Inferencialismo y Relevancia: el caso de la conexividad

This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance, through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.Este artículo proporciona una motivación inferencialista para una lógica perteneciente a la familia conexiva, tomando prestados elementos de la interpretación bilateralista de la Lógica Clásica sin la regla de Corte, propuesta por David Ripley. El artículo se centra en la relación entre inferencialismo y relevancia, a través de la exploración de lo que llamamos aserción y negación relevantes, mostrando que un sistema conexivo emerge como síntoma de este interesante vínculo. Con el presente intento, esperamos ampliar las interpretaciones disponibles para las lógicas conexivas, mostrando que pueden estar motivadas legítimamente en términos de ciertas restricciones relevantes impuestas a la aserción y la negación

Bridging the Two Plans in the Semantics for Relevant Logic

Part of the Synthese Library book series (SYLI, volume 418)This paper considers how the two plans in the semantics for relevant logic are related to each other. The so-called American plan, classical-style four-valued semantics, is intuitive, but weak. The so-called Australian plan, two-valued frame semantics, is very powerful, but the semantic devices employed need some explanation. Examining R. Routley’s 1984 paper ‘American plan completed, ’ this paper argues that the American plan provides an explanatory and ontological basis for the Australian plan, and that the latter is just a developed form of the former

The embodied grounding of focal parts in English wh-dialogues with negative answers

Negative answers are common types of responsive expressions to English wh-questions. From the theoretical perspective of Embodied-Cognitive Linguistics (Wang, 2014) and the dialogic view on meaning construction, this study takes English wh-dialogues with negative answers as the research object and regards such dialogues as a special group of dialogic constructions in conversation. This paper gives an account of the embodied properties of the semantic grounding process of dialogic focuses in such constructions, in terms of the types of the semantic grounding and the categories of semantic consistency of focal parts of wh-questions with those of negative answers, with the ultimate goal to decipher the cognitive mechanism by which wh-dialogues with negative answers are produced and construed in linguistic communication.  

From Excluded Middle to Homogenization in Plumwood’s Feminist Critique of Logic

A key facet of Valerie Plumwood’s feminist critique of logic is her analysis of classical negation. On Plumwood’s reading, the exclusionary features of classical negation generate hierarchical dualisms, i.e., dichotomies in which dominant groups’ primacy is reinforced while underprivileged groups are oppressed. For example, Plumwood identifies the system collapse following from ex contradictione quodlibet—that a theory including both φ and ∼φ trivializes—as a primary source of many of these features. Although Plumwood considers the principle of excluded middle to be compatible with her goals, that she identifies relevant logics as systems lacking a hierarchical negation—whose first-degree fragments are both paraconsistent and paracomplete—suggests that excluded middle plays some role in hierarchical dualisms as well. In these notes, I examine the role of excluded middle in generating oppressive homogenization and try to clarify the relationship between Plumwood’s critique and this principle from several contemporary perspectives. Finally, I examine the matter of whether Plumwood’s critique requires relevance or whether a non-relevant logic could satisfy her criteria and serve as a liberatory logic of difference

On the proof complexity of deep inference

International audienceWe obtain two results about the proof complexity of deep inference: (1) Deep-inference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; (2) there are analytic deep-inference proof systems that exhibit an exponential speedup over analytic Gentzen proof systems that they polynomially simulate

From proof complexity to circuit complexity via interactive protocols

Folklore in complexity theory suspects that circuit lower bounds against NC1 or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers

A Fortiori Logic: Innovations, History and Assessments

A Fortiori Logic: Innovations, History and Assessments is a wide-ranging and in-depth study of a fortiori reasoning, comprising a great many new theoretical insights into such argument, a history of its use and discussion from antiquity to the present day, and critical analyses of the main attempts at its elucidation. Its purpose is nothing less than to lay the foundations for a new branch of logic and greatly develop it; and thus to once and for all dispel the many fallacious ideas circulating regarding the nature of a fortiori reasoning. The work is divided into three parts. The first part, Formalities, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged. The second part of the work, Ancient and Medieval History, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah. The third part of the present work, Modern and Contemporary Authors, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is impressive, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed. This book also contains, in a final appendix, some valuable contributions to general logic, including new analyses of symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox