, Logic (from the Greek λογική logikē) is the formal systematic study of the principle
s of valid inference
and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy
, mathematics
, semantics
, and computer science
. It examines general forms which argument
s may take, which forms are valid, and which are fallacies. In philosophy, the study of logic is applied in most major areas: ontology
, epistemology, ethics
, metaphysics
.
Logic hasn't wholly dispelled the society of witches and prophets and sorcerers and soothsayers.
Logic is a large drawer, containing some useful instruments, and many more that are superfluous. A wise man will look into it for two purposes, to avail himself of those instruments that are really useful, and to admire the ingenuity with which those that are not so, are assorted and arranged.
Logic is logic. That's all I say.
Logic is one thing and commonsense another.
The want of logic annoys. Too much logic bores. Life eludes logic, and everything that logic alone constructs remains artificial and forced.
Logic, like whiskey, loses its beneficial effect when taken in too large quantities.
Metaphysics may be, after all, only the art of being sure of something that is not so, and logic only the art of going wrong with confidence.
Pure logic is the ruin of the spirit.
One cannot use one's logic to explain actions driven by others' logic.
, Logic (from the Greek λογική logikē) is the formal systematic study of the principle
s of valid inference
and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy
, mathematics
, semantics
, and computer science
. It examines general forms which argument
s may take, which forms are valid, and which are fallacies. In philosophy, the study of logic is applied in most major areas: ontology
, epistemology, ethics
, metaphysics
. In mathematics, it is the study of valid inference
s within some formal language
. Logic is also studied in argumentation theory
.
Logic was studied in several ancient civilizations, including the Indian subcontinent
, China
and Greece
. Logic was established as a discipline by Aristotle
, 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 reasoning
and deductive reasoning
.
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 logicand modern symbolic logic are examples of formal logics.
- Informal logicInformal logicInformal 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 languageNatural 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 PlatoPlatoPlato , 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 inferenceInferenceInference 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 AristotleAristotleAristotle 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 logicPredicate logicIn 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 logicMathematical 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...
is an extension of symbolic logic into other areas, in particular to the study of model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, proof theoryProof 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 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 theoryRecursion 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 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 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, 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 connectiveLogical connectiveIn 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 logicPredicate logicIn 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 quantificationUniversal quantificationIn predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
and implicationEntailmentIn 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 generalityProblem 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...
, 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 reasoningconcerns what follows necessarily from given premises (if a, then b). However, inductive reasoning
—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 if
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 semantics
. 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 model
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:- ConsistencyConsistency proofIn 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. - ValidityValidityIn 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 soundnessSoundnessIn 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). - CompletenessCompletenessIn 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). - SoundnessSoundnessIn 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 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 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 Philosophythat 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".
By contrast, Immanuel Kant
argued that logic should be conceived as the science of judgment, an idea taken up in Gottlob Frege
'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 judgment
s or thoughts.
History
The earliest sustained work on the subject of logic is that of Aristotle. 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 syllogism
, temporal
modal logic
, and inductive logic
. In Europe
during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian
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 logical
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 dynasty
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
, continued from ancient times into the early 18th century with the Navya-Nyaya
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
. 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 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. 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 departments, often as a compulsory discipline.
Syllogistic logic
The Organonwas 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 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 generality
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 theory
to help develop and critically question argumentation schemes that are used in artificial intelligence
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 logic, second-order logic
, many-sorted logic
, and infinitary logic
.
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 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 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 Hilbert
and Wilhelm Ackermann
in 1928. The analytical generality of predicate logic allowed the formalisation 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
.
Modal logic
In languages, modalitydeals 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 Aristotle
, who was concerned with the alethic modalities
of necessity
and possibility
, 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 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 revolutionised 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
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 Organontreat 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 will form the heart of a course in critical thinking
, a compulsory course at many universities.
Argumentation theory
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
and law
.
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 geometry
in relation to logic and philosophy goes back to the ancient Greeks such as Euclid
, Plato
, 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 undoubtedly the logicism
pioneered by philosopher-logicians such as Gottlob Frege
and Bertrand Russell
: the idea was that mathematical theories were logical tautologies
, 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 paradox
, to the defeat of Hilbert's program
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 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. 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 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 class
es — when is a problem efficiently solvable? — and the classification of degrees of unsolvability
.
Philosophical logic
Philosophical logicdeals 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 logic
s, tense logics) as well as various extensions of classical logic
(e.g., modal logic
s), and non-standard semantics for such logics (e.g., Kripke
'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 Turing's work on the Entscheidungsproblem
followed from Kurt Gödel
'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 notation
, 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 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 fields of Artificial Intelligence
, and Computer Science
, and these fields 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 languagesFormal semantics of programming languagesIn 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 methodsFormal 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 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 logicBoolean 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...
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 logicModal 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 default logicDefault 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 clauseHorn 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 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 proving
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 "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 which reject bivalence.
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
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 logic
, the first multi-valued logic
.
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.
Intuitionistic logic
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 formalisation in mathematics, but his student Arend Heyting
studied intuitionistic logic formally, as did Gerhard Gentzen
. 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 logic
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 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 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 metaphysics
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 ofproblems 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
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 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 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 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 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 Habermas
, 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 Russell
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 gateLogic gateA 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 puzzleLogic puzzleA 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
- MathematicsMathematicsMathematics 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 mathematicsOutline of mathematicsThe 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 :*...
- MetalogicMetalogicMetalogic 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
- PhilosophyPhilosophyPhilosophy 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
- ReasonReasonReason 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 ThinkingStraight and Crooked ThinkingStraight 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 symbolsTable of logic symbolsIn 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...
- TruthTruthTruth 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 RescherNicholas RescherNicholas 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.
- Nicholas Rescher
- Essays
- "Symbolic Logic" and "The Game of Logic", Lewis CarrollLewis CarrollCharles 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.
- "Symbolic Logic" and "The Game of Logic", Lewis Carroll
- Reference material
- Translation Tips, by Peter Suber, for translating from English into logical notation.
- Ontology and History of Logic. An Introduction with an annotated bibliography.
- Reading lists
- The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject: