Katana VentraIP

Law 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. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.

This article is about axiomatic rules due to various logicians and philosophers. For Boole's book on logic, see The Laws of Thought.

According to the 1999 Cambridge Dictionary of Philosophy,[1] 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 three traditional laws[edit]

History[edit]

Hamilton offers a history of the three traditional laws that begins with Plato, proceeds through Aristotle, and ends with the schoolmen of the Middle Ages; in addition he offers a fourth law (see entry below, under Hamilton):

Indian logic[edit]

The law of non-contradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini,[6] and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.[7]

Locke[edit]

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.[8]

Schopenhauer[edit]

Four laws[edit]

"The primary laws of thought, or the conditions of the thinkable, are four: – 1. The law of identity [A is A]. 2. The law of contradiction. 3. The law of exclusion; or excluded middle. 4. The law of sufficient reason." (Thomas Hughes, The Ideal Theory of Berkeley and the Real World, Part II, Section XV, Footnote, p. 38)


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) xy = yx [commutative law]

I. Leibniz' Law: x = y, if, and only if, x has every property which y has, and y has every property which x has.

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.

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:


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.

Algebra of concepts

"The Categories", Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Aristotle

Aristotle, "On Interpretation", Harold P. Cooke (trans.), pp. 111–179 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Aristotle, "", Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Prior Analytics

translated by Lydia Gillingham Robinson, 1914, The Algebra of Logic, The Open Court Publishing Company, Chicago and London. Downloaded via googlebooks.

Louis Couturat

Gödel 1944 Russell's mathematical logic in Kurt Gödel: Collected Works Volume II, Oxford University Press, New York, NY,  978-0-19-514721-6

ISBN

(Henry L. Mansel and John Veitch, ed.), 1860 Lectures on Metaphysics and Logic, in Two Volumes. Vol. II. Logic, Boston: Gould and Lincoln. Downloaded via googlebooks.

Sir William Hamilton, 9th Baronet

1967, Mathematical Logic reprint 2002, Dover Publications, Inc., Mineola, NY, ISBN 0-486-42533-9 (pbk.)

Stephen Cole Kleene

James R. Newman, 1958, Gödel's Proof, New York University Press, LCCCN: 58-5610.

Ernest Nagel

The Problems of Philosophy (1912), Oxford University Press, New York, 1997, ISBN 0-19-511552-X.

Bertrand Russell

The World as Will and Representation, Volume 2, Dover Publications, Mineola, New York, 1966, ISBN 0-486-21762-0

Arthur Schopenhauer

1946 (second edition), republished 1995, Introduction to Logic and to the Methodology of Deductive Sciences translated by Olaf Helmer, Dover Publications, Inc., New York, ISBN 0-486-28462-X (pbk.)

Alfred Tarski

1967, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, ISBN 978-0-674-32449-7 (pbk)

Jean van Heijenoort

James Danaher, "", The Philosopher, Volume LXXXXII No. 1

The Laws of Thought

Peter Suber, " Archived 2011-08-04 at the Wayback Machine", Earlham College

Non-Contradiction and Excluded Middle