In
philosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
,
Logic (from the Greek
λογική logikē) is the formal systematic study of the
principleA principle is a law or rule that has to be, or usually is to be followed, or can be desirably followed, or is an inevitable consequence of something, such as the laws observed in nature or the way that a system is constructed...
s of valid
inferenceInference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of
philosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
,
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
,
semanticsSemantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
, and
computer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. It examines general forms which
argumentIn philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
s may take, which forms are valid, and which are fallacies. In philosophy, the study of logic is applied in most major areas:
ontologyOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
,
epistemology,
ethicsEthics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
,
metaphysicsMetaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
. In mathematics, it is the study of valid
inferenceInference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
s within some
formal languageA formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
. Logic is also studied in
argumentation theoryArgumentation theory, or argumentation, is the interdisciplinary study of how humans should, can, and do reach conclusions through logical reasoning, that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion...
.
Logic was studied in several ancient civilizations, including the
Indian subcontinentThe Indian subcontinent, also Indian Subcontinent, Indo-Pak Subcontinent or South Asian Subcontinent is a region of the Asian continent on the Indian tectonic plate from the Hindu Kush or Hindu Koh, Himalayas and including the Kuen Lun and Karakoram ranges, forming a land mass which extends...
,
ChinaChinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
and
GreeceAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
. Logic was established as a discipline by
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric.
Logic is often divided into two parts,
inductive reasoningInductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
and
deductive reasoningDeductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
.
The Study of Logic
The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional
Aristotelian syllogistic logicA syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
and modern symbolic logic are examples of formal logics.
- Informal logic
Informal logic, intuitively, refers to the principles of logic and logical thought outside of a formal setting. However, perhaps because of the informal in the title, the precise definition of informal logic is matters of some dispute. Ralph H. Johnson and J...
is the study of natural languageIn the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...
arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
are good examples of informal logic.
- Formal logic is the study of inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuance of natural language.
- Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two branches: propositional logic and predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
.
- Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
is an extension of symbolic logic into other areas, in particular to the study of model theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, proof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, and recursion theoryComputability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
.
Logical form
Logic is generally accepted to be
formal, in that it aims to analyze and represent the
form (or
logical formIn logic, the logical form of a sentence or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form...
) of any valid argument type. The form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. If one considers the notion of form to be too philosophically loaded, one could say that formalizing is nothing else than translating English sentences into the language of logic.
This is known as showing the
logical form of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical. It requires, first, ignoring those grammatical features which are irrelevant to logic (such as gender and declension if the argument is in Latin), replacing conjunctions which are not relevant to logic (such as 'but') with logical conjunctions like 'and' and replacing ambiguous or alternative logical expressions ('any', 'every', etc.) with expressions of a standard type (such as 'all', or the universal quantifier ∀).
Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression 'all As are Bs' shows the logical form which is common to the sentences 'all men are mortals', 'all cats are carnivores', 'all Greeks are philosophers' and so on.
That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variable letters to represent valid inferences in
Prior AnalyticsThe Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....
, leading Jan Łukasiewicz to say that the introduction of variables was 'one of Aristotle's greatest inventions'. According to the followers of Aristotle (such as
AmmoniusAmmonius Saccas was a Greek philosopher from Alexandria who was often referred to as one of the founders of Neoplatonism. He is mainly known as the teacher of Plotinus, whom he taught for eleven years from 232 to 243. He was undoubtably the biggest influence on Plotinus in his development of...
), only the logical principles stated in schematic terms belong to logic, and not those given in concrete terms. The concrete terms 'man', 'mortal', etc., are analogous to the substitution values of the schematic placeholders 'A', 'B', 'C', which were called the 'matter' (Greek 'hyle') of the inference.
The fundamental difference between modern formal logic and traditional or Aristotelian logic lies in their differing analysis of the logical form of the sentences they treat.
- In the traditional view, the form of the sentence consists of (1) a subject (e.g. 'man') plus a sign of quantity ('all' or 'some' or 'no'); (2) the copula which is of the form 'is' or 'is not'; (3) a predicate (e.g. 'mortal'). Thus: all men are mortal. The logical constants such as 'all', 'no' and so on, plus sentential connectives such as 'and' and 'or' were called 'syncategorematic' terms (from the Greek 'kategorei' – to predicate, and 'syn' – together with). This is a fixed scheme, where each judgement has an identified quantity and copula, determining the logical form of the sentence.
