Principle of bivalence
The principle of bivalence, also called bivalence, is the property of a logic that is assigned to exactly one of two truth values semantically each formula. Often these truth values are called true and false.
Logic for which the principle of the two- value is met, is also called divalent logics. Is the principle of bivalence is not fulfilled, one speaks of multi-valued logic.
The principle of bivalence is to be distinguished from the valid even within multiple -valued logics law of excluded middle, which states that P ∨ ¬ P can be derived syntactically within the logical system and its calculus.
Clarification
If one sets up a formal semantics for a calculus, then you used for the assignment of truth values to formulas a function, the evaluation function is called (also Denotationsfunktion or truth-values function). For the evaluation function often the sign is used; to be evaluated formula is a top priority between the square brackets. If we denote the set of well-formed formulas of the calculus with, then states the principle:
In bivalence is neither implies that the amount or that the evaluation function is effectively determined in any way. This question is moved to the considered calculus.
Discussion of the principle
Since the evaluation function does not " actually determined " must be, it can also be in a logic that satisfies the principle of bivalence, give statements whose truth value (" for the instantaneous time " or even forever ) is unknown. A famous example of this discussion that this may be the case even within mathematics, is the so-called Goldbach's conjecture that every even number greater than 2 can be written as a sum of two primes. It is argued here: either there is a presumption for the "real natural numbers " or it does not apply; but perhaps must remain unexplained, which is the case of both.
Since the evaluation function for all statements provides a truth value that follows the " law of excluded middle " simply from the principle of bivalence.
The bivalence is not a normative principle, thus no requirement that logical systems must be bivalent, but descriptive semantic property of logical systems. Some logical systems have this property, eg classical logic: they are divalent. Other systems do not have this property: they are multi-valued.
- Three-valued logic, fuzzy logic, intuitionistic logic, multivalued logic
The bivalence about other issues in connection, especially with metaphysical or linguistic issues. An example would be the metaphysical question of whether reality can be adequately described by two-valued logic, ie whether a metaphysical principle of bivalence is true - if there is an absolute truth. Such issues are addressed in the philosophy of science and philosophy of language. The correspondence theory of truth is based on an objective, absolute truth and affirms such a metaphysical idea, while the coherence theory sees truth as a subjective social construct that exists only in relation to the social position of the observer.
In the philosophy of mathematics, the bivalence applies in particular to the question of whether mathematical propositions are just strings that are converted, or whether they make statements about objects in a mathematical world as the sentence " Today it is raining " after the realism of the common sense makes a statement about the real world. Plato was of the opinion that there is an objective ideal mathematical world that belongs to his doctrine of ideas to the world of ideas ( the intelligible world ) which exists independently of the thinking subject, but basically this on a purely intellectual way is recognizable. This will be discussed among other things in Plato's cave allegory. This view is especially rejected in intuitionism, where the truth or falsity of a sentence is reduced to the subjective experience of evidence at his deductive construction. Karl Popper tried in his pluralistic ontology ( three-world theory ), to combine both perspectives by recognizing though that mathematical worlds are created by man, but nevertheless took the view that the existence of the world and in particular the quality objective and is independent of the people. Mathematical theories thus belong in Popper's World 3, the world of objective contents of human culture.
Single Documents
Swell
- W. Gellert, H.Kästner, S.Neuber (ed.): Encyclopedia ABC mathematics. Thun and Frankfurt 1978, ISBN 3-87144-336-0. Article propositional calculus
- W. Stegmüller, MVvKibéd: structural types of logic, Volume III by W. Stegmüller, problems and results of the theory of science and analytic philosophy, Springer- Verlag, Berlin, Heidelberg, New York, Tokyo 1984, ISBN 3-540-12210-9 ( born ), ISBN 0-387-12210-9 ( hard cover ). Especially S 51 ff
- J. M. Bochenski: Formal Logic, Freiburg / Munich 1970 Chapter 43 the history of the formulation of this principle.
- K. Wuchterl: methods of contemporary philosophy, Bern and Stuttgart 1977 ( UTB paperbacks 646), ISBN 3-258-02606-8