Negation

Quantification, Negation, and Focus: Challenges at the Conceptual-Intentional Semantic Interface Tista Bagchi National Institute of Science, Technology, and Development Studies (NISTADS) and the University of Delhi Since the proposal of Logical Form (LF) was put forward by Robert May in his 1977 MIT doctoral dissertation and was subsequently adopted into the overall architecture of language as conceived under Government-Binding Theory (Chomsky 1981), there has been a steady research effort to determine the nature of LF in language in light of structurally (...) diverse languages around the world, which has ultimately contributed to the reinterpretation of LF as a Conceptual-Intentional (C-I) interface level between the computational syntactic component of the faculty of language and one or more interpretive faculties of the human mind. While this has opened up further possibilities of research in phenomena such as quantifier scope and scope interactions between negation, quantification, and focus, it has also given rise to a few real challenges to linguistic theory as well. Some of these are: (i) the split between lexical meaning – a matter supposedly belonging to the phase-wise selection of lexical arrays – and issues of semantic interpretation that arise purely from binding and scope phenomena (Mukherji 2010); (ii) partially relatedly, the level at which theta role assignment can be argued to take place, an issue that is taken up by me in Bagchi (2007); and (iii) how supposedly “pure” scopal phenomena relating to quantifiers, negation, and emphasizing expressions such as only and even (comparable to, e.g., Urdu/Hindi hii and bhii, Bangla –i and –o) also have dimensions of both focus and discourse reference. While recognizing all of these challenges, this talk aims to highlight particularly challenge (iii), both in terms of scholarship in the past and for the rich prospects for research on languages of south Asia with the semantics of quantification, negation, and focus in view. The scholarship of the past that I seek to relate this issue to is where, parallel to (and largely independently of) the research on LF that had been happening, Barwise and Cooper were developing their influential view of noun phrases as generalized quantifiers, culminating in their key 1981 article “Generalized Quantifiers and Natural Language” while, independently, McCawley, in his 1981 book Everything that Linguists have Always Wanted to Know about Logic, established through argumentation that all noun phrases semantically behave like generalized quantified expressions (further elaborated by him in the second – 1994 – revised edition of his book). I seek to demonstrate, based on limited data analysis from selected languages of south Asia, that our current understanding of quantification, negation, and focus under the Minimalist view owes something significant to the two major, but now largely marginalized, works of scholarship, and that for the way forward it is essential to adopt a more formal-semantic approach as adopted by them and also by later works such as Denis Bouchard’s (1995) The Semantics of Syntax, Mats Rooth’s work on focus (e.g., Rooth 1996, “Focus” in Shalom Lappin’s Handbook of Contemporary Semantic Theory), Heim and Kratzer’s Semantics in Generative Grammar (1998), and Yoad Winter’s (2002) Linguistic Inquiry article on semantic number, to cite just a few instances. (shrink)

In the history of philosophy, the distinction between positive and negative predication has been collapsed. The collapse has caused us to search for a way through Parmenides’ gate: we have constructed scaffolding to see over its boundaries. Kant gave us the distinction between conceptual and non-conceptual knowledge; Hegel gave us determinate negation; Frege gave us the negation stroke; Husserl gave us bracketing and disappointment; G. Spencer-Brown gave us a calculus of distinction. Despite this, we find ourselves—alongside Wittgenstein— wondering how it (...) can be the case that we are able to think what is not the case. We are puzzled by the fact that we can think nothing without the thought lacking content. I suggest that this confusion is unnecessary. Throughout this work, I exercise the thought that the distinction between positive and negative predication is mistaken. I rather suggest that to think what is or is not the case necessarily depends upon an exemplar distinction that is experientially recognized and learned in the world. All the same, our capacity to understand natural distinction likewise depends upon our practical and pervasively conceptual capacities to express it in language. I conclude by suggesting that what makes positive and negative predication distinct is a fundamental unity in conceptual form through its capacity to play distinct functional roles in linguistic practice. (shrink)

