Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this article is to compare Tonk with some hitherto unnoticed nasty connectives, which are defined in semantic terms. I first use many-valued truth-tables for classical sentential logic to define a nasty connective, Knot. I then argue that we should refuse to add Knot to our language. And I show that this reverses the standard dialectic surrounding Tonk, and yields a novel solution to (...) the problem of many-valued truth-tables for classical sentential logic. I close by outlining the technicalities surrounding nasty connectives on many-valued truth-tables. (shrink)

Natural language does not express all connectives definable in classical logic as simple lexical items. Coordination in English is expressed by conjunction and, disjunction or, and negated disjunction nor. Other languages pattern similarly. Non-lexicalized connectives are typically expressed compositionally: in English, negated conjunction is typically expressed by combining negation and conjunction (not both). This is surprising: if $$\wedge $$ ∧ and $$\vee $$ ∨ are duals, and the negation of the latter can be expressed lexically (nor), why not (...) the negation of the former? I present a two-tiered model of the semantics of the binary connectives. The first tier captures the expressive power of the lexicon: it is a bilateral state-based semantics that, under a restriction, can express all and only the distinctions that can be expressed by the lexicon of natural language (and, or, nor). This first tier is characterized by rejection as non-assertion and a Neglect Zero assumption. The second tier is obtained by dropping the Neglect Zero assumption and enforcing a stronger notion of rejection, thereby recovering classical logic and thus definitions for all Boolean connectives. On the two-tiered model, we distinguish the limited expressive resources of the lexicon and the greater combinatorial expressive power of the language as a whole. This gives us a logic-based account of compositionality for the Boolean fragment of the language. (shrink)

This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider (...) extensions of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property. (shrink)

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what (...) we call Gentzen-regular connectives. For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations. (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)

In this paper, I argue that the temporal connective prima (‘before’) is a comparative adverb. The argument is based on a number of grammatical facts from Italian, showing that there is an asymmetry between prima and dopo (‘after’). On the ground of their divergent behaviour, I suggest that dopo has a different grammatical status from prima. I propose a semantic treatment for prima that is based on an independently motivated analysis of comparatives which can be traced back to Seuren (in: (...) Kiefer and Ruwet (eds.) Generative grammar in Europe, 1973). Dopo is analyzed instead as an atomic two-place predicate which contributes a binary relation over events to the sentence meaning. The different semantic treatments of the two connectives provide an explanation for the grammatical asymmetries considered at the outset; interestingly, they also shed some light on other asymmetries between prima and dopo, which are known to hold for the English temporal connectives before and after as well: these asymmetries are related to the veridicality properties, the distribution of NPIs, and the logical properties of these connectives first described in Anscombe (Philos Rev 73:3–24, 1964). (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much (...) more, I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

While the role of discourse connectives has long been acknowledged in argumentative frameworks, these approaches often take a coarse-grained approach to connectives, treating them as a unified group having similar effects on argumentation. Based on an empirical study of the straw man fallacy, we argue that a more fine-grained approach is needed to explain the role of each connective and illustrate their specificities. We first present an original corpus study detailing the main features of four causal connectives (...) in French that speakers routinely use to attribute meaning to another speaker, which is a key element of straw man fallacies. We then assess the influence of each of these connectives in a series of controlled experiments. Our results indicate each connective has different effects for the persuasiveness of straw man fallacies, and that these effects can be explained by differences in their semantic profile, as evidenced in our corpus study. Taken together, our results demonstrate that connectives are important for argumentation but should be analyzed individually, and that the study of fallacies should include a fine-grained analysis of the linguistic elements typically used in their formulation. (shrink)

In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using $n$-sided sequent calculus. These systems are understood as a further (...) development of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt. (shrink)

The correct use of connectives has great influence on language learners’ writing proficiency, while errors of connectives are common in foreign learners’ interlanguages. This study examines the types of errors that occur in native English-speaking learners’ Chinese writing, the possible causes for the errors, and the learners’ consequent learning strategies. The present research adopted corpora investigation, questionnaire survey, and focus-group interviews to examine the error types, causes of identified errors, and related learning strategies. Data analysis indicated that: the (...) main error types made by native English-speaking learners from high to low are misuse, overuse, mismatch, misplacement, and underuse of connectives; causes related to intralingual transfer greatly contributes to the presence of errors; and memory, social, and cognitive strategies were the most preferred, followed by metacognitive and compensation strategies, and then by effective strategies which were the least preferred. These findings showed that different strategies can be employed to cope with different errors in writing. The study further suggests that teachers and educators need to help native English-speaking learners find strategies that work best for them in terms of learning Chinese connectives. (shrink)

