With logic (of ancient Greek λογικὴ τέχνη logiké téchnē , thinking art ',' Procedure ') or consistency is generally the rational reasoning and in particular the teaching of which - the inference teaching or even thought teaching - referred to. In logic, the structure of arguments is examined with regard to their validity , regardless of the content of the statements . In this sense, one speaks of “formal” logic. Traditionally, logic is part of philosophy . Originally, traditional logic developed alongside rhetoric . Since the 20th century, logic has mainly been understood as symbolic logic , which is also used as a basic structural science , e.g. B. within mathematics and theoretical computer science .
The modern symbolic logic used instead of natural language an artificial language (a sentence like The apple is red . B. is for the predicate calculus as a formalized, which for the apple and for red stands) and used strictly defined rules of inference . A simple example of such a formal system is propositional logic (so-called atomic propositions are replaced by letters). The symbolic logic is also called mathematical logic or formal logic in the narrower sense.
Different meanings of the word "logic"
The expression “logic”, in Greek logiké technē, stands for a doctrine of reasoning or reasoning in both the older Stoa and the older Peripatos , but was not used in this meaning before the 1st century BC. Occupied. The term was already coined by the ancient Stoic Zeno von Kition .
In German, the word “logic” is often used in the 19th century (for example by Immanuel Kant or Georg Wilhelm Friedrich Hegel ) in the sense of an epistemology , ontology or a general dialectic . On the other hand, logic in the modern sense was often referred to differently, for example as analytics, dialectics or logistics. Even today z. B. in sociology formulations such as logic of action or literary studies such as logic of poetry and the like. Ä. where “logic” is not understood as a theory of reasoning, but rather a doctrine of general “laws” or procedures that apply in a particular area. In the tradition of the philosophy of normal language in particular , a “logical” analysis was often understood to mean an analysis of conceptual relationships.
The use of the term “logic”, as described in the introduction, has been common since the beginning of the 20th century.
In colloquial language, expressions such as “logic” or “logical thinking” are also understood in a much broader or completely different sense and contrasted with “ lateral thinking ”, for example . Similarly, there is the concept of "women's logic," "Men logic" which "affect logic" and the concept of "everyday logic" - also known as " common sense " ( common sense ) - in the vernacular . In these areas, “logic” often refers to forms of action, pragmatics . An argument is colloquially referred to as "logical" if it appears sound, compelling, convincing, plausible and clear. The ability to think should be expressed in a logical argument.
Even in current debates it is largely undisputed that the theory of correct reasoning is at the core of logic; However, it is controversial which theories can still be included in logic and which are not. Contentious cases include the set theory , the reasoning theory (which is approximately at a pragmatic consideration with false conclusions employed) and the speech act .
History of logic
We speak of classical logic or a classical logical system when the following semantic conditions are met:
- Every statement has exactly one of two truth values , which are usually referred to as true and false . This principle is called the principle of bivalent or bivalence principle.
- The truth value of a compound statement is uniquely determined by the truth values of its partial statements and the way in which these are compounded. This principle is called the principle of extensionality or compositionality.
The term classical logic is to be understood more in the sense of established, fundamental logic, because the non-classical logics are based on it, than as a historical reference. It was rather the case that Aristotle , the classical representative of logic , so to speak , was very much concerned with multi-valued logic , i.e. non-classical logic.
The most important sub-areas of formal classical logic are classical propositional logic , first-level predicate logic and higher-level logic , as they were at the end of the 19th and beginning of the 20th centuries by Gottlob Frege , Charles Sanders Peirce , Bertrand Russell and Alfred North Whitehead were developed. In the propositional logic statements are examined to determine whether they are in turn re-assembled from statements by connectives (z. B. "and", "or") are connected together. If a statement does not consist of partial statements linked by connectives, then from the point of view of propositional logic it is atomic, i.e. H. cannot be further dismantled.
In predicate logic , the inner structure of sentences can also be represented, which cannot be further broken down from a propositional logic. The internal structure of the statements ( the apple is red. ) Is represented by predicates (also called statement functions) ( is red ) on the one hand and by their arguments on the other ( the apple ); The predicate expresses, for example, a property ( red ) that applies to its argument or a relation that exists between its arguments (x is greater than y). The concept of the statement function is derived from the mathematical concept of the function . A logical proposition function, just like a mathematical function, has a value that is not a numerical value, but a truth value.
