Concept

Leibniz operator

Overview

In abstract algebraic logic, a branch of mathematical logic, the Leibniz operator is a tool used to classify deductive systems, which have a precise technical definition and capture a large number of logics. The Leibniz operator was introduced by Wim Blok and Don Pigozzi, two of the founders of the field, as a means to abstract the well-known Lindenbaum–Tarski process, that leads to the association of Boolean algebras to classical propositional calculus, and make it applicable to as wide a variety of sentential logics as possible. It is an operator that assigns to a given theory of a given sentential logic, perceived as a term algebra with a consequence operation on its universe, the largest congruence on the algebra that is compatible with the theory.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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1. Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.

   Alfred North Whitehead, Science and the modern world

   States what Leibniz operator is.

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2. From this condensed sketch, taken from Leibniz himself, the main features of his ethical doctrine clearly appear. When we were studying freedom we saw that it was not so much a starting-point of the will as its goal and ideal. We saw also that true freedom is dependent upon knowledge, upon recognition of the eternal and universal.

   John Dewey, Leibniz's New Essays Concerning the Human Understanding

   States what Leibniz operator is.

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3. It is this parallelism that we find both in Leibniz and in Spinoza--in different forms, it is true, because of the unequal importance which they attach to extension. With Spinoza, the two terms Thought and Extension are placed, in principle at least, in the same rank.

   Henri Bergson, Creative Evolution

   States what Leibniz operator is.

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Related ideas

• If and only if
• Propositional logic
• Mathematical logic
• Boolean algebra (structure)
• Formal system
• Quantifier (logic)
• Signature (logic)
• Algebraic logic
• Congruence relation
• Abstract algebraic logic

Place in the tradition

Part of Non-classical logic.

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