Hintikka’s Independence-Friendly Logic Meets Nelson’s Realizability

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics (abbreviated GTS) for independence-friendly first-order logic (IF-FOL), but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation—which extends the famous Kleene’s realizability interpretation by adding ‘strong negation’—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called ‘trump semantics’ which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson’s restricted realizability interpretation for the implication-free first-order formulas.     

On Axiomatizing Shramko-Wansing’s Logic

This work treats the problem of axiomatizing the truth and falsity consequence relations, ⊨ _t and ⊨ _f , determined via truth and falsity orderings on the trilattice SIXTEEN ₃ (Shramko and Wansing, 2005). The approach is based on a representation of SIXTEEN ₃ as a twist-structure over the two-element Boolean algebra.     

Logic’s Tower of Babel

Late in his life, Jung speculated that the natural numbers, the integers, “contain the whole of mathematics and everything yet to be discovered in this field.” This article presents the attempts by mathematicians to address this question in their terms; that is, whether arithmetic (the mathematics of the natural numbers) was complete and consistent.

Early in the twentieth century, mathematicians began to seek a formalism that could provide a solid foundation for mathematics. The first important product of this new formalism was Giuseppe Peano’s Postulates: five axioms from which the full arithmetic of the natural numbers or integers (i.e., 0, 1, 2, 3, …) can be derived. Inspired by Peano’s achievement, philosopher and mathematician Bertrand Russell began a project to show that mathematics could be reduced to logic. His overweening aim was to eventually show that all science could be reduced to logic.

Logician Kurt Gödel realized that the goal of the formalists and logicians was impossible. He produced a logically impeccable proof that no system at least as complex as arithmetic could be proved both complete and consistent within the system. In essence, he proved that the core of mathematical discovery must be intuitive: direct perception of reality, which then clothes itself in mathematical garb. This accords closely with Jung’s own insight, which was based on the idea that each number is qualitatively different from every other number. To this day, Gödel’s proof stands unchallenged.     

The Tense Logic for Master Argument in Prior’s Reconstruction

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K _t 4 plus a new axiom (P): ‘p Λ G p ⊃ P G p’. This formula was used by Prior in his original analysis of Master Argument. (P) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by (P) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. Presented by Jacek Malinowski     

Buridan’s Secular Demarcation of Logic

Philosophia - In his Tractatus de Consequentiis, the 14th century logician John Buridan states a theorem, which reads like a refutation of Aristotelian Syllogistics. Syllogistics in its usual form,...     

Chateaubriand’s Realist Conception of Logic

Axiomathes - I present the realist conception of logic supported by Oswaldo Chateaubriand which integrates ontological and epistemological aspects, opposing it to mathematical and linguistic...     

Franco Montagna’s Work on Provability Logic and Many-valued Logic

Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.     

Some Modifications of Carnap’s Modal Logic

In this paper, Carnap??s modal logic C is reconstructed. It is shown that the Carnapian approach enables us to create some epistemic logics in a relatively straight-forward way. These epistemic modifications of C are axiomatized and one of them is compared with intuitionistic logic. At the end of the paper, some connections between this epistemic logic and Medvedev??s logic of finite problems and inquisitive semantics are shortly discussed.     

Rosenkranz’s Logic of Justification and Unprovability

Journal of Philosophical Logic - Rosenkranz has recently proposed a logic for propositional, non-factive, all-things-considered justification, which is based on a logic for the notion of being in a...     

Frege’s Anti-Psychologism about Logic : the Relationship between Logic and Judgment

Philosophia - Frege is an anti-psychologist about logic who takes logic to be sharply distinguished from psychology. However, Frege also takes judgment, which seems to be a subject of psychology,...     

Nelson Pike’s Contribution to the Philosophy of Religion

In this paper I attempt to capture the essence of Nelson Pike’s contribution to the philosophy of religion. My summary of his insights will revolve around three general topics: omniscience (and in particular its relation to human freedom), omnipotence (and in particular its relation to the existence of human suffering), and mysticism (with a focus on the question of whether and in what sense mystic visions can be sources of knowledge). Although the details vary in interesting ways, his work on these topics largely consists of recognizing an important challenge to the viability of the relevant doctrine or framework, sharpening that challenge by presenting it in a more forceful way, and then offering and assessing potential responses. Pike’s writings are characterized by exemplary rigor and relentless clarity, and together they constitute a rich (and under-appreciated) source of insight.     

