Logic

Logic is an interdisciplinary field which studies

truth

and

reasoning

. Informal logic seeks to characterize valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and argument patterns symbolically, using formal systems such as first order logic. Within formal logic, mathematical logic studies the mathematical characteristics of formal logical systems, while philosophical logic applies them to philosophical problems such as the nature of meaning,

knowledge

, and

existence

. Systems of formal logic are also applied in other fields including

linguistics

,

cognitive science

, and computer science.

Logic has been studied since

Antiquity

, early approaches including Aristotelian logic, Stoic logic, Anviksiki, and the mohists. Modern formal logic has its roots in the work of late 19th century mathematicians such as Gottlob Frege.

Definition

The word "logic" originates from the Greek word "logos", which has a variety of translations, such as reason,

discourse

, or

language

. Logic is traditionally defined as the study of the laws of thought or correct

reasoning

. This is usually understood in terms of

inference

s or arguments: reasoning may be seen as the activity of drawing inferences, whose outward expression is given in arguments. An inference or an argument is a set of premises together with a conclusion. Logic is interested in whether arguments are good or inferences are valid, i.e. whether the premises support their conclusions.

These general characterizations apply to logic in the widest sense since they are true both for formal and informal logic. But many definitions of logic focus on formal logic because it is the paradigmatic form of logic. In this narrower sense, logic is a formal science that studies how conclusions follow from premises in a topic-neutral way. As a formal science, it contrasts with empirical sciences, like

physics

or

biology

, because it tries to characterize the inferential relations between premises and conclusions only based on how they are structured. This means that the actual content of these propositions, i.e. their specific topic, is not important for whether the inference is valid or not. This can be expressed by distinguishing between logical and non-logical vocabulary: inferences are valid because of the logical terms used in them, independent of the meanings of the non-logical terms. Valid inferences are characterized by the fact that the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false. The general logical structures characterizing valid inferences are called rules of inference. In this sense, logic is often defined as the study of valid inference. This contrasts with another prominent characterization of logic as the science of logical truths. A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms. These two characterizations of logic are closely related to each other: an inference is valid if the material conditional from its premises to its conclusion is logically true.

The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, ''a logic'' is a logical formal system. Different logics differ from each other concerning the

formal languages

used to express them and, most importantly, concerning the rules of inference they accept as valid. Starting in the 20th century, many new formal systems have been proposed. There is an ongoing debate about which of these systems should be considered logics in the strict sense instead of non-logical formal systems. Suggested criteria for this distinction include logical completeness and proximity to the intuitions governing classical logic. According to these criteria, it has been argued, for example, that higher-order logics and

fuzzy logic

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

Formal and informal logic

Logic can be studied formally or informally. A formal approach is one that abstracts away from content, looking for patterns that arise from form alone. For instance, the formal rule of conjunction introduction states that any two propositions $P$ and $Q$ together imply their conjunction $P \& Q$. This rule is formal since the symbols $P$ and $Q$ can stand in for any two statements, regardless of their content. Formal logic is often considered the paradigmatic form of logic.

On an informal approach, inferences of this sort would have to be characterized using particular statements. Informal logic is often part of courses in critical thinking, while informal approaches such as dialectical logic and argumentation theory continue as areas of research.

Subfields

Philosophical logic

Philosophical logic is the study of logic within philosophy. It includes applications to problems in epistemology, ethics, philosophy of mathematics, and natural language semantics.

Mathematical logic

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and computability theory.

Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics. The latter was a major concern in early 20th century mathematical logic, which pursued the program of 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 theorems.

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 axioms.

Recursion theory 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 classes—when is a problem efficiently solvable?—and the classification of degrees of unsolvability.

Computational logic

In computer science, logic is studied as part of the theory of computation. Key areas of logic that are relevant to computing include computability theory, modal logic, and category theory. Early computer machinery was based on ideas from logic such as the lambda calculus. Computer scientists also apply concepts from logic to problems in computing and vice versa. For instance, modern

artificial intelligence

builds on logicians' work in argumentation theory, while automated theorem proving can assist logicians in finding and checking proofs. In logic programming languages such as

Prolog

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

Formal semantics of natural language

Formal semantics is a subfield of both

linguistics

and philosophy which uses logic to analyze meaning in natural language. It is an empirical field which seeks to characterize the denotations of linguistic expressions and explain how those denotations are composed from the meanings of their parts. The field was developed by Richard Montague and

Barbara Partee

in the 1970s, and remains an active area of research. Central questions include scope, binding, and linguistic modality.

Concepts

Varieties of reasoning

Arguments are often divided into those that are ''deductive'', '' inductive'', and '' abductive''. In the most prominent conception of logic, only deductive reasoning counts as logic in the strict sense.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. .

A ''deductive argument'' is one whose premises are intended to guarantee the truth of its conclusion. In other words, a deductive argument seeks to reach its conclusion by logical necessity. For instance, the following argument is deductive.

