This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

Can an omnipotent being create a stone too heavy for him to lift? If not, he is not omnipotent. But if so, he is not omnipotent either, since there is something he cannot lift. Hence there can be no omnipotent being. J.L. Cowan's recent reformulation of this paradox of omnipotence has been sharpened through a number of objections and clarifications, and, in its final form, constitutes a significant problem for the analysis of the concept of an omnipotent agent. I will (...) develop fragments of two systems in which the problem can be defined more exactly, and try to indicate some formal guidelines within which constructive steps towards a solution may be possible. I will argue that the paradox shows the need for a special kind of restriction on omnipotence that can be distinguished from some related restrictions. (shrink)

Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

The notions of ground and ontological dependence have made a prominent resurgence in much of contemporary metaphysics. However, objections have been raised. On the one hand, objections have been raised to the need for distinctively metaphysical notions of ground and ontological dependence. On the other, objections have been raised to the usefulness of adding ground and ontological dependence to the existing store of other metaphysical notions. Even the logical properties of ground and ontological dependence are under debate. In this article, (...) I focus on how to account for the judgements of non-symmetry in several of the cases that motivate the introduction of notions like ground and ontological dependence. By focusing on the notion of explanation relative to a theory, I conclude that we do not need to postulate a distinctively asymmetric metaphysical notion in order to account for these judgements. (shrink)

This is an extended version of the lectures given during the 12-thConference on Applications of Logic in Philosophy and in the Foundationsof Mathematics in Szklarska Poręba. It contains a surveyof modal hybrid logic, one of the branches of contemporary modal logic. Inthe first part a variety of hybrid languages and logics is presented with adiscussion of expressivity matters. The second part is devoted to thoroughexposition of proof methods for hybrid logics. The main point is to showthat application of hybrid logics (...) may remarkably improve the situation inmodal proof theory. (shrink)

Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives . The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect by (...) considering a Sheffer function for intuitionistic logic. (shrink)