TheInfoList

Logic is an interdisciplinary field which studies
truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

and
reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: ...

.
Informal logic Informal logic encompasses the principles of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: g ...
seeks to characterize valid arguments informally, for instance by listing varieties of
fallacies A fallacy is the use of invalid or otherwise faulty reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of ...
.
Formal logic Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...
represents statements and argument patterns symbolically, using formal systems such as
first order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, wh ...
. Within formal logic,
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
studies the mathematical characteristics of formal logical systems, while
philosophical logic Understood in a narrow sense, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophi ...
applies them to philosophical problems such as the nature of
meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discussed in philosophy * Meaning (non-linguistic), a general ter ...
,
knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to e ...

, and
existence Existence is the ability of an entity to interact with physical reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to re ...

. Systems of formal logic are also applied in other fields including
linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the methods for studying ...

,
cognitive science Cognitive science is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fie ...

, and
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
. Logic has been studied since
Antiquity Antiquity or Antiquities may refer to Historical objects or periods Artifacts * Antiquities, objects or artifacts surviving from ancient cultures Eras Any period before the European Middle Ages In the history of Europe, the Middle Ages ...

, early approaches including
Aristotelian logic In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, langu ...
,
Stoic logic Stoic logic is the system of propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying log ...
,
Anviksiki Ānvīkṣikī is a term in Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-Euro ...
, and the
mohists Mohism or Moism () was an ancient Chinese philosophy Chinese philosophy originates in the Spring and Autumn period () and Warring States period (), during a period known as the "Hundred Schools of Thought", which was characteri ...
. Modern formal logic has its roots in the work of late 19th century mathematicians such as
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
.

# Definition

The word "logic" originates from the Greek word "logos", which has a variety of translations, such as
reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: ...
,
discourse Discourse is a generalization of the notion of a conversation Conversation is interactive communication Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (philosophy) ...

, or
language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...

. Logic is traditionally defined as the study of the
laws of thought The laws of thought are fundamental axiom An axiom, postulate, or assumption is a Statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the G ...
or correct
reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: ...

. This is usually understood in terms of
inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and i ...

s or
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
s: reasoning may be seen as the activity of drawing inferences, whose outward expression is given in arguments. An inference or an argument is a set of premises together with a conclusion. Logic is interested in whether arguments are good or inferences are valid, i.e. whether the premises support their conclusions. These general characterizations apply to logic in the widest sense since they are true both for formal and informal logic. But many definitions of logic focus on formal logic because it is the paradigmatic form of logic. In this narrower sense, logic is a formal science that studies how conclusions follow from premises in a topic-neutral way. As a
formal science Formal science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable ...
, it contrasts with empirical sciences, like
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

or
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

, because it tries to characterize the inferential relations between premises and conclusions only based on how they are structured. This means that the actual content of these propositions, i.e. their specific topic, is not important for whether the inference is valid or not. This can be expressed by distinguishing between logical and non-logical vocabulary: inferences are valid because of the logical terms used in them, independent of the meanings of the non-logical terms. Valid inferences are characterized by the fact that the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false. The general logical structures characterizing valid inferences are called
rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their Syntax (logic), syntax, and returns a conclusion (or multiple-conclusion logic, ...
. In this sense, logic is often defined as the study of valid inference. This contrasts with another prominent characterization of logic as the science of
logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...
s. A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all
possible world A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously ...
s and under all interpretations of its non-logical terms. These two characterizations of logic are closely related to each other: an inference is valid if the
material conditional The material conditional (also known as material implication) is an binary operator, operation commonly used in mathematical logic, logic. When the conditional symbol \rightarrow is semantics of logic, interpreted as material implication, a fo ...
from its premises to its conclusion is logically true. The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, ''a logic'' is a logical
formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essen ...
. Different logics differ from each other concerning the
formal languages In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...

used to express them and, most importantly, concerning the rules of inference they accept as valid. Starting in the 20th century, many new formal systems have been proposed. There is an ongoing debate about which of these systems should be considered logics in the strict sense instead of non-logical formal systems. Suggested criteria for this distinction include logical completeness and proximity to the intuitions governing classical logic. According to these criteria, it has been argued, for example, that
higher-order logic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s and
fuzzy logic In fuzzy mathematics, fuzzy logic is a form of many-valued logic in which the truth value Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ...

should not be considered ''logics'' when understood in a strict sense.

## Formal and informal logic

Logic can be studied formally or informally. A formal approach is one that abstracts away from content, looking for patterns that arise from form alone. For instance, the formal rule of
conjunction introduction Conjunction introduction (often abbreviated simply as conjunction and also called and introduction)Moore and Parker is a Validity (logic), valid rule of inference of propositional calculus, propositional logic. The rule makes it possible to introdu ...
states that any two propositions $P$ and $Q$ together imply their
conjunction Conjunction may refer to: * Conjunction (astronomy), in which two astronomical bodies appear close together in the sky * Conjunction (astrology), astrological aspect in horoscopic astrology * Conjunction (grammar), a part of speech * Logical conjun ...
$P \& Q$. This rule is formal since the symbols $P$ and $Q$ can stand in for any two statements, regardless of their content. Formal logic is often considered the paradigmatic form of logic. On an informal approach, inferences of this sort would have to be characterized using particular statements. Informal logic is often part of courses in
critical thinking Critical thinking is the analysis of facts to form a judgment. The subject is complex; several different Critical thinking#Definitions, definitions exist, which generally include the rational, skepticism, skeptical, and unbiased analysis or eval ...
, while informal approaches such as
dialectical logic Dialectical logic is the system of laws of thought, developed within the Hegelian and Marxist traditions, which seeks to supplement or replace the laws of formal logic Mathematical logic, also called formal logic, is a subfield of mathematics ...
and
argumentation theory Two men argue at a political protest in New York City. Argumentation theory, or argumentation, is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one ac ...
continue as areas of research.

