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Intuitionistic firstorder 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

Firstorder modal logic has extra modal operators 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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