The laws of thought are fundamental
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
atic 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
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
and
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
. Generally they are taken as laws that guide and underlie everyone's thinking,
thoughts
In their most common sense, the terms thought and thinking refer to conscious cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, reasoning, concept formation, problem solving, an ...
, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as
intuitionistic logic
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 ...
,
dialetheism
Dialetheism (from Greek 'twice' and 'truth') is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true ...
and
fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
.
According to the 1999 ''
Cambridge Dictionary of Philosophy
''The Cambridge Dictionary of Philosophy'' (1995; second edition 1999; third edition 2015) is a dictionary of philosophy published by Cambridge University Press and edited by the philosopher Robert Audi
Robert N. Audi (born November 1941) is an ...
'', 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
In logic, the law of identity states that each thing is identical with itself. It is the first of the historical three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are bui ...
(ID), the
law of contradiction
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
(or non-contradiction; NC), and the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...
(EM).
Sometimes, these three expressions are taken as
propositions
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the n ...
of
formal ontology
In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or a ...
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
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
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
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
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
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
; 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
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
principles of
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
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
In Aristotelian logic, ''dictum de omni et nullo'' (Latin: "the maxim of all and none") is the principle that whatever is affirmed or denied of a whole kind K may be affirmed or denied (respectively) of any subkind of K. This principle is fundamen ...
attributed to
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
, the substitutivity of identicals (or equals) attributed to
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, the so-called
identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
attributed to
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
, 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
Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ...
, nor do they involve explicit reference to a thinker or knower as would be the case in
pragmatics
In linguistics and related fields, pragmatics is the study of how context contributes to meaning. The field of study evaluates how human language is utilized in social interactions, as well as the relationship between the interpreter and the int ...
or in
epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
. 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
History
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
offers a history of the three traditional laws that begins with
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, proceeds through Aristotle, and ends with the
schoolmen
Scholasticism was a medieval school of philosophy that employed a critical organic method of philosophical analysis predicated upon the Aristotelian 10 Categories. Christian scholasticism emerged within the monastic schools that translate ...
of the
Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
; in addition he offers a fourth law (see entry below, under Hamilton):
:"The principles of Contradiction and Excluded Middle can be traced back to Plato: The principles of Contradiction and of Excluded Middle can both be traced back to Plato, by whom they were enounced and frequently applied; though it was not till long after, that either of them obtained a distinctive appellation. To take the principle of Contradiction first. This law Plato frequently employs, but the most remarkable passages are found in the Phœdo, in the Sophista, and in the fourth and seventh books of the Republic.
amilton LECT. V. LOGIC. 62 Amilton may refer to:
*Amilton Prado (born 1979), Peruvian football defender
*Amílton (footballer, born 1981), Brazilian football striker
*Amilton (footballer, born 1989), Brazilian football right winger
*Amilton Filho (born 1992), Belizean footbal ...
:Law of Excluded Middle: The law of Excluded Middle between two contradictories remounts, as I have said, also to Plato, though the Second Alcibiades, the dialogue in which it is most clearly expressed, must be admitted to be spurious. It is also in the fragments of Pseudo-Archytas, to be found in
Stobæus.
amilton LECT. V. LOGIC. 65
:Hamilton further observes that "It is explicitly and emphatically enounced by Aristotle in many passages both of his Metaphysics (l. iii. (iv.) c.7.) and of his Analytics, both Prior (l. i. c. 2) and Posterior (1. i. c. 4). In the first of these, he says: "It is impossible that there should exist any medium between contradictory opposites, but it is necessary either to affirm or to deny everything of everything."
amilton LECT. V. LOGIC. 65
:"Law of Identity.
amilton also calls this "The principle of all logical affirmation and definition"Antonius Andreas: The law of Identity, I stated, was not explicated as a coordinate principle till a comparatively recent period. The earliest author in whom I have found this done, is
Antonius Andreas Antonius Andreas (c. 1280 in Tauste, Aragon – 1320) was a Spanish Franciscan theologian, a pupil of Duns Scotus.
He was teaching at the University of Lleida
The University of Lleida (officially in Catalan: ''Universitat de Lleida'') is a univ ...
