# Introduction to Philosophy/Logic/Modal Logic

Introduction to Philosophy > Logic > Modal Logic

## Modality[edit]

Still holding the copy of

The Grasshopper Lies Heavy, Robert said, 'What sort of alternate present does this book describe?'

Betty, after a moment, said, 'One in which Germany and Japan lost the war.'

They were all silent.

(Philip K. Dick,The Man in the High Castle)

This is the basic premise of modal logic: some things could have been otherwise.

In our everyday reasoning, we distinguish between *necessary* truths and statements that just happen to be true but could have been false. We say that the latter are *contingently* true, or that their negation is *possible*. The concepts of *necessary*, *contingent*, *possible* and *impossible* are very closely related: something is *necessary* if it is *impossible* for it to be false, and something is *possible* if its falsehood is not *necessary*. *Contingency* is the negation of *necessity*, and *impossibility* the negation of *possibility*.

Modal logic attempts to include the notions of modality (*necessary*, *contingent*, *possible* and *impossible*, among others) into the structure of *classical logic* (propositional logic and predicate calculus) and is therefore an extension of classical logic.

## Modal propositional calculus[edit]

### Syntax[edit]

Modal logic introduces three new symbols into classical logic: (necessary), (possible) and (if, then). Formulae are constructed in the usual way, with the following rules added *before the closure clause* (see the rules for Well-Formed Formulae in propositional calculus):

- If is a well-formed formula, then and are also well-formed formulae.
- If and are well-formed formulae, then is also a well formed formula.

Actually, one can define the modal propositional calculus axiomatically using just or . For example, if we wish to define and in terms of , can be defined simply as a abbreviated notation for and as another way of writing .

### Rules of inference[edit]

## Bibliography[edit]

**Hughes, G. E.**, &**Cresswell, M. J.**:*An Introduction to Modal Logic*, Methuen and Co Ltd, London (1972).**Hughes, G. E.**, &**Cresswell, M. J.**:*A Companion to Modal Logic*, Methuen and Co Ltd, London (1984).**Smullyan, R.**:*Forever Undecided*(1987). Part IX: Possible Worlds.