The difference between the first-level predicate logic and the higher-level predicate logic is what is quantified using the quantifiers (“all”, “at least one”): In the first-level predicate logic, only individuals are quantified (e.g. “All pigs are pink "), in the predicate logic of a higher level, predicates themselves are also quantified (e.g." There is a predicate that applies to Socrates ").
Formally, predicate logic requires a distinction between different expression categories such as terms , functors , predicators and quantifiers. This is overcome in the step logic , a form of the typified lambda calculus . This makes mathematical induction , for example, an ordinary, derivable formula.
The syllogistics that was dominant up to the 19th century and goes back to Aristotle can be understood as a forerunner of predicate logic. A basic term in syllogistics is the term "concepts"; it is not broken down further there. In predicate logic, terms are expressed as single-digit predicates; Multi-digit predicates can also be used to analyze the internal structure of terms and thus show the validity of arguments that cannot be syllogistically grasped. A frequently quoted intuitively catchy example is the argument “All horses are animals; so all horse heads are animal heads ”, which can only be derived in higher logics such as predicate logic.
It is technically possible to expand and change the formal syllogistics of Aristotle in such a way that the predicate logic results in calculi of equal power. Such undertakings were occasionally undertaken from a philosophical perspective in the 20th century and are philosophically motivated, for example out of the desire to be able to view purely formal terms as elementary components of statements and not have to break them down according to predicate logic. More about such calculations and the philosophical background can be found in the article on conceptual logic .
Calculus types and logical procedures
Modern formal logic is dedicated to the task of developing exact criteria for the validity of inferences and the logical validity of statements (semantically valid statements are called tautologies , syntactically valid statements are theorems ). Various methods have been developed for this purpose.
In particular in the area of propositional logic (but not only), semantic methods are used, i.e. those methods that are based on the statements being assigned a truth value. These include on the one hand:
While truth tables provide a complete listing of all truth value combinations (and can only be used in the propositional area), the other procedures (which can also be used in predicate logic) proceed according to the scheme of a reductio ad absurdum : If a tautology is to be proven, one starts from its negation and tries to derive a contradiction . Several variants are common here:
The logical calculi that get by without semantic evaluations include:
One speaks of non-classical logic or a non-classical logical system when at least one of the two above-mentioned classical principles (two-valued and / or extensionality) is abandoned. If the principle of duality is abandoned, multivalued logic emerges . If the principle of extensionality is given up, there arises dimensional logic. Intensional are, for example, modal logic and intuitionistic logic . If both principles are abandoned, multi-valued dimensional logic arises. ( See also: Category: Non-Classical Logic )
Philosophical logic is a fuzzy collective term for various formal logics that change or expand classical propositional and predicate logic in different ways, usually by enriching their language with additional operators for certain areas of speech. Philosophical logics are usually not of direct interest to mathematics, but are used, for example, in linguistics or computer science . They often deal with questions that go far back into the history of philosophy and that have been discussed in some cases since Aristotle, for example how to deal with modalities ( possibility and necessity ).
The following areas, among others, are assigned to philosophical logic:
- Modal logic introduces modal sentence operators like "it is possible that ..." or "it is necessary that ..." and examines the validity conditions of modal arguments;
- epistemic logic or doxastic logic examines and formalizes statements of belief, conviction and knowledge as well as arguments formed from them;
- Deontic logic or the logic of norms examines and formalizes commandments, prohibitions and concessions (“it is permitted that ...”) as well as arguments formed from them;
- Temporal logic of actions , quantum logic and other temporal logics examine and formalize statements and arguments in which reference is made to points in time or periods of time;
- Intensional logics do not only concern the extension (denotation; meaning in the sense of designated elements), but also their intension (meaning; meaning in the sense of designated properties) of concepts or sentences.
- Interrogative logic examines questions as well as the question of whether logical relationships can be established between questions;
- Conditional sentence logic examines “if-then” conditions that go beyond the material implication ;
- Paraconsistent logics are characterized by the fact that in them it is not possible to derive any statement from two contradicting statements. This also includes the
- Relevance logic that uses an implication instead of the material implication that is only true if its antecedent is relevant for its subsequent clause (see also the following chapter)
Intuitionism, relevance logic and connected logic
The most discussed deviations from classical logic are those logics that dispense with certain axioms of classical logic. The non-classical logics in the narrower sense are “weaker” than the classical logic, i. H. In these logics fewer statements are valid than in classical logic, but all statements valid there are also classically valid.
does not contain, whereby the sentence " tertium non datur " (for every statement p applies: p or not-p),
can not be derived, as well as subsequent thereto relevance logics , in which only those statements of the scheme are valid, where for relevant causal ( see implication # object language implications ). In the dialogic logic and in the sequence calculi, both the classical and the non-classical logics can be converted into one another by means of corresponding additional rules.
