Logic

Under Logic ( from ancient Greek λογικὴ τέχνη logiké Techne "thinking art", "Procedure" ) is the doctrine of rational reasoning. In the logic structure of arguments in view of their validity is tested, regardless of the content of the statements. Already in this sense one speaks of " formal " logic. Traditionally, logic is a part of philosophy. Originally, the traditional logic has developed adjacent to the rhetoric. Since the 20th century is meant by logic mainly symbolic logic that is also as a basic structure of science, such as within mathematics and theoretical computer science, treated.

The modern symbolic logic used instead of natural language artificial language ( A sentence like The apple is red is in the predicate logic formalized for example as f ( a), where a is for the apple and f for red is ) 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 narrow sense.

Different meanings of the term "logic"

The term "logic" in Greek logiké techne, is available in both the older Stoicism as in older Peripatos for a theory of argumentation or closing, but is not shown before the 1st century BC in this meaning. The term was coined by the ancient Stoic Zeno of Kition.

In German, the word " logic " in the 19th century in many cases (eg in Immanuel Kant and Georg Wilhelm Friedrich Hegel) is also used in the sense of epistemology, ontology, or a general dialectic. The logic in the modern sense was often called by other names on the other side, such as analysis, dialectic or logistics. Even today phrases like logic of literature are like in different disciplines spread, in which the " logic " no theory of reasoning is understood, but a doctrine of general "laws" or procedures that apply in a particular area.

In particular, in the tradition of the philosophy of ordinary language was a "logical" analysis in many cases an analysis of conceptual relationships understood.

The use manner illustrated introduction of the term " logic ", however, is common since the beginning of the 20th century.

In the vernacular expressions such as " logic " or " logical thinking " in addition be understood in a much wider or completely different meaning and as a " lateral thinking" compared. Similarly, there is the concept of " woman logic ", " men logic ", the " affect-logic " and the concept of "everyday logic " - also known as " common sense " ( common sense ) - in the vernacular. In these areas relates "logic" often on forms of action, the pragmatics. An argument is colloquially referred to as " logic", if this relevant, compelling, convincing, reasonable and clearly displayed. In a logical argument, the skill of thinking should be reflected.

Also in current debates is widely accepted that the theory of correct reasoning is at the heart of logic; controversial, however, is that theories are just expected more on logic and which are not. Disputed cases are about the set theory, the theory of argumentation ( which deals about a pragmatic consideration with fallacies ), and the speech act.

History of logic

Subregions

Classical logic

From classical logic or from a classical logical system is called if and only if the following semantic conditions are met:

The term classical logic is more to be understood in the sense of well-established, basic logic, because the non-classical logics based on them, because as a historical reference. Rather, it was so, that Aristotle, as it were, a classic of the logic that has been very well occupied with polyvalent logic, ie non-classical logic.

The most important areas of formal classical logic are the classical propositional logic, the first stage and logic higher -order predicate logic as in the late 19th and the early 20th century by Gottlob Frege, Charles Sanders Peirce, Bertrand Russell and Alfred North Whitehead have been developed. In propositional logic statements are examined to determine whether they are in turn reassembled from statements that are connected together by connectives (such as " and", " or "). If a statement is not affiliated with connectives partial statements, then it is atomic from the perspective of propositional logic, ie, not further decomposable.

In predicate logic, and the internal structure of sentences can be represented, which are evaluated logically not separable. (. The apple is red) represented by predicates ( also called propositional functions ), the internal structure of the statements here ( red ) on the one hand and by the arguments of the other part ( the apple ); it expresses the predicate, for example, a property (red ), which applies to his argument, or a relation that exists between its arguments (x is greater than y). The term of the message function is derived from the mathematical concept of the function. A logical propositional function has exactly like a mathematical function has a value that is not numeric, but a truth value but.

