Bacchus at 21:38, 20 January 2023

2023-01-20T21:38:34Z

@@ Line 18: / Line 18: @@
 More confusing issues come from cases where one modal operator ranges over another - does Lp imply LLp, for example? (is a necessary truth necessarily necessary?)
 ==External links==
@@ Line 23: / Line 35: @@
 * [http://plato.stanford.edu/entries/logic-modal/ Stanford Encyclopedia of Philosophy entry]
 * [http://www-formal.stanford.edu/jmc/mcchay69/node22.html A discussion of modal logic] by John McCarthy

Bacchus: Created page with "'''Modal logic''' is a form of logic which distinguishes between '''necessary truths''' and '''contingent truths'''. A truth is ''necessary'' if it cannot be avoided, such as 2 + 2 = 4; by contrast, a ''contingent'' truth just happens to be the case, for instance "more than half of the earth is covered by water". In the most common interpretation of modal logic, one considers "all possible worlds". If a statement is true in all possible worlds, then it is a necessar..."

2023-01-20T21:36:00Z

Created page with "'''Modal logic''' is a form of logic which distinguishes between '''necessary truths''' and '''contingent truths'''. A truth is ''necessary'' if it cannot be avoided, such as 2 + 2 = 4; by contrast, a ''contingent'' truth just happens to be the case, for instance "more than half of the earth is covered by water". In the most common interpretation of modal logic, one considers "all possible worlds". If a statement is true in all possible worlds, then it is a necessar..."

New page

'''Modal logic''' is a form of [[logic]] which distinguishes between '''necessary truths''' and '''contingent truths'''.

A truth is ''necessary'' if it cannot be avoided, such as 2 + 2 = 4; by contrast, a ''contingent'' truth just happens to be the case, for instance "more than half of the earth is covered by water". In the most common interpretation of modal logic, one considers "all possible worlds". If a statement is true in all possible worlds, then it is a necessary truth. If a statement happens to be true in our world, but is not true in all other worlds, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a ''possible'' truth.

Modal logic adds to the ''well formed formulae'' of [[propositional logic]] operators for necessity (L) and possibility (M). The two are definable in terms of each other:

* Lp (necessarily p) has the same meaning as -M-p (not possible that not-p)
* Mp (possibly p) has the same meaning as -L-p (not necessarily not-p)

Precisely what axioms must be added to propositional logic to create a usable system of modal logic has been the subject of much debate. One weak system, named K after Saul Kripke, adds only the following:

* Necessitation Rule: If p is a theorem of K, then so is Lp.
* Distribution Axiom: If L(p → q) then (Lp → Lq)

These rules lack an axiom to go from the necessity of p to p actually being the case, and therefore are usually supplimented with:

* Lp → p (If it's necessary that p, then p is the case)

More confusing issues come from cases where one modal operator ranges over another - does Lp imply LLp, for example? (is a necessary truth necessarily necessary?)

==External links==

* [http://plato.stanford.edu/entries/logic-modal/ Stanford Encyclopedia of Philosophy entry]
* [http://www-formal.stanford.edu/jmc/mcchay69/node22.html A discussion of modal logic] by John McCarthy

Modal Logic - Revision history

Bacchus at 21:38, 20 January 2023