Definition:Modal Logic

Definition

Modal logic is a branch of logic in which truth values are more complex than being merely true or false, and which distinguishes between different "modes" of truth.

There are two operators in classical modal logic, defined for some proposition $P$ dependent on some world $w$:

$(1): \quad$ Necessity, represented by $\nec$, defined by:

$\nec P : \iff \forall w: \map P w$

$(2): \quad$ Possibility, represented by $\pos$, defined by:

$\pos P: \iff \exists w: \map P w$

Modal logic may also have other operators, including:

Temporal logic, which uses several operators including present and future

Epistemic logic, which uses operators "an individual knows that" and "for all an individual knows it might be true that"

Multi-Modal logic, which uses more than two unary modal operators.

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Examples

$\text {S5}$ System

The $\text {S5}$ system of modal logic contains the axioms and rules of inference of propositional logic together with the axioms:

\((1)\)	$:$				\(\ds \nec A \implies A \)
\((2)\)	$:$				\(\ds \map \nec {A \implies B} \implies \paren {\nec A \implies \nec B} \)
\((3)\)	$:$				\(\ds \pos A \implies \nec \pos A \)

In order to interpret $\text {S5}$, we need to state the condition under which a WFF of the form $\nec A$ is assigned the truth value true.

This is done via:

$\nec A$ is true if and only if $A$ is true in all possible worlds.

This leads to:

$\pos A$ is true if and only if $A$ is true in some possible world.

Also see

• Results about modal logic can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): modal logic
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): modal logic