Uwagi o arytmetyce Grassmanna

Author(s): Jerzy Hanusek / Language(s): Polish Issue: 45/2015

Hermann Grassmann’s 1861 work [2] was probably the first attempt at an axiomatic approach to arithmetic (of integers with a distinguished subset of positive ones). The historical significance of this work is enormous, even though the set of axioms has proven to be incomplete. Basing on the interpretation of Grassmann’s theory provided by Hao Wang in [4], I present its detailed discussion, define the class of models of Grassmann’s arithmetic and discuss a certain axiom system for integers, modeled on Grassmann’s theory. At the end I propose to modify the set of axioms of Grassmann’s arithmetic, which consists in adding an elementary sentence and removing a non-elementary one. I prove that after this modification the only model of the theory up to isomorphism is the standard model.

Sprawozdanie z LX Konferencji Historii Logiki. Kraków, 21-22 października 2014 r.

Author(s): Łukasz Polkowski / Language(s): Polish Issue: 3/2014

Analiza dowodu pierwszej tezy Gorgiasza

Author(s): Andrzej Nowakowski / Language(s): Polish Issue: 48/2016

The proof of first thesis of Gorgias is an indirect deductive argument constructed by the multiple use of the modus tollens pattern. Those of its premises which are assumed without proofs are misleadingly similar to some logical or analytical truths. If they actually were such truths, some contradictions would have to appear in the proof. In the times of Gorgias the proof could be regarded as correct; the possibility of showing and clearly describing its defects emerged only after formal logic had been invented. Gorgias could announce his thesis seriously – as a thesis of extreme metaphysical nihilism – without being embarrassed. If the proof is a sophistry, then it is a masterpiece among sophistries, the most outstanding achievement of the sophists in this domain.

ПРОБЛЕМИ НА ПРЕХОДА ОТ ВЕРОЯТНОСТ КЪМ ДОСТОВЕРНОСТ ПРИ РАЗКРИВАНЕ НА ОБЕКТИВНАТА ИСТИНА ПРИ РАЗСЛЕДВАНЕТО

Author(s): Hristo Pavlov / Language(s): Bulgarian Issue: 1/2015

This article examines the nature of the concepts of „variability", probability” and „authenticity" in the process of discovery of the objective truth in the investigation. The article analyzes the main problems associated with the transition from probability to reliability in revealing the objective truth in the investigation.

HILBERT ALGEBRAS WITH A NECESSITY MODAL OPERATOR

Author(s): Sergio A. Celani,Daniela Montangie / Language(s): English Issue: 49/2014

We introduce the variety of Hilbert algebras with a modal operator _, called H-algebras. The variety of H-algebras is the algebraic counterpart of the {-_}-fragment of the intuitionitic modal logic IntK. We will study the theory of representation and we will give a topological duality for the variety of H-algebras. We are going to use these results to prove that the basic implicative modal logic IntK! and some axiomatic extensions are canonical. We shall also to determine the simple and subdirectly irreducible algebras in some subvarieties of H-algebras.

RECURSIVELY ENUMERABLE SETS AND WELL-ORDERING OF THEIR ENUMERATIONS

Author(s): Jerzy Mycka / Language(s): English Issue: 49/2014

We will introduce the special kind of the order relations into recursively enumerable sets and prove that they can be used to distinguish (albeit in a non-constructive way) between recursive and non-recursive sets.

COMPLEXITY OF COVER-PRESERVING EMBEDDINGS OF BIPARTITE ORDERS INTO BOOLEAN LATTICES

Author(s): Grzegorz Herman / Language(s): English Issue: 49/2014

We study the problem of deciding, whether a given partial order is embeddable into two consecutive layers of a Boolean lattice. Employing an equivalent condition for such embeddability similar to the one given by J. Mittas and K. Reuter, we prove that the decision problem is NP-complete by showing a polynomial-time reduction from the not-all-equal variant of the Satisfiability problem.

Tichý and Fictional Names

Author(s): Daniela Glavaničová / Language(s): English Issue: 3/2017

The paper examines two possible analyses of fictional names within Pavel Tichý’s Transparent Intensional Logic. The first of them is the analysis actually proposed by Tichý in his (1988) book The Foundations of Frege’s Logic. He analysed fictional names in terms of free variables. I will introduce, explain, and assess this analysis. Subsequently, I will explain Tichý’s notion of individual role (office, thing-to-be). On the basis of this notion, I will outline and defend the second analysis of fictional names. This analysis is close to the approach known in the literature as role realism (the most prominent advocates of this position are Nicholas Wolterstorff, Gregory Currie, and Peter Lamarque).

