Propositional computability logic I

In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The present contribution gives a detailed exposition of a soundness and completeness proof for an axiomatization of one of the most basic fragments of computability logic. The logical vocabulary of this fragment contains operators for the so called parallel and choice operations, and its atoms represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper "Introduction to computability logic" [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, -- the context that determines the value of the present results. Online introduction to the subject is available at http://www.cis.upenn.edu/~giorgi/cl.html and http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi

Entropic Geometry from Logic

We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy. That is: 1. A finitary probability space \Delta^n (=all probability measures on {1,...,n}) can be fully and faithfully represented by the pair consisting of the abstraction D^n (=the object up to isomorphism) of a partially ordered set (\Delta^n,\sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via a systematic purely order-theoretic procedure (which embodies introduction of partiality of knowledge) on an (algebraic) logic. This procedure applies to any poset A; D_A\cong(\Delta^n,\sqsubseteq) when A is the n-element powerset and D_A\cong(\Omega^n,\sqsubseteq), the domain of mixed quantum states, when A is the lattice of subspaces of a Hilbert space. (We refer to http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html for a domain-theoretic context providing the notions of approximation and content.)Comment: 19 pages, 8 figure

Connexive logics. An overview and current trends

In this introduction, we offer an overview of main systems developed in the growing literature on connexive logic, and also point to a few topics that seem to be collecting attention of many of those interested in connexive logic. We will also make clear the context to which the papers in this special issue belong and contribute

Context-dependent Trust Decisions with Subjective Logic

A decision procedure implemented over a computational trust mechanism aims to allow for decisions to be made regarding whether some entity or information should be trusted. As recognised in the literature, trust is contextual, and we describe how such a context often translates into a confidence level which should be used to modify an underlying trust value. J{\o}sang's Subjective Logic has long been used in the trust domain, and we show that its operators are insufficient to address this problem. We therefore provide a decision-making approach about trust which also considers the notion of confidence (based on context) through the introduction of a new operator. In particular, we introduce general requirements that must be respected when combining trustworthiness and confidence degree, and demonstrate the soundness of our new operator with respect to these properties.Comment: 19 pages, 4 figures, technical report of the University of Aberdeen (preprint version

Relations and Non-commutative Linear Logic

Linear logic differs from intuitionistic logic primarily in the absence of the structural rules of weakening and contraction. Weakening allows us to prove a proposition in the context of irrelevant or unused premises, while contraction allows us to use a premise an arbitrary number of times. Linear logic has been called a ''resource-conscious'' logic, since the premises of a sequent must appear exactly as many times as they are used.In this paper, we address this ''experimental nature'' by presenting a non-commutative intuitionistic linear logic with several attractive properties. Our logic discards even the limited commutativityof Yetter's logic in which cyclic permutations of formulae are permitted. It arises in a natural way from the system of intuitionistic linear logic presented by Girard and Lafont, and we prove a cut elimination theorem. In linear logic, the rules of weakening and contraction are recovered in a restricted sense by the introduction of the exponential modality(!). This recaptures the expressive power of intuitionistic logic. In our logic the modality ! recovers the non-commutative analogues of these structural rules. However, the most appealing property of our logic is that it is both sound and complete with respect to interpretation in a natural class of models which we call relational quantales

Reasoning about obligations in Obligationes : a formal approach.

Despite the appearance of `obligation' in their name, medieval obligational dispu- tations between an Opponent and a Respondent seem to many to be unrelated to deontic logic. However, given that some of the example disputations found in me- dieval texts involve Respondent reasoning about his obligations within the context of the disputation, it is clear that some sort of deontic reasoning is involved. In this paper, we explain how the reasoning diers from that in ordinary basic deontic logic, and dene dynamic epistemic semantics within which the medieval obligations can be expressed and the examples evaluated. Obligations in this framework are history- based and closely connected to action, thus allowing for comparisons with, e.g., the knowledge-based obligations of Pacuit, Parikh, and Cogan, and stit-theory. The con- tributions of this paper are twofold: The introduction of a new type of obligation into the deontic logic family, and an explanation of the precise deontic concepts involved in obligationes

Reasoning about Obligations in Obligationes: A Formal Approach

Despite the appearance of `obligation' in their name, medieval obligational dispu- tations between an Opponent and a Respondent seem to many to be unrelated to deontic logic. However, given that some of the example disputations found in me- dieval texts involve Respondent reasoning about his obligations within the context of the disputation, it is clear that some sort of deontic reasoning is involved. In this paper, we explain how the reasoning diers from that in ordinary basic deontic logic, and dene dynamic epistemic semantics within which the medieval obligations can be expressed and the examples evaluated. Obligations in this framework are history- based and closely connected to action, thus allowing for comparisons with, e.g., the knowledge-based obligations of Pacuit, Parikh, and Cogan, and stit-theory. The con- tributions of this paper are twofold: The introduction of a new type of obligation into the deontic logic family, and an explanation of the precise deontic concepts involved in obligationes