Logic (from Greek
|label=none|lit=possessed of reason
[Also related to (''logos''), "word, thought, idea, argument, account, reason, or principle." (Liddell and Scott, 1999).]
is the systematic study of valid rules of inference
, i.e. the relations that lead to the acceptance of one proposition (the conclusion
) on the basis of a set of other propositions (premise
s). More broadly, logic is the analysis and appraisal of argument
There is no universal agreement as to the exact definition and boundaries of logic (see ). However, it has traditionally included the classification of arguments; the systematic exposition of the logical form
s; the validity
of deductive reasoning
; the strength
of inductive reasoning
; the study of formal proof
s and inference
es and fallacies
); and the study of syntax
A good argument not only possesses validity and soundness (or strength, in induction), but it also avoids circular dependencies
, is clearly stated, relevant, and consistent
; otherwise it is useless for reasoning and persuasion, and is classified as a fallacy
In ordinary discourse, inferences may be signified by words such as ''therefore'', ''thus'', ''hence'', ''ergo'', and so on.
Historically, logic has been studied in philosophy
(since ancient times) and mathematics
(since the mid-19th century). More recently, logic has been studied in cognitive science
, which draws on computer science
, philosophy and psychology
, among other disciplines. A logician is any person, often a philosopher or mathematician, whose topic of scholarly study is logic.
Types of logic
'' is an area of philosophy. It's a set of methods used to solve philosophical problems and a fundamental tool for the advancement of metaphilosophy
'' is the study of natural language argument
s. The study of fallacies
is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. (See .)
'' is the study of inference
with purely formal content. An inference possesses a ''purely formal and explicit content'' (i.e. it can be expressed as a particular application of a wholly abstract rule) such as, a rule that is not about any particular thing or property. In many definitions of logic, logical consequence
and inference with purely formal content are the same.
Examples of formal logic include (1) traditional syllogistic logic
(a.k.a. term logic) and (2) modern symbolic Logic
* ''Syllogistic logic'' can be found in the works of Aristotle
, making it the earliest known formal study and studies types of syllogism
. Modern formal logic follows and expands on Aristotle.
* ''Symbolic logic'' is the study of symbolic abstractions that capture the formal features of logical inference,
often divided into two main branches: propositional logic
and predicate logic
'' is an extension of symbolic logic into other areas, in particular to the study of model theory
, proof theory
, set theory
, and computability theory
The concepts of logical form
are central to logic.
An argument is constructed by applying one of the forms of the different types of logical reasoning
, and abductive
. In deduction, the ''validity
'' of an argument is determined solely by its logical form, not its content, whereas the ''soundness
'' requires both validity and that all the given premises are actually true.
Completeness, consistency, decidability, and expressivity, are further fundamental concepts in logic. The categorization of the logical systems
and of their properties has led to the emergence of a metatheory
of logic known as metalogic
. However, agreement on what ''logic'' actually is has remained elusive, although the field of universal logic
has studied the common structure of logics.
Logic is generally considered formal when it analyzes and represents the ''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. Simply put, to formalize simply means to translate English sentences into the language of logic.
This is called 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 irrelevant to logic (such as gender and declension, if the argument is in Latin), replacing conjunctions irrelevant to logic (e.g. "but") with logical conjunction
s like "and" and replacing ambiguous, or alternative logical expressions ("any", "every", etc.) with expressions of a standard type (e.g. "all", or the universal quantifier
Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression "all Ps are Qs" shows the logical form common to the sentences "all men are mortals", "all cats are carnivores", "all Greeks are philosophers", and so on. The schema can further be condensed into the formula
''A(P,Q)'', where the letter ''A'' indicates the judgement 'all – are –'.
The importance of form was recognised from ancient times. Aristotle uses variable letters to represent valid inferences in ''Prior Analytics
'', 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 Ammonius
), only the logical principles stated in schematic terms belong to logic, not those given in concrete terms. The concrete terms 'man', 'mortal', etc., are analogous to the substitution values of the schematic placeholders ''P'', ''Q'', ''R'', which were called the 'matter' (Greek
: ) of the inference.
