Logic

under the logic (griech. λογική [τέχνη] „the thinking [art, proceeding] “) is partly understood today generally one in philosophy, partly in mathematics and in computer science settled theory, which concerns itself primarily with the standards of correct (conclusion) concluding.

Today one understands predominantly formal logic (also symbolic logic or mathematical logic called) by logic, as one finds it for example in the propositional calculus and in the formal systems. The philosophical logic did not have however always this structure.

[to work on]

different meanings of „logic “

in the history of philosophy is the use way of the expression represented above „logic “only since beginning 20. Century usually, although this term had been already coined/shaped by the antique Stoiker Zenon of Kition. Before the expression was often used (for instance with Immanuel Kant , Georg William Friedrich Hegel or Karl Christian Friedrich ruffle) very many far in the sense of a theory of knowledge , Ontologie or a general dialectic. The logic in the modern sense was frequently differently designated on the other side, about than analytics, dialectic or logistics. Also today still idioms are such as logic of the research, logic of the seal among other things in philosophy and the Geisteswissenschaften spread, with which by „logic “no theory of concluding is understood, but teachings more generally „laws “, which apply within a certain range.

In particular in the tradition of the Ordinary LANGUAGE Philosophy by one „logical “analysis an analysis of conceptual connections was often understood.

In the colloquial language expressions are understood beyond that like „logic “or „logical thinking “in a very much more further or completely different senses and confronted about to one „lateral thinking “. Likewise there is the term „of the woman logic “, „man logic “, „the affect logic “and the term „of the everyday life logic “- admits also as „healthy human understanding “(common scythe) - in the colloquial language. Within these ranges the logic is regarded to the pragmatics often as logic of acting. An argument is called „logically “, if this is soundly, compellingly, convincingly, plausible and clear. In a logical argument the talent of thinking and justifying is to come to the expression.

Also in the present debate it is clear that the theory of correct concluding constitutes the core of the logic; disputed is however, which are to be counted theories exactly still to the logic and which not. Contentious cases are for instance the theory of sets, the argumentation theory (for instance concerned under pragmatic consideration with false conclusions) and the speech act theory.

[Work on]

history of the logic

[work on]

antique one

as founders of the logic applies for Aristoteles. Particularly to call its Syllogistik , a formal logical system, is in which arguments of rigid structure, syllogisms are called, examined. The statements, which arise within syllogisms, set terms to each other in relationship (e.g. “All S is P”, i.e. everything that falls under the term S, falls also under the term P). Logical systems, in whose statements terms are set to each other in relationship, are called term logics.

Also on Aristoteles back (however not in its analytics , but in its Metaphysik developed ) the science of some fundamental principles of human thinking goes. For this about the sentence third impossible from the contradiction and the sentence of count.

In different other works (in De interpretatione and the 2. Analytics) have themselves Aristoteles besides with central languagephilosophical-logical terms such as judgement and term employs and with general rules of proving and disproving.

By work of Łukasiewicz starting from 1923 and Mates starting from 1949 one knows that already the Stoa had a fully prepared junctor logic. Because of the large authority, which enjoyed Aristoteles, one considered the stoische logic in the twentieth century not or did not understand her at least not, because one could not set her with the Syllogistik to coincide.

[Work on]

the Middle Ages

a further important epoch for the logic are the Middle Ages. In the medieval university enterprise the logic has its place in the so-called “Artistenfakultät “(facultas artium). The study that artes is a condition for the study at all other faculties. Starting from for instance the center 13. Century covers the instruction material of the logic three separate Textkorpora. With logica vetus and logica the new facts concerns it delivered logical writings, in particular the Organon of the Aristoteles and the comments of the Boëthius and the Porphyrius. Parva logicalia one can regard as self-creation of the medieval logic. Here off the antique collecting mains a whole set of new problem definitions from the frontier between logic and semantics are developed and discussed in from each other independent treatises. Some usual treatise types are briefly presented:

De proprietatibus terminorum is concerned with the characteristics of the materialen (non-logical) terms.
De syncategorematicis examines against it the formal, i.e. logical expressions.
De suppositio terminorum formulates the medieval Suppositionstheorie
with De consequentiis goes it around consequences.
De insolubilibus has paradoxes and fallacies to the article.
De Relativis is concerned with the characteristics of anaphorischer expressions.
De Modalibus examines Modal expressions.
With De Obligationibus it concerns the logical conditions of a coherent dispute.