# Subfields

## Philosophical logic

Philosophical logic is the study of logic within philosophy. It includes applications to problems in
epistemology Epistemology (; ) is the concerned with . Epistemologists study the nature, origin, and scope of knowledge, epistemic , the of , and various related issues. Epistemology is considered a major subfield of philosophy, along with other major ...
,
ethics Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong action (philosophy), behavior".''Internet Encyclopedia of Philosophy'"Ethics"/ref> The field of ethics, al ...
,
philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bra ...
, and
natural language semantics Semantics (from grc, wikt:σημαντικός, σημαντικός ''sēmantikós'', "significant") is the study of reference, Meaning (philosophy), meaning, or truth. The term can be used to refer to subfields of several distinct discipline ...
.

## Mathematical logic

Mathematical logic is the study of formal logic within mathematics. Major subareas include
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
,
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
,
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, and
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based
foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
. The latter was a major concern in early 20th century mathematical logic, which pursued the program of
logicism In the philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is ...
pioneered by philosopher-logicians such as
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...
. Mathematical theories were supposed to be logical tautologies, and the programme was to show this by means of a reduction of mathematics to logic. The various attempts to carry this out met with failure, from the crippling of Frege's project in his ''Grundgesetze'' by
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

, to the defeat of
Hilbert's programIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
by Gödel's incompleteness theorems.
Set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
originated in the study of the infinite by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
, and it has been the source of many of the most challenging and important issues in mathematical logic, from
Cantor's theorem In elementary set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. ...
, through the status of the
Axiom of Choice In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

and the question of the independence of the
continuum hypothesis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, to the modern debate on
large cardinal In the mathematical field of set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections o ...
axioms.
Recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...
captures the idea of computation in logical and
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
terms; its most classical achievements are the undecidability of the
Entscheidungsproblem In mathematics and computer science, the ' (, German language, German for "decision problem") is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers ...

by
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

, and his presentation of the
Church–Turing thesis In computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degr ...
. Today recursion theory is mostly concerned with the more refined problem of
complexity class Complexity characterises the behaviour of a system or model (disambiguation), model whose components interaction, interact in multiple ways and follow local rules, meaning there is no reasonable higher instruction to define the various possible i ...
es—when is a problem efficiently solvable?—and the classification of degrees of unsolvability.

## Computational logic

In computer science, logic is studied as part of the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
. Key areas of logic that are relevant to computing include
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
,
modal logic Modal logic is a collection of s originally developed and still widely used to represent statements about . The basic (1-place) modal operators are most often interpreted "□" for "Necessarily" and "◇" for "Possibly". In a , each can be expr ...
, and
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. Early computer machinery was based on ideas from logic such as the
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable N ...
. Computer scientists also apply concepts from logic to problems in computing and vice versa. For instance, modern
artificial intelligence Artificial intelligence (AI) is intelligence Intelligence has been defined in many ways: the capacity for abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstra ...

builds on logicians' work in
argumentation theory Two men argue at a political protest in New York City. Argumentation theory, or argumentation, is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one ac ...
, while
automated theorem provingAutomated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Gree ...
can assist logicians in finding and checking proofs. In
logic programming Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic prog ...
languages such as
Prolog Prolog is a logic programming Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some ...

, a program computes the consequences of logical axioms and rules to answer a query.

## Formal semantics of natural language

Formal semantics is a subfield of both
linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the methods for studying ...

and philosophy which uses logic to analyze meaning in
natural language In neuropsychology Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...
. It is an empirical field which seeks to characterize the
denotation The denotation of a word is its central sense A sense is a biological system used by an organism for sensation, the process of gathering information about the world and responding to Stimulus (physiology), stimuli. (For example, in the human bod ...
s of linguistic expressions and explain how those denotations are composed from the meanings of their parts. The field was developed by
Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such t ...
and
Barbara Partee Barbara Hall Partee (born June 23, 1940) is a Distinguished University Professor Professors in the United States commonly occupy any of several positions in academia. In the U.S., the word "professor" informally refers collectively to the Acade ...

in the 1970s, and remains an active area of research. Central questions include scope, binding, and
linguistic modality In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis inclu ...
.