Intensional evidence is any reason to accept a proposition that is not the truth values of the proposition accepted or, if it is a complex proposition, is not the truth values of its propositional contents. Extensional evidence is non-intensional evidence. Someone can accept a complex proposition, but deny its logical consequences when her acceptance is based on intensional evidence, while the logical consequences of the proposition presuppose the acceptance of extensional evidence, e.g., she can refuse the logical consequence of a (...) proposition she accepts because she doesn’t know what are the truth-values of its propositional contents. This tension motivates counterexamples to the negation of conditionals, the propositional analysis of conditionals, hypothetical syllogism, contraposition and or-to-if. It is argued that these counterexamples are non-starters because they rely on a mix of intensionally based premises and extensionally based conclusions. Instead, a genuine counterexample to classical argumentative forms should present circumstances where an intuitively true and extensionally based premise leads to an intuitively false conclusion that is also extensionally based. The other point is that evidentiary concerns about intensionally based beliefs should be constrained by the truth conditions of propositions presented by classical logic, which are nothing more than requirements of coherence in distributions of truth value. It is argued that this restriction also dissolves some known puzzles such as conditional stand-offs, the Adams pair, the opt-out property and the burglar’s puzzle. (shrink)

The principle of Conditional Excluded Middle has been a matter of longstanding controversy in both semantics and metaphysics. The principle suggests (among other things) that for any coin that isn't flipped, there is a fact of the matter about how it would have landed if it had been flipped: either it would have landed heads, or it would have landed tails. This view has gained support from linguistic evidence indicating that ‘would’ commutes with negation (e.g., ‘not: if A, would C’ (...) is equivalent to ‘if A, would not C’). There is, however, a long list of operators that similarly appear to commute with negation, even though the corresponding excluded middle principles are indefensible. We suggest that the data supporting Conditional Excluded Middle is best explained as a pragmatic effect. (shrink)

According to perspectival realism, reality is (at least partially) constituted by “purely perspectival” facts, that is, facts that appear to describe reality only from a given “perspective”. Fragmentalism is a form of perspectival realism that maintains both that no perspective is privileged and that perspectival facts constitute reality absolutely. Assuming that reality is sufficiently variegated, fragmentalism entails that reality is absolutely constituted by incompatible facts. Given that incompatible facts can never obtain together, reality must be divided into a plurality of (...) “fragments” that never compose a coherent, unitary whole. The main aim of this paper is to provide the first rigorous, detailed map of fragmentalism and its rivals. First, we will argue that fragmentalism is best construed as having two “unitist” (that is, non-fragmentalist) boundaries, which we will call ‘coherent’ and ‘incoherent unitism’. Second, we will present what we take to be the six main viable versions of fragmentalism and show how they can be categorized according to whether they take the logical operations of conjunction and negation to be either “local” or “global”. (shrink)

This paper investigates the logic of compatibility as a ground for the logic of conditionals. We identify a family of principles expressing key properties of compatibility, which can be coherently ordered. Assuming that conditionals are definable in terms of incompatibility — the negation of compatibility — each of the principles identified yields corresponding principles governing conditionals. Clarifying these derivability relations provides a new perspective on several existing accounts of conditionals.

Fragmentalism allows incompatible facts to constitute reality in an absolute manner, provided that they fail to obtain together. In recent years, the view has been extensively discussed, with a focus on its formalisation in model-theoretic terms. This paper focuses on three formalisations: Lipman’s approach, the subvaluationist interpretation, and a novel view that has been so far overlooked. The aim of the paper is to explore the application of these formalisations to the alethic modal case. This logical exploration will allow us (...) to study (i) cases of metaphysical incompatibility between modal facts and (ii) cases of modal dialetheias. In turn, this will enrich our understanding of the role of impossibility in the fragmentalist framework. (shrink)

In their terrific book Reasons for Logic, Logic for Reasons, Ulf Hlobil and Robert Brandom defend a normative-pragmatic interpretation of implication and incompatibility cashed out in bilateral terms. Using epistemic modal cases, I argue that Hlobil and Brandom’s normative-pragmatic interpretation of implication fails to account for the force of consequence and to provide an extensionally adequate characterization of implication. I also show that the same cases cause trouble for Hlobil and Brandom’s condition relating negation to incompatibility. I argue, however, that (...) these problems can be avoided by switching from a bilateral to a multilateral setting. (shrink)

In this paper, I distinguish and compare three kinds of logical expressivism. The first, reminiscent of attitude expressivism in meta-ethics, holds that logic is expressive in that logical vocabulary serves to express attitudes. For instance, traditional attitude expressivism about negation, going back to the work of Frank Plumpton Ramsey, Huw Price and others, holds that 'not' expresses disbelief. The second kind of logical expressivism, reminiscent of deflationism about truth and championed by Robert Brandom, holds that logic is expressive in that (...) logical vocabulary serves to make explicit—typically, by expressing them as contents of assertions—the commitments that are implicit in our discursive practices. For instance, content expressivism about the conditional holds that 'if' expresses as content commitment to the goodness of certain inferential moves. The third kind of logical expressivism, and the one I will be arguing for, holds that, in a sense, logic is expressive in both ways: logical vocabulary serves to make explicit commitments to expressions of attitudes. (shrink)

We develop a novel solution to the negation version of the Frege-Geach problem by taking up recent insights from the bilateral programme in logic. Bilateralists derive the meaning of negation from a primitive *B-type* inconsistency involving the attitudes of assent and dissent. Some may demand an explanation of this inconsistency in simpler terms, but we argue that bilateralism’s assumptions are no less explanatory than those of *A-type* semantics that only require a single primitive attitude, but must stipulate inconsistency elsewhere. Based (...) on these insights, we develop a version of B-type expressivism called *inferential expressivism*. This is a novel semantic framework that characterises meanings by inferential roles that define which *attitudes* one can *infer* from the use of terms. We apply this framework to normative vocabulary, thereby solving the Frege-Geach problem generally and comprehensively. Our account moreover includes a semantics for epistemic modals, thereby also explaining normative terms under epistemic modals. (shrink)

This book brings together contributions from philosophers interested in logic and logicians with a philosophical orientation to address a variety of logical and philosophical topics of current interest, including modal logic, intuitionistic logic, relevant logic, substructural logic, many-value logic, formal semantics, proof theory, and paradox. In particular, it provides the state of the art in three major aspects of current research in philosophical logic: negation, modality, and proof. New frameworks are proposed, e.g., meta sequent, unified inductive logic, generalized Clemens semantics, (...) etc. The variety of topics and issues discussed in this volume is of interest to readers from a wide range of disciplines, such as logic, semantics, and computer science, philosophy, linguistics, artificial intelligence, and beyond. (shrink)

Jc Beall argues that if FDE is logic proper, then there is no logical negation. This claim is largely based on the fact that, in standard proof systems for FDE, there are no stand-alone negation rules that suffice to capture the behavior of negation. In this paper, I show that by adopting a bilateral proof system for FDE, one can maintain that there is a logical negation, it is the very same logical negation that belongs to classical logic, and its (...) basic function is to flip-flop between assertion and denial. After laying out the bilateral proof systems on which this claim is technically based, I develop the conception of assertion and denial on which this claim is philosophically based, responding to a number of objections. I conclude by considering the possibility of a different, so-called "Boolean" negation, in the context of this bilateral framework. (shrink)

Avowal expressivism holds that serious and competent utterances of first-person, present-tense ascriptions of mental states – e.g. “I’m in pain,” “I love you,” “I believe that p” – characteristically function as explicit expressions of the very states they mention. I argue that this stance commits its adherents to a matching treatment of negative avowals (“disavowals”) such as utterances of “I’m not in pain” and “I don’t love you.” Drawing on parity with positive avowals and on the behavior of pure disavowals (...) that resist contrary-state and neg-raising analyses, I defend expressive denegation: disavowals can be expressive of the absence of the named state. The paper diagnoses why standard alternatives (hidden-avowal and purely descriptivist accounts) misclassify these cases, and shows how first-person authority and groundlessness extend to the negative domain. The result is a conditional commitment: endorsing avowal expressivism commits one to expressive denegation for negative avowals. (shrink)

