An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

The Relationship Between Belief and Credence

Sometimes epistemologists theorize about belief, a tripartite attitude on which one can believe, withhold belief, or disbelieve a proposition. In other cases, epistemologists theorize about credence, a fine-grained attitude that represents one’s subjective probability or confidence level toward a proposition. How do these two attitudes relate to each other? This article explores the relationship between belief and credence in two categories: descriptive and normative. It then explains the broader significance of the belief-credence connection and concludes with general lessons from the debate thus far

Philosophical Theology and Christian Doctrines

This contribution discusses Leibniz’s views on key Christian doctrines which were surrounded, in the early modern period, by particularly lively debates. The first section delves into his defence of the Trinity and the Incarnation against the charge of contradiction, and his exploration of metaphysical models capacious enough to accommodate these mysteries. The second section focuses on the resurrection and the Eucharist with special regard to their connections with Leibniz’s metaphysics of bodies. The third section investigates Leibniz’s position on predestination, grace, salvation, and damnation. It comes to the conclusion that salvation, for Leibniz, does not ultimately depend on believing a set of true doctrines, but on a practical attitude: the love of God above all things. Leibniz’s theology is thus fundamentally a theology of love which is ultimately practical, and tries to be both universalist and Christian

Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s

As analytic philosophy is becoming increasingly aware of and interested in its own history, the study of that field is broadening to include, not just its earliest beginnings, but also the mid-twentieth century. One of the towering figures of this epoch is W.V. Quine (1908-2000), champion of naturalism in philosophy of science, pioneer of mathematical logic, trying to unite an austerely physicalist theory of the world with the truths of mathematics, psychology, and linguistics. Quine's posthumous papers, notes, and drafts revealing the development of his views in the forties have recently begun to be published, as well as careful philosophical studies of, for instance, the evolution of his key doctrine that mathematical and logical truth are continuous with, not divorced from, the truths of natural science. But one central text has remained unexplored: Quine's Portuguese-language book on logic, his 'farewell for now' to the discipline as he embarked on an assignment in the Navy in WWII. Anglophone philosophers have neglected this book because they could not read it. Jointly with colleagues, I have completed the first full English translation of this book. In this accompanying paper I draw out the main philosophical contributions Quine made in the book, placing them in their historical context and relating them to Quine's overall philosophical development during the period. Besides significant developments in the evolution of Quine's views on meaning and analyticity, I argue, this book is also driven by Quine's indebtedness to Russell and Whitehead, Tarski, and Frege, and contains crucial developments in his thinking on philosophy of logic and ontology. This includes early versions of some arguments from 'On What There Is', four-dimensionalism, and virtual set theory

Neutrality and Many-Valued Logics

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.Comment: 119 page

Domesticating Descartes, Renovating Scholasticism: Johann Clauberg And The German Reception Of Cartesianism

This article studies the academic context in which Cartesianism was absorbed in Germany in the mid-seventeenth century. It focuses on the role of Johann Clauberg (1622-1665), first rector of the new University of Duisburg, in adjusting scholastic tradition to accommodate Descartes’ philosophy, thereby making the latter suitable for teaching in universities. It highlights contextual motivations behind Clauberg’s synthesis of Cartesianism with the existing framework such as a pedagogical interest in Descartes as offering a simpler method, and a systematic concern to disentangle philosophy from theological disputes. These motivations are brought into view by situating Clauberg in the closely-linked contexts of Protestant educational reforms in the seventeenth century, and debates around the proper relation between philosophy and theology. In this background, it argues that Clauberg nevertheless retains an Aristotelian conception of ontology for purely philosophical reasons, specifically, to give objective foundations to Descartes’s metaphysics of substance. In conclusion, Clauberg should not be assimilated either to Aristotelianism or to Cartesianism or, indeed, to syncretic labels such as ‘Cartesian Scholastic’. Instead, he should be read as transforming both schools by drawing on a variety of elements in order to address issues local to the academic milieu of his time