- According to the modern view, the fundamental form of a simple sentence is given by a recursive schema, involving logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s, such as a quantifier with its bound variable, which are joined to by juxtaposition to other sentences, which in turn may have logical structure.
- The modern view is more complex, since a single judgement of Aristotle's system will involve two or more logical connectives. For example, the sentence "All men are mortal" involves in term logic two non-logical terms "is a man" (here M) and "is mortal" (here D): the sentence is given by the judgement A(M,D). In predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
the sentence involves the same two non-logical concepts, here analyzed as and , and the sentence is given by , involving the logical connectives for universal quantificationIn predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
and implicationIn logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
.
- But equally, the modern view is more powerful: medieval logicians recognized the problem of multiple generality
The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:then it follows logically that:The syntax of traditional logic permits exactly four sentence types: "All As are Bs", "No As are...
, where Aristotelean logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure.
Deductive and inductive reasoning, and retroductive inference
Deductive reasoningDeductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
concerns what follows necessarily from given premises (if a, then b). However,
inductive reasoningInductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
—the process of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Correspondingly, we must distinguish between deductive validity and inductive validity (called "
cogency"). An inference is deductively valid
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there is no possible situation in which all the premises are true but the conclusion false. An inductive argument can be neither valid nor invalid; its premises give only some degree of probability, but not certainty, to its conclusion.
The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of
semanticsSemantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use
mathematical modelA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
s of probability. For the most part this discussion of logic deals only with deductive logic.
Retroductive inference is a mode of reasoning that Peirce proposed as operating over and above induction and deduction to “open up new ground” in processes of theorizing (1911, p. 2). He defines retroduction as a logical inference that allows us to "render comprehensible" some observations/events which we perceive, by relating these back to a posited state of affairs that would help to shed light on the observations
(Peirce, 1911, p. 2). He remarks that the “characteristic formula” of reasoning that he calls retroduction is that it involves reasoning from a consequent (any observed/experienced phenomena that confront us) to an antecedent (that is, a posited state of things that helps us to render comprehensible the observed phenomena). Or, as he otherwise puts it, it can be considered as “regressing from a consequent to a hypothetical antecedent” (1911, p. 4). See for instance, the discussion at: http://www.helsinki.fi/science/commens/dictionary.html
Some authors have suggested that this mode of inference can be used within social theorizing to postulate social structures/mechanisms that explain the way that social outcomes arise in social life and that in turn also indicate that these structures/mechanisms are alterable with sufficient social will (and visioning of alternatives). In other words, this logic is specifically liberative in that it can be used to point to transformative potential in our way of organizing our social existence by our re-examining/exploring the deep structures that generate outcomes (and life chances for people). In her book on New Racism (2010) Norma Romm offers an account of various interpretations of what can be said to be involved in retroduction as a form of inference and how this can also be seen to be linked to a style of theorizing (and caring) where processes of knowing (which she sees as dialogically rooted) are linked to social justice projects (http://www.springer.com/978-90-481-8727-0)
Consistency, validity, soundness, and completeness
Among the important properties that logical systems can have:
- Consistency
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...
, which means that no theorem of the system contradicts another.
- Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
, which means that the system's rules of proof will never allow a false inference from true premises. A logical system has the property of soundnessIn mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...
when the logical system has the property of validity and only uses premises that prove true (or, in the case of axioms, are true by definition).
- Completeness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
, of a logical system, which means that if a formula is true, it can be proven (if it is true, it is a theorem of the system).
- Soundness
In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...
, the term soundness has multiple separate meanings, which creates a bit of confusion throughout the literature. Most commonly, soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. A distinct, peripheral use of soundness refers to arguments, which means that the premises of a valid argument are true in the actual world.
Some logical systems do not have all four properties. As an example,
Kurt GödelKurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
's
incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
show that sufficiently complex formal systems of arithmetic cannot be consistent and complete; however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.
Rival conceptions of logic
Logic arose (see below) from a concern with correctness of argumentation. 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 PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of
rationalityIn philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one's beliefs with one's reasons for belief, or with one's actions with one's reasons for action...
