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Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ
Infinitary logic allows infinitely long sentences; for example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables
First-order modal logic has extra
with meanings which can be characterised informally as, for example "it is necessary that φ" and "it is possible that φ“
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