Language employs various coordinators to connect propositions, a subset of which are “logical” in nature and thus analogous to the truth operators of formal logic. We here focus on two linguistic connectives and their negations: conjunction _and_ and (inclusive) disjunction _or_. Linguistic connectives exhibit a truth-conditional component as part of their meaning (their semantics), but their use in context can give rise to various implicatures and presuppositions (the domain of pragmatics) as well as to inferences that go beyond (...) semantic/pragmatic properties (the result of reasoning processes). We provide a comprehensive review of the role of the logical connectives in language and argue that three sets of factors—semantic, pragmatic, and those related to reasoning—are separate and separable, though some details may differ cross-linguistically. As a way to showcase the argument, we present two experiments in language comprehension in Spanish wherein pragmatic content was minimised and reasoning processes neutered, thus potentially highlighting what might be the default meanings of the connectives under study. In Experiment 1 we show that the conjunctive reading of inclusive disjunction is available in positive contexts other than in syntactically intricate cases such as downward entailing and free choice contexts, contrary to what has been claimed in the literature. In Experiment 2 we show that negated conjunctions and disjunctions in Spanish can easily receive the same interpretation when contrasted against the same context and, moreover, that these interpretations match those available in English, despite claims from the literature that linguistic connectives and local negation interact differently in English and Romance languages. (shrink)

In this paper, we show (among other things) that the conditional in Frege's Begriffsschrift is ambiguous.

Challenges to classical logic have emerged from several sources. According to recent work, the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's (...) recent approach to vagueness. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the non-classicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic. (shrink)

This paper focuses on the interpretation of the Italian approximative adverb quasi 'almost' by primarily looking at cases in which it modifies temporal connectives, a domain which, to our knowledge, has been largely unexplored thus far. Consideration of this domain supports the need for a scalar account of the semantics of quasi (close in spirit to Hitzeman's semantic analysis of almost, in: Canakis et al. (eds) Papers from the 28th regional meeting of the Chicago Linguistic Society, 1992). When paired (...) with suitable analyses of temporal connectives, such an account can provide a simple explanation of the patterns of implication that are observed when quasi modifies locational (e. g. quando 'when'), directional (e. g. fino 'until' and da 'since'), and event-sequencing temporal connectives (e. g. prima 'before' and dopo ' after'). A challenging empirical phenomenon that is observed is a contrast between the modification of fino and da by quasi, on the one hand, and the modification of prima and dopo by the same adverb, on the other. While quasi fino and quasi da behave symmetrically, a puzzling asymmetry is observed between quasi prima and quasi dopo. To explain the asymmetry, we propose an analysis of prima and dopo on which the former has the meaning of the temporal comparative più presto 'earlier', while the latter is seen as an atomic predicate denoting temporal succession between events (Del Prete, Nat Lang Semantics 16: 157-203, 2008). We show that the same pattern of implication observed for quasi prima is attested when quasi modifies overt comparatives, and propose a pragmatic analysis of this pattern that uniformly applies to both cases, thus providing new evidence for the claim that prima is underlyingly a comparative. A major point of this paper is a discussion of the notion of scale which is relevant for the semantics of quasi', in particular, we show that the notion of Horn (entailment-based) scale is not well-suited for handling modification of temporal connectives, and that a more general notion of scale is required in order to provide a uniform analysis of quasi as a cross-categorial modifier. (shrink)

A set of propositional connectives is said to be functionally complete if all propositional formulae can be expressed using only connectives from that set. In this paper we give sufficient and necessary conditions for a one-element set of propositional connectives to be functionally complete. These conditions provide a simple and elegant characterization of functionally complete one-element sets of propositional connectives.

Logic, Logical connectives, Verbal disputes, Proof theory, Non-classical logic.

The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some (...) additional remarks on matrix methods and on alternative terminology. Background: The size of this lattice was underestimated in [3]; some missing cases were noted in [4] in the course of correcting an example from [3] purporting to show the non-distributivity of the lattice. All the 'missing cases' (as well as those originally noted) are covered here. (The present discussion is self-contained.). (shrink)

While corpus studies have shown that discourse connectives that convey the same coherence relation can display subtle differences, research on online discourse processing has only focused on a rather limited set of connectives. Yet, different connectives – for example, rare or polyfunctional ones – might elicit different reading patterns. In order to explore this assumption, we test the robustness of discourse processing for French native speakers by measuring the way they process causal and concessive sentences that are (...) conveyed by either an appropriate or inappropriate connective. Throughout three experiments, we change important characteristics of the connectives: we first test frequently used connectives, secondly less frequent ones, and finally less frequent connectives that are polyfunctional and for which different functions clearly compete. Our results show that the processing for incoherent items was affected for all connectives, however readers showed altered reading fluency when infrequent connectives were used. We conclude that discourse processing is quite robust and that readers are able to insert meaning conveyed by rare connectives while still showing the highest reading ease with frequent connectives. (shrink)

