This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

Modal propositional logic; Modal predicate logic; A survey of modal logic.

Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two (...) or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £. (shrink)

This paper criticises necessitarianism, the thesis that there is at least one necessary truth; and defends possibilism, the thesis that all propositions are contingent, or that anything is possible. The second section maintains that no good conventionalist account of necessity is available, while the third section criticises model theoretic necessitarianism. The fourth section sketches some recent technical work on nonclassical logic, with the aim of weakening necessitarian intuitions and strengthening possibilist intuitions. The fifth section considers several a prioristic attempts at (...) demonstrating that there is at least one necessary proposition and finds them inadequate. The final section emphasises the epistemic aspect of possibilism. (shrink)

ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.