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The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic.

quine --- logic --- ontology --- multiple conclusion rule --- disjunction property --- metadisjunction --- axiomatizations of arithmetic of natural and integers numbers --- second-order theories --- Peano’s axioms --- Wilkosz’s axioms --- axioms of integer arithmetic modeled on Peano and Wilkosz axioms --- equivalent axiomatizations --- metalogic --- categoricity --- independence --- consistency --- logic of typical and atypical instances (LTA) --- logic of determination of objects (LDO) --- quasi topology structure (QTS) --- concept --- object --- typical object --- atypical object --- lattice --- filter --- ideal --- discussive logics --- the smallest discussive logic --- discussive operators --- seriality --- accessibility relation --- Kotas’ method --- modal logic --- deontic logic --- ontology of situations --- semantics of law --- formal theory of law --- Wittgenstein --- Wolniewicz --- non-Fregean logic --- identity connective --- sentential calculus with identity --- situational semantics --- deduction --- (dual) tableau --- Gentzen system --- deductive refutability --- refutation systems --- hybrid deduction–refutation rules --- derivative hybrid rules --- soundness --- completeness --- natural deduction --- meta-proof theory --- synthetic tableaux --- principle of bivalence --- cut --- first-order theory --- universal axiom --- Peano’s axiomatics of natural numbers --- Leśniewski’s elementary ontology --- Frege’s predication scheme --- Frege’s Zahl-Anzahl distinction --- term logic --- Franz Brentano --- Lewis Carroll --- logic trees --- logic diagrams --- paraconsistent logic --- paraconsistency --- Sette’s calculus --- the law of explosion --- the principle of ex contradictione sequitur quodlibet --- semantic tree --- distribution --- Aristotle’s logic --- syllogistic --- Jan Łukasiewicz --- axiomatic system --- axiomatic refutation --- temporal logic --- intuitionistic logic --- minimal system --- knowledge --- sequent-type calculi --- nonmonotonic logics --- default logic --- rejection systems --- Kripke models --- logics of evidence and truth --- n/a --- Peano's axioms --- Wilkosz's axioms --- Kotas' method --- hybrid deduction-refutation rules --- Peano's axiomatics of natural numbers --- Leśniewski's elementary ontology --- Frege's predication scheme --- Frege's Zahl-Anzahl distinction --- Sette's calculus --- Aristotle's logic