Editing First-order logic (section)

===Non-classical and modal logics===

{{main|Non-classical logic}} 

*''[[intuitionistic logic|Intuitionistic first-order logic]]'' uses intuitionistic rather than classical reasoning; for example, ¬¬φ need not be equivalent to φ and ¬ ∀x.φ is in general not equivalent to ∃ x.¬φ .
*First-order ''[[modal logic]]'' allows one to describe other possible worlds as well as this contingently true world which we inhabit. In some versions, the set of possible worlds varies depending on which possible world one inhabits. Modal logic has extra ''modal operators'' with meanings which can be characterized informally as, for example "it is necessary that φ" (true in all possible worlds) and "it is possible that φ" (true in some possible world). With standard first-order logic we have a single domain, and each predicate is assigned one extension. With first-order modal logic we have a ''domain function'' that assigns each possible world its own domain, so that each predicate gets an extension only relative to these possible worlds. This allows us to model cases where, for example, Alex is a philosopher, but might have been a mathematician, and might not have existed at all. In the first possible world ''P''(''a'') is true, in the second ''P''(''a'') is false, and in the third possible world there is no ''a'' in the domain at all.
*''[[t-norm fuzzy logics|First-order fuzzy logics]]'' are first-order extensions of propositional fuzzy logics rather than classical [[propositional calculus]].