Work in the field of feminist logic is still rather scarce and the field itself remains a contested area of study, but still, it is developing. One approach concentrates on analyzing logical systems with respect to structural features that may perpetuate sexism and oppression or, on the other hand, features that may be helpful for resisting and opposing these social phenomena. Upon this assumption, I want to investigate possible applications of queer feminist views on (philosophy of) logic with respect to (...) a very specific group, namely contradictory logics, i.e., logical systems containing contradictions in their set of theorems. I want to show that, on the one hand, the formal set-up of contradictory logics makes them well-suited from the perspectives of feminist logic and, on the other hand, that queer feminist theories provide a relevant, and so far undeveloped, conceptual motivation for contradictory logics. Thus, bringing together contradictory logics and queer feminist theories may prove fruitful both as a ‘real-life’ motivation for these peripheral logical systems and as a formal basis for a philosophical field that is still characterized by a distrust of formalism. (shrink)

Emerging single-process propositional perspectives in psychology and philosophy have introduced the key idea that, much like their explicit (deliberately retrieved) counterparts, implicit (automatically retrieved) attitudes should be sensitive to logical operations such as negation. In the present project, we subject this idea to a particularly stringent test by probing not only whether the formation of implicit attitudes is sensitive to negation but also whether such sensitivity additionally reflects the distinction between easy-to-negate bipolar adjectives (those with clear antonyms, e.g., strong) and (...) difficult-to-negate unipolar adjectives (those without clear antonyms, e.g., unique). Across four experiments (three preregistered; total N = 6,707), we found (a) no significant difference in the formation of implicit attitudes in response to affirmed bipolar versus affirmed unipolar adjectives (β = 0.07); (b) modulation of the formation of implicit attitudes by whether adjectives were affirmed or negated (β = 0.67); and, critically, (c) stronger modulation of the formation of implicit attitudes by the negation of bipolar relative to unipolar adjectives (β = 0.23). These results generalized across three sets of adjectives, two implicit attitude measures (the Implicit Association Test and the Affect Misattribution Procedure), and two novel targets (social and nonsocial). Together, these data provide strong evidence to suggest that the formation of implicit attitudes is sensitive to logical operations, including not only whether an adjective is affirmed or negated but even whether adjectives are relatively easy or difficult to negate. We discuss theoretical implications for single-process and dual-process accounts of attitude formation and higher-order human cognition. (shrink)

The propositional connectives – and, or, not, if–then, etc. – are fundamental building blocks in formal as well as natural languages. In the Western tradition, they were first studied as such by the Stoics, and Propositional Logic is the fundament of practically all current systems of logic; every beginning logic course starts with it. Still, the proof theory and semantics of systems of propositional logic are far from trivial, and have been studied intensely by logicians in the last one and (...) a half centuries, not least in recent decades. It is actually a vast area of research, as witnessed by Lloyd Humberstone’s 1500 page tome The Connectives (2011), which overviews much of that research. Perhaps the most familiar recent work in this area concerns conditionals in formal and natural languages. The 4th Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning (TLLM 2024) also focused on the apparently simpler connectives expressing (various versions of) conjunction, disjunction, and negation. -/- Researchers working from a cross-linguistic perspective also study how the connectives are encoded in different languages, and ask whether classical logic is capable of capturing the variations and universals exhibited. Even in well-studied languages such as English, there are intricate phenomena that remain challenging for classical logic, including free choice disjunction, non-Boolean conjunction, and metalinguistic negation, to name just a few. There is also growing interest in the acquisition and processing of natural language connectives. In the context of the hotly discussed Large Language Models (LLMs), understanding connectives presents novel challenges that deserve in-depth exploration. (shrink)

The term 'covert mixed quotation' describes cases in which linguistic material is interpreted in the manner of mixed quotation — that is, used in addition to being mentioned — despite the superficial absence of any commonly recognized conventional devices indicating quotation. After developing a novel theory of mixed quotation, I show that positing covert mixed quotation allows us to give simple and unified treatments of a number of puzzling semantic phenomena, including the projective behavior of conventional implicature items embedded in (...) indirect speech reports and propositional attitude ascriptions, so-called 'c-monsters,' metalinguistic negation, metalinguistic negotiation, and 'in a sense' constructions. (shrink)

The ability to express negation in language may have been the result of an adaptive process. However, there are different accounts of adaptation in linguistics, and more than one of them may describe the case of negation. In this paper, I distinguish different versions of the claim that negation is adaptive and defend a proposal, based on recent work by Steinert-Threlkeld ( 2016 ) and Incurvati and Sbardolini ( 2021 ), on which negation is an indirect adaptation.

