Logical Connectives

Logical connectives (otherwise known as 'logical constants' or 'logical particles') have seemed challenging to philosophers of language. This article gives a concise account of logical connectives.

This paper formulates a bilateral account of harmony that is an alternative to one proposed by Francez. It builds on an account of harmony for unilateral logic proposed by Kürbis and the observation that reading the rules for the connectives of bilateral logic bottom up gives the grounds and consequences of formulas with the opposite speech act. I formulate a process I call 'inversion' which allows the determination of assertive elimination rules from assertive introduction rules, and rejective elimination rules from (...) rejective introduction rules, and conversely. It corresponds to Francez's notion of vertical harmony. I also formulate a process I call 'conversion', which allows the determination of rejective introduction rules from assertive elimination rules and conversely, and the determination of assertive introduction rules from rejective elimination rules and conversely. It corresponds to Francez's notion of horizontal harmony. The account has a number of features that distinguish it from Francez's. (shrink)

Arguments have always played a central role within logic and philosophy. But little attention has been paid to arguments as a distinctive kind of discourse, with its own semantics and pragmatics. The goal of this essay is to study the mechanisms by means of which we make arguments in discourse, starting from the semantics of argument connectives such as `therefore'. While some proposals have been made in the literature, they fail to account for the distinctive anaphoric behavior of `therefore', as (...) well as uses of argument connectives in complex arguments, suppositional arguments, arguments with non-declarative conclusions, as well as arguments with parenthetical remarks. A comprehensive account of arguments requires imposing a distinctive tree-like structure on contexts. We show how to extend our account to accommodate modal subordination and and di fferent flavors of `therefore'. (shrink)

Graham Priest has recently proposed a solution to the problem of the One and the Many which involves inconsistent objects and a non-transitive identity relation. We show that his solution entails either that the object everything is identical with the object nothing or that they are mutual parts; depending on whether Priest goes for an extensional or a non-extensional mereology.

There is a profound, but frequently ignored relationship between the classical notion of logical consequence (formal implication) and material implication. The first repeats the patterns of the latter, but with a wider modal reach. It is argued that this kinship between formal and material implication simply means that they express the same variety of implication, but differ in scope. Formal implication is unrestricted material implication. This apparently innocuous observation has some significant corollaries: (1) conditionals are not connectives, but arguments; (2) (...) the traditional examples of valid argumentative forms are metalogical principles that express the properties of logical consequence; (3) formal logic is not a useful guide to detect valid arguments in the real world; (4) it is incoherent to propose alternatives to the material implication while accepting the classical properties of formal implication; (5) the counter-examples to classical argumentative forms and conditional puzzles are unsound. (shrink)

View more Abstract If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. ‘Televisions are televisions’ and ‘TVs are televisions’ neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical (...) sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The ‘=’ of mathematics is an objectual relation between numbers; the ‘=’ of logic marks a syntactic relation of coreferring terms. -/- . (shrink)

Este texto presenta, y en cierta medida analiza, ambigüedades existentes en textos de lógica y filosofía de la lógica (como la interpretación de los llamados principios, postulados, leyes o verdades lógicas, la coexistencia de la tesis de que toda relación presupone la existencia de al menos dos relata y la de que una cosa puede relacionarse consigo misma, o la llamada "paradoja del mentiroso") bajo el supuesto de que, dado que la lógica no es anterior a la semántica, un análisis (...) semántico puede mostrar no solo que tal polisemia sugiere que ni la disciplina ni las teorías lógicas son tan rigurosas como se ha afirmado, sino también que varias afirmaciones de esas disciplinas carecen de sentido. (shrink)

This paper develops a probabilistic analysis of conditionals which hinges on a quantitative measure of evidential support. In order to spell out the interpreta- tion of ‘if’ suggested, we will compare it with two more familiar interpretations, the suppositional interpretation and the strict interpretation, within a formal framework which rests on fairly uncontroversial assumptions. As it will emerge, each of the three interpretations considered exhibits specific logical features that deserve separate consideration.

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)

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.

We argue that distinct conditionals—conditionals that are governed by different logics—are needed to formalize the rules of Truth Introduction and Truth Elimination. We show that revision theory, when enriched with the new conditionals, yields an attractive theory of truth. We go on to compare this theory with one recently proposed by Hartry Field.

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.