There is a big difference between the kinds of formulas seen in traditional term logic and the predicate calculus
that is the fundamental advance of modern logic. The formula ''A(P,Q)'' (all Ps are Qs) of traditional logic corresponds to the more complex formula
in predicate logic, involving the logical connectives for universal quantification
rather than just the predicate letter ''A'' and using variable arguments
where traditional logic uses just the term letter ''P''. With the complexity comes power, and the advent of the predicate calculus inaugurated revolutionary growth of the subject.
The validity of an argument depends upon the meaning, or ''semantics'', of the sentences that make it up.
Aristotle's six ''Organon
'', especially ''De Interpretatione
'', gives a cursory outline of semantics which the scholastic logic
ians, particularly in the thirteenth and fourteenth century, developed into a complex and sophisticated theory, called ''supposition theory
''. This showed how the truth of simple sentences, expressed schematically, depend on how the terms 'supposit', or stand for, certain extra-linguistic items. For example, in part II of his ''Summa Logicae''
, William of Ockham
presents a comprehensive account of the necessary and sufficient conditions for the truth
of simple sentences, in order to show which arguments are valid and which are not. Thus "every A is B' is true if and only if there is something for which 'A' stands, and there is nothing for which 'A' stands, for which 'B' does not also stand."
Early modern logic defined semantics purely as a relation between ideas. Antoine Arnauld
in the ''Port Royal-Logic
'', says that after conceiving things by our ideas, we compare these ideas, and, finding that some belong together and some do not, we unite or separate them. This is called ''affirming'' or ''denying'', and in general ''judging''. Thus truth and falsity are no more than the agreement or disagreement of ideas. This suggests obvious difficulties, leading Locke
to distinguish between 'real' truth, when our ideas have 'real existence' and 'imaginary' or 'verbal' truth, where ideas like harpies or centaurs exist only in the mind. This view, known as ''psychologism
'', was taken to the extreme in the nineteenth century, and is generally held by modern logicians to signify a low point in the decline of logic before the twentieth century.
is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of quantification
, needed to solve the problem of multiple generality
, rendered impossible the kind of subject-predicate analysis that underlies medieval semantics. The main modern approach is ''model-theoretic semantics'', based on Alfred Tarski
's ''semantic theory of truth
''. The approach assumes that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation function
s from them to some predefined domain of discourse
: an interpretation of first-order predicate logic
is given by a mapping from terms to a universe of individual
s, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics is one of the fundamental concepts of model theory
. Modern semantics also admits rival approaches, such as the proof-theoretic semantics
that associates the meaning of propositions with the roles that they can play in inferences, an approach that ultimately derives from the work of Gerhard Gentzen
on structural proof theory
and is heavily influenced by Ludwig Wittgenstein
's later philosophy, especially his aphorism
"meaning is use
''Inference'' is not to be confused with ''implication
''. An ''implication'' is a sentence of the form 'If p then q', and can be true or false. The stoic logic
ian Philo of Megara
was the first to define the truth conditions of such an ''implication'': false only when the antecedent p is true and the consequent q is false, in all other cases true. An ''inference'', on the other hand, consists of two separately asserted propositions of the form 'p therefore q'. An inference is not true or false, but valid or invalid. However, there is a connection between implication and inference, as follows: if the implication 'if p then q' is ''true'', the inference 'p therefore q' is ''valid''. This was given an apparently paradoxical formulation by Philo, who said that the implication 'if it is day, it is night' is true only at night, so the inference 'it is day, therefore it is night' is valid in the night, but not in the day.
The theory of inference (or '''consequences''
') was systematically developed in medieval times by logicians such as William of Ockham
and Walter Burley
. It is uniquely medieval, though it has its origins in Aristotle's ''Topica''
' ''De Syllogismis hypotheticis.'' Many terms in logic, for this reason, are in Latin. For instance, the rule that licenses the move from the implication 'if p then q' plus the assertion of its antecedent p, to the assertion of the consequent q, is known as ('mode of positing')—from . The Latin formulations of many other rules such as ('from falsehood, anything ollows
), and ''reductio ad absurdum
'' ('reduction to absurdity'; i.e. to disprove by showing the consequence as absurd), also date from this period.