, a scholar of Scotus, who flourished at the end of the thirteenth and beginning of the fourteenth century. The schoolman, in the fourth book of his Commentary of Aristotle's Metaphysics – a commentary which is full of the most ingenious and original views, – not only asserts to the law of Identity a coordinate dignity with the law of Contradiction, but, against Aristotle, he maintains that the principle of Identity, and not the principle of Contradiction, is the one absolutely first. The formula in which Andreas expressed it was ''Ens est ens''. Subsequently to this author, the question concerning the relative priority of the two laws of Identity and of Contradiction became one much agitated in the schools; though there were also found some who asserted to the law of Excluded Middle this supreme rank."
rom Hamilton LECT. V. LOGIC. 65–66
Three traditional laws: identity, non-contradiction, excluded middle
The following states the three traditional "laws" in the words of Bertrand Russell (1912):
The law of identity
The
law of identity
In logic, the law of identity states that each thing is identical with itself. It is the first of the historical three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are bui ...
: 'Whatever is, is.'
[Russell 1912:72,1997 edition.]
For all a: a = a.
Regarding this law, Aristotle wrote:
More than two millennia later,
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ire ...
alluded to the very same principle as did Aristotle when Boole made the following observation with respect to the nature of
language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
and those principles that must inhere naturally within them:
The law of non-contradiction
The
law of non-contradiction
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
(alternately the 'law of contradiction'
[Russell 1912:72, 1997 edition.]): 'Nothing can both be and not be.'
In other words: "two or more contradictory statements cannot both be true in the same sense at the same time":
¬(A
∧
Wedge (∧) is a symbol that looks similar to an in-line caret (^). It is used to represent various operations. In Unicode, the symbol is encoded and by \wedge and \land in TeX. The opposite symbol (∨) is called a vel, or sometimes a (descend ...
¬A).
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:
The law of excluded middle
The law of excluded middle: 'Everything must either be or not be.'
In accordance with the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...
or excluded third, for every proposition, either its positive or negative form is true: A
∨
The descending wedge symbol ∨ may represent:
* Logical disjunction in propositional logic
* Join in lattice theory
* The wedge sum in topology
The vertically reflected symbol, ∧, is a wedge, and often denotes related or dual operators.
The ...
¬A.
Regarding the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...
, Aristotle wrote:
Rationale
As the quotations from Hamilton above indicate, in particular the "law of identity" entry, the rationale for and expression of the "laws of thought" have been fertile ground for philosophic debate since Plato. Today the debate—about how we "come to know" the world of things and our thoughts—continues; for examples of rationales see the entries, below.
Plato
In one of Plato's
Socratic dialogue
Socratic dialogue ( grc, Σωκρατικὸς λόγος) is a genre of literary prose developed in Greece at the turn of the fourth century BC. The earliest ones are preserved in the works of Plato and Xenophon and all involve Socrates as the p ...
s,
Socrates
Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...
described three
principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a Legal rule, rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, suc ...
s derived from
introspection
Introspection is the examination of one's own conscious thoughts and feelings. In psychology, the process of introspection relies on the observation of one's mental state, while in a spiritual context it may refer to the examination of one's s ...
:
Indian logic
The
law of non-contradiction
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
is found in ancient
Indian logic
The development of Indian logic dates back to the ''anviksiki'' of Medhatithi Gautama (c. 6th century BCE); the Sanskrit grammar rules of Pāṇini (c. 5th century BCE); the Vaisheshika school's analysis of atomism (c. 6th century BCE to 2nd centur ...
as a meta-rule in the ''
Shrauta Sutras
Kalpa ( sa, कल्प) means "proper, fit" and is one of the six disciplines of the Vedānga, or ancillary science connected with the Vedas – the scriptures of Hinduism. This field of study is focused on the procedures and ceremonies associ ...
'', the grammar of
Pāṇini
, era = ;;6th–5th century BCE
, region = Indian philosophy
, main_interests = Grammar, linguistics
, notable_works = ' (Sanskrit#Classical Sanskrit, Classical Sanskrit)
, influenced=
, notable_ideas=Descript ...