. 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".
By contrast,
Immanuel KantImmanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
argued that logic should be conceived as the science of judgment, an idea taken up in
Gottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
's logical and philosophical work, where thought (German:
Gedanke) is substituted for judgment (German:
Urteil). On this conception, the valid inferences of logic follow from the structural features of
judgmentA judgment , in a legal context, is synonymous with the formal decision made by a court following a lawsuit. At the same time the court may also make a range of court orders, such as imposing a sentence upon a guilty defendant in a criminal matter, or providing a remedy for the plaintiff in a civil...
s or thoughts.
History
The earliest sustained work on the subject of logic is that of
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
. Aristotelian logic became widely accepted in science and mathematics and remained in wide use in the West until the early 19th century. Aristotle's system of logic was responsible for the introduction of
hypothetical syllogismIn logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.-Propositional logic:...
,
temporalIn logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
modal logicModal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, and
inductive logicInductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
. In
EuropeEurope is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with
ChristianA Christian is a person who adheres to Christianity, an Abrahamic, monotheistic religion based on the life and teachings of Jesus of Nazareth as recorded in the Canonical gospels and the letters of the New Testament...
faith. During the later period of the Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.
The
Chinese logicalIn the history of logic, logic in China plays a particularly interesting role due to its length and relative isolation from the strong current of development of the study of logic in Europe and the Islamic world, though it may have some influence from Indian logic due to the spread of...
philosopher Gongsun Long (ca. 325–250 BC) proposed the paradox "One and one cannot become two, since neither becomes two." In China, the tradition of scholarly investigation into logic, however, was repressed by the
Qin dynastyThe Qin Dynasty was the first imperial dynasty of China, lasting from 221 to 207 BC. The Qin state derived its name from its heartland of Qin, in modern-day Shaanxi. The strength of the Qin state was greatly increased by the legalist reforms of Shang Yang in the 4th century BC, during the Warring...
following the legalist philosophy of Han Feizi.
Logic in Islamic philosophy, particularly Avicenna's logic, was heavily influenced by Aristotelian logic.
In India, innovations in the scholastic school, called
Nyaya' is the name given to one of the six orthodox or astika schools of Hindu philosophy—specifically the school of logic...
, continued from ancient times into the early 18th century with the
Navya-NyayaThe Navya-Nyāya or Neo-Logical darśana of Indian logic and Indian philosophy was founded in the 13th century CE by the philosopher Gangeśa Upādhyāya of Mithila. It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati...
school. By the 16th century, it developed theories resembling modern logic, such as
Gottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
'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 theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as
Charles BabbageCharles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...
,
Augustus De MorganAugustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
, and
George BooleGeorge Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively.
The
syllogisticA syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the
foundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
stimulated the development of symbolic logic (now called
mathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
). In 1854, George Boole published
An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and ProbabilitiesThe Laws of Thought, more precisely, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, is a very influential 19th century book on logic by George Boole, the second of his two monographs on algebraic logic...
, introducing symbolic logic and the principles of what is now known as
Boolean logicBoolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
. In 1879, Gottlob Frege published
BegriffsschriftBegriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...
which inaugurated modern logic with the invention of
quantifierQuantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...
notation. From 1910 to 1913,
Alfred North WhiteheadAlfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
and
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
published
Principia MathematicaThe Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
on the foundations of mathematics, attempting to derive mathematical truths from
axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s and inference rules in symbolic logic. In 1931,
GödelGodel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*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 philosophyAnalytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
), and
Philosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to
computer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. Logic is commonly taught by university philosophy departments, often as a compulsory discipline.
Syllogistic logic
The
OrganonThe Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
was
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's body of work on logic, with the
Prior AnalyticsThe Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....
constituting the first explicit work in formal logic, introducing the syllogistic. The parts of syllogistic logic, also known by the name
term logicIn philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
, are the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of
syllogismA syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
s that consist of two propositions sharing a common term as premise, and a conclusion which is a proposition involving the two unrelated terms from the premises.
Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. However, it was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians. Also, the
problem of multiple generalityThe problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:then it follows logically that:The syntax of traditional logic permits exactly four sentence types: "All As are Bs", "No As are...
was recognised in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.
Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of propositional logic and the predicate calculus. Others use Aristotle in
argumentation theoryArgumentation theory, or argumentation, is the interdisciplinary study of how humans should, can, and do reach conclusions through logical reasoning, that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion...
to help develop and critically question argumentation schemes that are used in
artificial intelligenceArtificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
and legal arguments.
Propositional logic (sentential logic)
A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and in which a system of formal proof rules allows certain formulae to be established as "theorems".
Predicate logic
Predicate logic is the generic term for symbolic formal systems such as
first-order logicFirst-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
,
second-order logicIn logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
,
many-sorted logicMany-sorted logic can reflect formally our intention, not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming...
, and
infinitary logicAn infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and...
.
Predicate logic provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take. Predicate logic allows sentences to be analysed into subject and argument in several additional ways, thus allowing predicate logic to solve the
problem of multiple generalityThe problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:then it follows logically that:The syntax of traditional logic permits exactly four sentence types: "All As are Bs", "No As are...
that had perplexed medieval logicians.
The development of predicate logic is usually attributed to
Gottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Mathematical Logic by
David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and
Wilhelm AckermannWilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
in 1928. The analytical generality of predicate logic allowed the formalisation of mathematics, drove the investigation of
set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, and allowed the development of
Alfred TarskiAlfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
's approach to
model theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. It provides the foundation of modern
mathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
.
Frege's original system of predicate logic was second-order, rather than first-order.
Second-order logicIn logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
is most prominently defended (against the criticism of
Willard Van Orman QuineWillard Van Orman Quine was an American philosopher and logician in the analytic tradition...
and others) by
George BoolosGeorge Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
and
Stewart ShapiroStewart Shapiro is O'Donnell Professor of Philosophy at the Ohio State University and a regular visiting professor at the University of St Andrews in Scotland. He is an important contemporary figure in the philosophy of mathematics where he defends a version of structuralism. He studied...
.
Modal logic
In languages,
modalityIn linguistics, modality is what allows speakers to evaluate a proposition relative to a set of other propositions.In standard formal approaches to modality, an utterance expressing modality can always roughly be paraphrased to fit the following template:...
deals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles. For example, "
We go to the games" can be modified to give "
We should go to the games", and "
We can go to the games"" and perhaps "
We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.
The logical study of modality dates back to
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, who was concerned with the
alethic modalitiesAlethic modality is a linguistic modality which indicates logical necessity, possibility or impossibility.Alethic modality is often associated with epistemic modality in research. However, it has been questioned whether this modality should be considered distinct from epistemic modality which...
of
necessityIn U.S. criminal law, necessity may be either a possible justification or an exculpation for breaking the law. Defendants seeking to rely on this defense argue that they should not be held liable for their actions as a crime because their conduct was necessary to prevent some greater harm and when...
and
possibilityA logically possible proposition is one that can be asserted without implying a logical contradiction. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true...
, which he observed to be dual in the sense of De Morgan duality. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of
Clarence Irving LewisClarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in...
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 logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts...
and
epistemic logicEpistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy,...
. The seminal work of
Arthur PriorArthur Norman Prior was a noted logician and philosopher. Prior founded tense logic, now also known as temporal logic, and made important contributions to intensional logic, particularly in Prior .-Biography:Prior was entirely educated in New Zealand, where he was fortunate to have come under the...
applied the same formal language to treat
temporal logicIn logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
and paved the way for the marriage of the two subjects.
Saul KripkeSaul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center...
discovered (contemporaneously with rivals) his theory of
frame semanticsKripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
which revolutionised the formal technology available to modal logicians and gave a new
graph-theoreticIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
way of looking at modality that has driven many applications in
computational linguisticsComputational linguistics is an interdisciplinary field dealing with the statistical or rule-based modeling of natural language from a computational perspective....
and
computer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, such as
dynamic logicDynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general complex behaviors arising in linguistics, philosophy, AI, and other fields.-Language:...
.
Informal reasoning
The motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also to become a better person. Half of the works of Aristotle's
OrganonThe Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
treat inference as it occurs in an informal setting, side by side with the development of the syllogistic, and in the Aristotelian school, these informal works on logic were seen as complementary to Aristotle's treatment of
rhetoricRhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...