However, the ''theory of consequences'', or the so-called ''hypothetical'' syllogism
, was never fully integrated into the theory of the ''categorical'' syllogism. This was partly because of the resistance to reducing the categorical judgment 'every s is p' to the so-called hypothetical judgment 'if anything is s, it is p'. The first was thought to imply 'some s is p', the latter was not, and as late as 1911 in the Encyclopædia Britannica
article on "Logic", we find the Oxford logician T. H. Case arguing against Sigwart's and Brentano's modern analysis of the universal proposition.
A formal system is an organization
of terms used for the analysis of deduction. It consists of an alphabet, a language over the alphabet to construct sentences, and a rule for deriving sentences. Among the important properties that logical system
s can have are:
: no theorem of the system contradicts another.
: the system's rules of proof never allow a false inference from true premises.
: if a formula is true, it can be proven, i.e. is a ''theorem'' of the system.
: if any formula is a theorem of the system, it is true. This is the converse of completeness. (Note that in a distinct philosophical use of the term, an argument is sound when it is both valid and its premises are true.)
: what concepts can be expressed in the system.
Some logical systems do not have all these properties. As an example, Kurt Gödel
's incompleteness theorems
show that sufficiently complex formal systems of arithmetic cannot be consistent and complete;
however, first-order predicate
logics not extended by specific axiom
s to be arithmetic formal systems with equality can be complete and consistent.
Logic and rationality
As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential importance to rationality
. Here we have defined logic to be "the systematic study of the form of arguments;" the reasoning behind argument is of several sorts, but only some of these arguments fall under the aegis of logic proper.
concerns the logical consequence
of given premises and is the form of reasoning most closely connected to logic. On a narrow conception of logic (see below) logic concerns just deductive reasoning, although such a narrow conception controversially excludes most of what is called informal logic from the discipline.
There are other forms of reasoning that are rational but that are generally not taken to be part of logic. These include inductive reasoning
, which covers forms of inference that move from collections of particular judgements to universal judgements, and abductive reasoning
[On abductive reasoning, see:
* Magnani, L. 2001. ''Abduction, Reason, and Science: Processes of Discovery and Explanation''. New York: Kluwer Academic Plenum Publishers. xvii. .
* Josephson, John R., and Susan G. Josephson. 1994. ''Abductive Inference: Computation, Philosophy, Technology''. New York: Cambridge University Press. viii. .
* Bunt, H. and W. Black. 2000. ''Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics'', (''Natural Language Processing'' 1). Amsterdam: John Benjamins. vi. .]
which is a form of inference that goes from observation to a hypothesis that accounts for the reliable data (''observation'') and seeks to explain relevant evidence. American philosopher Charles Sanders Peirce
(1839–1914) first introduced the term as ''guessing''.
[Bergman, Mats, and Saami Paavola, eds.]
" ''The Commens Dictionary: Peirce's Terms in His Own Words'' (new ed.) Retrieved 10 May 2020. . Retrieved 10 May 2020.
Peirce said that to ''abduce'' a hypothetical explanation
from an observed surprising circumstance
is to surmise that
may be true because then
would be a matter of course.
[Peirce, Charles Sanders. 1903. "Lectures on Pragmatism." Pp. 14–212 in ''Collected Papers of Charles Sanders Peirce'' 5]
Thus, to abduce
involves determining that
is sufficient (or nearly sufficient), but not necessary
While inductive and abductive inference are not part of logic proper, the methodology of logic has been applied to them with some degree of success. For example, the notion of deductive validity (where an inference is deductively valid if and only if
there is no possible situation in which all the premises are true but the conclusion false) exists in an analogy to the notion of inductive validity, or "strength", where an inference is inductively strong if and only if its premises give some degree of probability to its conclusion. Whereas the notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics
, inductive validity 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 logical association rule induction
, while others may use mathematical model
s of probability such as decision tree
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 Philosophy
'' that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality
. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations."
The idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to logicism
in mathematics (19th and 20th centuries) and the advent of the influence of mathematical logic on philosophy. A consequence of taking logic to treat special kinds of argument is that it leads to identification of special kinds of truth, the logical truths (with logic equivalently being the study of logical truth), and excludes many of the original objects of study of logic that are treated as informal logic. Robert Brandom
has argued against the idea that logic is the study of a special kind of logical truth, arguing that instead one can talk of the logic of material inference
(in the terminology of Wilfred Sellars
), with logic making explicit the commitments that were originally implicit in informal inference.
Logic comes from the Greek word ''logos'', originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason". In the Western World, logic was first developed by Aristotle
, who called the subject 'analytics'.
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 syllogism
temporal modal logic
and inductive logic
as well as influential vocabulary such as terms
s and proposition
s. There was also the rival Stoic logic
during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian
faith. During the High Middle Ages
, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of scholasticism
. In 1323, William of Ockham
's influential ''Summa Logicae
'' was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Holberg
's satirical play ''Erasmus Montanus
The Chinese logical
philosopher Gongsun Long
() proposed the paradox "One and one cannot become two, since neither becomes two."
[The four Catuṣkoṭi logical divisions are formally very close to the four opposed propositions of the Greek ''tetralemma'', which in turn are analogous to the four truth values of modern relevance logic.
(cf. Belnap, Nuel. 1977. "A useful four-valued logic." In ''Modern Uses of Multiple-Valued Logic'', edited by Dunn and Eppstein. Boston: Reidel;
Jayatilleke, K. N.. 1967. "The Logic of Four Alternatives." In ''Philosophy East and West''. University of Hawaii Press.)]
In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty
following the legalist philosophy of Han Feizi
In India, the Anviksiki
school of logic was founded by Medhātithi
(c. 6th century BCE). Innovations in the scholastic school, called Nyaya
, continued from ancient times into the early 18th century with the Navya-Nyāya
school. By the 16th century, it developed theories resembling modern logic, such as Gottlob Frege
's "distinction between sense and reference of proper names" and his "definition of number", as well as the theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory
[Chakrabarti, Kisor Kumar. 1976. "Some Comparisons Between Frege's Logic and Navya-Nyaya Logic." ''Philosophy and Phenomenological Research'' 36(4):554–63. .
"This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory."]
Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage
, Augustus De Morgan
, and George Boole
In the 20th century, Western philosophers like Stanislaw Schayer
and Klaus Glashoff have explored Indian logic more extensively.
logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics
stimulated the development of symbolic logic (now called mathematical logic
). In 1854, George Boole published ''The Laws of Thought
'', introducing symbolic logic and the principles of what is now known as Boolean logic
. In 1879, Gottlob Frege published ''Begriffsschrift
'', which inaugurated modern logic with the invention of quantifier
notation, reconciling the Aristotelian and Stoic logics in a broader system, and solving such problems for which Aristotelian logic was impotent, such as the problem of multiple generality
. From 1910 to 1913, Alfred North Whitehead
and Bertrand Russell
published ''Principia Mathematica
on the foundations of mathematics, attempting to derive mathematical truths from axiom
s and inference rule
s in symbolic logic. In 1931, Gödel
raised serious problems with the foundationalist program and logic ceased to focus on such issues.
The development of logic since Frege, Russell, and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see analytic philosophy
) and philosophy of mathematics
. Logic, especially sentential logic, is implemented in computer logic circuits
and is fundamental to computer science
. Logic is commonly taught by university philosophy, sociology, advertising and literature departments, often as a compulsory discipline.
'' was Aristotle
's body of work on logic, with the ''Prior Analytics
'' constituting the first explicit work in formal logic, introducing the syllogistic.
The parts of syllogistic logic, also known by the name term logic
, 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 syllogism
s that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises.
Aristotle's 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 generality
was recognized 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 theory
to help develop and critically question argumentation scheme
s that are used in artificial intelligence
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
(usually represented with p, q, etc.) using logical connectives
etc.); these propositions and connectives are the only elements of a standard propositional calculus.