The Role of Disjunction in Some Alleged Non-Monotonic Inferences

Author(s): Miguel López-Astorga / Language(s): English Issue: 1/2017

Lukowski has argued that, if it is the case that there are actual non-monotonic inferences, they are very hard to find. In this paper, a representative kind of inference that is often considered to be non-monotonic is addressed. Likewise, certain arguments provided by Lukowski to demonstrate that that type of inference is not really non-monotonic are reviewed too. Finally, I propose an explanation of why, despite the fact that the arguments given by him seem to be convincing, it is usually thought that those inferences are not monotonic. In this way, I also try to account for the role that disjunction has in this issue and argue in favor of the idea that we can continue to suppose that the human mind does not ignore the essential requirements of classical logic.

A Valid Rule of β-conversion for the Logic of Partial Functions

Author(s): Marie Duží,Miloš Kosterec / Language(s): English Issue: 1/2017

The goal of this paper is to examine the conditions of validity for the rule of β-conversion in TIL, which is a hyperintensional, typed λ-calculus of partial functions. The rule of β-reduction is a fundamental computational rule of the λ-calculi and functional programming languages. However, it is a well-known fact that the specification of this rule is ambiguous (see, e.g., Plotkin 1975 or Chang & Felleisen 2012). There are two procedurally non-equivalent ways of executing the rule, namely β-conversion ‘by name’ and β-conversion ‘by value’. In the λ-calculi conversion by name is usually applied, though it is known that such a conversion is not unconditionally valid when partial functions are involved. If a procedure that is typed to produce an argument value is improper by failing to produce one, conversion by name cannot be validly applied. On the other hand, conversion by value is valid even in the case of improperness. Moreover, we show that in a typed λ-calculus the specification of λ-closure is also not unambiguous. There is an interpretation of this specification under which β-reduction by name is not valid even when the argument procedure does not fail to produce a value. As a result, we present a universally valid rule of β-reduction by value.

Toward a Demarcation of Forms of Determinism

Author(s): Vladimir Marko / Language(s): English Issue: 1/2017

In the current philosophical literature, determinism is rarely defined explicitly. This paper attempts to show that there are in fact many forms of determinism, most of which are familiar, and that these can be differentiated according to their particular components. Recognizing the composite character of determinism is thus central to demarcating its various forms.

In Defence of Δ-TIL

Author(s): Daniela Glavaničová / Language(s): English Issue: 1/2017

In 2015-2016, my two papers on deontic modalities analysed in terms of Transparent Intensional Logic (TIL) were published in Organon F. The first of them, Glavaničová (2015), is based on the results of my bachelor thesis. This paper stands at the beginning of my (ongoing) research into deontic logic. The second one, Glavaničová (2016), suggests a small amendment to the analysis provided in the first paper.

Quantifier Domain Restriction, Hidden Variables and Variadic Functions

Author(s): Andrei Moldovan / Language(s): English Issue: 3/2016

In this paper I discuss two objections raised against von Fintel’s (1994) and Stanley and Szabó’s (2000a) hidden variable approach to quantifier domain restriction (QDR). One of them concerns utterances of sentences involving quantifiers for which no contextual domain restriction is needed, and the other concerns multiple quantified contexts. I look at various ways in which the approaches could be amended to avoid these problems, and I argue that they fail. I conclude that we need a more flexible account of QDR, one that allows for the hidden variables in the LF responsible for QDR to vary in number. Recanati’s (2002; 2004) approach to QDR, which makes use of the apparatus of “variadic functions”, is flexible enough to account successfully for the two phenomena discussed. I end with a few comments on what I take to be the most promising way to construe variadic functions.

What Is (Modern) Logic Taken to Be About and What It Is About

Author(s): Jaroslav Peregrin / Language(s): English Issue: 2/2016

Since Antiquity, logic has always enjoyed a status of something crucially important, because it shows us how to reason, if we are to reason correctly. Yet the twentieth century fostered an unprecedented boost in logical studies and delivered a wealth of results, most of which are not only not understandable by non-specialists, but their very connection with the original agenda of logic is far from clear. In this paper, I survey how the achievements of modern logic are construed by non-specialists and subject their construals to critical scrutiny. I argue that logic cannot be taken as a theory of the limits of our world and that its prima facie most plausible construal as a theory of reasoning is too unclear to be taken at face value. I argue that the viable construal of logic takes it to be explicative of the constitutive (rather than strategic) rules of reasoning, not of the rules that tell us how to reason, but rather of rules that make up the tools with which (or in terms of which) we reason.