As important medieval logicians can be called: Petrus Abaelardus, William of Ockham and Johannes Buridan.

[Work on]

modern times

in the modern times erlahmt first the interest in the logic. Common the opinion Immanuel Kant is far that the system of the logic with the Aristoteli Syllogistik to the conclusion would have come, and that there would be nothing else to therefore discover here. Accordingly little outstanding logicians are brought out, an exception are however Gottfried William Leibniz.

Only in the middle of the nineteenth century finds the logic again broader attention, first particularly in England. Pointing the way here George Boole with the shorter treatise is „The Mathematical analysis OF logic “(1847) and its later Hauptwerk „Laws OF Thought “(1854). Booles idea is it to understand logic as mathematics which is limited to the values 0 and 1 (truely and wrongly). On class symbols as algebraic operations can as addition, multiplication etc. are implemented. In this way Boole develops a complete system of the one-digit predicate calculus, which contains the Syllogistik as subsystem. At the same time with Boole Augustus De Morgan formally publishes its work „logic “ 1847. De Morgan is interested here among other things for a Verallgemeinerung of the Syllogistik on statements of the form „most A are B “. A further logician in England is John Venn, which publishes its book „symbolism logic “with the famous Venn diagrams 1881. In addition at the logical research are involved in America Charles of Sanders Peirce and in Germany Ernst Schröder.

The actual break-through to the modern logic succeeds however to God praise Frege, which must probably be regarded as the logicians most meaning beside Aristoteles at all. In its ideography (1879) he presents for the first time a full predicate calculus of second stage. In addition it develops the idea of a formal language here and whereupon constructing the idea of the formal proof, in which after Freges words nothing remains left „for the Errathen “. Straight these ideas form a completely substantial theoretical basis for the development of the modern computer engineering and computer science. Freges work is first hardly noticed however by its contemporaries; this likes among other things because of its logical notation which can be read very with difficulty are (see also ideography notation). Into the 1893 and 1903 published both volumes „the Basic Laws of arithmetic “Frege entire mathematics in a kind theory of sets tries to axiomatisieren . This system contains however a contradiction (the so-called Russell Antinomie), as Frege in famous a letter of Bertrand Russell, become , must experience from the year 1902.

Russell remains reserving it to submit the first assertible set-theoretical foundation of mathematics together with Alfred North Whitehead in the Principia Mathematica (1910). The authors appreciate Frege in the preface, it owed it most in „logical-analytic questions “. Contrary to Freges work becomes the Principia Mathematica a piercing success. For this one knows a reason and others see in the notation used by Russell/Whitehead, which is still usual to far parts today. Impacts to this notation supplied Giuseppe Peano, a further important logician to outgoing 19 with. Century, Russell learned which to know in the year 1900 at a congress. Apart from its thoughts on the logical notation Peano is particularly well-known for its Axiomatisierung of the number theory ( the so-called Peano axioms).

[Work on]

modern trend

the Boolean fragment „of the Principia Mathematica “serves a whole number of metalogischer terms as starting point for the development. In its Habilitationsschrift of 1918 Paul Bernays ( constructing on the work David Hilberts) shows compatibility, syntactic and semantic completeness and Entscheidbarkeit and examines the independence of the axioms (whereby he states that one of the axioms actually dependently, thus redundant, is).

Apart from the axiomatic method „of the Principia “further calculation types are developed. Gerhard Gentzen its system of natural reasoning and the sequence calculation presents 1934. Constructing Evert Willem Beth 1959 on that develops the tablet calculation. Again at this Paul Lorenzen orients itself with his dia.-logical logic.

