Concept
Dynamic logic (modal logic)
Overview
In logic, philosophy, and theoretical computer science, dynamic logic is an extension of modal logic capable of encoding properties of computer programs.
A simple example of a statement in dynamic logic is
The ground is dry
→
[
It rains
]
The ground is wet
,
{\displaystyle {\text{The ground is dry}}\to [{\text{It rains}}]{\text{The ground is wet}},}
which states that if the ground is currently dry and it rains, then afterwards the ground will be wet.
The syntax of dynamic logic contains a language of propositions (like "the ground is dry") and a language of actions (like "it rains"). The core modal constructs are
[
a
]
p
{\displaystyle [a]p}
, which states that after performing action a the proposition p should hold, and
⟨
a
⟩
p
{\displaystyle \langle a\rangle p}
, which states that after performing action a it is possible that p holds.
The action language supports operations
a
;
b
{\displaystyle a\mathbin {;} b}
(doing one action followed by another),
a
∪
b
{\displaystyle a\cup b}
(doing one action or another), and iteration
a
∗
{\displaystyle a{*}}
(doing one action zero or more times). The proposition language supports Boolean operations (and, or, and not). The action logic is expressive enough to encode programs. For an arbitrary program
P
{\displaystyle P}
, precondition
φ
{\displaystyle \varphi }
, and postcondition
φ
′
{\displaystyle \varphi '}
, the dynamic logic statement
φ
→
[
P
]
φ
′
{\displaystyle \varphi \to [P]\varphi '}
encodes the correctness of the program, making dynamic logic more general than Hoare logic.
Beyond its use in formal verification of programs, dynamic logic has been applied to describe complex behaviors arising in linguistics, philosophy, AI, and other fields.
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.
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 Dynamic logic is.
Little time is left to speak of the account of the nature of truth given by the experimental and functional type of logic. This is less to be regretted because this account is completely a corollary from the nature of thinking and ideas. If the view held as to the latter is understood, the conception of truth follows as a matter of course. If it be not understood, any attempt to present the theory of truth is bound…
John Dewey, Reconstruction in Philosophy
States what Dynamic logic is.
And this is the true business of logic: to show that those thoughts, which as usually employed merely float before consciousness neither understood nor demonstrated, are really grades in the self-determination of thought. It is by this means that they are understood and demonstrated. In common life, and it is the same in the finite sciences, this reflective form is often employed as a key to the secret of the real…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Dynamic logic is.
Related ideas
- First-order logic
- Artificial intelligence
- Philosophy
- Counterexample
- If and only if
- Propositional variable
- Logical conjunction
- Propositional logic
- Mathematical induction
- Modus ponens
- Natural number
- Linguistics
- Formal verification
- Temporal logic
- Theoretical computer science
- Modal logic
- Modal operator
- Logic
- Boolean algebra
- Rigid designator
- Computational complexity
- Possible world
- Kripke semantics
Place in the tradition
Part of Description logic.
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