The difference between predicate logic of first stage and predicate logic higher level is what means the quantifiers ( "all", "at least one " ) is quantified: in the first order predicate logic is quantified only on individuals ( for example, " All pigs are pink " ), in which higher -order predicate logic level is quantified via predicates itself (eg, " There is a predicate that applies to Socrates ").

Formal predicate logic requires a distinction between different expression categories such terms, functors, and quantifiers predictors. This is the stage logic, a form of typed lambda - calculus, to overcome. This will, for example, an ordinary mathematical induction, derivable formula.

The dominant until the 19th century syllogistic, which goes back to Aristotle, can be understood as a precursor of predicate logic. A fundamental concept of syllogism, the term " terms"; He is there not broken down further. In predicate logic terms are expressed as predicates; with more -place predicates can additionally analyze the internal structure of words, and thus demonstrate the validity of arguments that are syllogistically intangible. A frequently cited example is the argument intuitively catchy " All horses are animals; therefore all horses heads animal heads ", which can be derived only in higher logics such as predicate logic.

It is technically possible to extend the formal syllogistic of Aristotle so and change that the predicate logic equally powerful calculi arise. Such things have been made sporadically from the philosophical side in the 20th century and are philosophically motivated, for example, out of a desire, even a purely formal to view concepts as elementary components of statements and not to have to disassemble predicate logic. More on such calculations, and the philosophical backgrounds can be found in the article on the concept of logic.

Calculus types and logical method

The modern formal logic deals with the task of precise criteria for the validity of conclusions and the logical validity of statements ( semantically valid statements are called tautologies syntactically valid statements theorems ) to develop. Various methods have been developed.

Particularly in the area of propositional logic ( but not only) semantic methods are commonly used, ie, those methods that are based on the statements that a truth value is attributed. These include, on the one hand:

Truth Tables

While truth tables to make a complete listing of all truth-value combinations ( and have so far used only in the propositional logic field ), go the other (also in predicate logic recyclable ) method according to the scheme of a reductio ad absurdum: If a tautology is to prove one goes from its negation and attempts to derive a contradiction. Here are several variants can be used:

Resolution,
Tree calculus or Beth- Tableaux (after Evert Willem Beth )

The logical calculi that do not require semantic rating include:

Axiomatic logic calculi
Natural deduction systems
Sequences calculi
Dialogical logics

Non-classical logics

From non-classical logic and a non- classical logic system is used when at least one of the above two classical principles ( bivalence and / or extensionality ) is abandoned. The principle of two-valued abandoned generated multi-valued logic. If the principle of extensionality abandoned arises intensional logic. Intensional are for example modal logic and intuitionistic logic. If both principles abandoned, produced multi-valued intensional logic. ( See also: Category: Non-classical logic)

Philosophical logic

Philosophical logic is a fuzzy collective term for various formal logics that change or expand, usually by enriching their language to other operators for certain speech areas of the classical propositional and predicate logic in different ways. Philosophical logic is usually not of direct interest for mathematics, but are used for example in linguistics or computer science. They treat many questions that go far back in the history of philosophy and partially discussed since Aristotle, for example, dealing with modalities ( possibility and necessity ).

Be attributed to the philosophical logic include the following areas:

Modal logic leads modal set operators such as " it is possible, dass .. " or " it is necessary, dass .. " and examines the conditions of validity of modal arguments;
Epistemic logic and doxastic logic are studied and formalized statements of faith, belief and knowledge, and formed from them arguments;
Examined deontic logic or standard logic and formalized commandments, prohibitions and concessions ( " it is allowed dass .. ") and formed from them arguments;
Temporal logic of actions, examine the quantum logic and other temporal logics and formalize statements and arguments, in which reference is made to points in time or periods of time;
Interrogativlogik investigated interrogative sentences and the question whether it is possible between interrogative sentences establish logical relationships;
Konditionalsatzlogik examined on the material implication beyond " if-then " conditions;
Para Consistent logics are characterized by the fact that it is not possible in them, of two contradictory statements derive any statement. This includes the
Relevance logic that uses an implication rather than the material implication which is true only if their antecedent is relevant to their trailer (see also the next chapter )

Intuitionism, relevance logic and connected forms of logic

The most discussed deviations from classical logic represent such logics that without certain axioms of classical logic. The strictly speaking non-classical logics are " weaker " than the classic logic, ie in these logics fewer statements are valid in classical logic, but they are all valid statements there also classically valid.