In addition the modern logic brings the development to a semantics of the predicate calculus with itself. An important preliminary work for this represents the famous Löwenheim Skolem theorem (first proven of Leopold lion home in the year 1915, a more general result shows Albert Thoralf Skolem 1920). Gödel proves briefly the completeness of the predicate calculus of first stage (Gödel completeness set) to 1929, 1930 the incompleteness of Peano arithmetic (Gödel incompleteness set). Alfred Tarski a truth theory for the predicate calculus formulates 1933.

Further highlights in the history of the modern logic are the development of the Intuitionisti logic, Modallogik, the Lambda calculation, the type theory as well as the stage logic (logic of higher stage). An important trend in the modern logic is also the development of theorem proofs as well as the use of logic in computer science by formal methods.

[Work on]

subsections

[work on]

classical logic

major item: Classical logic

of classical logic and/or. of a classical logical system one speaks exactly if the following semantic conditions are fulfilled:

Each statement has exactly one of exactly two logical values, which are usually called true and wrongly. (Principle of bivalence or Bivalenzprinzip)
the logical value of a compound statement is clearly certain by the logical values of their Teilaussagen. (Principle of the Extensionalität or Kompositionalität).

The term classical logic is more in the sense of more establish to understand fundamental logic on which not-classical logics develop, because as historical reference; rather was it like that that already Aristoteles, as it were the classical representative of the logic, very probably concerned itself with multi-valued logic, thus not-classical logic.

The most important subsections of the formal classical logic are the classical propositional calculus, predicate calculus of the first stage and higher stage as well as stage logic. In the propositional calculus statements become thereupon examined, whether they are compound for their part for statements, those by junctors (e.g. “are connected and”, “or”). If a statement does not exist connected Teilaussagen out by junctors, then it is from view of the propositional calculus atomically, i.e. not further detachable.

In the predicate calculus also the internal structure of sentences can be represented, which are Boolean not further detachable. The difference between predicate calculus of the first stage and predicate calculus of higher stage consists of about what by means of the Quantoren (“all”, “at least”) it is quantified: In the predicate calculus of first stage only over individuals one quantifies (e.g. “All pigs are pink”), in the predicate calculus of higher stage also over descriptors themselves are quantified (“there is a descriptor, which applies to Sokrates”).

The predicate calculus formally requires a distinction between different expression categories such as terms, radio gates, Prädikatoren and Quantoren. This is overcome in the stage logic, a form of the typed Lambda calculation. Thus for example the mathematical induction becomes a usual, derivable formula.

Up to 19. Century dominant factor Syllogistik, which decreases/goes back on Aristoteles, can be understood as a forerunner of the predicate calculus. A fundamental idea of the Syllogistik is the term “term”; it is not continued to divide there. In the predicate calculus terms are expressed as one-digit descriptors; with descriptors of several characters additionally the internal structure of terms can be analyzed and to be shown thus the validity of arguments, which are syllogistisch not understandable. A frequently quoted example is the argument “all horses is animals. Thus all horse heads are animal heads, “despite its intuitive a freedom of movement with the development of the predicate calculus to only deduce left themselves.

It is technically possible to extend and change the formal Syllogistik of the Aristoteles in such a way that equivalent calculations develop for the predicate calculus. Such enterprises were made particularly from philosophical side and are philosophically motivated, for example from the desire to be able and them descriptor-logically to divide not have to also purely formally regard terms as elementary components of statements. More to such calculations and the philosophical would background are in the article to the term logic.

[Work on]

dedicates themselves calculation types and logical

procedures the modern formal logic to the task to develop accurate criteria for the validity of conclusions and the logical validity of statements (semantically valid statements hot Tautologien, syntactically valid statements of theorems). For this different procedures were developed.