, and the ''
Brahma Sutras
The ''Brahma Sūtras'' ( sa, ब्रह्मसूत्राणि) is a Sanskrit text, attributed to the sage bādarāyaṇa or sage Vyāsa, estimated to have been completed in its surviving form in approx. 400–450 CE,, Quote: "...we can ...
'' attributed to
Vyasa
Krishna Dvaipayana ( sa, कृष्णद्वैपायन, Kṛṣṇadvaipāyana), better known as Vyasa (; sa, व्यासः, Vyāsaḥ, compiler) or Vedavyasa (वेदव्यासः, ''Veda-vyāsaḥ'', "the one who cl ...
. It was later elaborated on by medieval commentators such as
Madhvacharya
Madhvacharya (; ; CE 1199-1278 or CE 1238–1317), sometimes Anglicisation, anglicised as Madhva Acharya, and also known as Purna Prajna () and Ānanda Tīrtha, was an Indian philosopher, theologian and the chief proponent of the ''Dvaita'' ...
.
Locke
John Locke
John Locke (; 29 August 1632 – 28 October 1704) was an English philosopher and physician, widely regarded as one of the most influential of Age of Enlightenment, Enlightenment thinkers and commonly known as the "father of liberalism ...
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
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'' principles.
Leibniz
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
formulated two additional principles, either or both of which may sometimes be counted as a law of thought:
:*
principle of sufficient reason
The principle of sufficient reason states that everything must have a reason or a cause. The principle was articulated and made prominent by Gottfried Wilhelm Leibniz, with many antecedents, and was further used and developed by Arthur Schopenhau ...
:*
identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
In Leibniz's thought, as well as generally in the approach of
rationalism
In philosophy, rationalism is the epistemological view that "regards reason as the chief source and test of knowledge" or "any view appealing to reason as a source of knowledge or justification".Lacey, A.R. (1996), ''A Dictionary of Philosophy' ...
, the latter two principles are regarded as clear and incontestable
axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
. They were widely recognized in
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
an 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
The law of continuity is a heuristic principle introduced by Gottfried 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". Kepler used ...
, these two laws involve matters which, in contemporary terms, are subject to much debate and analysis (respectively on
determinism
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
and
extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
). Leibniz's principles were particularly influential in German thought. In France, the ''
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 Jan ...
'' was less swayed by them.
Hegel
Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends a ...
quarrelled with the
identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
in his ''
Science of Logic
''Science of Logic'' (''SL''; german: Wissenschaft der Logik, ''WdL''), first published between 1812 and 1816, is the work in which Georg Wilhelm Friedrich Hegel outlined his vision of logic. Hegel's logic is a system of '' dialectics'', i.e., ...
'' (1812–1816).
Schopenhauer
Four laws
"The primary laws of thought, or the conditions of the thinkable, are four: – 1. The law of identity
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
Arthur Schopenhauer ( , ; 22 February 1788 – 21 September 1860) was a German philosopher. He is best known for his 1818 work ''The World as Will and Representation'' (expanded in 1844), which characterizes the phenomenal world as the prod ...
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'' (german: Ueber die vierfache Wurzel des Satzes vom zureichenden Grunde) is an elaboration on the classical Principle of Sufficient Reason, written by German philosopher Arthur Schope ...
'', §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.
Also:
To show that they are the foundation of
reason
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, ...
, he gave the following explanation:
Schopenhauer's four laws can be schematically presented in the following manner:
#A is A.
#A is not not-A.
#X 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. In the ninth chapter of the second volume of ''
The World as Will and Representation
''The World as Will and Representation'' (''WWR''; german: Die Welt als Wille und Vorstellung, ''WWV''), sometimes translated as ''The World as Will and Idea'', is the central work of the German philosopher Arthur Schopenhauer. The first edition ...
'', he wrote:
Boole (1854): From his "laws of the mind" Boole derives Aristotle's "Law of contradiction"
The title of
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ire ...
'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
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
.
