Modal logic
Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional logic and predicate logic to include operators expressing modality.
(TO DO: fix logic template.)
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Description
Modals -- words that express modalities -- qualify a statement.
For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal.
Traditional alethic modalities
The traditional alethic modalities, or modalities of truth, include:
- Possibility ("Possibly, p", "It is possible that p")
- Necessity ("Necessarily, p", "It is necessary that p")
- Impossibility ("Impossibly, p", "It is impossible that p").
Other modalities
Other modalities that have been formalized in modal logic include:
- Temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p")
- Deontic modalities (notably, "It is obligatory that p", and "It is permissible that p")
- Epistemic modalities, or modalities of knowledge ("It is known that p")
- Doxastic modalities, or modalities of belief ("It is believed that p")
Modal operators
A formal modal logic represents modalities using modal operators.
For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility.
In a modal logic this is represented as an operator, Possibly, attached to the sentence "It will rain today".
Representation
The basic unary (1-place) modal operators are usually written □ for Necessarily and ◇ for Possibly. In a classical modal logic, each can be expressed by the other with negation:
\Diamond P \leftrightarrow \lnot \Box \lnot P; \Box P \leftrightarrow \lnot \Diamond \lnot P.
Thus it is possible that it will rain today if and only if it is not necessary that it will not rain today; and it is necessary that it will rain today if and only if it is not possible that it will not rain today.
Alternative symbols
Alternative symbols used for the modal operators are "L" for Necessarily and "M" for Possibly.
See also
External links
- Modal logic @ Wikipedia
