Logical Connectives

In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.

A globally expressivist analysis of the indicative conditional based on the Ramsey Test is presented. The analysis is a form of ‘global’ expressivism in that it supplies acceptance and rejection conditions for all the sentence forming connectives of propositional logic and so allows the conditional to embed in arbitrarily complex sentences. The expressivist framework is semantically characterized in a restrictor semantics due to Vann McGee, and is completely axiomatized in a logic dubbed ICL. The expressivist framework extends the AGM framework (...) for belief revision and so provides a categorical epistemology for conditionals that complements McGee’s probabilistic framework while drawing on the same semantics. The result is an account of the semantics and acceptability conditions of the indicative conditional that fits well with the linguistic data while integrating both expressivist and semanticist perspectives. (shrink)

Is identity fundamental to every conceptual systems? In this article I intend to present reasons against the claim that every conceptual system presupposes the notion of identity. To address this debate I analyze the positions of Bueno (2014; 2016) and Krause and Arenhart (2015). While Bueno argues that identity is necessary for all conceptual systems, Krause and Arenhart present a series of objections against such position, thus defending that identity is not fundamental. I intend to show that the main objections (...) presented by Krause and Arenhart do not affect Bueno’s theory because of a misinterpretation of his arguments. Finally, I will present some objections against Bueno that, hopefully, does not miss the target. (shrink)

Is identity fundamental to formal systems? Even if a system have no the identity relation, is that concept is not assumed in any way – whether in a metalinguistic or intuitive level? In this paper we shall discuss this issue. Otávio Bueno (2014, 2016) argues against the elimination of identity, holding that this concept is fundamental and non-eliminable (even in does systems that claim to do so). Décio Arenhart Krause and Jonas (2015), by the other hand, have a number of (...) objections to Bueno’s thesis. Firstly, we look at how the concept of identity was accounted in the philosophical tradition, as well as part of its formal account. We will also examine the notion of indiscernibility that, according to the traditional approach to identity, is equivalent to the notion of identity, but (arguably) is not it logically equivalent. Thus, we shall state Krause and Arenhart’s strategy, according to, in a formal system we can eliminate identity in favor of the indiscernibility notion. Finally, we shall expose Bueno’s criticism to such strategy, and then to put forward some objections against it. (shrink)

Identity is traditionally taken to be a fundamental notion of our conceptual framework as well as a fundamental metaphysical component of entities. But as far as we make this claim we face ourselves with two problems: what is identity? And why would it be fundamental? These questions will guide us towards a discussion put forward by Bueno (2014), Krause and Arenhart (2015). Bueno holds that there are four aspects that make identity being fundamental: (1) identity is assumed in every conceptual (...) system; (2) it is required for a minimal characterisation of being an individual; (3) it cannot be defined; and (4) identity is required for quantification. On the other hand, Krause and Arenhart refuse the thesis that identity is fundamental replying to Bueno's arguments. In this dissertation we will deal with this debate. In the introduction we will deal with the first problem what is identity? , showing how this concept is traditionally understood, either for its metaphysical characteristics as for its formal account. After that we will deal with each of the four aspects defended by Bueno and challenged by Krause and Arenhart. After a critical presentation of each position we will also provide other arguments for the current debate. Finally we will outline an alternative view to those defended throughout this work. -/- . (shrink)

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with two RM-phenomena, namely, splittings and disjunctions. As to splittings, there are some examples in RM of theorems A, B, C such that A↔, that is, A can be split into two (...) independent parts B and C. As to disjunctions, there are examples in RM of theorems D, E, F such that D↔, that is, D can be written as the disjunction of two independent parts E and F. By contrast, we show in this paper that there is a plethora of splittings and disjunctions in Kohlenbach’s higher-order RM. (shrink)

This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.

It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories always consist of (...) an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations. (shrink)

Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermann's strenge Implikation ([1]). Interest in Ackermann's system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present (...) some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermann's system is identical with the structure of modalities in Lewis's S4, and secondly that (Ackermann's apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication. (shrink)

There is a well-documented paradigm-shift in eighteenth century Jesuit philosophy and science, at the very least in Central Europe: traditional scholastic version of Aristotelianism were replaced by early modern rationalism and early modern science and mathematics. In the field of probability, this meant that the traditional Jesuit engagement with probability, uncertainty, and truthlikeness could translate into mathematical language, and can be analysed against the background of the accounts of probability, pre-mathematical Jesuit logic, Wolff's conceptual analysis, and Bernoullian mathematisation. The works (...) of two Jesuit philosophers, Berthold Hauser and Sigismund Storchenau, can be related to this context. The core of their logic of probability is the account of negation and implication, in particular, the algorithms for computing the reliability of one piece of evidence when compared to the respective counter-evidence and for computing the probability of a conclusion given the probability of its premises. (shrink)

Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode the meanings of the connectives (not by determining a range of admissible valuations, but in their own way), and I demonstrate (...) a kind of converse to Carnap’s result. (shrink)

This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question–Answer Semantics. We develop a generalized account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression ‘No’ is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators.

Prior’s arguments for and against seeing ‘ought’ as a copula and his considerations about normative negation are applied to the case of responsibility judgments. My thesis will be that responsibility judgments, even though often expressed by using the verb ‘to be’, are in fact normative judgments. This is shown by analyzing their negation, which parallels the behavior of ought negation.

One of Da Costa’s motives when he constructed the paraconsistent logic C! was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to C!. Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic structures. The paper (...) then investigates dualising the intuitionistic conditional in the same way. This establishes various connections between the logic, and a logic called in the literature ‘Brouwerian logic’ or ‘closed-set logic’. (shrink)

In discussing questions of free will, Soviet philosophers fail to distinguish conditions from causes. This makes them unable to understand the very opponents they like to criticize.

Unrestricted inferentialism holds both that any collection of inference rules can determine a meaning for an expression and meaning constituting rules are automatically valid. Prior's infamous tonk connective refuted unrestricted inferentialism, or so it is universally thought. This paper argues against this consensus. I start by formulating the metasemantic theses of inferentialism with more care than they have hitherto received; I then consider a tonk language — Tonklish — and argue that the unrestricted inferentialist's treatment of this language is only (...) problematic if it is mistakenly assumed that Tonklish can be homophonically translated into English. Next, I discuss the proper, non-homophonic, translation of Tonklish into English, rebut various objections, and consider several variants of Tonklish. The paper closes by highlighting the philosophical advantages that unrestricted inferentialism has over its competitors once the terrors of tonk have been tamed. (shrink)

Is modus ponens valid for the indicative conditional? McGee [1985] famously presents several alleged counterexamples to this inference rule. More recently, Kolodny and MacFarlane [2010] and Willer [2010] argue that modus ponens is unreliable in certain hypothetical contexts. However, none of these attacks undermines an informational conception of logic on which modus ponens is valid.

For the hypoelliptic differential operators $P={\partial ^2_ x}+\left^2$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic =$), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

This essay treats semantical negation from the standpoint of game-theoretical semantics. It consists of an exposition and critique of the one previous treatment of negation from this viewpoint, a formulation of negation-formation rules with special attention to the interaction of quantifier scopes and the scope of negation, and applications of this treatment to a number of problems: negation of sentences containing definite descriptions, sentences having contextually determined restrictions on quantifier domains, reciprocal quantification, negative polarity, negative raising, term negation of quantifier (...) phrases, no, the noneffectiveness of negation in general, quantificational only, and tag questions. (shrink)