Rationale: How the "laws of the mind" are to be distinguished
Boole begins his chapter I "Nature and design of this Work" with a discussion of what characteristic distinguishes, generally, "laws of the mind" from "laws of nature":
: "The general laws of Nature are not, for the most part, immediate objects of perception. They are either inductive inferences from a large body of facts, the common truth in which they express, or, in their origin at least, physical hypotheses of a causal nature. ... They are in all cases, and in the strictest sense of the term, probable conclusions, approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience. ..."
Contrasted with this are what he calls "laws of the mind": Boole asserts these are known in their first instance, without need of repetition:
:"On the other hand, the knowledge of the laws of the mind does not require as its basis any extensive collection of observations. The general truth is seen in the particular instance, and it is not confirmed by the repetition of instances. ... we not only see in the particular example the general truth, but we see it also as a certain truth – a truth, our confidence in which will not continue to increase with increasing experience of its practical verification." (Boole 1854:4)
Boole's signs and their laws
Boole begins with the notion of "signs" representing "classes", "operations" and "identity":
:"All the signs of Language, as an instrument of reasoning may be conducted by a system of signs composed of the following elements
::"1st Literal symbols as x, y, etc representing things as subjects of our conceptions,
::"2nd Signs of operation, as +, −, x standing for those operations of the mind by which conceptions of things are combined or resolved so as to form new conceptions involving the same elements,
::"3rd The sign of identity, =.
:And these symbols of Logic are in their use subject to definite laws, partly agreeing with and partly differing from the laws of the corresponding symbols in the science of Algebra. (Boole 1854:27)
Boole then clarifies what a "literal symbol" e.g. x, y, z,... represents—a name applied to a collection of instances into "classes". For example, "bird" represents the entire class of feathered winged warm-blooded creatures. For his purposes he extends the notion of class to represent membership of "one", or "nothing", or "the universe" i.e. totality of all individuals:
:"Let us then agree to represent the class of individuals to which a particular name or description is applicable, by a single letter, as z. ... By a class is usually meant a collection of individuals, to each of which a particular name or description may be applied; but in this work the meaning of the term will be extended so as to include the case in which but a single individual exists, answering to the required name or description, as well as the cases denoted by the terms "nothing" and "universe," which as "classes" should be understood to comprise respectively 'no beings,' 'all beings.'" (Boole 1854:28)
He then defines what the string of symbols e.g. xy means
odern logical &, conjunction
:"Let it further be agreed, that by the combination xy shall be represented that class of things to which the names or descriptions represented by x and y are simultaneously, applicable. Thus, if x alone stands for "white things," and y for "sheep," let xy stand for 'white Sheep;'" (Boole 1854:28)
Given these definitions he now lists his laws with their justification plus examples (derived from Boole):
* (1) xy = yx
ommutative law:: "x represents 'estuaries,' and y 'rivers,' the expressions xy and yx will indifferently represent" 'rivers that are estuaries,' or 'estuaries that are rivers,'"
* (2) xx = x, alternately x
2 = x
bsolute identity of meaning, Boole's "fundamental law of thought" cf page 49:: "Thus 'good, good' men, is equivalent to 'good' men".
Logical OR: Boole defines the "collecting of parts into a whole or separate a whole into its parts" (Boole 1854:32). Here the connective "and" is used disjunctively, as is "or"; he presents a commutative law (3) and a distributive law (4) for the notion of "collecting". The notion of ''separating'' a part from the whole he symbolizes with the "-" operation; he defines a commutative (5) and distributive law (6) for this notion:
* (3) y + x = x + y
ommutative law:: "Thus the expression 'men and women' is ... equivalent with the expression" women and men. Let x represent 'men,' y, 'women' and let + stand for 'and' and 'or' ..."
* (4) z(x + y) = zx + zy
istributive law:: z = European, (x = "men, y = women): European men and women = European men and European women
* (5) x − y = −y + x
ommutation law: separating a part from the whole
:: "All men (x) except Asiatics (y)" is represented by x − y. "All states (x) except monarchical states (y)" is represented by x − y
* (6) z(x − y) = zx − zy
istributive lawLastly is a notion of "identity" symbolized by "=". This allows for two axioms: (axiom 1): equals added to equals results in equals, (axiom 2): equals subtracted from equals results in equals.
