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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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Existence property for second-order propositional logic
The answer to your question, strictly speaking, is negative. For instance take the sequent $P \vee Q \vdash \exists X ((X \wedge P) \vee (\neg X \wedge Q))$, which is (intuitionistically) derivable si …