Using 129 natural conversations and 185 episodes of television drama conversations as well as the theoretical frameworks of usage-based theory and grammaticalization, I investigate two forms of ‘or’ in Korean, tunci and kena. Generally believed to be largely interchangeable, these two forms’ actual usages have never been compared. I demonstrate that the two are selectively used in conversation, and propose that three types of factor influence the selection. The first factors are genre and setting. In formal settings and formal descriptive (...) writing, kena is preferred. The second factor is the presence of negative markers: while 43% of kena tokens were used in negative sentences, only one token of tunci was. The third factor is the distinction between ‘propositive’ tunci and ‘descriptive’ kena. In conversation, tunci occurs mainly in making suggestions and presenting plans, and hence is oriented toward the future tense and the irrealis domain. Kena occurs mainly in describing present and past situations, facts, or premises, and hence is oriented toward the present and past tense and the realis domain. This study confirms the usage-based theory’s argument that speakers possess extensive procedural knowledge of language: Korean speakers may not be conscious of it, but they differentiate tunci and kena with high regularity. Lastly, I explain their differences based on their history. The older form, kena, dates back to the 8th century, whereas tunci became widespread only in the 19th century. Because tunci retains its original semantics of probability, tunci tends to occur in presenting suggestions and plans and not in factual descriptions. (shrink)

Dans cet article, nous appuyant sur un corpus de 3 444 075 tweets correspondant à 44 107 210 tokens (mots, signes de ponctuation, émojis, émoticônes, etc.) recueillis en décembre 2016, nous nous intéressons aux procédés de segmentation à l’œuvre dans les tweets. Après avoir évoqué certaines caractéristiques de ces écrits particuliers, nous rappelons les procédés généraux de segmentation à l’écrit : les signes de ponctuation et les connecteurs. Nous nous penchons ensuite sur la segmentation opérée dans les tweets par ces (...) deux procédés généraux. Enfin, nous montrons que les émoticônes et les émojis constituent des procédés spécifiques permettant de diversifier les stratégies de segmentation des utilisateurs de tweets (et d’autres écrits numériques, tels les SMS et les courriels). (shrink)

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in (...) all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (...) (§ 2) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective). (shrink)

ABSTRACT In this paper we present a new semantic approach for propositional linear temporal logic with discrete time, strongly based in the well-order of IN (the set of natural numbers). We consider temporal connectives which express precedence, posteriority and simultaneity, and they provide a family of expressively complete temporal logics. The selection of the new semantics and connectives used in this work was principally to obtain a suitable executable temporal logic, which can be used for the specification and (...) control of process behaviour in discrete time in a similar way to thai presented by D. Gabbay in [GAB 89], Our new approach has two advantages: firstly, the connectives are defined intuitively so that they have interpretations which relate to properties of interest in real systems; secondly, it provides a new semantics that facilitates simpler proofs of many valid formulas and metatheorems. To confirm this second advantage, we use our semantics to give a formal proof of the Separation Theorem for one of these logics (specifically LN3). The great interest of this Separation Theorem for the executable temporal logic is emphasised by D. Gabbay in [GAB 89], [GAB 88]. (shrink)

The choice of connectives for many valued propositional logics suitable for theoretical and applicational interests in most cases is an open problem up to now. We will not offer a general solution here, but support the point of view of some recent developments in fuzzy set theory that the triangular norms – t-norms for short – of Schweizer/Sklar [3] and the ϕ-operators of Pedrycz [2] represent quite general classes of connectives at least for many valued logics with truth (...) value set the real interval [0,1] . Special classes of t-norms and general properties of them have been discussed in some papers just recently, e.g. by Dombi [1], Weber [4], Yager [5] to mention only a few of them. (shrink)

T?s?, a thirteenth century logician writing in Arabic, uses two logical connectives to build up molecular propositions: ?if-then?, and ?either-or?. By referring to a dichotomous Tree, T?s? shows how to choose the proper disjunction relative to the terms in the disjuncts. He also discusses the disjunctive propositions which follow from a conditional proposition.

In a previous paper we investigated the extraction of proof-theoretic properties of natural deduction derivations from their impredicative translation into System F. Our key idea was to introduce an extended equational theory for System F codifying at a syntactic level some properties found in parametric models of polymorphic type theory. A different approach to extract proof-theoretic properties of natural deduction derivations was proposed in a recent series of papers on the basis of an embedding of intuitionistic propositional logic into a (...) predicative fragment of System F, called atomic System F. In this paper we show that this approach finds a general explanation within our equational study of second-order natural deduction, and a clear semantic justification in terms of parametricity. (shrink)

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex (...) and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions. (shrink)

Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two (...) implications in the more familiar language of modal logic. In §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)