* (7) Identity ("is", "are") e.g. x = y + z, "stars" = "suns" and "the planets"
Nothing "0" and Universe "1": He observes that the only two numbers that satisfy xx = x are 0 and 1. He then observes that 0 represents "Nothing" while "1" represents the "Universe" (of discourse).
The logical NOT: Boole defines the contrary (logical NOT) as follows (his Proposition III):
:"If x represent any class of objects, then will 1 − x represent the contrary or supplementary class of objects, i.e. the class including all objects which are not comprehended in the class x" (Boole 1854:48)
::If x = "men" then "1 − x" represents the "universe" less "men", i.e. "not-men".
The notion of a particular as opposed to a universal: To represent the notion of "some men", Boole writes the small letter "v" before the predicate-symbol "vx" some men.
Exclusive- and inclusive-OR: Boole does not use these modern names, but he defines these as follows x(1-y) + y(1-x) and x + y(1-x), respectively; these agree with the formulas derived by means of the modern Boolean algebra.
Boole derives the law of contradiction
Armed with his "system" he derives the "principle of
onontradiction" starting with his law of identity: x
2 = x. He subtracts x from both sides (his axiom 2), yielding x
2 − 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
(1 − x)=0thus 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)
Boole defines the notion "domain (universe) of discourse"
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
omainhas been specified explicitly or implicitly, or the specification of a D
omainis too vague (Kleene 1967:84).
Hamilton (1837–38 lectures on Logic, published 1860): a 4th "Law of Reason and Consequent"
As noted above,
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
specifies ''four'' laws—the three traditional plus the fourth "Law of Reason and Consequent"—as follows:
:"XIII. The Fundamental Laws of Thought, or the conditions of the thinkable, as commonly received, are four: – 1. The Law of Identity; 2. The Law of Contradiction; 3. The Law of Exclusion or of Excluded Middle; and, 4. The Law of Reason and Consequent, or of
Sufficient Reason."
Rationale: "Logic is the science of the Laws of Thought as Thought"
Hamilton opines that thought comes in two forms: "necessary" and "contingent" (Hamilton 1860:17). With regards the "necessary" form he defines its study as "logic": "Logic is the science of the necessary forms of thought" (Hamilton 1860:17). To define "necessary" he asserts that it implies the following four "qualities":
:(1) "determined or necessitated by the nature of the thinking subject itself ... it is subjectively, not objectively, determined;
:(2) "original and not acquired;
:(3) "universal; that is, it cannot be that it necessitates on some occasions, and does not necessitate on others.
:(4) "it must be a law; for a law is that which applies to all cases without exception, and from which a deviation is ever, and everywhere, impossible, or, at least, unallowed. ... This last condition, likewise, enables us to give the most explicit enunciation of the object-matter of Logic, in saying that Logic is the science of the Laws of Thought as Thought, or the science of the Formal Laws of Thought, or the science of the Laws of the Form of Thought; for all these are merely various expressions of the same thing."
Hamilton's 4th law: "Infer nothing without ground or reason"
Here's Hamilton's fourth law from his LECT. V. LOGIC. 60–61:
:"I now go on to the fourth law.
:"Par. XVII. Law of Sufficient Reason, or of Reason and Consequent:
:"XVII. The thinking of an object, as actually characterized by positive or by negative attributes, is not left to the caprice of Understanding – the faculty of thought; but that faculty must be necessitated to this or that determinate act of thinking by a knowledge of something different from, and independent of; the process of thinking itself. This condition of our understanding is expressed by the law, as it is called, of Sufficient Reason (principium Rationis Sufficientis); but it is more properly denominated the law of Reason and Consequent (principium Rationis et Consecutionis). That knowledge by which the mind is necessitated to affirm or posit something else, is called the ''logical reason ground,'' or ''antecedent''; that something else which the mind is necessitated to affirm or posit, is called the ''logical consequent''; and the relation between the reason and consequent, is called the ''logical connection or consequence''. This law is expressed in the formula – Infer nothing without a ground or reason.