Is logic empirical? Is logic to be found in the world? Or is logic rather a convention, a product of conventions, part of the many rules that regulate the language game? Answers fall in either camp. We like the linguistic answer. In this paper, we want to analyze how a linguistic community would tackle the problem of developing a logic and show how the linguistic conventions adopted by the community determine the properties of the local logic. Then show how to (...) move from a notion of logic that varies from community to community to a notion of logic that is in a sense universal. The framework is conventional up to a point: we have sentences, atomic and composite, the connectives are interpreted, values are computed, and the value of a composite sentence is a function of the values of its subsentences. Less conventional is the use of a plurality of truth values, and the sharp distinction we draw between sentences and statements, in the spirit of the distinction between proposition and judgment that one may find in proof theory. The linguistic community will face many choices. What are the good ones, the ones to avoid? Are there, in some sense, optimal choices? These are the kind of issues we are addressing. Where do we end up? With some kind of universal bivalent logic, ironically enough. We start from an arbitrarily large number of truth values, atomic sentences and connectives, construct a generic many-valued logic, recover more or less the usual results and issues, and in the end it all comes down to a positive bivalent logic with two connectives, `and' and `or', as if logic is nothing more than a mere accounting of possibilities. (shrink)

We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .

Like '&', '=' is no term; it represents no extrasentential property. It marks an atomic, nonpredicative, declarative structure, sentences true solely by codesignation. Identity (its necessity and total reflexivity, its substitution rule, its metaphysical vacuity) is the objectual face of codesignation. The syntax demands pure reference, without predicative import for the asserted fact. 'Twain is Clemens' is about Twain, but nothing is predicated of him. Its informational value is in its 'metailed' semantic content: the fact of codesignation (that 'Twain' names (...) Clemens) that explains what fact it asserts and why it is necessary. Critiques of concepts of rigidity and elimination of singular terms result. (shrink)

We discuss a well-known puzzle about the lexicalization of logical operators in natural language, in particular connectives and quantifiers. Of the many logically possible operators, only few appear in the lexicon of natural languages: the connectives in English, for example, are conjunction _and_, disjunction _or_, and negated disjunction _nor_; the lexical quantifiers are _all, some_ and _no_. The logically possible nand (negated conjunction) and Nall (negated universal) are not expressed by lexical entries in English, nor in any natural language. Moreover, (...) the lexicalized operators are all upward or downward monotone, an observation known as the Monotonicity Universal. We propose a logical explanation of lexical gaps and of the Monotonicity Universal, based on the dynamic behaviour of connectives and quantifiers. We define update potentials for logical operators as procedures to modify the context, under the assumption that an update by \( \phi \) depends on the logical form of \( \phi \) and on the speech act performed: assertion or rejection. We conjecture that the adequacy of update potentials determines the limits of lexicalizability for logical operators in natural language. Finally, we show that on this framework the Monotonicity Universal follows from the logical properties of the updates that correspond to each operator. (shrink)

In this article, we provide three generators of propositional formulae for arbitrary languages, which uniformly sample three different formulae spaces. They take the same three parameters as input, namely, a desired depth, a set of atomics and a set of logical constants (with specified arities). The first generator returns formulae of exactly the given depth, using all or some of the propositional letters. The second does the same but samples up-to the given depth. The third generator outputs formulae with exactly (...) the desired depth and all the atomics in the set. To make the generators uniform (i.e. to make them return every formula in their space with the same probability), we will prove various cardinality results about those spaces. (shrink)

The use of the symbol ∨ for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or”, vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead (...) and Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

I argue against the skeptical epistemological view exemplified by the Groarkes that “all theories of informal argument must face the regress problem.” It is true that in our theoretical representations of reasoning, infinite regresses of self-justification regularly and inadvertently arise with respect to each of the RSA criteria for argument cogency (the premises are to be relevant, sufficient, and acceptable). But they arise needlessly, by confusing an RSA criterion with argument content, usually premise material.

We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical.

In this paper, we approach the problem of classical recapture for LP and K3 by using normality operators. These generalize the consistency and determinedness operators from Logics of Formal Inconsistency and Underterminedness, by expressing, in any many-valued logic, that a given formula has a classical truth value (0 or 1). In particular, in the rst part of the paper we introduce the logics LPe and Ke3 , which extends LP and K3 with normality operators, and we establish a classical recapture (...) result based on the two logics. In the second part of the paper, we compare the approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. Finally, we discuss technical issues connecting LPe and Ke3 to the tradition of Logics of Formal Inconsistency and Underterminedness. (shrink)

In 1926, Ernst Mally, an Austrian logician, has introduced a system of deontic logic in which he has proposed three fundamental distinctions which proved to be important in the context of the further development of the logic of norms. It is argued that in his philosophical considerations Mally has introduced a number of important distinctions concerning the very concept of norm, but by getting them confused in introducing the subsequent formalisms he failed to formally preserve them. In some of his (...) philosophically made distinctions Mally apparently foresaw contemporary trends in logic of norms. To some extent this particular feature of Mally’s system open wide opportunities to reconstruct –– with the corresponding renovations — his illformed Deontik into many nowadays known systems of logic of norms and thus provides a fertile ground for this kind of research. (shrink)

Despite the wide acceptance of standard modal logic, there has always been a temptation to think that ordinary modal discourse may be correctly analyzed and adequately represented in terms of predicates rather than in terms of operators. The aim of the formal model outlined in this paper is to capture what I take to be the only plausible sense in which ‘possible’ and ‘necessary’ can be treated as predicates. The model is built by enriching the language of standard modal logic (...) with a quantificational apparatus that is “substitutional” rather than “objectual”, and by obtaining from the language so enriched another language in which constants for such predicates apply to singular terms that stand for propositions. (shrink)

One logic or many? I say—many. Or rather, I say there is one logic for each way of specifying the class of all possible circumstances, or models, i.e., all ways of interpreting a given language. But because there is no unique way of doing this, I say there is no unique logic except in a relative sense. Indeed, given any two competing logical theories T1 and T2 (in the same language) one could always consider their common core, T, and settle (...) on that theory. So, given any language L, one could settle on the minimal logic T0 corresponding to the common core shared by all competitors. That would be a way of resisting relativism, as long as one is willing to redraw the bounds of logic accordingly. However, such a minimal theory T0 may be empty if the syntax of L contains no special ingredients the interpretation of which is independent of the specification of the relevant L-models. And generally—I argue—this is indeed the case. (shrink)

This self-contained lecture examines uses and misuses of the adverb conversely with special attention to logic and logic-related fields. Sometimes adding conversely after a conjunction such as and signals redundantly that a converse of what preceded will follow. -/- (1) Tarski read Church and, conversely, Church read Tarski. -/- In such cases, conversely serves as an extrapropositional constituent of the sentence in which it occurs: deleting conversely doesn’t change the proposition expressed. Nevertheless it does introduce new implicatures: a speaker would (...) implicate belief that the second sentence expresses a converse of what the first expresses. Perhaps because such usage is familiar, the word conversely can be used as “sentential pronoun”—or prosentence—representing a sentence expressing a converse of what the preceding sentence expresses. -/- (2) Tarski read Church and conversely. -/- This would be understood as expressing the proposition expressed by (1). Prosentential usage introduces ambiguity when the initial proposition has more than one converse. Confusion can occur if the initial proposition has non-equivalent converses. -/- Every proposition that is the negation of a false proposition is true and conversely. -/- One sense implies that every proposition that is the negation of a true proposition is false, which is true of course. But another sense, probably more likely, implies that every proposition that is true is the negation of a false proposition, which is false: the proposition that one precedes two is not a negation and thus is not the negation of a false proposition. The above also applies to synonyms of conversely such as vice versa. Although prosentence has no synonym, extrapropositional constituents are sometimes called redundant rhetoric, filler, or expletive. Authors discussed include Aristotle, Boole, De Morgan, Peirce, Frege, Russell, Tarski, and Church. END OF PUBLISHED ABSTRACT -/- See also: Corcoran, John. 2015. Converses, inner and outer. 2015. Cambridge Dictionary of Philosophy, third edition, Robert Audi (editor). Cambridge: Cambridge UP. https://www.academia.edu/10396347/Corcoran_s_27_entries_in_the_1999_second_edition_Audi_s_Cambridge_ Dictionary_of_Philosophy . (shrink)

This paper consider Prior's connective Tonk from a particular bilateralist perspective. I show that there is a natural perspective from which we can see Tonk and its ilk as perfectly well-defined pieces of vocabulary; there is no need for restrictions to bar things like Tonk.

In this essay I give a complete join semi-lattice of possible display-equivalence schemes for Display Logic, using the standard connectives, and leaving fixed only the schemes governing the star. In addition to proving the completeness of this list, I offer a discussion of the basic properties of these schemes.