*Deductive argument:
*# ''Victoria is tall.''
*# ''Victoria has brown hair.''
*# ''Therefore, Victoria is tall and has brown hair.''

''Inductive arguments'' are those in which the premises are merely evidence for the conclusion.

*Inductive argument:
*# ''Victoria is tall.''
*# ''Tall people are generally good at basketball.''
*# ''Therefore, Victoria is good at basketball.''

''Abductive reasoning'' involves reasoning to the most likely explanation.

*Abductive argument:
*# ''Victoria is tall.''
*# ''Victoria has brown hair.''
*# ''Therefore, Victoria must have a tall or brown-haired ancestor .''

Logical truth

A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some modal logics, this notion can be understood equivalently as truth at all possible worlds.

Formal system

A formal system of logic consists of a language, a proof system, and a semantics. A system's language and proof system are sometimes grouped together as the system's ''syntax'', since they both concern the form rather than the content of the system's expressions.

The term "a logic" is often used a countable noun to refer to a particular formal system of logic. Different logics can differ from each other in their language, proof system, or their semantics. Starting in the 20th century, many new formal systems have been proposed.

Formal language

A ''language'' is a set of well formed formulas. For instance, in propositional logic, $P \& Q$ is a formula but $P Q \& \& \&$ is not. Languages are typically defined by providing an ''alphabet'' of basic expressions and

recursive

syntactic rules which build them into formulas.

Proof system

A ''proof system'' is a collection of formal rules which define when a conclusion follows from given premises. For instance, the classical rule of conjunction introduction states that $P \& Q$ follows from the premises $P$ and $Q$. Rules in a proof systems are always defined in terms of formulas' syntactic form, never in terms of their meanings. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are a number of different types of proof systems including natural deduction and sequent calculi. Proof systems are closely linked to philosophical work which characterizes logic as the study of valid inference.

Semantics

A ''semantics'' is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance, the semantics for classical propositional logic assigns the formula $P \& Q$ the denotation "true" whenever $P$ is true and $Q$ is too. Entailment is a semantic relation which holds between formulas when the first cannot be true without the second being true as well. Semantics is closely tied to the philosophical characterization of logic as the study of logical truth.

Metalogic

Metalogic is the study of properties of formal systems. Two central metalogical properties are ''

soundness

'' and '' completeness''. A system of logic is ''sound'' when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is ''complete'' when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly.

Other important metalogical properties include ''

consistency

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

Propositional logic

Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives. For instance, propositional logic represents the conjunction of two atomic propositions $P$ and $Q$ as the complex formula $P \& Q$. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones; it cannot represent inferences that results from the inner structure of a proposition.

Predicate logic

Predicate logic is the generic term for symbolic formal systems such as first-order logic, 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 $(\exists x ) (\text(x) \wedge (\forall y) (\text(y) \rightarrow (\text(x, y) \leftrightarrow \neg \text(y, y))))$, using the non-logical predicate $\text(x)$ to indicate that ''x'' is a man, and the non-logical relation $\text(x, y)$ to indicate that ''x'' shaves ''y''; all other symbols of the formulae are logical, expressing the universal and existential quantifiers, conjunction, implication,

negation

and biconditional.

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.

Modal logic

Modal logic is the study of formal systems originally developed to represent statements about necessity and possibility. For instance the modal formula $\Diamond P$ can be read as "possibly $P$" while $\Box P$ can be read as "necessarily $P$". Modal logics can be used to represent different phenomena depending on what ''flavor'' of necessity and possibility is under consideration. When $\Box$ is used to represent epistemic necessity, $\Box P$ states that $P$ is known. When $\Box$ is used to represent deontic necessity, $\Box P$ states that $P$ is a moral or legal obligation. Within philosophy, modal logics are widely used in formal epistemology, formal ethics, and

metaphysics

. Within linguistic semantics, systems based on modal logic are used to analyze linguistic modality in natural languages.

The earliest formal system of modal logic was developed by

Avicenna

, who ultimately developed a theory of " temporally modalized" syllogistic. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of C. I. Lewis in 1918, who formulated a family of rival axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic 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

and

Jaakko Hintikka

built on this work to develop the frame semantics, which is now the standard semantics for modal logic. This graph-theoretic way of looking at modality has driven many applications in computational linguistics and computer science, such as dynamic logic.

Non-classical logic

Non-classical logics are systems that reject various rules of classical logic. They are motivated by the view that classical logic does not accurately represent the nature of truth and reasoning.

One major non-classical paradigm is intuitionistic logic, which rejects the law of the excluded middle. Intuitionism was developed by the Dutch mathematicians L.E.J. Brouwer and Arend Heyting to underpin their constructive approach to mathematics, in which the existence of a mathematical object can only be proven by constructing it. Intuitionism was further pursued by

Gerhard Gentzen

, Kurt Gödel, Michael Dummett, among others. 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. It is closely related to nonclassical systems such as Gödel–Dummett logic and inquisitive logic.

Multi-valued logics depart from classicality by rejecting the principle of bivalence which requires all propositions to be either true or false. For instance,

Jan Łukasiewicz

and Stephen Cole Kleene both proposed ternary logics which have a third truth value representing that a statement's truth value is indeterminate. These logics have seen applications including to presupposition in linguistics. Fuzzy logics are multivalued logics that have an infinite number of "degrees of truth", represented by a real number between 0 and 1.

Controversies

"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?"

Hilary Putnam

, 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.

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.

Conceptions of logic

Logic arose 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.

Rejection of logical truth

The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, sometimes leading to the conclusion that there are no logical truths. 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.

Friedrich Nietzsche

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 utof illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished". Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: "Logic, too, also rests on assumptions that do not correspond to anything in the real world".

This position held by Nietzsche however, has come under extreme scrutiny for several reasons. Some philosophers, such as Jürgen Habermas, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge. Georg Lukács, in his book ''The Destruction of Reason'', asserts that, "Were we to study Nietzsche's statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible." Bertrand Russell described Nietzsche's irrational claims with "He is fond of expressing himself paradoxically and with a view to shocking conventional readers" in his book ''A History of Western Philosophy''.

History

Logic was developed independently in several cultures during antiquity. One major early contributor was

Aristotle

, who developed term logic in his ''

Organon

'' and ''Prior Analytics''. In this approach, ''judgements'' are broken down into ''propositions'' consisting of two terms that are related by one of a fixed number of relation. Inferences are expressed by means of syllogisms that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises. Aristotle's monumental insight was the notion that arguments can be characterized in terms of their form. The later logician

Łukasiewicz

described this insight as "one of Aristotle's greatest inventions". Aristotle's system of logic was also responsible for the introduction of hypothetical syllogism, temporal modal logic, and inductive logic, as well as influential vocabulary such as

terms

, predicables, syllogisms and propositions. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century. It has now been superseded by later work, though many of its key insights live on in modern systems of logic.

Ibn Sina

(Avicenna) (980–1037 CE) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world, and also had an important influence on Western medieval writers such as

Albertus Magnus

and

William of Ockham

. Avicenna wrote on the hypothetical syllogism and on the propositional calculus. He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic. He also made use of inductive logic, such as the methods of agreement, difference, and concomitant variation which are critical to the

scientific method

.Goodman, Lenn Evan (2003), ''Islamic Humanism'', p. 155,

Oxford University Press

, . Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's " first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873).Muhammad Iqbal, ''The Reconstruction of Religious Thought in Islam'', "The Spirit of Muslim Culture" (cf.br>
an

In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the High Middle Ages, 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. Initially, medieval Christian scholars drew on the classics that had been preserved in Latin through commentaries by such figures such as

Boethius

, later the work of Islamic philosophers such as

Avicenna

and

Averroes

were drawn on, which expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries. In 1323,

William of Ockham

'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.

The syllogistic 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'',

Boole, George

. 1854. '' An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities.'' introducing symbolic logic and the principles of what is now known as Boolean logic. In 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 rules 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.

See also

Notes

References

Bibliography

* Barwise, J. (1982). ''Handbook of Mathematical Logic''. Elsevier. .
* Belnap, N. (1977). "A useful four-valued logic". In Dunn & Eppstein, ''Modern uses of multiple-valued logic''. Reidel: Boston.
* Bocheński, J.M. (1959). ''A précis of mathematical logic''. 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''). Springer-Verlag.
* Hodges, W. (2001). ''Logic. An introduction to Elementary Logic'', Penguin Books.
* Hofweber, T. (2004)
Logic and Ontology
''Stanford Encyclopedia of Philosophy''. Edward N. Zalta (ed.).
* Hughes, R.I.G. (1993, ed.). ''A Philosophical Companion to First-Order Logic''. Hackett Publishing.
*
* Kneale, William, and Kneale, Martha, (1962). ''The Development of Logic''. Oxford University Press, London, UK.
*
* Mendelson, Elliott, (1964). ''Introduction to Mathematical Logic''. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif.
* Smith, B. (1989). "Logic and the Sachverhalt". ''The Monist'' 72(1): 52–69.
* Whitehead, Alfred North and Bertrand Russell (1910). ''Principia Mathematica''. Cambridge University Press: Cambridge, England.

External links

*
*
*
*

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.
* Essays
*
"Symbolic Logic"
an
"The Game of Logic"

Lewis Carroll

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

by

Peter Suber

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

**

**

*

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Abstraction
Critical thinking
Formal sciences
Philosophical logic
Philosophy of logic
Reasoning
Thought