Joanna Golinska-Pilarek

We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some heuristics, such as the so called phantom variables, which improve the efficiency of the selection of variables used un the proof.

We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of $\mathsf{SCI}_{\mathsf{Q}}$.

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other.