1
:Relations between Reason and Consequent: The relations between Reason and Consequent, when comprehended in a pure thought, are the following:
:1. When a reason is explicitly or implicitly given, then there must ¶ exist a consequent; and, ''vice versa'', when a consequent is given, there must also exist a reason.
::
1 See Schulze, ''Logik'', §19, and Krug, ''Logik'', §20, – ED.
:2. Where there is no reason there can be no consequent; and, ''vice versa'', where there is no consequent (either implicitly or explicitly) there can be no reason. That is, the concepts of reason and of consequent, as reciprocally relative, involve and suppose each other.
:The logical significance of this law: The logical significance of the law of Reason and Consequent lies in this, – That in virtue of it, thought is constituted into a series of acts all indissolubly connected; each necessarily inferring the other. Thus it is that the distinction and opposition of possible, actual and necessary matter, which has been introduced into Logic, is a doctrine wholly extraneous to this science.
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:
Russell (1903–1927)
The sequel to
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
's 1903 "The Principles of Mathematics" became the three-volume work named ''
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' (hereafter ''PM''), written jointly with
Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...
. 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".
''The Principles of Mathematics'' (1903)
In his 1903 "Principles" Russell defines Symbolic or Formal Logic (he uses the terms synonymously) as "the study of the various general types of deduction" (Russell 1903:11). He asserts that "Symbolic Logic is essentially concerned with inference in general" (Russell 1903:12) and with a footnote indicates that he does not distinguish between inference and
deduction; moreover he considers
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
"to be either disguised deduction or a mere method of making plausible guesses" (Russell 1903:11). This opinion will change by 1912, when he deems his "principle of induction" to be par with the various "logical principles" that include the "Laws of Thought".
In his Part I "The Indefinables of Mathematics" Chapter II "Symbolic Logic" Part A "The Propositional Calculus" Russell reduces deduction ("propositional calculus") to 2 "indefinables" and 10 axioms:
:"17. We require, then, in the propositional calculus, no indefinable except the two kinds of implication
imple aka "material" and "formal"- remembering, however, that formal implication is a complex notion, whose analysis remains to be undertaken. As regards our two indefinables, we require certain indemonstrable propositions, which hitherto I have not succeeded in reducing to less ten (Russell 1903:15).
From these he ''claims'' to be able to ''derive'' the law of excluded middle and the law of contradiction but does not exhibit his derivations (Russell 1903:17). Subsequently, he and Whitehead honed these "primitive principles" and axioms into the nine found in ''PM'', and here Russell actually ''exhibits'' these two derivations at ❋1.71 and ❋3.24, respectively.
''The Problems of Philosophy'' (1912)
By 1912 Russell in his "Problems" pays close attention to "induction" (inductive reasoning) as well as "deduction" (inference), both of which represent just two ''examples'' of "self-evident logical principles" that include the "Laws of Thought."
Induction principle: Russell devotes a chapter to his "induction principle". He describes it as coming in two parts: firstly, as a repeated collection of evidence (with no failures of association known) and therefore increasing probability that whenever A happens B follows; secondly, in a fresh instance when indeed A happens, B will indeed follow: i.e. "a sufficient number of cases of association will make the probability of a fresh association nearly a certainty, and will make it approach certainty without limit."
He then collects all the cases (instances) of the induction principle (e.g. case 1: A
1 = "the rising sun", B
1 = "the eastern sky"; case 2: A
2 = "the setting sun", B
2 = "the western sky"; case 3: etc.) into a "general" law of induction which he expresses as follows:
:"(a) The greater the number of cases in which a thing of the sort A has been found associated with a thing of the sort B, the more probable it is (if cases of failure of association are known) that A is always associated with B;
:"(b) Under the same circumstances, a sufficient number of cases of the association of A with B will make it nearly certain that A is always associated with B, and will make this general law approach certainty without limit."
He makes an argument that this induction principle can neither be disproved or proved by experience, the failure of disproof occurring because the law deals with ''probability'' of success rather than certainty; the failure of proof occurring because of unexamined cases that are yet to be experienced, i.e. they will occur (or not) in the future. "Thus we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future".
