Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Staying Faithful to the Standards of Proof

Academics have never quite understood the standards of proof or, indeed, much about the theory of proof Their formulations beget probabilistic musings, which beget all sorts of paradoxes, which in turn beget radical reconceptions and proposals for reform. The theoretical radicals argue that the law needs some basic reconception such as recognizing the aim of legal proof as not at all a search for truth but rather the production of an acceptable result, or that the law needs some shattering reform such as greatly heightening the civil standard of proof on each part of the case to ensure a more-likely than- not overall result. This Article refutes all those baroque rereadings. It shows that the standards of proof properly understood on the law\u27s own terms without a probabilistic overlay, work just fine. The law tells factfinders to compare their degree of belief in the alleged fact to their degree of contradictory disbelief. Obeying that instruction resolves mathematically the paradoxes that traditional probability theory creates for itself Most surprising, the burden of proof by which the proponent must prove all the elements and the opponent need disprove only one, does not produce an asymmetry between the parties. The law\u27s standards of proof need no drastic reconception or reform, because the law knew what it was doing all along. It deals with factual beliefs in a world that will remain uncertain, not with the odds of the facts becoming certain. And the well-established mathematics of beliefs are not the mathematics of odds

Towards scientific metaphysics.Vol. 1In the circle of the scientific metaphysics of Zygmunt Zawirski : development and comments on Zawirski's concepts and their philosophical context

Lively discussions about developing a general theory of reality are magnified at the time of the emergence of new scientific theories. This situation occurred in the first half of the 20th century: in the natural sciences there appeared two fundamental theories of macro- and microcosmos: theory of relativity and quantum mechanics; whereas, in the formal sciences, a breakthrough was to be observed – among others manyvalued logic systems and the so-called limitation theorems were elaborated. Groundbreaking achievements in detailed sciences have changed the view of the world of natural phenomena. The mechanic image of reality was removed and in its place new insights into traditional problems that carried a philosophical charge began to be introduced. What we are referring here to is not only the change in the understanding of space, time or matter, the principles of causality, but also the development of systems distant from thermodynamic equilibrium and basic natural sciences research. The testimonies of the philosophical struggle with the development of specific sciences and new concepts appearing in them can be, developed by Zygmunt Zawirski and Benedykt Bornstein, two different concepts of scientific metaphysics[...

Imprecise probability in epistemology

There is a growing interest in the foundations as well as the application of imprecise probability in contemporary epistemology. This dissertation is concerned with the application. In particular, the research presented concerns ways in which imprecise probability, i.e. sets of probability measures, may helpfully address certain philosophical problems pertaining to rational belief. The issues I consider are disagreement among epistemic peers, complete ignorance, and inductive reasoning with imprecise priors. For each of these topics, it is assumed that belief can be modeled with imprecise probability, and thus there is a non-classical solution to be given to each problem. I argue that this is the case for peer disagreement and complete ignorance. However, I discovered that the approach has its shortcomings, too, specifically in regard to inductive reasoning with imprecise priors. Nevertheless, the dissertation ultimately illustrates that imprecise probability as a model of rational belief has a lot of promise, but one should be aware of its limitations also