According to some critics, Aristotle’s elenctic defence (elenchos, elenchus) of the Law of Non-Contradiction (Metaphysics IV) would be ineffective because it viciously begs the question. After briefly recalling the elenctic refutation of the denier of the Law of Non-Contradiction, I will first focus on Filippo Costantini’s objection to the elenchus, which, in turn, is based on the dialetheic account of negation developed by Graham Priest. Then, I will argue that there is at least one reading of the elenchus that might (...) not be viciously question-begging. In doing so, I will leverage, reinterpret and adjust the distinction between two senses of epistemic dependence, offered by Noah Lemos and originally based on some thoughts about George Edward Moore’s ‘proof of an external world.’ The key point of my counter-objection to recover the elenchus is to use the distinction between a necessary-condition relation between propositions (p only if q) and a grounding relation between facts (the fact that an epistemic agent S believes that p is grounded in the fact that S believes that q), where p and q are the content of S’s beliefs. (shrink)

A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and a characterizing (...) frame property. While the identification of negation as impossibility and negation as unnecessity does solve Onishi’s problems, in general it nevertheless seems desirable to keep the two notions separate. The present paper addresses the first problem by introducing a reformulation of Onishi’s display sequent calculus in a language of sequents that incorporates Boolean negation. Instead of identifying negation as impossibility and negation as unnecessity, a notion of weak frame correspondence is defined. It is observed that DNE weakly corresponds with a certain frame property, and two structural sequent rules are presented that together also weakly correspond with that frame property and allow one to derive DNE. Moreover, the reformulated display calculus has an independent motivation by considerations on proof-theoretic semantics. (shrink)

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; (...) if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

There are essentially two ways to develop the Peircean idea that future contingents are all false. One is to provide a quantificational semantics for "will," as is usually done. The other is to define a quantificational postsemantics based on a linear semantics for "will." As we will suggest, the second option, although less conventional, is more plausible than the first in some crucial respects. The postsemantic approach overcomes three major troubles that have been raised in connection with Peirceanism: the apparent (...) scopelessness of "will" with respect to negation, the failure of Future Excluded Middle, and the so-called zero credence problem. (shrink)

This paper investigates a particular philosophical puzzle via an examination of its status in the writings of Wittgenstein. The puzzle concerns negation and can take on three interrelated guises. The first puzzle is how not-p can so much as negate p at all – for if p is not the case, then nothing corresponds to p. The second puzzle is how not-p can so much as negate p at all when not-p rejects p not as false but as unintelligible – (...) for if p is unintelligible, then p is nothing but scratches and sounds and does not seem apt for negation. And the third puzzle is how “not” could be anything but hopelessly equivocal if it sometimes (per the first puzzle) requires, and sometimes (per the second puzzle) precludes the intelligibility of p. The paper investigates these three puzzles, their respective structures, and their relations to each other. The second puzzle is expounded as the centre of gravity, and in countering two objections to the threefold puzzle, a special predicament is expounded with regard to the second puzzle’s concern with unipolar propositions – propositions that do not admit of an intelligible negation. The text concludes by indicating the first steps that could potentially lead us out of the threefold puzzle. (shrink)

Deontic necessity modals (e.g. 'have to', 'ought to', 'must', 'need to', 'should', etc.) seem to vary in how they interact with negation. According to some accounts, what forces modals like 'ought' and 'should' to outscope negation is their polarity sensitivity -- modals that scope over negation do so because they are positive polarity items. But there is a conflict between this account and a widely assumed theory of if-clauses, namely the restrictor analysis. In particular, the conflict arises for constructions containing (...) a bound pronoun in the if-clause. This note spells out the core conflict. (shrink)

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...) extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics. (shrink)

In this paper we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics. We give a characterization of the degree of non-functionality which is compatible with the propositional structure of quantum theory, showing that having truth-functional connectives, together with some assumptions regarding the relation of logical consequence, commits us to the adequacy of the interpretation sets of these connectives. An advantage of our proof is that it is independent of (...) the number of truth values involved, generalizing previous works. We also show the failure of the adequacy of every Nmatrix that is a model of the non-distributive lattice of quantum propositions. (shrink)

