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At the turn of the 17th century, the German mathematician Gottfried Wilhelm von Leibniz postulated that Every judgment is either true or false. This was what later became known as the Law of Excluded Middle and in some of its many versions, the word “judgment” was replaced by the term “proposition”. We will learn propositional logic later on as one of the most important formal systems because it is a subset of almost any formal logical calculus and thus a kind of a “common logic” intrinsic to these logical systems.

But what is significant about this Leibniz’s simple postulate?

The significance of it is a little bit subtle: It is Leibniz’s assertion that the truth can only have two values – true and false, but nothing in-between. This is what is known as the law of bivalence.

Now, why is this assertion so significant? Because it is the foundation of almost all formal systems and, in fact, almost the entire mathematics. Every modern mathematical proof implies the law of bivalence, even proofs of theories involving multi-valent logics1. Therefore, we introduce the law of bivalence as an axiom, based on which we will define the semantics of formal languages.

1 e.g. Fuzzy Logic, which allows vague truth values, and considers the values “true” and “false” only as extremes of a whole scale of possible values.

Axiom: Bivalence of Truth

We postulate that the concept of truth in a given formal system allows only two values – “true” and “false”.

| | | | | created: 2014-06-21 19:09:48 | modified: 2018-02-02 23:34:15 | by: bookofproofs | references: [593]

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Bibliography (further reading)

[593] Cryan Dan, Shatil Sharron, Mayblin Bill: “Logic. A Graphic Guide”, Icon Books Ltd., London, 2001

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