# Concepts

## Varieties of reasoning

Argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
s are often divided into those that are ''
deductive Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
'', '' inductive'', and '' abductive''. In the most prominent conception of logic, only deductive reasoning counts as logic in the strict sense.On
abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infe ...
, see: * Magnani, L. 2001. ''Abduction, Reason, and Science: Processes of Discovery and Explanation''. New York: Kluwer Academic Plenum Publishers. xvii. . * Josephson, John R., and Susan G. Josephson. 1994. ''Abductive Inference: Computation, Philosophy, Technology''. New York: Cambridge University Press. viii. . * Bunt, H. and W. Black. 2000. ''Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics'', (''Natural Language Processing'' 1). Amsterdam:
John Benjamins John Benjamins Publishing Company is an independent academic publisher in social sciences and humanities Humanities are academic disciplines An academic discipline or academic field is a subdivision of knowledge that is Education, taugh ...
. vi. .
A ''deductive argument'' is one whose premises are intended to guarantee the truth of its conclusion. In other words, a deductive argument seeks to reach its conclusion by
logical necessity Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a Statement (logic), statement which is truth, true regardless of the truth or falsity of its constituent propositions. In other words, a logical tr ...
. For instance, the following argument is deductive. *Deductive argument: *# ''Victoria is tall.'' *# ''Victoria has brown hair.'' *# ''Therefore, Victoria is tall and has brown hair.'' ''Inductive arguments'' are those in which the premises are merely evidence for the conclusion. *Inductive argument: *# ''Victoria is tall.'' *# ''Tall people are generally good at basketball.'' *# ''Therefore, Victoria is good at basketball.'' ''Abductive reasoning'' involves reasoning to the most likely explanation. *Abductive argument: *# ''Victoria is tall.'' *# ''Victoria has brown hair.'' *# ''Therefore, Victoria must have a tall or brown-haired ancestor .''

## Logical truth

A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some
modal logic Modal logic is a collection of s originally developed and still widely used to represent statements about . The basic (1-place) modal operators are most often interpreted "□" for "Necessarily" and "◇" for "Possibly". In a , each can be expr ...
s, this notion can be understood equivalently as truth at all
possible world A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously ...
s.

## Formal system

A formal system of logic consists of a
language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...
, a proof system, and a
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
. A system's language and proof system are sometimes grouped together as the system's ''
syntax In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the ...
'', since they both concern the form rather than the content of the system's expressions. The term "a logic" is often used a
countable noun In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include ...
to refer to a particular formal system of logic. Different logics can differ from each other in their language, proof system, or their semantics. Starting in the 20th century, many new formal systems have been proposed.

### Formal language

A ''language'' is a set of
well formed formula In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
s. For instance, in
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, ...
, $P \& Q$ is a formula but $P Q \& \& \&$ is not. Languages are typically defined by providing an ''alphabet'' of basic expressions and
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the science, scientific study of language. It e ...

syntactic rules which build them into formulas.

### Proof system

A ''proof system'' is a collection of formal rules which define when a conclusion follows from given premises. For instance, the classical rule of
conjunction introduction Conjunction introduction (often abbreviated simply as conjunction and also called and introduction)Moore and Parker is a Validity (logic), valid rule of inference of propositional calculus, propositional logic. The rule makes it possible to introdu ...
states that $P \& Q$ follows from the premises $P$ and $Q$. Rules in a proof systems are always defined in terms of formulas' syntactic form, never in terms of their meanings. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are a number of different types of proof systems including
natural deductionIn logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
and sequent calculi. Proof systems are closely linked to philosophical work which characterizes logic as the study of valid inference.

### Semantics

A ''semantics'' is a system for
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
expressions of a formal language to their denotations. In many systems of logic, denotations are
truth value In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
s. For instance, the semantics for
classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, ...
assigns the formula $P \& Q$ the denotation "true" whenever $P$ is true and $Q$ is too.
Entailment Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies th ...
is a semantic relation which holds between formulas when the first cannot be true without the second being true as well. Semantics is closely tied to the philosophical characterization of logic as the study of
logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...
.

### Metalogic

Metalogic Metalogic is the study of the metatheory A metatheory or meta-theory is a theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational th ...
is the study of properties of formal systems. Two central metalogical properties are ''
soundness In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

'' and '' completeness''. A system of logic is ''sound'' when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is ''complete'' when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. Other important metalogical properties include ''
consistency In classical logic, classical deductive logic, a consistent theory (mathematical logic), theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic def ...

'', '' decidability'', and '' expressive power''.

## Propositional logic

Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives. For instance, propositional logic represents the
conjunction Conjunction may refer to: * Conjunction (astronomy), in which two astronomical bodies appear close together in the sky * Conjunction (astrology), astrological aspect in horoscopic astrology * Conjunction (grammar), a part of speech * Logical conjun ...
of two atomic propositions $P$ and $Q$ as the complex formula $P \& Q$. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones; it cannot represent inferences that results from the inner structure of a proposition.

## Predicate logic

Predicate logic is the generic term for symbolic formal systems such as
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantificat ...
,
second-order logic In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
,
many-sorted logic Many-sorted logic can reflect formally our intention not to handle the universe as a homogeneous collection of objects, but to partition (number theory), partition it in a way that is similar to types in type system, typeful programming. Both functi ...
, and
infinitary logicAn infinitary logic is a logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, though ...
. It provides an account of
quantifiers Quantifier may refer to: * Quantifier (linguistics), an indicator of quantity * Quantifier (logic) * Quantification (science) See also

*Quantification (disambiguation) {{disambiguation ...
general enough to express a wide set of arguments occurring in natural language. For example,
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...
's famous
barber paradox The barber paradox is a puzzle A puzzle is a game, Problem solving, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logical way, in order to arrive at the correct o ...
, "there is a man who shaves all and only men who do not shave themselves" can be formalised by the sentence $\left(\exists x \right) \left(\text\left(x\right) \wedge \left(\forall y\right) \left(\text\left(y\right) \rightarrow \left(\text\left(x, y\right) \leftrightarrow \neg \text\left(y, y\right)\right)\right)\right)$, using the non-logical predicate $\text\left(x\right)$ to indicate that ''x'' is a man, and the non-logical relation $\text\left(x, y\right)$ to indicate that ''x'' shaves ''y''; all other symbols of the formulae are logical, expressing the universal and existential
quantifiers Quantifier may refer to: * Quantifier (linguistics), an indicator of quantity * Quantifier (logic) * Quantification (science) See also

