Concept
Modal algebra
Overview
In algebra and logic, a modal algebra is a structure
⟨
A
,
∧
,
∨
,
−
,
0
,
1
,
◻
⟩
{\displaystyle \langle A,\land ,\lor ,-,0,1,\Box \rangle }
such that
⟨
A
,
∧
,
∨
,
−
,
0
,
1
⟩
{\displaystyle \langle A,\land ,\lor ,-,0,1\rangle }
is a Boolean algebra,
◻
{\displaystyle \Box }
is a unary operation on A satisfying
◻
1
=
1
{\displaystyle \Box 1=1}
and
◻
(
x
∧
y
)
=
◻
x
∧
◻
y
{\displaystyle \Box (x\land y)=\Box x\land \Box y}
for all x, y in A.
Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics.
Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame.
A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying
◻
(
−
◻
x
∨
x
)
=
◻
x
{\displaystyle \Box (-\Box x\lor x)=\Box x}
. Magari algebras correspond to provability logic and are named after Roberto Magari.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
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 Modal algebra is.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Modal algebra is.
The transcendental conception of reason is therefore nothing else than the conception of the totality of the conditions of a given conditioned. Now as the unconditioned alone renders possible totality of conditions, and, conversely, the totality of conditions is itself always unconditioned; a pure rational conception in general can be defined and explained by means of the conception of the unconditioned, in so far…
Immanuel Kant, The Critique of Pure Reason
States what Modal algebra is.
Related ideas
- Isomorphism
- Propositional logic
- Modal logic
- Boolean algebra (structure)
- Logic
- Classical logic
- Normal modal logic
- Abstract algebraic logic
Place in the tradition
Part of Outline of logic.