# Principle of duality

The principle of bivalence , also known as the bivalence principle , is the property of a logic that semantically every formula is assigned exactly one of two truth values . Often these truth values ​​are referred to as true and false .

Logics for which the principle of two-valued is fulfilled are also called two-valued logics. If the principle of two- valued is not fulfilled, one speaks of multi-valued logic .

The principle of two-valued is to be distinguished from the sentence of the excluded third , which is also valid within several multivalued logics , which states that P ∨ ¬P can be syntactically derived within the logical system or its calculus .

## Clarification

If you set up formal semantics for a calculus , then you use a function for the assignment of truth values ​​to formulas , which is called the evaluation function (also called denotation function or truth value function ). The symbol is often used for the evaluation function; the formula to be evaluated is written between the square brackets. If one denotes the set of well-formed formulas of the calculus , then the principle says: ${\ displaystyle [[\ cdot]]}$${\ displaystyle M}$

${\ displaystyle [[\ cdot]]}$is a function in the mathematical sense that is (at least) fully defined and that delivers exactly one of the truth values ​​“true” or “false” for every well-formed formula.${\ displaystyle M}$

The bivalence principle does not imply that the quantity or that the evaluation function can be determined effectively in any way. This question is postponed to the calculation under consideration. ${\ displaystyle M}$

## Discussion of the principle

Since the evaluation function does not have to be “actually determinable”, there can also be statements in a logic that fulfills the bivalence principle, the truth of which is unknown (“at the current point in time” or even forever). A famous example in discussion that this can also be the case in mathematics is the so-called Goldbach hypothesis that every even number greater than 2 can be written as the sum of two prime numbers. It is argued here: either the conjecture holds for the “real natural numbers” or it does not hold; but perhaps it must remain unclear which of the two is the case.

Since the evaluation function provides a truth value for all statements, the “proposition of the excluded third party” simply follows from the bivalence principle.

The bivalence principle is not a normative principle, i.e. not a requirement that logical systems must be two-valued , but rather a descriptive semantic property of logical systems. Some logical systems have this property, e.g. B. the classical logic : They are two-valued. Other systems do not have this property: they are multivalued.

The bivalence principle is related to other issues, especially metaphysical or linguistic issues. An example would be the metaphysical question of whether reality can be adequately described by two-valued logic, i.e. whether a metaphysical bivalence principle applies - whether there is an absolute truth. Such questions are dealt with in the philosophy of science and philosophy of language . The correspondence theory of truth starts from an objective, absolute truth and affirms such a metaphysical idea , while the coherence theory understands truth as a subjective social construction that only exists relative to the social position of the viewer.

In the philosophy of mathematics, the principle of bivalence refers in particular to the question of whether mathematical sentences are only strings of characters that are transformed, or whether they make statements about objects in a mathematical world, like the sentence "today it is raining" according to 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 which, according to his doctrine of ideas, belongs to the world of ideas ( intelligible world ), which exists independently of the thinking subject, but which is basically recognizable in a purely spiritual way. This is discussed , among other things, in Plato's allegory of the cave . This view is particularly rejected in intuitionism , where the truth and falsity of a sentence is reduced to the subjective experience of evidence in its deductive construction. Karl Popper tried in his pluralistic ontology ( three worlds doctrine ) to combine both perspectives by recognizing that mathematical worlds are created by humans, but nevertheless took the position that the existence of the world and especially its properties are objective and is independent of humans. Mathematical theories thus belong in Popper's World 3, the world of the objective contents of human culture.

## Single receipts

1. K.Wuchterl, Methods of Contemporary Philosophy , p. 53.
2. Karl R. Popper: Collected Works, Volume 12, Knowledge and the Body-Soul Problem, Tübingen, Mohr Siebeck (2012). The book contains in a new translation Knowledge and the Body-Mind Problem (1994) and the Popp part from Karl R. Popper, John C. Eccles: Das Ich und seineirn (1977), editorial remarks and an afterword by the editor with an overview of approx. 40 further works on the theory of three worlds.

## Individual evidence

• Walter Gellert, Herbert Kästner , Siegfried Neuber (Hrsg.): Fachlexikon ABC Mathematik . 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 Philosophy of Science and Analytical Philosophy , Springer-Verlag, Berlin, Heidelberg, New York, Tokyo 1984, ISBN 3-540-12210-9 , ISBN 0-387-12210-9 . Especially p 51 ff.
• JM Bochenski: Formal Logic , Freiburg / Munich 1970. Chapter 43 on the history of the formulation of this principle
• K. Wuchterl: Methods of Contemporary Philosophy , Bern and Stuttgart 1977, (UTB Taschenbücher 646), ISBN 3-258-02606-8