.
This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically
dialecticDialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
al logic will form the heart of a course in
critical thinkingCritical thinking is the process or method of thinking that questions assumptions. It is a way of deciding whether a claim is true, false, or sometimes true and sometimes false, or partly true and partly false. The origins of critical thinking can be traced in Western thought to the Socratic...
, a compulsory course at many universities.
Argumentation theoryArgumentation theory, or argumentation, is the interdisciplinary study of how humans should, can, and do reach conclusions through logical reasoning, that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion...
is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in
artificial intelligenceArtificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
and
lawLaw is a system of rules and guidelines which are enforced through social institutions to govern behavior, wherever possible. It shapes politics, economics and society in numerous ways and serves as a social mediator of relations between people. Contract law regulates everything from buying a bus...
.
Mathematical logic
Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.
The earliest use of mathematics and
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
in relation to logic and philosophy goes back to the ancient Greeks such as
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
,
PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
, and
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.
One of the boldest attempts to apply logic to mathematics was undoubtedly the
logicismLogicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
pioneered by philosopher-logicians such as
Gottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
and
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
: the idea was that mathematical theories were logical
tautologiesIn logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
, and the programme was to show this by means to a reduction of mathematics to logic. The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his
Grundgesetze by
Russell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
, to the defeat of
Hilbert's programIn mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
by Gödel's incompleteness theorems.
Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of
proof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
. Despite the negative nature of the incompleteness theorems,
Gödel's completeness theoremGödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
, a result in
model theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's
proof calculusIn mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules...
is enough to
describe the whole of mathematics, though not
equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.
If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject.
Set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
originated in the study of the infinite by
Georg CantorGeorg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
, and it has been the source of many of the most challenging and important issues in mathematical logic, from
Cantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
, through the status of the
Axiom of Choice and the question of the independence of the
continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
, to the modern debate on large cardinal axioms.
Recursion theoryComputability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
captures the idea of computation in logical and
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
terms; its most classical achievements are the undecidability of the
EntscheidungsproblemIn mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...
by
Alan TuringAlan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
, and his presentation of the Church-Turing thesis. Today recursion theory is mostly concerned with the more refined problem of
complexity classIn computational complexity theory, a complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form:...
es — when is a problem efficiently solvable? — and the classification of
degrees of unsolvabilityIn computer science and mathematical logic the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set...
.
Philosophical logic
Philosophical logicPhilosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g.,
free logicA free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain...
s, tense logics) as well as various extensions of
classical logicClassical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
(e.g.,
modal logicModal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
s), and non-standard semantics for such logics (e.g.,
KripkeSaul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center...
's technique of supervaluations in the semantics of logic).
Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure his own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to formulate an argument correctly.
Logic and computation
Logic cut to the heart of computer science as it emerged as a discipline:
Alan TuringAlan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
's work on the
EntscheidungsproblemIn mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...
followed from
Kurt GödelKurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
's work on the incompleteness theorems, and the notion of general purpose computers that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s.
In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with
mathematical notationMathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
, it would be possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of human reasoning. In
logic programmingLogic programming is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy's [1958] advice-taker proposal, logic is used as a purely declarative representation language, and a...
, a program consists of a set of axioms and rules. Logic programming systems such as
PrologProlog is a general purpose logic programming language associated with artificial intelligence and computational linguistics.Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of...
compute the consequences of the axioms and rules in order to answer a query.
Today, logic is extensively applied in the fields of
Artificial IntelligenceArtificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
, and
Computer ScienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, and these fields provide a rich source of problems in formal and informal logic.
Argumentation theoryArgumentation theory, or argumentation, is the interdisciplinary study of how humans should, can, and do reach conclusions through logical reasoning, that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion...
is one good example of how logic is being applied to artificial intelligence. The
ACM Computing Classification SystemThe ACM Computing Classification System is a subject classification system for computer science devised by the Association for Computing Machinery...
in particular regards:
- Section F.3 on Logics and meanings of programs and F.4 on Mathematical logic and formal languages as part of the theory of computer science: this work covers formal semantics of programming languages
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation...