No existing conditional semantics captures the dual role of 'if' in embedded interrogatives — 'X wonders if p' — and conditionals. This paper presses the importance and extent of this challenge, linking it to cross-linguistic patterns and other phenomena involving conditionals. Among these other phenomena are conditionals with multiple 'if'-clauses in the antecedent — 'if p and if q, then r' — and relevance conditionals — 'if you are hungry, there is food in the cupboard'. Both phenomena are shown to (...) be problematic for existing analyses. Surprisingly, the decomposition of conditionals needed to capture the link with interrogatives provides a new analysis that captures all three phenomena. The model-theoretic semantics offered here relies on a dynamic conception of meaning and compositionality, a feature discussed throughout. (shrink)

Motivated by H. Curry’s well-known objection and by a proposal of L. Henkin, this article introduces the positive tableaux, a form of tableau calculus without refutation based upon the idea of implicational triviality. The completeness of the method is proven, which establishes a new decision procedure for the (classical) positive propositional logic. We also introduce the concept of paratriviality in order to contribute to the question of paradoxes and limitations imposed by the behavior of classical implication.

The aim of this paper is to clarify the truth-functional interpretation of the logical connective of the material implication. The importance of such clarification lies in the fact that it allows avoiding the supposed paradoxes introduced by C. I. Lewis (1918). I argue that an adequate understanding of the history and purposes of logic is enough to dissolve them away. The defense is based on an exposition of propositional compositionalism. To compare, I also present Stalnaker’s (1968) alternative that seeks to (...) explain the implicit concept of conditionality of conditionals assertions. In order to reinforce my aim I expound the twelve different meanings of ‘implication’ stated by Corcoran (1993) in order to hold that a proper understanding of these meanings allows us to avoid ambiguities and confusions. Finally, I expect to show that despite the possible alternatives, compositionalism is still the best explanation for the truth-functionality account of logical connectives. (shrink)

This article has one aim, to reject the claim that negation is semantically ambiguous. The first section presents the putative incompatibility between truth-value gaps and the truth-schema; the second section presents the motivation for the ambiguity thesis; the third section summarizes arguments against the claim that natural language negation is semantically ambiguous; and the fourth section indicates the problems of an introduction of two distinct negation operators in natural language.

Proves two simple identities relating the biconditional and exclusive disjunction. -/- The PDF has been made available gratis by the publisher.

I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.

The theory of imperatives is philosophically relevant since in building it — some of the long standing problems need to be addressed, and presumably some new ones are waiting to be discovered. The relevance of the theory of imperatives for philosophical research is remarkable, but usually recognized only within the field of practical philosophy. Nevertheless, the emphasis can be put on problems of theoretical philosophy. Proper understanding of imperatives is likely to raise doubts about some of our deeply entrenched and (...) tacit presumptions. In philosophy of language it is the presumption that declaratives provide the paradigm for sentence form; in philosophy of science it is the belief that theory construction is independent from the language practice, in logic it is the conviction that logical meaning relations are constituted out of logical terminology, in ontology it is the view that language use is free from ontological commitments. The list is not exhaustive; it includes only those presumptions that this paper concerns. (shrink)

This paper defines the form of prior knowledge that is required for sound inferences by analogy and single-instance generalizations, in both logical and probabilistic reasoning. In the logical case, the first order determination rule defined in Davies (1985) is shown to solve both the justification and non-redundancy problems for analogical inference. The statistical analogue of determination that is put forward is termed 'uniformity'. Based on the semantics of determination and uniformity, a third notion of "relevance" is defined, both logically and (...) probabilistically. The statistical relevance of one function in determining another is put forward as a way of defining the value of information: The statistical relevance of a function F to a function G is the absolute value of the change in one's information about the value of G afforded by specifying the value of F. This theory provides normative justifications for conclusions projected by analogy from one case to another, and for generalization from an instance to a rule. The soundness of such conclusions, in either the logical or the probabilistic case, can be identified with the extent to which the corresponding criteria (determination and uniformity) actually hold for the features being related. (shrink)

Italian translation of "On Logical Relativity" (2002), by Luca Morena.

This work gives an extended presentation of the treatment of variable-binding operators adumbrated in [3:1993d]. Illustrative examples include elementary languages with quantifiers and lambda-equipped categorial languages. Some remarks are also offered to illustrate the philosophical import of the resulting picture. Particularly, a certain conception of logic emerges from the account: the view that logics are true theories in the model-theoretic sense, i.e. the result of selecting a certain class of models as the only “admissible” interpretation structures (for a given language).