In particular within the range of the propositional calculus (however not only) semantic procedures are common, thus such procedures, which are based on the fact that a logical value is attributed to the statements. For this count on the one hand:

Truth tables

during truth tables a complete listing of all logical value combinations make (and to that extent also only in the Boolean range are usable), proceed the remaining (also descriptor-logically usable) procedures according to the pattern of a Reductio ad absurdum : If a tautology is to be proven, one proceeds from its negation and tries a contradiction to derive. Here three variants are common:

Resolution,
semantic trees and
Beth Tableaux (after: Evert Willem Beth)

to the logical calculations, which get along without semantic evaluations, count:

[work on]

Nichtklassi logics

of not-classical logic and/or. a not-classical logical system speaks one, if at least one of the two classical principles specified above (bivalence and/or Extensionalität) is given up. If the principle of bivalence is given up, multi-valued logic develops. If the principle of the Extensionalität is given up, intensionale logic develops. Intensional are for example the Modallogik and the intuitionistic logic. If both principles are given up, multi-valued intensionale logic develops.

[Work on]

philosophical logics

the classical stating and predicate calculus can be modified on the one hand, by enriching the language around further operators for certain speech ranges. Thus the Modallogik concerns itself with expressions as “necessarily” or “possible”; the deontische logic with “required” or “permits”; the epistemische logic with “know” and “believe”. These logics are called frequently philosophical logics.

[Work on]

Intuitionism, relevance logic and konnexe logic

the most-discussed deviations from the classical logic represent such logics, which do without certain axioms of the classical logic. The strictly speaking non-classical logics are „more weakly “than the classical logic, i.e. in these logics are fewer arguments valid than in the classical logic, there are however all there valid arguments also classically valid.

To it belong from L. E. J. Brouwer developed Intuitionisti logic, which “duplex negatio” - the axiom (from the double negation of a statement p follows p)

(DN) <math> \ neg \ neg p \ Rightarrow p< /math>

does not contain, whereupon the sentence “tertium non datur “(to each statement p applies: p or not p),

(TND) <math> \ neg p \ or p< /math>

, that is no longer derivable minimum insignificant I. Johanssons, with which the set of “ex quodlibet falso “(from a contradiction any statement follows),

(EFQ) <math> \ neg p \ Rightarrow (p \ Rightarrow q)< /math>

no more not to be derived can, as well as hieran the following relevance logics, in which only such implications are valid, in which the Antezedens for the Sukzedens is relevant. In the dia.-logical logic and in the sequence calculations both the classical ones and the non-classical logics are into one another transferable by appropriate auxiliary rules.

On the other side logics are to be mentioned, the principles contained, which are classically not valid. Thus for instance /math applies in a konnexen <logic> math \ neg (p \ Rightarrow \ neg p<)> - a sentence, which does not represent a classical tautology despite its high plausibility. If the classical logic is maximumconsistent, i.e. if each genuine reinforcement of a classical calculation became to lead to a contradiction, this sentence not a classical calculation could be added as the further axiom; rather would have to be made weaker classical calculations first.

Connexive logic. Entry (English) in the Stanford Encyclopedia OF Philosophy (inclusive Literature data)

[work on]

multi-valued ones and Fuzzy logic

major item: Multi-valued logic

crosswise for this the multi-valued logics stand, in which the principle of bivalence and often also the aristotelische “sentence of impossible third “is repealed set, how the trivalent logic and the unendlichwertige logic of January Łukasiewicz (“Warsaw school”), which in the Fuzzy logic practical application find, and which endlichwertige logic of God hard Günther (“Günther logic”), which is applied to problems fulfilling forecasts in the sociology. Albert my and Niels Öffenberger developed a multi-valued logic (more than two logical values) while maintaining the aristotelischen conception.

[Work on]

Nichtmonotone logics

classical logics such as stating and predicate calculus have characteristic the Monotonie -. This essentially means that by conclusions only new knowledge can be revised be won, not however already existing knowledge. Which was once proven, remains always valid in a monotonous logic, even if one has at a later time new information.

