Sentence of the excluded third party

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The sentence of the excluded third ( Latin tertium non datur literally “a third is not given” or “a third does not exist”; English Law of the Excluded Middle , LEM ) or the principle of the excluded middle between two contradictory opposites (Latin principium exclusi tertii sive medii inter duo contradictoria ) is a logical basic principle or axiom that says that for any statement only the statement itself or its opposite can apply: A third possibility, i.e. that only something middle applies that is neither the statement , nor its opposite, but somewhere in between, cannot exist. In modern formal logic, the principle of the excluded third means that the statement (" or not ") applies to any statement .

This principle has to be distinguished from the principle of two-valence , which states that every statement is either true or false, i.e. This means that semantically every formula is assigned exactly one of two truth values ​​(in contrast to multi-valued logic ).

The principle of excluded third party must not be confused with the principle of contradiction , which states that a statement and its opposite cannot apply at the same time, i.e. This means that the statement ("not simultaneously and not ") applies to any statement . The principle of the excluded third party in itself is neutral to this assertion. If, however, the inference rules of classical logic and in particular the law of double negation are available, one sentence follows trivially from the other and vice versa.

logic

In modern formal logic the proposition of the excluded third refers to a statement and its proposition negation. It says that for any statement P the statement ( P or not P ) holds. The best-known logical system in which the principle of the excluded third applies is classical logic .

If z. B. P the statement

Hans is blonde.

then the disjunction holds

Hans is blond or it is not the case that Hans is blond.

However, the theorem of the excluded third party says nothing about whether P itself is true or not.

The excluded third theorem is not limited to two-valued logics; there are also some multivalued logics in which it holds. Conversely, however, there are also two-valued and multi-valued logics in which it does not apply. Some final rule calculi in which it does not apply replace the rule with the weaker one ( ex falso quodlibet ).

interpretation

If the theorem of the excluded third party is interpreted within classical logic (with a two-valued Boolean algebra ), then it is a tautology , i.e. it is true regardless of the choice of and regardless of its internal structure.

In logical systems in which the atomic sets and the logical connectives (connectives) are interpreted differently, this is not necessarily the case. For example, interprets intuitionistic logic the statement as the existence of a proof or disproof of the statement G . Since a large number of concrete statements (e.g. the continuum hypothesis ) can neither be proven nor refuted, Tertium non datur does not generally apply to this interpretation. Correspondingly, calculi for such logical systems are constructed in such a way that the theorem does not apply there.

Conversely, one can additionally presuppose the theorem of the excluded third party in such logics if necessary. Such a logic is thus more general and allows more interpretations than the classical one. One of these interpretations is the Curry-Howard isomorphism , which has also proven to be practically viable , especially in the area of machine-aided proof .

Demarcation

Whether the principle of the excluded third party applies within a certain logical system can be examined purely formally on the basis of the underlying calculus.

A clear distinction must be made between these purely formal questions and philosophical questions, e.g. B. the metaphysical question of what kind of logical system (with or without tertium non datur) reality can be described; or the pragmatic question of what kind of logical system can be used to advance mathematics as easily as possible. With regard to these questions, lively discussions were held , among other things in the fundamental dispute.

philosophy

The principle of the excluded third has a long tradition in the history of philosophy; In traditional logic it is considered the generally recognized third law of thought and is seen partly as an ontological , partly as an epistemological principle.

As an ontological principle, it means that there is no third between being and not-being.

The first well-known objection to the general validity of the principle of the excluded third was provided by Aristotle De interpretatione , chapters 7–9. His argument is that the principle of the excluded third party does not apply to statements about the future such as the sentence "Tomorrow there will be a sea battle" because the course of the future is still open and a statement about the future can therefore neither be true nor false.

Rejection

Anyone who rejects or criticizes the proposition (or the principle) of the excluded third party does not necessarily claim that there is something third, but rejects logical inferences that can be derived from logic and not from the facts about the respective scientific object true or existent. Such a criticism was expressed very polemically at the beginning of the 20th century. The mathematician, logician and philosopher Luitzen Egbertus Jan Brouwer particularly criticized statements of the form that can be derived from the sentence about the excluded third:

If for no x: not A (x), then for all x: A (x)

Brouwer set up intuitionist logic calculi in which the theorem of the excluded third cannot be derived. A rejection of the proposition regarding mathematics becomes relevant for statements about infinity and outside of mathematics regarding future or past events, if one assumes truth as certain knowledge (see also methodical constructivism ) . An example is the assertion: “Either the world has always been there or it started at some point”, which needs the sentence of the excluded third party to be true according to this understanding of truth.

See also

literature

  • Luitzen Egbertus Jan Brouwer: Justification of the set theory independent of the logical proposition of the excluded third party. First part, general set theory . In Koninklijke Akademie van Wetenschappen te Amsterdam, Verhandelingen 1e sectie, deel XII, no 5 (1918), 1:43

Individual evidence

  1. See also Friedrich Kirchner , Dictionary of Basic Philosophical Concepts (1907) : Principium exclusi tertii seu medii inter duo contradictoria .
  2. See Thomas Zoglauer, Introduction to formal logic for philosophers (1999), p. 25.