*Quantification (disambiguation) {{disambiguation ...
,
conjunction Conjunction may refer to: * Conjunction (astronomy), in which two astronomical bodies appear close together in the sky * Conjunction (astrology), astrological aspect in horoscopic astrology * Conjunction (grammar), a part of speech * Logical conjun ...
, implication,
negation In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

and
biconditional In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
. The development of predicate logic is usually attributed to
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
, who is also credited as one of the founders of
analytic philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality ...
, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Mathematical Logic by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gr ...
and
Wilhelm Ackermann Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens ...
in 1928. The analytical generality of predicate logic allowed the formalization of mathematics, drove the investigation of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, and allowed the development of
Alfred Tarski Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician ...
's approach to
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
. It provides the foundation of modern
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
.

## Modal logic

Modal logic is the study of formal systems originally developed to represent statements about necessity and possibility. For instance the modal formula $\Diamond P$ can be read as "possibly $P$" while $\Box P$ can be read as "necessarily $P$". Modal logics can be used to represent different phenomena depending on what ''flavor'' of necessity and possibility is under consideration. When $\Box$ is used to represent epistemic necessity, $\Box P$ states that $P$ is known. When $\Box$ is used to represent deontic necessity, $\Box P$ states that $P$ is a moral or legal obligation. Within philosophy, modal logics are widely used in
formal epistemology Formal epistemology uses formal methods from decision theory Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent's choices. Decision theory can be broken into two branches: normative decision ...
, formal ethics, and
metaphysics Metaphysics is the branch of philosophy that studies the first principles of being, identity and change, space and time, causality, necessity and possibility. It includes questions about the nature of consciousness and the relationship between ...

. Within
linguistic semantics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include phonetics, phonet ...
, systems based on modal logic are used to analyze
linguistic modality In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis inclu ...
in natural languages. The earliest formal system of modal logic was developed by
Avicenna Ibn Sina ( fa, ابن سینا), also known as Abu Ali Sina (), Pur Sina (), and often known in the West as Avicenna (;  – June 1037), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, t ...

, who ultimately developed a theory of " temporally modalized" syllogistic. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of C. I. Lewis in 1918, who formulated a family of rival axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include
deontic logicDeontic logic is the field of philosophical logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...
and
epistemic logicEpistemic modal logic is a subfield of modal logic Modal logic is a collection of formal systems originally developed and still widely used to represent statements about Linguistic modality, necessity and possibility. The basic Unary operation, unar ...
. The seminal work of
Arthur Prior Arthur Norman Prior (4 December 1914 – 6 October 1969), usually cited as A. N. Prior, was a New Zealand–born logician and philosopher. Prior (1957) founded tense logic, now also known as temporal logic, and made important contribution ...
applied the same formal language to treat
temporal logicIn logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
and paved the way for the marriage of the two subjects.
Saul Kripke Saul Aaron Kripke (; born November 13, 1940) is an American philosopher American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States ** Americans, citizens and nationa ...

and
Jaakko Hintikka Kaarlo Jaakko Juhani Hintikka (12 January 1929 – 12 August 2015) was a Finnish philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'l ...

built on this work to develop the frame semantics, which is now the standard semantics for modal logic. This
graph-theoretic In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' ( ...
way of looking at modality has driven many applications in
computational linguistics Computational linguistics is an interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, computational linguistics ...
and
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, such as dynamic logic.

## Non-classical logic

Non-classical logicNon-classical logics (and sometimes alternative logics) are formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are ...
s are systems that reject various rules of
classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Char ...
. They are motivated by the view that classical logic does not accurately represent the nature of truth and reasoning. One major non-classical paradigm is
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 of ...
, which rejects the
law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, Exclusive or, either this proposition or its negation is Truth value, true. It is one of the so called Law_of_thought#The_three_traditi ...
. Intuitionism was developed by the Dutch mathematicians L.E.J. Brouwer and
Arend Heyting __notoc__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
to underpin their constructive approach to mathematics, in which the existence of a mathematical object can only be proven by constructing it. Intuitionism was further pursued by
Gerhard Gentzen Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such t ...

,
Kurt Gödel Kurt Friedrich Gödel (; ; April 28, 1906 – January 14, 1978) was a Austrian-German-American logician, mathematician, and analytic philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians ...
,
Michael Dummett Sir Michael Anthony Eardley Dummett (1925–2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He was, until 1992, Wykeham ...
, among others. Intuitionistic logic is of great interest to computer scientists, as it is a
constructive logic Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has a ...
and sees many applications, such as extracting verified programs from proofs and influencing the design of
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s through the formulae-as-types correspondence. It is closely related to nonclassical systems such as
Gödel–Dummett logicIn mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate lo ...
and inquisitive logic. Multi-valued logics depart from classicality by rejecting the
principle of bivalence In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...
which requires all propositions to be either true or false. For instance,
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. He was born in Lemberg, a city in the Austrian Galicia, Galician Kingdom of Austria-Hungar ...

and
Stephen Cole Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
both proposed
ternary logic In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
s which have a third truth value representing that a statement's truth value is indeterminate. These logics have seen applications including to
presupposition In the branch of linguistics known as pragmatics In linguistics and related fields, pragmatics is the study of how context (language use), context contributes to meaning. The field of study evaluates how human language is utilized in social int ...
in linguistics. Fuzzy logics are multivalued logics that have an infinite number of "degrees of truth", represented by a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
between 0 and 1.