Nichtmonotone logics make a revising possible of won realizations. If we have from the statements “Tux are closed a bird” and “birds can fly” that Tux can fly, then we revise this conclusion, if we the additional information “Tux is a penguin” received. This is certainly possible only if we use another consequence operation than in a classical logic. A usual beginning consists of using defaults so mentioned. A default conclusion is valid if from a classical-logical conclusion a contradiction does not result to it.

The conclusion from the given example would look in such a way then: “Tux is a bird” remains the condition (prerequisite). We combine these now with a reason in such a way specified (justification): “Birds can normally fly.” From this reason we conclude that Tux can fly, as long as nothing speaks against it. The consequence reads thus “Tux can fly.” If we receive now the information “Tux are a penguin” and “penguins cannot not fly”, then a contradiction results. Over the default conclusion we arrived at the consequence that Tux can fly. With a classical-logical conclusion way however we could prove that Tux cannot fly. In this case the default is revised and the consequence of the classical-logical conclusion is re-used. This - here roughly described - procedure is called also Reiter' default logic.

Entry (English) in the Stanford Encyclopedia OF Philosophy (inclusive Literature data)< small/>

[Work on]

Possibilisti logic

in a possibilistischen logic classical-logical statements with possibility and necessity degrees are quantified. One can use then possibilistische resolution procedures, in order to derive from a quantity of possibilistischer formulas new possibilistische statements.

[Work on]

important authors

Aristoteles (384-322 v.u.Z.):

In the Analytica Priora development to in 19. Syllogistik, a gathering mold of the predicate calculus used century.

Cicero (106-43 v.u.Z.):

He transferred the science of the logic of Aristoteles and transferred her as acre logica to latin: De finibus bonorum et malorum.

Gottfried William Leibniz (1646-1716):

First beginnings to a symbolic logic

George Boole (1815-1864):

Development of algebra.

George CAN gate (1845-1918):

Development of the set theory.

God praise Frege (1848-1925):

Development of the modern stating and predicate calculus.

Edmund Husserl (1859-1938):

Criticism of the psychology mash in the logic

Bertrand Russell (1872-1970):

Russell Antinomie.

Kurt Gödel (1906-1978):

Completeness of the predicate calculus. Incompleteness of Peano arithmetic.

Charles of Sanders Peirce (1839 - 1914)
January Lukasiewicz (1878 - 1956)
Alfred Tarski (1901 - 1983)

further logicians are in the category: Logicians (see below)

[to work on]

important ones of works

Aristoteles: Theory of the Schlusz or first analytics, Hamburg: My 1922 3. Aufl. 1992 ISBN 3-7873-1092-4
Frege, God praise: Ideography, one the arithmetic copied Formelsprache of pure thinking, resounds/to Saale: 1879, in part printed e.g. in: Berka, Karel; Kreiser, Lothar: Logic texts. Commentated selection for the history of the modern logic, Berlin: Academy publishing house 4. Aufl. 1986
Frege, God praise; Patzig, Günther (Hg.): Logical investigations, Goettingen: Vandenhoeck & Ruprecht 1966, 3. Aufl. 1986 (=Kleine Vandenhoeck row 1219) ISBN 3-525-33518-0
Łukasiewicz, January: Logika dwuwartościowa, Przegląd Filosoficzny, 23, 1921, side 189ff
Łukasiewicz, January: Borkowski, L. (Hg.): Selected Works, Warsaw: PWN 1970
Peano, Giuseppe: Notation de logique mathématique, Turin: 1894
Peirce, Charles of Sanders: On the algebra OF logic. A contribution ton the philosophy OF notation, The American journal OF Mathematics 7, 1885
Quine, wanting pool of broadcasting corporations Van Orman: Fundamentals of the logic, Frankfurt/Main: Suhrkamp 1969, 6. Aufl. 1988 (=Suhrkamp paperback science) ISBN 3-518-27665-4
Tarski, Alfreds: Introduction to the mathematical logic, Goettingen: Vandenhoeck & Ruprecht 1966, 5. Aufl. 1977 (=Moderne mathematics in elementary representation 5) ISBN 3-525-40540-5
Whithehead, Alfred North; Russell, Bertrand: Principia Mathematica, Cambridge: 1910-1913 2. Aufl. 1925-1927