In this paper we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics. We give a characterization of the degree of non-functionality which is compatible with the propositional structure of quantum theory, showing that having truth-functional connectives, together with some assumptions regarding the relation of logical consequence, commits us to the adequacy of the interpretation sets of these connectives. An advantage of our proof is that it is independent of (...) the number of truth values involved, generalizing previous works. We also show the failure of the adequacy of every Nmatrix that is a model of the non-distributive lattice of quantum propositions. En este trabajo discutimos las nociones de adecuación y veritativo-funcionalidad en la lógica cuántica, desde el punto de vista de una semántica no determinista. Damos una caracterización del grado de no-funcionalidad compatible con la estructura proposicional de la teoría cuántica, mostrando que tener conectivos veritativo-funcionales, junto con algunos presupuestos relativos a la relación de consecuencia lógica, nos compromete con la adecuación de los conjuntos de interpretación de estos conectivos. Un punto ventajoso de nuestra prueba es que se independiza de la cantidad de valores de verdad involucrados, generalizando trabajos previos. Probamos también que falla la adecuación de cualquier Nmatriz que sirva como modelo para el retículo no distributivo de proposiciones cuánticas. (shrink)

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)

The ability to express negation in language may have been the result of an adaptive process. However, there are different accounts of adaptation in linguistics, and more than one of them may describe the case of negation. In this paper, I distinguish different versions of the claim that negation is adaptive and defend a proposal, based on recent work by Steinert-Threlkeld (2016) and Incurvati and Sbardolini (2021), on which negation is an indirect adaptation.

I argue that rejection is a speech act that cannot be reduced to assertion. Adapting an argument by Huw Price, I conclude that rejection is best conceived of as the speech act that is used to register that some other speech act is (or would be) violating a rule of the conversation game. This can be naturally understood as registering *norm violations* where speech acts are characterised by their essential norms. However, I argue that rejection itself is not to be (...) characterised by a norm. Instead, registering violations is a necessary condition for understanding the normative framework in the first place. The core observation is that the concept of an 'illegal move' is intelligible, so a speech act can be (say) an assertion, despite violating the essential norm of asserting. Rejection has the function of pointing out that a move is illegal. But registering rule violations is a precondition of playing games with rules (it is part of the concept 'game'), not itself a rule in a game. A similar special role of rejection (that it is not explicable in the terms provided by a framework, but needed to conceptualise these terms) likely occurs in other frameworks as well, e.g. characterising speech acts by commitments or their effect on a common ground. (shrink)

In one of their papers, Michael De and Hitoshi Omori observed that the notion of classical negation is not uniquely determined in the context of so-called Belnap-Dunn logic, and in fact there are 16 unary operations that qualify to be called classical negation. These varieties are due to different falsity conditions one may assume for classical negation. The aim of this paper is to observe that there is an interesting way to make sense of classical negation independent of falsity conditions. (...) We discuss two equivalent semantics, and offer a Hilbert-style system that is sound and complete with respect to the semantics. (shrink)

Frame semantics for negation on the Australian Plan accommodates many different negations, but it falls short on accommodating subminimal negation when the language contains conjunction and disjunction. In this paper, I will present a multi-relational frame semantics –multi-incompatibility frame semantics– that can accommodate subminimal negation. I will first argue that multi-incompatibility frames are in accordance with the philosophical motivations behind negation on the Australian Plan, namely its modal and exclusion-expressing nature. Then, I will prove the soundness and completeness results of (...) a subminimal logic that consists of the multi-incompatibility semantics and a proof system with operational rules that characterize subminimal negation, conjunction and disjunction. Lastly, I will prove some key correspondence theorems that relate frame conditions to certain principles that are associated with stronger negations, which will give rise to a new kite of negations that includes subminimal negation. (shrink)

In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of (...) the systems have the variable sharing property). I emphasize the intuitive content of the semantics and discuss how natural informational considerations underly each of the examined systems. (shrink)

Nothing new here. Really. The identification (and correct representation) of space.

Negation is a sophisticated form of duplication. Proving (and demonstrating) 'The Singularity.' Eliminating the need for proof.