# Controversies

## "Is Logic Empirical?"

What is the
epistemological Epistemology (; ) is the branch of philosophy concerned with knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. The usual test for a statement of fact ...

status of the laws of logic? What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled "
Is Logic Empirical?"Is Logic Empirical?" is the title of two articles (one by Hilary Putnam and another by Michael Dummett) that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the questio ...
"
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'love ...

, building on a suggestion of W. V. Quine, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of
mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ...

or of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity, substituting for classical logic the
quantum logic In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper by Garrett Birkhoff and John von Neumann, w ...
proposed by
Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
and
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

. Another paper of the same name by
Michael Dummett Sir Michael Anthony Eardley Dummett (1925–2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He was, until 1992, Wykeham ...
argues that Putnam's desire for realism mandates the law of distributivity. Distributivity of logic is essential for the realist's understanding of how propositions are true of the world in just the same way as he has argued the principle of bivalence is. In this way, the question, "Is Logic Empirical?" can be seen to lead naturally into the fundamental controversy in
metaphysics Metaphysics is the branch of philosophy that studies the first principles of being, identity and change, space and time, causality, necessity and possibility. It includes questions about the nature of consciousness and the relationship between ...

on realism versus anti-realism.

## Tolerating the impossible

Georg Wilhelm Friedrich Hegel Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For cit ...
was deeply critical of any simplified notion of the
law of non-contradiction In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
. It was based on
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ...

's idea that this law of logic also requires a sufficient ground to specify from what point of view (or time) one says that something cannot contradict itself. A building, for example, both moves and does not move; the ground for the first is our solar system and for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable. Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency.
Relevance logicRelevance logic, also called relevant logic, is a kind of non-classical logicClassical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, ...
and
paraconsistent logic A paraconsistent logic is an attempt at a logical system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical cal ...
are the most important approaches here, though the concerns are different: a key consequence of
classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Char ...
and some of its rivals, 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 of ...
, is that they respect the
principle of explosion In classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to r ...
, which means that the logic collapses if it is capable of deriving a contradiction.
Graham Priest Graham Priest (born 1948) is Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne where he was Boyce Gibson Professor of Philosophy and also at the University of St And ...

, the main proponent of
dialetheismDialetheism (from Ancient Greek, Greek 'twice' and 'truth') is the view that there are statement (logic), statements which are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. S ...
, has argued for paraconsistency on the grounds that there are in fact, true contradictions.

## Conceptions of logic

Logic arose from a concern with correctness of
argumentation Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be reached through logical reasoning; that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, co ...
. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the ''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia An online encyclopedia, also called an Internet encyclopedia, or a digital encyclopedia, is an encyclopedia An encyclopedia or encyclopaedia (British E ...
'' that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of
rationality Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with t ...

. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations." The idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to
logicism In the philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is ...
in mathematics (19th and 20th centuries) and the advent of the influence of mathematical logic on philosophy. A consequence of taking logic to treat special kinds of argument is that it leads to identification of special kinds of truth, the logical truths (with logic equivalently being the study of logical truth), and excludes many of the original objects of study of logic that are treated as informal logic.
Robert Brandom Robert Boyce Brandom (born March 13, 1950) is an American philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The co ...

has argued against the idea that logic is the study of a special kind of logical truth, arguing that instead one can talk of the logic of material inference (in the terminology of Wilfred Sellars), with logic making explicit the commitments that were originally implicit in informal inference.

## Rejection of logical truth

The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, sometimes leading to the conclusion that there are no
logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...
s. This is in contrast with the usual views in
philosophical skepticism Philosophical skepticism (American and British English spelling differences, UK spelling: scepticism; from Ancient Greek, Greek σκέψις ''skepsis'', "inquiry") is a family of Philosophy, philosophical views that question the possibility o ...
, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of
Sextus Empiricus Sextus Empiricus ( grc-gre, Σέξτος Ἐμπειρικός; c. 160 – c. 210 AD) was a Ancient Greece, Greek Pyrrhonism, Pyrrhonist philosopher and a physician. His philosophical works are the most complete surviving account of ancient Gree ...
.
Friedrich Nietzsche Friedrich Wilhelm Nietzsche (; or ; 15 October 1844 – 25 August 1900) was a German philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, me ...

provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a "... mobile army of metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins". His rejection of truth did not lead him to reject the idea of either inference or logic completely but rather suggested that "logic
ame American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States. Currently, American English is the mo ...

into existence in man's head utof illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished". Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: "Logic, too, also rests on assumptions that do not correspond to anything in the real world". This position held by Nietzsche however, has come under extreme scrutiny for several reasons. Some philosophers, such as
Jürgen Habermas Jürgen Habermas (, ; ; born 18 June 1929) is a German philosopher and sociologist in the tradition of critical theory and pragmatism. His work addresses communicative rationality and the public sphere. Associated with the Frankfurt School, Habe ...
, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge.
Georg Lukács Georg may refer to: * Georg (film), ''Georg'' (film), 1997 *Georg (musical), Estonian musical * Georg (given name) * Georg (surname) * , a Kriegsmarine coastal tanker See also

* George (disambiguation) {{disambiguation ...
, in his book ''The Destruction of Reason'', asserts that, "Were we to study Nietzsche's statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible."
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...
described Nietzsche's irrational claims with "He is fond of expressing himself paradoxically and with a view to shocking conventional readers" in his book ''A History of Western Philosophy''.