[work on]

Artikelanthologie to the logic

Berka, Karel; Kreiser, Lothar: Logic texts. Commentated selection for the history of the modern logic, Berlin: Academy publishing house 4. Aufl. 1986

[work on]

literature

Philosophy bibliography: Logic - to this topic there are additional reference works

history of the logic

Kneale, William; Kneale, Martha: The development OF logic, Clarendon press 1962 ISBN 0-19-824773-7
Mates, Benson: Stoic logic, Berkeley: University OF California press 1953 (=University OF California Publications in Philosophy 26) ISBN 0-608-11119-8 (in English language) Books on and

logical Propädeutik

Ernst Tugendhat, Ursula wolf: Logical-semantic Propädeutik. Reproduction. Reclam, Stuttgart 2001, ISBN 3-15-008206-4 (RUB 8206)
William Kamlah, Paul Lorenzen: Logical Propädeutik. Preparatory school of reasonable talking. 3. Aufl. Metzler, Stuttgart and others 1996, ISBN 3-476-01371-5

philosophy

Ansgar Beckermann: Introduction to the logic. 2. Aufl. De Gruyter, Berlin and others 2003, ISBN 3-11-017965-2
Ruediger Inhetveen: Logic. An interactive introduction. OD. at the good mountain place, Leipzig 2003, ISBN 3-937219-02-1
Thomas Zoglauer: Introduction to the formal logic for philosophers. 3. Aufl. Vandenhoeck & Ruprecht, Goettingen 2005. ISBN 3-8252-1999-2, ISBN 3-525-03293-5 (UTB for science Bd. 1999)

Logic

bar meadow, Jon; Etchemendy, John: The LANGUAGE OF roofridge order logic, CSLI center for the Study OF LANGUAGE and information Leland Stanford junior University 1991 (=CSLI Lecture Notes 23) ISBN 0-937073-74-1
Kutschera, Franz v., & broad head, Alfred: Introduction to the modern logic. Freiburg: Talk nonsense 2000 (7. Aufl.). ISBN 3495479775
Lemmon, E. J.: Beginning logic, London: Chapman and resound to London 1965, 2. Aufl. 1987 ISBN 0-412-38090-0
Mates, Benson: Elementary logic. Predicate calculus of the first stage with identity, Goettingen: Vandenhoeck & Ruprecht 2. Aufl. 1978 (=Moderne mathematics in elementary representation 9) ISBN 3525405413
Salmon, Wesley C.: Logic, Stuttgart: Reclam 1983 (=Universal-Bibliothek) ISBN 3-15-007996-9

mathematics

Heinz Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to the mathematical logic. 4. Aufl. Spectrum, academy, Heidelberg and others 1998, ISBN 3-8274-0130-5 (spectrum university paperback)
Wolfgang lozenge mountain: Introduction to the mathematical logic. A text book. 2. Aufl. Vieweg, Braunschweig and others 2002, ISBN 3-528-16754-8
Donald W Barnes, John M. Mack: On algebraic Introduction ton of Mathematical logic GTiM Springer publishing house Berlin 1975, ISBN 3-540-90109-4. A very mathematical entrance to the logic.

Computer science

Uwe Schöning: Logic for computer scientists. 5. Aufl. Spectrum, academy, Heidelberg and others 2000, ISBN 3-8274-1005-3 (spectrum university paperback)
Bernhard Heinemann, Klaus Weihrauch: Logic for computer scientists. An introduction. 2. Aufl. Teubner, Stuttgart 1992, ISBN of 3-519-12248-0 (manuals and Monographien of computer science)