Unlike predicate logic or syllogistic logic where individual subjects and predicates (which do not have truth values) are the smallest unit, propositional logic takes full propositions with truth values as its most basic component.
) are included in extended propositional calculus, but they only quantify over full propositions, not individual subjects or predicates.
A given propositional logic is a system of formal proof with rules that establish which well-formed formulae of a given language are "theorems" by proving them from axioms which are assumed without proof.
Predicate logic is the generic term for symbolic formal systems such as first-order logic
, second-order logic
, many-sorted logic
, and infinitary logic
. It provides an account of quantifiers
general enough to express a wide set of arguments occurring in natural language. For example, Bertrand Russell
's famous barber paradox
, "there is a man who shaves all and only men who do not shave themselves" can be formalised by the sentence
, using the non-logical predicate
to indicate that ''x'' is a man, and the non-logical relation
to indicate that ''x'' shaves ''y''; all other symbols of the formulae are logical, expressing the universal and existential quantifiers
Whilst 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—allowing predicate logic to solve the problem of multiple generality
that had perplexed medieval logicians.
The development of predicate logic is usually attributed to Gottlob Frege
, who is also credited as one of the founders of analytic philosophy
, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Mathematical Logic
by David Hilbert
and Wilhelm Ackermann
in 1928. The analytical generality of predicate logic allowed the formalization of mathematics, drove the investigation of set theory
, and allowed the development of Alfred Tarski
's approach to model theory
. It provides the foundation of modern mathematical logic
Frege's original system of predicate logic was second-order, rather than first-order. Second-order logic
is most prominently defended (against the criticism of Willard Van Orman Quine
and others) by George Boolos
and Stewart Shapiro
In languages, modality
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. Confusing modality is known as the modal fallacy
's logic is in large parts concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous sea-battle argument
in ''De Interpretatione
'' § 9, that are now seen as anticipations of modal logic and its connection with potentiality
and time, the earliest formal system of modal logic was developed by Avicenna
, who ultimately developed a theory of "temporally modalized
While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of C. I. Lewis
in 1918, who formulated a family of rival axiomatizations of the alethic modalities
. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic
and epistemic logic
. The seminal work of Arthur Prior
applied the same formal language to treat temporal logic
and paved the way for the marriage of the two subjects. Saul Kripke
discovered (contemporaneously with rivals) his theory of frame semantics
, which revolutionized the formal technology available to modal logicians and gave a new graph-theoretic
way of looking at modality that has driven many applications in computational linguistics
and computer science
, such as dynamic logic
Informal reasoning and dialectic
The motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good arguments 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 Organon
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 rhetoric
This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectic
al logic forms the heart of a course in critical thinking
, a compulsory course at many universities. Dialectic has been linked to logic since ancient times, but it has not been until recent decades that European and American logicians have attempted to provide mathematical foundations for logic and dialectic by formalising dialectical logic. Dialectical logic
is also the name given to the special treatment of dialectic in Hegel
ian and Marx
ist thought. There have been pre-formal treatises on argument and dialectic, from authors such as Stephen Toulmin
(''The Uses of Argument''), Nicholas Rescher
(''Dialectics''), and van Eemeren and Grootendorst (Pragma-dialectics
). Theories of defeasible reasoning
can provide a foundation for the formalisation of dialectical logic and dialectic itself can be formalised as moves in a game, where an advocate for the truth of a proposition and an opponent argue. Such games can provide a formal game semantics
for many logics.
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 intelligence
Mathematical logic comprises 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 geometry
in relation to logic and philosophy goes back to the ancient Greeks such as Euclid
, and Aristotle
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 the logicism
pioneered by philosopher-logicians such as Gottlob Frege
and Bertrand Russell
. Mathematical theories were supposed to be logical tautologies
, and the programme was to show this by means of a reduction of mathematics to logic.
The various attempts to carry this out met with failure, from the crippling of Frege's project in his ''Grundgesetze'' by Russell's paradox
, to the defeat of Hilbert's program
by Gödel's incompleteness theorem
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 theory
Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem
, a result in model theory
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 calculus
is enough to ''describe'' the whole of mathematics, though not ''equivalent'' to it.