# History

Logic was developed independently in several cultures during antiquity. One major early contributor was
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

, who developed
term logic In philosophy, term logic, also known as traditional logic, Syllogism, syllogistic logic or Aristotelianism, Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and was developed further in ancient history mostly ...
in his ''
Organon The ''Organon'' ( grc, Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics. They are as follows: Constitut ...

'' and ''
Prior Analytics The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by on , known as his , composed around 350 BCE. Being one of the six extant Aristotelian writings on logic and scientific method, it is pa ...
''. In this approach, ''judgements'' are broken down into ''propositions'' consisting of two terms that are related by one of a fixed number of relation. Inferences are expressed by means of
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...
s that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises. Aristotle's monumental insight was the notion that arguments can be characterized in terms of their form. The later logician described this insight as "one of Aristotle's greatest inventions". Aristotle's system of logic was also responsible for the introduction of
hypothetical syllogism In classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach ...
, temporal
modal logic Modal logic is a collection of s originally developed and still widely used to represent statements about . The basic (1-place) modal operators are most often interpreted "□" for "Necessarily" and "◇" for "Possibly". In a , each can be expr ...
, and
inductive logic Inductive reasoning is a method of reasoning in which the premises are viewed as supplying ''some'' evidence, but not full assurance, of the truth of the conclusion. It is also described as a method where one's experiences and observations, in ...
, as well as influential vocabulary such as , predicables,
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...
s and
proposition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
s.
Aristotelian logic In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, langu ...
was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century. It has now been superseded by later work, though many of its key insights live on in modern systems of logic.
Ibn Sina Ibn Sina ( fa, ابن سینا), also known as Abu Ali Sina (), Pur Sina (), and often known in the West as Avicenna (;  – June 1037), was a Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' ...

(Avicenna) (980–1037 CE) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world, and also had an important influence on Western medieval writers such as
Albertus Magnus Albertus Magnus (c. 1200 – November 15, 1280), also known as Saint Albert the Great or Albert of Cologne, was a German Catholic The Catholic Church, often referred to as the Roman Catholic Church, is the List of Christian denominat ...

and
William of Ockham William of Ockham (; also Occam, from la, Gulielmus Occamus; 1287 – 1347) was an English Franciscan friar, Scholasticism, scholastic philosopher, and theologian, who is believed to have been born in Ockham, Surrey, Ockham, a small village in ...

. Avicenna wrote on the
hypothetical syllogism In classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach ...
and on the
propositional calculus Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
. He developed an original "temporally modalized" syllogistic theory, involving
temporal logicIn logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
and
modal logic Modal logic is a collection of s originally developed and still widely used to represent statements about . The basic (1-place) modal operators are most often interpreted "□" for "Necessarily" and "◇" for "Possibly". In a , each can be expr ...
. He also made use of
inductive logic Inductive reasoning is a method of reasoning in which the premises are viewed as supplying ''some'' evidence, but not full assurance, of the truth of the conclusion. It is also described as a method where one's experiences and observations, in ...
, such as the methods of agreement, difference, and concomitant variation which are critical to the
scientific method The scientific method is an empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence ...

.Goodman, Lenn Evan (2003), ''Islamic Humanism'', p. 155,
Oxford University Press Oxford University Press (OUP) is the university press A university press is an academic publishing Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for fre ...

, .
Fakhr al-Din al-Razi Fakhr al-Dīn al-Rāzī or Fakhruddin Razi ( fa, فخر الدين رازي) (26 January 1150 - 29 March 1210) often known by the sobriquet Sultan of the theologians, was a Persian polymath, Islamic scholar and a pioneer of inductive logic. He w ...
(b. 1149) criticised Aristotle's " first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by
John Stuart Mill John Stuart Mill (20 May 1806 – 7 May 1873), also cited as J. S. Mill, was an English philosopher, Political economy, political economist, Member of Parliament (United Kingdom), Member of Parliament (MP) and civil servant. One of the most i ...
(1806–1873).
Muhammad Iqbal Sir Muhammad Iqbal ( ur, ; 9 November 187721 April 1938), was a South Asian Muslim writer, philosopher, Quote: "In Persian, ... he published six volumes of mainly long poems between 1915 and 1936, ... more or less complete works on philosop ...
, ''
The Reconstruction of Religious Thought in Islam ''The Reconstruction of Religious Thought in Islam'' is a compilation of lectures delivered by Muhammad Iqbal on Islamic philosophy and published in 1930. These lectures were delivered by Iqbal in Madras, Hyderabad Hyderabad ( , , ) ...
'', "The Spirit of Muslim Culture" (
cf. The abbreviation ''cf.'' (short for the la, confer/conferatur, both meaning 'compare') is used in writing to refer the reader to other material to make a comparison with the topic being discussed. Style guides recommend that ''cf.'' be used only ...
br>
an

In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the
High Middle Ages The High Middle Ages, or High Medieval Period, was the period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical c ...
, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of
scholasticism Scholasticism was a medieval In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people ...
. Initially, medieval Christian scholars drew on the classics that had been preserved in Latin through commentaries by such figures such as
Boethius Anicius Manlius Severinus Boëthius, commonly called Boethius (; also Boetius ; 477 – 524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', and philosopher of the early 6th century. He was born about a ye ...

