern symbols]
The latter asserts that the logical sum (i. e. ⋁, OR) of a simple proposition p and a predicate ∀xf(x) implies the logical sum of each separately. But PM derives both of these from six primitive propositions of ❋9, which in the second edition of PM is discarded and replaced with four new "Pp" (primitive principles) of ❋8 (see in particular ❋8.2, and Hilbert derives the first from his "logical ε axiom" in his 1927 and does not mention the second. How Hilbert and Gödel came to adopt these two as axioms is unclear. Also required are two more "rules" of detachment ("modus ponens") applicable to predicates.

George Spencer Brown in his 1969 "Laws of Form" (LoF) begins by first taking as given that "we cannot make an indication without drawing a distinction". This, therefore, presupposes the law of excluded middle. He then goes on to define two axioms, which describe how distinctions (a "boundary") and indications (a "call") work:
Axiom 1. The law of calling: The value of a call made again is the value of the call. Axiom 2. The law of crossing: The value of a (boundary) crossing made again is not the value of the crossing. These axioms bare a resemblance to the "law of identity" and the "law of non contradiction" respectively. However, the law of identity is proven as a theorem (Theorem 4.5 in "Laws of Form") within the framework of LoF. In general, LoF can be reinterpreted as First order logic, propositional logic, and second order logic by assigning specific interpretations to the symbols and values of LoF.

All of the above "systems of logic" are considered to be "classical" meaning propositions and predicate expressions are two valued, with either the truth value "truth" or "falsity" but not both(Kleene 1967:8 and 83). While intuitionistic logic falls into the "classical" category, it objects to extending the "for all" operator to the Law of Excluded Middle; it allows instances of the "Law", but not its generalization to an infinite domain of discourse.

'Intuitionistic logic', sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

'Paraconsistent logic' refers to so called contradiction tolerant logical systems in which a contradiction does not necessarily result in trivialism. In other words, the principle of explosion is not valid in such logics. Some (namely the dialetheists) argue that the law of non contradiction is denied by dialetheic logic. They are motivated by certain paradoxes which seem to imply a limit of the law of non contradiction, namely the liar paradox. In order to avoid a trivial logical system and still allow certain contradictions to be true, dialetheists will employ a paraconsistent logic of some kind.

(cf Kleene 1967:49): These "calculi" include the symbols ⎕A, meaning "A is necessary" and ◊A meaning "A is possible". Kleene states that:
"These notions enter in domains of thinking where there are understood to be two different kinds of "truth", one more universal or compelling than the other A zoologist might declare that it is impossible that salamanders or any other living creatures can survive fire; but possible (though untrue) that unicorns exist, and possible (though improbable) that abominable snowmen exist."

'Fuzzy logic' is a form of many valued logic; it deals with reasoning that is approximate rather than fixed and exact.