Concept
Non-normal modal logic
Overview
A non-normal modal logic is a variant of modal logic that deviates from the basic principles of normal modal logics.
Normal modal logics adhere to the distributivity axiom (
◻
(
p
→
q
)
→
(
◻
p
→
◻
q
)
{\displaystyle \Box (p\to q)\to (\Box p\to \Box q)}
) and the necessitation principle which states that "a tautology must be necessarily true" (
⊢
A
{\displaystyle \vdash A}
entails
⊢
◻
A
{\displaystyle \vdash \Box A}
). On the other hand, non-normal modal logics do not always have such requirements. The minimal variant of non-normal modal logics is logic E, which contains the congruence rule in its Hilbert calculus or the E rule in its sequent calculus upon the corresponding proof systems for classical propositional logic. Additional axioms, namely axioms M, C and N, can be added to form stronger logic systems. With all three axioms added to logic E, a logic system equivalent to normal modal logic K is obtained.
Whilst Kripke semantics is the most common formal semantics for normal modal logics (e.g., logic K), non-normal modal logics are often interpreted with neighbourhood semantics.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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This is nothing else than the necessity of abstract logic, which is the presupposition involved in the very fact of interrelated existence as disclosed in each immediate occasion of experience. The key to the pattern means this fact:—that from a select set of those general conditions, exemplified in any one and the same occasion, a pattern involving an infinite variety of other such conditions, also exemplified in…
Alfred North Whitehead, Science and the modern world
States what Non-normal modal logic is.
To the number of modes of relation which the understanding cogitates by means of the categories, the number of pure rational conceptions will correspond. We must therefore seek for, first, an unconditioned of the categorical synthesis in a subject; secondly, of the hypothetical synthesis of the members of a series; thirdly, of the disjunctive synthesis of parts in a system.
Immanuel Kant, The Critique of Pure Reason
States what Non-normal modal logic is.
It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions--or axioms as they are called--which this relation satisfies. It has then been proved that there are alternative relations which satisfy these conditions equally well and that there…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Non-normal modal logic is.
Related ideas
- Formal grammar
- Power set
- Rule of inference
- Sequent calculus
- Modal logic
- Normal modal logic
- Kripke semantics
Place in the tradition
Part of Modal logic.