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 theory
originated in the study of the infinite by Georg Cantor
, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem
, through the status of the Axiom of Choice
and the question of the independence of the continuum hypothesis
, to the modern debate on large cardinal
captures the idea of computation in logical and arithmetic
terms; its most classical achievements are the undecidability of the Entscheidungsproblem
by Alan Turing
, and his presentation of the Church–Turing thesis
Today recursion theory is mostly concerned with the more refined problem of complexity class
es—when is a problem efficiently solvable?—and the classification of degrees of unsolvability
deals with formal descriptions of ordinary, non-specialist ("natural") language
, that is strictly only about the arguments within philosophy's other branches. Most philosophers assume that the bulk of everyday reasoning can be captured in logic if a method or methods to translate ordinary language into that logic can be found. Philosophical logic is essentially a continuation of the traditional discipline 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 logic
s, tense logic
s) as well as various extensions of classical logic
(e.g. modal logic
s) and non-standard semantics for such logics (e.g. Kripke
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 cut to the heart of computer science as it emerged as a discipline: Alan Turing
's work on the ''Entscheidungsproblem
'' followed from Kurt Gödel
's work on the incompleteness theorems
. The notion of the general purpose computer 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 notation
, it would be possible to create a machine that mimics the problem-solving skills of a human being. This was more difficult than expected because of the complexity of human reasoning. In the summer of 1956, John McCarthy
, Marvin Minsky
, Claude Shannon
and Nathan Rochester
organized a conference on the subject of what they called "artificial intelligence
" (a term coined by McCarthy for the occasion). Newell and Simon proudly presented the group with the Logic Theorist
and were somewhat surprised when the program received a lukewarm reception.
In logic programming
, a program consists of a set of axioms and rules. Logic programming systems such as Prolog
compute the consequences of the axioms and rules in order to answer a query.
Today, logic is extensively applied in the field of artificial intelligence, and this field provide a rich source of problems in formal and informal logic. Argumentation theory
is one good example of how logic is being applied to artificial intelligence. The ACM Computing Classification System
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
, as well as work of formal methods
such as Hoare logic
* Boolean logic
as fundamental to computer hardware: particularly, the system's section B.2 on "Arithmetic and logic structures
", relating to operatives AND
, and OR
* Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example modal logic
and default logic
in Knowledge representation formalisms and methods
, Horn clause
s in logic programming, and description 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 proving
, the machines can find and check proofs, as well as work with proofs too lengthy to write out by hand.
The logics discussed above are all "bivalent
" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false propositions. Non-classical logic
s are those systems that reject various rules of Classical logic
Hegel developed his own dialectic logic
that extended Kant
'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. Vasiliev
extended 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" (or an indeterminate, a hypothesis) so inventing ternary logic
, the first multi-valued logic
in the Western tradition.
A minor modification of the ternary logic was later introduced in a sibling ternary logic model proposed by Stephen Cole Kleene
. Kleene's system differs from the Łukasiewicz's logic with respect to an outcome of the implication. The former assumes that the operator of implication
between two hypotheses produces a hypothesis.
Logics such as fuzzy logic
have since been devised with an infinite number of "degrees of truth", represented by a real number
between 0 and 1.
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 intuitionism
. Brouwer rejected formalization in mathematics, but his student Arend Heyting
studied intuitionistic logic formally, as did Gerhard Gentzen
. Intuitionistic logic is of great interest to computer scientists, as it is a constructive logic
and sees many applications, such as extracting verified programs from proofs and influencing the design of programming language
s through the formulae-as-types correspondence
is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalized 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 logic
? What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled "Is Logic Empirical?
, building on a suggestion of W. V. Quine
, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics
or of general relativity
, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists
about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity
, substituting for classical logic the quantum logic
proposed by Garrett Birkhoff
and John von Neumann
Another paper of the same name by 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 metaphysics
on realism versus anti-realism
Implication: strict or material
The notion of implication formalized 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 explosion
. 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 logic
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 entailment
, such as relevance logic.