The Many Inadequate Justification of Methodological Naturalism

Author(s): Robert Larmer / Language(s): English Issue: 1/2019

Contrary to proponents’ claims, methodological naturalism is not metaphysically neutral. Consequently, its acceptance as a practice requires justification. Unfortunately for its advocates, attempts to justify it are failures. It cannot be defended as a definition, or a self-imposed limitation, of science, nor, more modestly, as an inductively justified commitment to natural causes. As a practice, it functions not to further scientific investigation, but rather to impose an explanatory straitjacket.

Prospects for Experimental Philosophical Logic

Author(s): Jeremiah Joven Joaquin / Language(s): English Issue: 2/2019

This paper focuses on two interrelated issues about the prospects for research projects in experimental philosophical logic. The first issue is about the role that logic plays in such projects; the second involves the role that experimental results from the cognitive sciences play in them. I argue that some notion of logic plays a crucial role in these research projects, and, in turn, the results of these projects might inform substantive debates in the philosophy of logic.

C.I. Lewis, E.J. Nelson, and the Modern Origins of Connexive Logic

Author(s): Edwin Mares,Francesco Paoli / Language(s): English Issue: 3/2019

Modern logic owes an important debt to C.I. Lewis and his students. In addition to Lewis’s five modal logics, they are responsible for the creation (or discovery) of the logic of analytic implication and connexive logic. In this paper, we examine E.J. Nelson’s connexive logic as an attempt to formalise the notion of entailment while avoiding the paradoxes of strict implication. We also look briefly at the reception of Nelson’s logic and at Lewis’s reply to it.

Alternative Axiomatizations of the Conditional System VC

Author(s): Claudio Pizzi / Language(s): English Issue: 3/2019

The central result of the paper is an alternative axiomatization of the conditional system VC which does not make use of Conditional Modus Ponens: (A > B) ⊃ (A ⊃ B) and of the axiom-schema CS: (A ∧ B) ⊃ (A > B). Essential use is made of two schemata, i.e. X1: (A ∧ ♢A) ⊃ (♢A >< A) and T: □A ⊃ A, which are subjoined to a basic principle named Int: (A ∧ B) ⊃ (♢A > ♢B). A hierarchy of extensions of the basic system V called VInt, VInt1, VInt1T is then construed and submitted to a semantic analysis. In Section 3 VInt1T is shown to be deductively equivalent to VC. Section 4 shows that in VC the thesis X1 is equivalent to X1∨: (♢A >< A) ∨ (♢¬A >< ¬A), so that VC is also equivalent to a variant of VInt1T here called VInt1To. In Section 6 both X1 and X1∨ offer the basis for a discussion on systems containing CS, in which it is argued that they cannot avoid various kinds of partial or full trivialization of some non truth-functional operators.

Wiredu contra Lewis on the Right Modal Logic

Author(s): David Martens / Language(s): English Issue: 3/2019

This paper is a critical study of an argument put forward by Kwasi Wiredu in his engagement with C. I. Lewis on choosing the right modal logic for logical necessity. Wiredu argues that Lewis “could have been more adventurous modally with perfect logicality” and could justifiably have accepted S4 rather than being “to the last cautious of any system stronger than S2” (Wiredu 1979). I address terse, incomplete, and provocatively incongruous notes on Wiredu’s paper by (Makinson 1980) and (Humberstone 2011), as well as a paper by (Cresswell 1965) that Humberstone cites, and I draw on recent work by (Lewitzka 2015; 2016). I conclude that Wiredu’s argument cannot be accepted as sound but a variant argument can be accepted as sound.

Modelul predicativ pentru silogistica categorică al lui Stanislav Jaskowski

Author(s): Georgeta Cuculeanu / Language(s): Romanian Issue: 1/2024

In this paper, a detailed analysis of Jaskowski’s predicative model is presented. By this model, Jaskowski showed that all categorical judgements can have a uniform interpretation. For this, in the present paper, the relationships between the universal and particular affirmative judgements are established. Also, all theorems are thoroughly demonstrated; one of them was divided into two theorems for those demonstrations that use different hypotheses.