In his next chapter ("On Our Knowledge of General Principles") Russell offers other principles that have this similar property: "which cannot be proved or disproved by experience, but are used in arguments which start from what is experienced." He asserts that these "have even greater evidence than the principle of induction ... the knowledge of them has the same degree of certainty as the knowledge of the existence of sense-data. They constitute the means of drawing inferences from what is given in sensation".
[Russell 1912:70, 1997 edition]
Inference principle: Russell then offers an example that he calls a "logical" principle. Twice previously he has asserted this principle, first as the 4th axiom in his 1903 and then as his first "primitive proposition" of ''PM'': "❋1.1 Anything implied by a true elementary proposition is true". Now he repeats it in his 1912 in a refined form: "Thus our principle states that if this implies that, and this is true, then that is true. In other words, 'anything implied by a true proposition is true', or 'whatever follows from a true proposition is true'. This principle he places great stress upon, stating that "this principle is really involved – at least, concrete instances of it are involved – in all demonstrations".
He does not call his inference principle ''
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
'', but his formal, symbolic expression of it in ''PM'' (2nd edition 1927) is that of ''modus ponens''; modern logic calls this a "rule" as opposed to a "law". In the quotation that follows, the symbol "⊦" is the "assertion-sign" (cf ''PM'':92); "⊦" means "it is true that", therefore "⊦p" where "p" is "the sun is rising" means "it is true that the sun is rising", alternately "The statement 'The sun is rising' is true". The "implication" symbol "⊃" is commonly read "if p then q", or "p implies q" (cf ''PM'':7). Embedded in this notion of "implication" are two "primitive ideas", "the Contradictory Function" (symbolized by NOT, "~") and "the Logical Sum or Disjunction" (symbolized by OR, "⋁"); these appear as "primitive propositions" ❋1.7 and ❋1.71 in ''PM'' (PM:97). With these two "primitive propositions" Russell defines "p ⊃ q" to have the formal logical equivalence "NOT-p OR q" symbolized by "~p ⋁ q":
:"''Inference''. The process of inference is as follows: a proposition "p" is asserted, and a proposition "p implies q" is asserted, and then as a sequel the proposition "q" is asserted. The trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error. Accordingly, whenever, in symbols, where p and q have of course special determination
::" "⊦p" and "⊦(p ⊃ q)"
:" have occurred, then "⊦q" will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of "⊦q". ... An inference is the dropping of a true premiss; it is the dissolution of an implication".
In other words, in a long "string" of inferences, after each inference we can detach the "consequent" "⊦q" from the symbol string "⊦p, ⊦(p⊃q)" and not carry these symbols forward in an ever-lengthening string of symbols.
The three traditional "laws" (principles) of thought: Russell goes on to assert other principles, of which the above logical principle is "only one". He asserts that "some of these must be granted before any argument or proof becomes possible. When some of them have been granted, others can be proved." Of these various "laws" he asserts that "for no very good reason, three of these principles have been singled out by tradition under the name of 'Laws of Thought'.
[Russell 1912:72, 1997 edition.] And these he lists as follows:
: "(1) ''The law of identity'': 'Whatever is, is.'
: (2) ''The law of contradiction'': 'Nothing can both be and not be.'
: (3) ''The law of excluded middle'': 'Everything must either be or not be.'"
Rationale: Russell opines that "the name 'laws of thought' is ... 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''." But he rates this a "large question" and expands it in two following chapters where he begins with an investigation of the notion of "a priori" (innate, built-in) knowledge, and ultimately arrives at his acceptance of the Platonic "world of universals". In his investigation he comes back now and then to the three traditional laws of thought, singling out the law of contradiction in particular: "The conclusion that the law of contradiction is a law of ''thought'' is nevertheless erroneous ...
ather the law of contradiction is about things, and not merely about thoughts ... a fact concerning the things in the world."
His argument begins with the statement that the three traditional laws of thought are "samples of self-evident principles". For Russell the matter of "self-evident" merely introduces the larger question of how we derive our knowledge of the world. He cites the "historic controversy ... between the two schools called respectively 'empiricists'
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