Logical realism is a view about the metaphysical status of logic. Common to most if not all the views captured by the label ‘logical realism’ is that logical facts are mind- and language-independent. But that does not tell us anything about the nature of logical facts or about our epistemic access to them. The goal of this paper is to outline and systematize the different ways that logical realism could be entertained and to examine some of the challenges that these (...) views face. It will be suggested that logical realism is best understood as a metaphysical view about the logical structure of the world, but this raises an important question: does logical realism collapse into standard metaphysical realism? It will be argued that this result can be accommodated, even if it cannot be altogether avoided. (shrink)

In this article I consider some recent objections raised against the syntactic treatment of negation in English multiclausal structures, in particular what has been called NEGraising. I argue that the objections based on pronominalisation and ellipsis presented in the recent literature do pose a problem for syntactic accounts of the mechanisms of so-called NOT-transportation that rely on a rule of leftwards movement, as is customary in generative grammar. However, there is an alternative syntactic treatment that assumes that negation originates as (...) a higher predicate and is subject to a rule of lowering. I show that a syntactic theory of NOT-transportation is tenable and accounts for the problematic data if NEGraising is replaced, in the analysis of the cases considered here, by a rule of NEG-lowering. (shrink)

This is a commentary on MM McCabe's "First Chop your logos... Socrates and the sophists on language, logic, and development". In her paper MM analyses Plato's Euthydemos, in which Plato tackles the problem of falsity in a way that takes into account the speaker and complements the Sophist's discussion of what is said. The dialogue looks as if it is merely a demonstration of the silly consequences of eristic combat. And so it is. But a main point of MM's paper (...) is that there is serious philosophy in the Euthydemos, too. MM argues that to counter the sophist brothers Euthydemos and Dionysodoros, Socrates points out that that there are different aspects to the verb 'to say' that run in parallel to the different aspects of the very 'to learn'. So just as there is continuity rather than ambiguity between 'to learn' and 'to understand', so there is continuity between the different aspects of saying. Thus Socrates puts forward a teleological account of both learning and meaning. Following up on some of MM's thoughts, I argue that the sophists subscribe, despite appearance, to a theory of meaning that respects serious and widely accepted philosophical theses on meaning. -/- Forthcoming in the Australasian Philosophical Review. The curator of the volume is Fiona Leigh, and the committee also has Hugh Benson and Tim Clarke. You can find MM's paper as well as the commentaries by Nicholas Denyer and Russell E. Jones and Ravi Sharma (and myself) by registering. (shrink)

A general framework for translating various logical systems is presented, including a set of partial unary operators of affirmation and negation. Despite its usual reading, affirmation is not redundant in any domain of values and whenever it does not behave like a full mapping. After depicting the process of partial functions, a number of logics are translated through a variety of affirmations and a unique pair of negations. This relies upon two preconditions: a deconstruction of truth-values as ordered and structured (...) objects, unlike its mainstream presentation as a simple object; a redefinition of the Principle of Bivalence as a set of four independent properties, such that its definition does not equate with normality. (shrink)

It is known that many relevant logics can be conservatively extended by the truth constant known as the Ackermann constant. It is also known that many relevant logics can be conservatively extended by Boolean negation. This essay, however, shows that a range of relevant logics with the Ackermann constant cannot be conservatively extended by a Boolean negation.

Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean extended (...) logic. (shrink)

Many relevant logics can be conservatively extended by Boolean negation. Mares showed, however, that E is a notable exception. Mares’ proof is by and large a rather involved model-theoretic one. This paper presents a much easier proof-theoretic proof which not only covers E but also generalizes so as to also cover relevant logics with a primitive modal operator added. It is shown that from even very weak relevant logics augmented by a weak K-ish modal operator, and up to the strong (...) relevant logic R with a S5 modal operator, all fail to be conservatively extended by Boolean negation. The proof, therefore, also covers Meyer and Mares’ proof that NR—R with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation. (shrink)

This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.

