We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionsitic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for ...

We introduce a substructural propositional calculus of Sequential Dynamic Logic (SDL) that subsumes a propositional part of dynamic predicate logic (DPL), and is shown to be expressively equivalent to propositional dynamic logic (PDL). Completeness of the calculus with respect to the intended relational semantics is established.

In this paper it is shown that addition of certain reductions to the standard cut removing reductions of deductions in propositional logic makes propositional logic non-normalizable. From this follows that some provable propositions in propositional logic has no direct proof.

This paper presents matching logic, a first-order logic (FOL) variant for specifying and reasoning about structure by means of patterns and pattern matching. Its sentences, the patterns, are constructed using variables, symbols, connectives and quantifiers, but no difference is made between function and predicate symbols. In models, a pattern evaluates into a power-set domain (the set of values...

The two main founders of paraconsistent logic, Stanis law Jaśkowski and Newton da Costa, built their systems on distinct grounds. Starting from different projects, they used different tools and ultimately designed quite different calculi to attend their needs. How successful were their enterprises? Here we discuss the problem of defining paraconsistent logics following the original instructions...

We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the e...

The exogenous semantics approach to enriching a logic consists in defining each model in the enrichment as a set of models in the original logic plus some relevant structure. We illustrate the approach by probabilizing classical propositional logic, including a novel global propositional logic. A model of the probability logic is a probability space where the outcomes are classical valuations. ...

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen cate...

—In this paper, we examine and compare De Morgan-, Kleene-, and Boolean-disjunctive and conjunctive normal forms and consider their role in fuzzy settings. In particular, we show that there are normal forms and truth tables for classical fuzzy propositional logic and intervalvalued fuzzy propositional logic that are completely analogous to those for Boolean propositional logic. Thus, determinin...

There are many reasons why a computer scientist should study logic. For example, logic underlies the reasoning in mathematical statements. In particular, proving that a program meets its specification is argued with the help of logical deductions. We begin by studying propositional logic, which is one of the simplest logical theories. Nevertheless, propositional logic has a large number of appl...