Tolerating the impossible
Georg Wilhelm Friedrich Hegel
was deeply critical of any simplified notion of the law of non-contradiction
. It was based on Gottfried Wilhelm 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 and for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable.
Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency
. Relevance logic
and paraconsistent logic
are the most important approaches here, though the concerns are different: a key consequence of classical logic
and some of its rivals, such as intuitionistic logic
, is that they respect the principle of explosion
, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest
, the main proponent of dialetheism
, 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 on which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion that there are no logical truth
s. This is in contrast with the usual views in philosophical skepticism
, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus
provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a "... mobile army of metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins".
His rejection of truth did not lead him to reject the idea of either inference or logic completely but rather suggested that "logic ame
into existence in man's head ut
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. Some philosophers, such as Jürgen Habermas
, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge.
, in his book ''The Destruction of Reason'', asserts that, "Were we to study Nietzsche's statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible."
described Nietzsche's irrational claims with "He is fond of expressing himself paradoxically and with a view to shocking conventional readers" in his book ''A History of Western Philosophy''.
* Barwise, J.
(1982). ''Handbook of Mathematical Logic''. Elsevier. .
* Belnap, N.
(1977). "A useful four-valued logic". In Dunn & Eppstein, ''Modern uses of multiple-valued logic''. Reidel: Boston.
* Bocheński, J.M.
(1959). ''A précis of mathematical logic
''. Translated from the French and German editions by Otto Bird. D. Reidel, Dordrecht, South Holland.
* Bocheński, J.M.
(1970). ''A history of formal logic
''. 2nd Edition. Translated and edited from the German edition by Ivo Thomas. Chelsea Publishing, New York.
* Cohen, R.S, and Wartofsky, M.W. (1974). ''Logical and Epistemological Studies in Contemporary Physics''. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. .
* Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974).
* Gabbay, D.M.
, and Guenthner, F. (eds., 2001–2005). ''Handbook of Philosophical Logic''. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.
* Haack, Susan
(1996).'' Deviant Logic, Fuzzy Logic: Beyond the Formalism'', University of Chicago Press.
* Hilbert, D.
, and Ackermann, W
, (1928). ''Grundzüge der theoretischen Logik'' (''Principles of Mathematical Logic
* Hodges, W.
(2001). ''Logic. An introduction to Elementary Logic'', Penguin Books.
* Hofweber, T. (2004)Logic and Ontology
''Stanford Encyclopedia of Philosophy
''. Edward N. Zalta
* Hughes, R.I.G. (1993, ed.). ''A Philosophical Companion to First-Order Logic''. Hackett Publishing.
* Kneale, William
, and Kneale, Martha, (1962). ''The Development of Logic''. Oxford University Press, London, UK.
* Mendelson, Elliott
, (1964). ''Introduction to Mathematical Logic''. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif.
* Smith, B.
(1989). "Logic and the Sachverhalt". ''The Monist'' 72(1): 52–69.
* Whitehead, Alfred North
and Bertrand Russell
(1910). ''Principia Mathematica
''. Cambridge University Press: Cambridge, England.
An Outline for Verbal Logic
* Introductions and tutorials
** aimed at beginners.
*forall x: an introduction to formal logic
by P.D. Magnus
, covers sentential and quantified logic.
*Logic Self-Taught: A Workbook
(originally prepared for on-line logic instruction).
*** Nicholas Rescher
. (1964). ''Introduction to Logic'', St. Martin's Press.
an"The Game of Logic" Lewis Carroll
*Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas.
In ''The Dictionary of the History of Ideas.''
* Online Tools
*Interactive Syllogistic Machine
A web-based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms.
* A Logic Calculator
A web-based application for evaluating simple statements in symbolic logic.
* Reference material
by Peter Suber, for translating from English into logical notation.
with an annotated bibliography.
* Reading lists
** ThLondon Philosophy Study Guide
offers many suggestions on what to read, depending on the student's familiarity with the subject:
Category:Philosophy of logic