Modal Logic

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?)

Other logical systems

• Classical logic
• Intermediate logic
• Functional logic
• Intuitionistic logic
• Minimal logic
• Relevant logic
• Affine logic
• Linear logic
• Ordered logic
• Dialetheism

External links

• Stanford Encyclopedia of Philosophy entry
• A discussion of modal logic by John McCarthy