These include the development of LEJ Brouwer Intuitionistic logic, which ( it follows from the double negation of a statement p p ) the "duplex - negatio " axiom

Does not contain, thus the phrase " tertium non datur " ( for each statement p: p or not -p),

Is not derivable, the minimal calculus I. Johansson, thus the phrase " ex falso quodlibet " ( follows from a contradiction any statement ),

Can not be derived more, as well as the adjoining this relevance logics in which only such statements of the scheme are valid, where for causally relevant (see implication # Objektsprachliche_Implikationen ). In the dialogical logic and in the sequences calculi are both the classical and the non -classical logics by appropriate additional rules interconvertible.

On the other hand, logics may be mentioned include the principles that are classically not valid. The first sentence seems to express an intuitively plausible logical principle: For if p is true, then p, it seems not be more wrong. Nevertheless, the sentence is not a valid theorem in classical logic. In this respect, classical logic is maximal - consistent, ie would thus result in any real gain of a classical calculus to a contradiction, this sentence could not be added as an additional axiom. The connected forms of logic that wants to be the front -formal intuition, which expresses the sentence, just, by awarding him as a theorem, therefore, must reject other classical logical theorems. Thus, while intuitionistic, minimal and relevant logic, the provable formulas are a proper subset of the classically provable formulas, respectively, whereas the ratio of konnexer and classical logic is that in both formulas are provable, which are not considered in the other logic.

Multi-valued and fuzzy logic

Transverse to this are the many-valued logics in which do not apply the principle of bivalence and often the Aristotelian principle of excluded middle, including the trivalent and the unendlichwertige logic by Jan Łukasiewicz ( "Warsaw school "). Numerous applications in control engineering is the unendlichwertige fuzzy logic, while about the logic of finitely Gotthard Günther ( " Günther - logic " ) to problems of self-fulfilling predictions has been applied in sociology.

Nonmonotonic logics

This is called a logical system monotonic if there are valid arguments also remains valid if one adds additional assumptions: What was once proved remains always valid, so even if one has at a later date on new information in a monotonic logic. Many logical systems have this monotonicity property, including all classical logics such as the propositional and predicate logic.

In everyday and scientific Close but preliminary conclusions are often drawn that are not valid in the strict logical sense and the need to be revised under certain circumstances at a later date. For example, could be made the statements " Tux is a bird. " And " Most birds can fly. " Provisionally conclude that Tux can fly. Now, if we but the additional information " Tux is a penguin. " Get, then we must correct this conclusion, because penguins are not capable of flying birds. This type of closure map, non-monotonic logics have been developed: you renounce the monotonicity property, that is a valid argument can be invalidated by adding additional premises.

This is of course only possible if another consequence operation is used than in a classical logic. A common approach is to use so-called defaults. A default conclusion is valid if not from a classic logical conclusion is a contradiction to him.

The conclusion from the given example would look like this: " Tux is a bird. " Remains the requirement ( prerequisite ). We now combine this with a so-called justification ( justification ): " birds can normally fly. " From this reasoning, we conclude that Tux can fly, as long as there is nothing. The consequence is thus " Tux can fly. " We now obtain the information " Tux is a penguin. " And " Penguins can not fly. ", So there is a contradiction. We have arrived at the result that Tux can fly over the default circuit. With a classic logical conclusion way but we were able to demonstrate that Tux can not fly. In this case, the default is revised and used the consequence of the classical logical inference. This - roughly described here - process is also known as Reitersche default logic.