, as well as work of formal methodsIn computer science and software engineering, formal methods are a particular kind of mathematically-based techniques for the specification, development and verification of software and hardware systems...
such as Hoare logicHoare logic is a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in 1969 by the British computer scientist and logician C. A. R. Hoare, and subsequently refined by Hoare and other researchers...
- Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
as fundamental to computer hardware: particularly, the system's section B.2 on Arithmetic and logic structures, relating to operatives AND, NOT, and OR;
- Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
and default logicDefault logic is a non-monotonic logic proposed by Raymond Reiter to formalize reasoning with default assumptions.Default logic can express facts like “by default, something is true”; by contrast, standard logic can only express that something is true or that something is false...
in Knowledge representation formalisms and methods, Horn clauseIn mathematical logic, a Horn clause is a clause with at most one positive literal. They are named after the logician Alfred Horn, who first pointed out the significance of such clauses in 1951...
s in logic programming, and description logicDescription logic is a family of formal knowledge representation languages. It is more expressive than propositional logic but has more efficient decision problems than first-order predicate logic....
.
Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using
automated theorem provingAutomated theorem proving or automated deduction, currently the most well-developed subfield of automated reasoning , is the proving of mathematical theorems by a computer program.- Decidability of the problem :...
the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.
Controversies
Just as there is disagreement over what logic is about, so there is disagreement about what logical truths there are.
Bivalence and the law of the excluded middle
The logics discussed above are all "
bivalentIn logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false propositions.
Non-classical logicNon-classical logics is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations...
s are those systems which reject bivalence.
Hegel developed his own dialectic logic that extended
KantKANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
's transcendental logic but also brought it back to ground by assuring us that "neither in heaven nor in earth, neither in the world of mind nor of nature, is there anywhere such an abstract 'either–or' as the understanding maintains. Whatever exists is concrete, with difference and opposition in itself".
In 1910
Nicolai A. VasilievNicolai Alexandrovich Vasiliev , also Vasil'ev, Vassilieff, Wassilieff was a Russian logician, philosopher, psychologist, poet, the forerunner of paraconsistent and multi-valued logics.-Early years:...
rejected the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction. In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing
ternary logicIn logic, a three-valued logic is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value...
, the first
multi-valued logicIn logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
.
Logics such as
fuzzy logicFuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
have since been devised with an infinite number of "degrees of truth", represented by a
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
between 0 and 1.
Intuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
was proposed by L.E.J. Brouwer as the correct logic for reasoning about mathematics, based upon his rejection of the law of the excluded middle as part of his
intuitionismIn the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
. Brouwer rejected formalisation in mathematics, but his student
Arend HeytingArend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
studied intuitionistic logic formally, as did
Gerhard GentzenGerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
. Intuitionistic logic has come to be of great interest to computer scientists, as it is a constructive logic, and is hence a logic of what computers can do.
Modal logicModal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalised with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable.
"Is logic empirical?"
What is the
epistemological status of the
laws of logicClassical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
? What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled "Is logic empirical?"
Hilary PutnamHilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
, 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
mechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
or of
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, 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
realistsContemporary philosophical realism is the belief that our reality, or some aspect of it, is ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc....
about the physical phenomena described by quantum theory, then we should abandon the
principle of distributivityThe principle of distributivity states that the algebraic distributive law is valid for classical logic, where both logical conjunction and logical disjunction are distributive over each other so that for any propositions A, B and C the equivalencesA \land \iff \lor andA \lor \iff \land...
, substituting for classical logic the
quantum logicIn quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
proposed by
Garrett BirkhoffGarrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
and
John von NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
.
Another paper by the same name by Sir Michael Dummett 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
metaphysicsMetaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
on realism versus anti-realism.
Implication: strict or material?
It is obvious that the notion of implication formalised in classical logic does not comfortably translate into natural language by means of "if… then…", due to a number of
problems called the
paradoxes of material implication.
The first class of paradoxes involves counterfactuals, such as "If the moon is made of green cheese, then 2+2=5", which are puzzling because natural language does not support the
principle of explosionThe principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of strict implication, which eventually led to more radically revisionist logics such as
relevance logicRelevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
.