Benedykt Bornstein’s Philosophy of Logic and Mathematics

The aim of this paper is to present and discuss main philosophical ideas concerning logic and mathematics of a significant but forgotten Polish philosopher Benedykt Bornstein. He received his doctoral degree with Kazimierz Twardowski but is not included into the Lvov–Warsaw School of Philosophy founded by the latter. His philosophical views were unique and quite different from the views of main representatives of Lvov–Warsaw School. We shall discuss Bornstein’s considerations on the philosophy of geometry, on the infinity, on the foundations of set theory and his polemics with Stanis?aw Le?niewski as well as his conception of a geometrization of logic, of the categorial logic and of the mathematics of quality.     

An Intuitionistic Reformulation of Mally’s Deontic Logic

In 1926, Ernst Mally proposed a number of deontic postulates. He added them as axioms to classical propositional logic. The resulting system was unsatisfactory because it had the consequence that A is the case if and only if it is obligatory that A. We present an intuitionistic reformulation of Mally’s deontic logic. We show that this system does not provide the just-mentioned objectionable theorem while most of the theorems that Mally considered acceptable are still derivable. The resulting system is unacceptable as a deontic logic, but it does make sense as a lax logic in the modern sense of the word.     

Carroll’s Regress and the Epistemology of Logic

On an internalist account of logical inference, we are warranted in drawing conclusions from accepted premises on the basis of our knowledge of logical laws. Lewis Carroll’s regress challenges internalism by purporting to show that this kind of warrant cannot ground the move from premises to conclusion. Carroll’s regress vindicates a repudiation of internalism and leads to the espousal of a standpoint that regards our inferential practice as not being grounded on our knowledge of logical laws. Such a standpoint can take two forms. One can adopt either a broadly externalist model of inference or a sceptical stance. I will attempt, in what follows, to defend a version of internalism which is not affected by the regress. The main strategy will be to show that externalism and scepticism are not satisfying standpoints to adopt with regard to our inferential practice, and then to suggest an internalist alternative.     

Moment/History Duality in Prior’s Logics of Branching-Time

The basic notions in Prior’s Ockhamist and Peircean logics of branching-time are the notion of moment and that of history (or course of events). In the tree semantics, histories are defined as maximal linearly ordered sets of moments. In the geometrical approach, both moments and histories are primitive entities and there is no set theoretical (and ontological) dependency of the latter on the former. In the topological approach, moments can be defined as the elements of a rank 1 base of a non-Archimedean topology on the set of histories. In this paper, it will be shown that the topological approach, and hence the other approaches, can be reconstructed in a framework in which the basic notions are those of history and of relative closeness relation among histories.     

A Simple Logical Matrix and Sequent Calculus for Parry’s Logic of Analytic Implication

Studia Logica - We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve...     

A Peculiarity in Pearl’s Logic of Interventionist Counterfactuals

We examine a formal semantics for counterfactual conditionals due to Judea Pearl, which formalizes the interventionist interpretation of counterfactuals central to the interventionist accounts of causation and explanation. We show that a characteristic principle validated by Pearl’s semantics, known as the principle of reversibility, states a kind of irreversibility: counterfactual dependence (in David Lewis’s sense) between two distinct events is irreversible. Moreover, we show that Pearl’s semantics rules out only mutual counterfactual dependence, not cyclic dependence in general. This, we argue, suggests that Pearl’s logic is either too weak or too strong.     

Predicate Metric Tense Logic for ‘Now’ and ‘Then’

In a number of publications A.N. Prior considered the use of what he called ‘metric tense logic’. This is a tense logic in which the past and future operators P and F have an index representing a temporal distance, so that Pnα means that α was true n-much ago, and Fnα means that α will be true n-much hence. The paper investigates the use of metric predicate tense logic in formalising phenomena ormally treated by such devices as multiple indexing or quantification over times.