From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Genuine Process Logic

The Genuine Process Logic described here (abbreviation: GPL) places the object-bound process itself at the center of formalism. It should be suitable for everyday use, i.e. it is not primarily intended for the formalization of computer programs, but instead, as a counter-conception to the classical state logics. The new and central operator of the GPL is an action symbol replacing the classical state symbols, e.g. of equivalence or identity. The complete renunciation of object-language state expressions also results in a completely new metalinguistic framework, both regarding the axioms and the expressive possibilities of this system. A mixture with state logical terms is readily possible

Fredkin Gates for Finite-valued Reversible and Conservative Logics

The basic principles and results of Conservative Logic introduced by Fredkin and Toffoli on the basis of a seminal paper of Landauer are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives. In particular, we consider the typical connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As a result, some possible three-valued and d-valued universal gates are described which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type

Relevant Logics Obeying Component Homogeneity

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues

Correctness and completeness of logic programs

We discuss proving correctness and completeness of definite clause logic programs. We propose a method for proving completeness, while for proving correctness we employ a method which should be well known but is often neglected. Also, we show how to prove completeness and correctness in the presence of SLD-tree pruning, and point out that approximate specifications simplify specifications and proofs. We compare the proof methods to declarative diagnosis (algorithmic debugging), showing that approximate specifications eliminate a major drawback of the latter. We argue that our proof methods reflect natural declarative thinking about programs, and that they can be used, formally or informally, in every-day programming.Comment: 29 pages, 2 figures; with editorial modifications, small corrections and extensions. arXiv admin note: text overlap with arXiv:1411.3015. Overlaps explained in "Related Work" (p. 21