The second class of paradoxes involves redundant premises, falsely suggesting that we know the succedent because of the antecedent: thus "if that man gets elected, granny will die" is materially true since granny is mortal, regardless of the man's election prospects. Such sentences violate the Gricean maxim of relevance, and can be modelled by logics that reject the principle of
monotonicity of entailmentMonotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such...
, such as relevance logic.
Tolerating the impossible
Hegel was deeply critical of any simplified notion of the Law of Non-Contradiction. It was based on Leibniz'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 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 logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
and
paraconsistent logicA paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
are the most important approaches here, though the concerns are different: a key consequence of
classical logicClassical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
and some of its rivals, such as
intuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
, is that they respect the
principle of explosionThe principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
, which means that the logic collapses if it is capable of deriving a contradiction.
Graham PriestGraham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
, the main proponent of
dialetheismDialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
, has argued for paraconsistency on the grounds that there are in fact, true contradictions.
Rejection of logical truth
The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases upon which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion that there are no logical truths. Observe that this is opposite to the usual views in
philosophical skepticismPhilosophical skepticism is both a philosophical school of thought and a method that crosses disciplines and cultures. Many skeptics critically examine the meaning systems of their times, and this examination often results in a position of ambiguity or doubt...
, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of
Sextus EmpiricusSextus Empiricus , was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism....
.
Friedrich NietzscheFriedrich Wilhelm Nietzsche was a 19th-century German philosopher, poet, composer and classical philologist...
provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealisation 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 [came] into existence in man's head [out] of 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. He fails to demonstrate the validity of his claims and merely asserts them rhetorically. Furthermore, his position has been claimed to be self-refuting by philosophers, such as
Jürgen HabermasJürgen Habermas is a German sociologist and philosopher in the tradition of critical theory and pragmatism. He is perhaps best known for his theory on the concepts of 'communicative rationality' and the 'public sphere'...
, who have accused Nietszche of not even having a coherent perspective let alone a theory of knowledge. George Lukacs in his book
The Destruction of Reason has asserted 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". Extreme skepticism such as that displayed by Nietzsche has not been met with much seriousness by analytic philosophers in the 20th century.
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
famously referred to Nietzsche's claims as "empty words" in his book
A History of Western Philosophy.
See also
- Digital electronics (also known as digital logic or logic gate
A logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation on one or more logic inputs and produces a single logic output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and...
s)
- Fallacies
- Logic puzzle
A logic puzzle is a puzzle deriving from the mathematics field of deduction.-History:The logic puzzle was first produced by Charles Lutwidge Dodgson, who is better known under his pen name Lewis Carroll, the author of Alice's Adventures in Wonderland...
- Logic symbols
- Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
- List of mathematics articles
- Outline of mathematics
The following outline is provided as an overview of and topical guide to mathematics:Mathematics – the search for fundamental truths in pattern, quantity, and change. For more on the relationship between mathematics and science, refer to the article on science.- Nature of mathematics :*...
- Metalogic
Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves...
- Outline of logic
- List of logic journals
- Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
- List of philosophy topics
- Outline of philosophy
- Reason
Reason is a term that refers to the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions, and beliefs. It is closely associated with such characteristically human activities as philosophy, science, language, ...
- Straight and Crooked Thinking
Straight and Crooked Thinking, first published in 1930 and revised in 1953, is a book by Robert H. Thouless which describes, assesses and critically analyses flaws in reasoning and argument. Thouless describes it as a practical manual, rather than a theoretical one.-Synopsis:*No. 3. proof by...
(book)
- Table of logic symbols
In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation and related...
- Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
External links and further readings
- Introductions and tutorials
- An Introduction to Philosophical Logic, by Paul Newall, 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 is an American philosopher at the University of Pittsburgh. In a productive research career extending over six decades, Rescher has established himself as a systematic philosopher of the old style and author of a system of pragmatic idealism which weaves together threads of...
. (1964). Introduction to Logic, St. Martin's Press.
- Essays
- "Symbolic Logic" and "The Game of Logic", Lewis Carroll
Charles Lutwidge Dodgson , better known by the pseudonym Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer. His most famous writings are Alice's Adventures in Wonderland and its sequel Through the Looking-Glass, as well as the poems "The Hunting of the...
, 1896.
- Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
- Reference material
- Reading lists