On the other hand, we should mention logics that contain principles that are classically not valid. The proposition initially seems to express an intuitively plausible logical principle: because if p holds, then p, it seems, can no longer be false. However, this theorem is not a valid theorem in classical logic . Insofar as the classical logic is maximally consistent , i.e. H. in so far as any genuine reinforcement of a classical calculus would lead to a contradiction, this theorem could not be added as a further axiom . The connected forms logic , which is to meet the formal pre-intuition which expresses the sentence by awarding him as a theorem, must therefore reject other classic logical theorems. So while with intuitionistic, minimal and relevant logic the provable formulas are each a real subset of the classically provable formulas, on the other hand, the relationship between connected and classic logic is such that formulas can also be proven in both that do not apply to the other logic.
Multi-valued logic and fuzzy logic
This is crossed by the multivalued logics, in which the principle of two- valued and often also the Aristotelian principle of the excluded third do not apply, including the three-valued and infinite logic of Jan Łukasiewicz ("Warsaw School"). The infinite fuzzy logic has numerous applications in control technology , while Gotthard Günther's finite logic ("Günther logic") was applied to problems of self-fulfilling predictions in sociology .
A logical system is called monotonic if every valid argument remains valid even if additional premises are added: What has been proven once remains valid in a monotonic logic, i.e. even if new information is available at a later point in time . Many logical systems have this monotony property, including all classical logics such as propositional and predicate logic.
In everyday and scientific reasoning, however, provisional conclusions are often drawn which are not valid in a strictly logical sense and which may have to be revised at a later point in time. For example, the statements "Tux is a bird." And "Most birds can fly." Could provisionally conclude that Tux can fly. But if we now receive the additional information "Tux is a penguin.", Then we have to correct this conclusion, because penguins are not airworthy birds. In order to map this type of reasoning, non-monotonic logics were developed: They dispense with the monotony property, that is, a valid argument can become invalid by adding further premises.
Of course, this is only possible if a different consequence operation is used than in classical logic. A common approach is to use so-called defaults . A default conclusion is valid if there is no contradiction to it from a classic-logical conclusion.
The conclusion from the example given would then look like this: “Tux is a bird.” The prerequisite remains . We now combine this with a so-called justification : "Birds can normally fly." From this reason we conclude that Tux can fly as long as nothing speaks against it. The consequence is “Tux can fly.” If we now receive the information “Tux is a penguin.” And “Penguins cannot fly”, there is a contradiction. Using the default conclusion, we came to the conclusion that Tux can fly. With a classic-logical conclusion, however, we were able to prove that Tux cannot fly. In this case the default is revised and the consequence of the classic-logical conclusion is used. This method - roughly described here - is also referred to as Rider's default logic . (See also non-monotonic inductive Bayesian logic .)
- Aristotle (384–322 BC):
- In the Analytica priora : Development of the syllogistics used until the 19th century , a pre-form of predicate logic .
- Chrysippos of Soloi (281 / 76–208 / 4 BC):
- Development of the stoic syllogistics, a preliminary form of the propositional calculus.
- Cicero (106–43 BC):
- Translated Greek logic into Latin.
- Gottfried Wilhelm Leibniz (1646–1716):
- First approaches to a symbolic logic.
- George Boole (1815–1864):
- Development of Boolean Algebra .
- Charles Sanders Peirce (1839-1914):
- First approaches to quantifier logic, introduction of relational logic, formulation of a theory of abduction .
- Georg Cantor (1845–1918):
- Development of set theory .
- Gottlob Frege (1848–1925):
- Edmund Husserl (1859–1938):
- Critique of Psychologism in Logic.
- Bertrand Russell (1872–1970):
- Discovered Russell's antinomy .
- Jan Łukasiewicz (1878–1956):
- Developed the Polish notation , dealt with multi-valued logic.
- Alfred Tarski (1901–1983):
- Kurt Gödel (1906–1978):
- Completeness of the predicate logic. Incomplete Peano arithmetic .
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