Law of excluded middle

The law of excluded middle ( tertium non datur Latin, literally, " a third is not given " or " a third thing does not exist"; engl Law of the Excluded Middle. ) Or principle of standing between two contradictory opposites Excluded Middle (lat. principium exclusi tertii immersive medii inter duo contradictoria ) is a logical principle or axiom which states that for any statement at least the statement itself or its opposite must be true: a third option, ie, that only applies somewhat Medium, neither the statement is, nor its opposite, but somewhere in between, it can not give.

This principle is to be distinguished from the principle of bivalence, which states that every statement is either true or false. He should not be confused with the principle of contradiction, which states that a statement and its opposite can not be considered simultaneously ( the law of excluded middle in itself is neutral to this statement also; are, however, in addition, the inference rules of classical logic to available, it follows a set of trivial from the other and vice versa).

Logic

In modern formal logic, the law of excluded middle applies to a sentence and its negation sentence. He states that for any proposition P is true, the statement (P or not P). The most common logical system in which applies the law of excluded middle, is the classical logic.

For example, if P is the statement

Designated, shall be considered inclusive disjunction

However, the law of excluded middle does not tell us whether P itself is or not.

The law of excluded middle is not limited to bivalent logic, there are also some multi-valued logics in which it applies. Conversely, there are also two - and many-valued logics in which it does not apply. Some rule of inference calculi, in which he does not hold, replace the rule (see truth value ).

Interpretation

If one interprets the law of excluded middle in classical logic ( with a bivalent Boolean algebra ), then it is a tautology, ie independent of the choice of and independent of its internal structure true.

In logical systems, in which the atomic sentences and the connectives ( connectives ) can be interpreted differently, this is not necessarily the case. For example, the intuitionistic logic interprets the statement as the existence of a proof or a refutation of the statement G. Since many concrete statements (eg, the continuum hypothesis ) are neither provable nor refutable, is not generally valid in this interpretation tertium non datur. According to calculi for logical systems are designed such that the sentence does not apply there.

Demarcation

Whether within a specific logical system of the law of excluded middle, can be studied purely formal basis of the underlying calculus. Therefore, no disagreements are possible in this regard.

Clearly distinguishable from the purely logical questions are philosophical questions, such as the metaphysical question of which kind of logical system (with or without tertium non datur ) can describe reality; or the pragmatic question of what sort of logical system, such as the mathematics to be easy to push. With regard to these questions were, among others, in the basic dispute lively discussions in progress.

Philosophy

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

As an ontological principle, it means that there is no third between being and non-being.

The first well- known objection to the generality of the principle of excluded middle gave Aristotle De interpretatione, chapters 7-9. His argument is that for statements about the future as the sentence "Tomorrow there will be a sea battle " the principle of the excluded middle does not apply, because the shape of the future is still open and a statement about Future therefore can be neither true nor false.

Rejection

Who the sentence ( or principle ) of the excluded middle rejects or criticizes, does not purport necessary that there is a third thing, but he rejects logical conclusions, with which one out of the logic and not from the facts of the particular scientific subject for some holds true or existent. Such criticism has been expressed very polemic in the early 20th century. The mathematician, logician and philosopher Luitzen Egbertus January Brouwer criticized particularly from the law of excluded middle derivable statements of the form:

Brouwer put on intuitionistic logic calculi in which the law of excluded middle is not derivable. Relevant is a rejection of the sentence with respect to mathematics, statements concerning infinite and outside mathematics with respect to future or past events, assuming truth as established knowledge (see also Methodological constructivism). An example is the statement: "Either the world has always been there or she eventually started." Who needs the law of excluded middle, to be true according to this understanding of truth.