Proof-theoretic methods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and (...) used to conclude that the considered logical systems are PSPACE-complete. (shrink)

Many think that expressivists have a special problem with negation. I disagree. For if there is a problem with negation, I argue, it is a problem shared by those who accept some plausible claims about the nature of intentionality. Whether there is any special problem for expressivists turns, I will argue, on whether facts about what truth-conditions beliefs have can explain facts about basic inferential relations among those beliefs. And I will suggest that the answer to this last question is, (...) on most plausible attempts at solving the problem of intentionality, ‘no’. (shrink)

The recent literature abounds with accounts of the semantics and pragmatics of so-called predicates of personal taste, i.e. predicates whose application is, in some sense or other, a subjective matter. Relativism and contextualism are the major types of theories. One crucial difference between these theories concerns how we should assess previous taste claims. Relativism predicts that we should assess them in the light of the taste standard governing the context of assessment. Contextualism predicts that we should assess them in the (...) light of the taste standard governing the context of use. We show in a range of experiments that neither prediction is correct. People have no clear preferences either way and which taste standard they choose in evaluating a previous taste claim crucially depends on whether they start out with a favorable attitude towards the object in question and then come to have an unfavorable attitude or vice versa. We suggest an account of the data in terms of what we call hybrid relativism. (shrink)

In the context of modal logics one standardly considers two modal operators: possibility ) and necessity ) [see for example Chellas ]. If the classical negation is present these operators can be treated as inter-definable. However, negative modalities ) and ) are also considered in the literature [see for example Béziau ; Došen :3–14, 1984); Gödel, in: Feferman, Collected works, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300; Lewis and Langford ]. Both of them can be (...) treated as negations. In Béziau a logic \ has been defined on the basis of the modal logic \. \ is proposed as a solution of so-called Jaśkowski’s problem [see also Jaśkowski ]. The only negation considered in the language of \ is ‘it is not necessary’. It appears that logic \ and \ inter-definable. This initial correspondence result between \ and \ has been generalised for the case of normal logics, in particular soundness-completeness results were obtained [see Marcos :279–300, 2005); Mruczek-Nasieniewska and Nasieniewski :229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski it has been proved that there is a correspondence between \-like logics and regular extensions of the smallest deontic logic. To obtain this result both negative modalities were used. This result has been strengthened in Mruczek-Nasieniewska and Nasieniewski :261–280, 2017) since on the basis of classical positive logic it is enough to solely use \ to equivalently express both positive modalities and negation. Here we strengthen results given in Mruczek-Nasieniewska and Nasieniewski by showing correspondence for the smallest regular logic. In particular we give a syntactic formulation of a logic that corresponds to the smallest regular logic. As a result we characterise all logics that arise from regular logics. From this follows via respective translations a characterisation of a class of logics corresponding to some quasi-regular logics where \ is the smallest element. Moreover, if a given quasi-regular logic is characterised by some class of models, the same class can be used to semantically characterise the logic obtained by our translation. (shrink)

Various philosophers have long since been attracted to the doctrine that future contingent propositions systematically fail to be true—what is sometimes called the doctrine of the open future. However, open futurists have always struggled to articulate how their view interacts with standard principles of classical logic—most notably, with the Law of Excluded Middle. For consider the following two claims: Trump will be impeached tomorrow; Trump will not be impeached tomorrow. According to the kind of open futurist at issue, both of (...) these claims may well fail to be true. According to many, however, the disjunction of these claims can be represented as p ∨ ~p—that is, as an instance of LEM. In this essay, however, I wish to defend the view that the disjunction these claims cannot be represented as an instance of p ∨ ~p. And this is for the following reason: the latter claim is not, in fact, the strict negation of the former. More particularly, there is an important semantic distinction between the strict negation of the first claim [~] and the latter claim. However, the viability of this approach has been denied by Thomason, and more recently by MacFarlane and Cariani and Santorio, the latter of whom call the denial of the given semantic distinction “scopelessness”. According to these authors, that is, will is “scopeless” with respect to negation; whereas there is perhaps a syntactic distinction between ‘~Will p’ and ‘Will ~p’, there is no corresponding semantic distinction. And if this is so, the approach in question fails. In this paper, then, I criticize the claim that will is “scopeless” with respect to negation. I argue that will is a so-called “neg-raising” predicate—and that, in this light, we can see that the requisite scope distinctions aren’t missing, but are simply being masked. The result: a under-appreciated solution to the problem of future contingents that sees and as contraries, not contradictories. (shrink)