, later the work of Islamic philosophers such as
Avicenna Ibn Sina ( fa, ابن سینا), also known as Abu Ali Sina (), Pur Sina (), and often known in the West as Avicenna (;  – June 1037), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, t ...

and
Averroes Ibn Rushd ( ar, ; full name Image:FML names-2.png, 300px, First/given, middle and last/family/surname with John Fitzgerald Kennedy as example. This shows a structure typical for the Anglosphere, among others. Other cultures use other struc ...

were drawn on, which expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries. In 1323,
William of Ockham William of Ockham (; also Occam, from la, Gulielmus Occamus; 1287 – 1347) was an English Franciscan friar, Scholasticism, scholastic philosopher, and theologian, who is believed to have been born in Ockham, Surrey, Ockham, a small village in ...

's influential '' Summa Logicae'' was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in 's satirical play ''
Erasmus Montanus ''Erasmus Montanus'' is a satirical play about academic An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning, research, or honorary membership. Academia is ...
''. The Chinese logical philosopher
Gongsun Long Gongsun Long (, BCLiu 2004, p. 336), courtesy name Zibing (子秉), was a Chinese philosopher and writer who was a member of the School of Names The School of Names (), sometimes called the School of Forms and Names (), was a school of Chinese ...
() proposed the paradox "One and one cannot become two, since neither becomes two."The four
Catuṣkoṭi''Catuṣkoṭi'' (Sanskrit; Devanagari: चतुष्कोटि, , Sinhala language, Sinhalese:චතුස්කෝටිකය) is a logical argument(s) of a 'suite of four discrete functions' or 'an indivisible quaternity' that has multiple a ...
logical divisions are formally very close to the four opposed propositions of the Greek ''
tetralemma The tetralemma is a figure that features prominently in the Indian logic, logic of India. Definition It states that with reference to any a logical proposition X, there are four possibilities: : X (affirmation) : \neg X (negation) : X \land\neg X ...
'', which in turn are analogous to the four
truth value In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
s of modern
relevance logicRelevance logic, also called relevant logic, is a kind of non-classical logicClassical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, ...
. (
cf. The abbreviation ''cf.'' (short for the la, confer/conferatur, both meaning 'compare') is used in writing to refer the reader to other material to make a comparison with the topic being discussed. Style guides recommend that ''cf.'' be used only ...
Belnap, Nuel. 1977. "A useful four-valued logic." In ''Modern Uses of Multiple-Valued Logic'', edited by Dunn and Eppstein. Boston: Reidel; Jayatilleke, K. N.. 1967. "The Logic of Four Alternatives." In ''Philosophy East and West''.
University of Hawaii Press The University of Hawaii Press is a university press that is part of the University of Hawaii, University of Hawaii. The University of Hawaii Press was founded in 1947, publishing research in all disciplines of the humanities and natural and soc ...
.)
In China, the tradition of scholarly investigation into logic, however, was repressed by the
Qin dynasty The Qin dynasty, or Ch'in dynasty in Wade–Giles Wade–Giles () is a romanization Romanization or romanisation, in linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of ever ...

following the legalist philosophy of
Han Feizi The ''Han Feizi'' () is an ancient Chinese text attributed to foundational political philosopher Han Fei Han Fei (; ; 233 BC), also known as Han Fei Zi, was a Chinese philosopher or statesman of the "Legalist Legalist, Inc. is a Legal finan ...
. In India, the
Anviksiki Ānvīkṣikī is a term in Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-Euro ...
school of logic was founded by Medhātithi (c. 6th century BCE). Innovations in the scholastic school, called
Nyaya (Sanskrit Sanskrit (; attributively , ; , , ) is a of that belongs to the branch of the . It arose in South Asia after its predecessor languages had there from the northwest in the late . Sanskrit is the of , the language of clas ...
, continued from ancient times into the early 18th century with the
Navya-Nyāya The Navya-Nyāya or Neo-Logical ''Darshana, darśana'' (view, system, or school) of Indian logic and Indian philosophy was founded in the 13th century Common Era, CE by the philosopher Gangesha Upadhyaya, Gangeśa Upādhyāya of Mithila (ancient), ...
school. By the 16th century, it developed theories resembling modern logic, such as
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
's "distinction between sense and reference of proper names" and his "definition of number", as well as the theory of "restrictive conditions for universals" anticipating some of the developments in modern
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
.Chakrabarti, Kisor Kumar. 1976. "Some Comparisons Between Frege's Logic and Navya-Nyaya Logic." ''
Philosophy and Phenomenological Research ''Philosophy and Phenomenological Research'' (''PPR'') is a bimonthly philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, ...
'' 36(4):554–63. . "This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory."
Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as
Charles Babbage Charles Babbage (; 26 December 1791 – 18 October 1871) was an English polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a subs ...

,
Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

, and
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 Irelan ...

. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively. The
syllogistic A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, ...
logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the
foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
stimulated the development of symbolic logic (now called
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
). In 1854, George Boole published ''
The Laws of Thought ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'' by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathem ...
'',. 1854. '' An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities.'' introducing symbolic logic and the principles of what is now known as
Boolean logic In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, respectively. Instead of elementary ...
. In 1879, Gottlob Frege published ''
Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation ...
'', which inaugurated modern logic with the invention of quantifier notation, reconciling the Aristotelian and Stoic logics in a broader system, and solving such problems for which Aristotelian logic was impotent, such as the problem of multiple generality. From 1910 to 1913,
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...
published ''
Principia Mathematica Image:Principia Mathematica 54-43.png, 500px, ✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...
'' on the foundations of mathematics, attempting to derive mathematical truths from
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s and
inference rule In the philosophy of logic Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to ...
s in symbolic logic. In 1931, Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues. The development of logic since Frege, Russell, and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see
analytic philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality ...
) and
philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bra ...
. Logic, especially sentential logic, is implemented in computer
logic circuits A logic gate is an idealized model of computation or physical electronics, electronic device implementing a Boolean function, a logical operation performed on one or more Binary number, binary inputs that produces a single binary output. Depending ...
and is fundamental to
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
. Logic is commonly taught by university philosophy, sociology, advertising and literature departments, often as a compulsory discipline.

# References

## Bibliography

* Barwise, J. (1982). ''Handbook of Mathematical Logic''. Elsevier. . * Belnap, N. (1977). "A useful four-valued logic". In Dunn & Eppstein, ''Modern uses of multiple-valued logic''. Reidel: Boston. * Bocheński, J.M. (1959). ''A précis of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
''. Translated from the French and German editions by Otto Bird. D. Reidel, Dordrecht, South Holland. * Bocheński, J.M. (1970). ''A history of
formal logic Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...
''. 2nd Edition. Translated and edited from the German edition by Ivo Thomas. Chelsea Publishing, New York. * * Cohen, R.S, and Wartofsky, M.W. (1974). ''Logical and Epistemological Studies in Contemporary Physics''. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. . * Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974). * Gabbay, D.M., and Guenthner, F. (eds., 2001–2005). ''Handbook of Philosophical Logic''. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht. * Haack, Susan (1996).'' Deviant Logic, Fuzzy Logic: Beyond the Formalism'', University of Chicago Press. * * Hilbert, D., and Ackermann, W, (1928). ''Grundzüge der theoretischen Logik'' ('' Principles of Mathematical Logic''). Springer-Verlag. * Hodges, W. (2001). ''Logic. An introduction to Elementary Logic'', Penguin Books. * Hofweber, T. (2004)
Logic and Ontology
''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia An online encyclopedia, also called an Internet encyclopedia, or a digital encyclopedia, is an encyclopedia An encyclopedia or encyclopaedia (British E ...
''.
Edward N. Zalta Edward Nouri Zalta (; born March 16, 1952) is an American philosopher who is a senior research scholar at the Center for the Study of Language and Information at Stanford University , mottoeng = "The wind of freedom blows" , type = Private un ...
(ed.). * Hughes, R.I.G. (1993, ed.). ''A Philosophical Companion to First-Order Logic''. Hackett Publishing. * * Kneale, William, and Kneale, Martha, (1962). ''The Development of Logic''. Oxford University Press, London, UK. * * Mendelson, Elliott, (1964). ''Introduction to Mathematical Logic''. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. * Smith, B. (1989). "Logic and the Sachverhalt". ''The Monist'' 72(1): 52–69. * Whitehead, Alfred North and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...
(1910). ''
Principia Mathematica Image:Principia Mathematica 54-43.png, 500px, ✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...
''. Cambridge University Press: Cambridge, England.

* * * *
An Outline for Verbal Logic
* Introductions and tutorials ** aimed at beginners. *
forall x: an introduction to formal logic
by P.D. Magnus, covers sentential and quantified logic. *
Logic Self-Taught: A Workbook
(originally prepared for on-line logic instruction). ***
Nicholas Rescher Nicholas Rescher (; ; born 15 July 1928) is a United States, German-American philosophy, philosopher, polymath, and author, teaching at the University of Pittsburgh. He is chairman of the Center for Philosophy of Science and was formerly chairman of ...

. (1964). ''Introduction to Logic'', St. Martin's Press. * Essays *
"Symbolic Logic"
an
"The Game of Logic"
Lewis Carroll Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English writer of children's fiction, notably ''Alice's Adventures in Wonderland'' and its sequel ''Through the Looking-Glass'' ...

, 1896. *
Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas.
In ''The Dictionary of the History of Ideas.'' * Online Tools *
Interactive Syllogistic Machine
A web-based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms. *
A Logic Calculator
A web-based application for evaluating simple statements in symbolic logic. * Reference material *

by
Peter Suber Peter Dain Suber (born November 8, 1951) is a philosopher specializing in the philosophy of law and open access to knowledge. He is a Senior Researcher at the Berkman Klein Center for Internet & Society, Director of the Harvard Office for Scholarly ...

, for translating from English into logical notation. *
Ontology and History of Logic. An Introduction
with an annotated bibliography. * Reading lists ** Th
London Philosophy Study Guide
offers many suggestions on what to read, depending on the student's familiarity with the subject: **

**

**

* {{Authority control Abstraction Critical thinking Formal sciences Philosophical logic Philosophy of logic Reasoning Thought