Logical Consequence

The thesis is an investigation into the logical pluralism debate, aiming to understand how the philosophical commitments sustaining each side to the debate connects to more general issues connected to the foundations of logic. My investigation centers on the following three notions: (1) Epistemic justification, (2) The metaphysical "ground" for logical truth, and (3) Normativity. Chapter 1 traces the monistic and pluralistic conception of logic back to its philosophical/mathematical roots, which we find in the writings of Rudolf Carnap and Gottlob Frege. I argue that logical pluralism - in its more plausible, epistemic (rather than ontic) form - was enabled by the semantic shift which Carnap seems to have anticipated and that, from a conventionalist perspective, his 'Principle of Tolerance' follows as a consequence of that shift. Chapter 2 concerns the issues ensuing from Willard V. O Quine’s critique of Carnap's conventionalism, which had a devastating effect for his foundationalist project. The aim is in particular to address the issue of meaning-variance, a crucial assumption for the conventionalist approach to pluralism. In chapter 3, I present another framework for pluralism, due to Stewart Shapiro’s [2014] ‘modelling’ conception of logic, according which logic is conceived as a mathematical model of natural language. Shapiro argues that our concept of logical consequence is vague and in need of a sharpening to attain a fixed meaning. Pluralism follows from there being two or more equally "correct" such sharpenings; i.e., relative to our theoretical aims. I argue that the modelling-conception is the best way to approach a justification of basic logical laws. However, since that conception also grounds Timothy Williamson’s [2017] argument for monism, I argue that the conception ultimately fails to establish logical pluralism. Since both Williamson and Shapiro take a pragmatic approach to justification, I conclude that the question of pluralism does not turn on epistemological commitments (i.e., on (1)), and suggest instead that it is a matter of (2), i.e., of one's conception of the "ground" for logical truth.

In their book EVALUATING CRITICAL THINKING Stephen Norris and Robert Ennis say: “Although it is tempting to think that certain [unstated] assumptions are logically necessary for an argument or position, they are not. So do not ask for them.” Numerous writers of introductory logic texts as well as various highly visible standardized tests (e.g., the LSAT and GRE) presume that the Norris/Ennis view is wrong; the presumption is that many arguments have (unstated) necessary assumptions and that readers and test takers can reasonably be expected to identify such assumptions. This paper proposes and defends criteria for determining necessary assumptions of arguments. Both theoretical and empirical considerations are brought to bear.

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.

Logical pluralism is the view that there is more than one correct logic. This very general characterization gives rise to a whole family of positions. I argue that not all of them are stable. The main argument in the paper is inspired by considerations in Keefe (2014), Priest (2006a), Read (2006), and Williamson (1988), and it aims at the most popular form of logical pluralism advocated by Beall and Restall (2000, 2006). I argue that there is a more general argument available that challenges all variants of logical pluralism that meet the following three conditions: (i) that there are at least two correct logical systems characterized in terms of different consequence relations, (ii) that there is some sort of rivalry among the correct logics, and (iii) that logical consequence is normative. The hypothesis I argue for amounts to what Caret (2016) calls a ‘collapse problem’ in form the of a conditional claim: If a position satisfies all these conditions, then that position is unstable in the sense that it collapses into competing positions.

There is a profound, but frequently ignored relationship between 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 kind 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) some of the counter-examples to classical argumentative forms and known conditional puzzles are unsound.

In a recent article, "Logical Consequence and Natural Language", Michael Glanzberg claims that there is no relation of logical consequence in natural language (Glanzberg, 2015). The present paper counters that claim. I shall discuss Glanzberg's arguments and show why they don't hold. I further show how Glanzberg's claims may be used to rather support the existence of logical consequence in natural language.

The well-known necessary-consequence/necessary-consequent ambiguity—a species of segmentation, scope, and structural ambiguities—is found in conditional sentences such as:

(A) If zero is oblong, then necessarily some square is oblong.

A can read as expressing a proposition to the effect that “some square is oblong” is a necessary consequence of “zero is oblong”. This necessary-consequence segmentation parses A with the adverb as part of the discontinuous modal connective ‘if … then necessarily’. The consequent ‘some square is oblong’ is non-modal. Alternatively, the adverb can be taken instead as part of the consequent yielding the necessary-consequent segmentation: A can read as expressing roughly that the modal proposition “necessarily some square is oblong” is a material consequence of “zero is oblong”. Brackets indicate segmentation:

Necessary-consequence segmentation
[If] zero is oblong, [then necessarily] some square is oblong.

Necessary-consequent segmentation
If zero is oblong, then [necessarily some square is oblong].

Similar confusing ambiguities arise in logic texts where ‘valid’, ‘logical’, ‘necessary’, ‘sound’, ‘derivable’, and the like occur before ‘conclusion’, ‘inference’, ‘deduction’ and the like.

This paper, supplementing [1], compares corresponding segmentations of various conditional sentences. E.g., taking both “necessities” as logical, the following have different truth-values—first true, second false. The second implies the first.

[If] Twain is Clemens, [then necessarily] Clemens is Twain.
If Twain is Clemens, then [necessarily Clemens is Twain].

The first says that “Clemens is Twain” follows logically from “Twain is Clemens”; the second that “Clemens is Twain” is logically necessary if Twain is Clemens.

This dissertation is concerned with logical consequence and the idea that it is a formal relation. Logical terms are considered by many the key to the notion of form, and it is generally assumed that there is a strict division of terms into those that are logical (which are completely fixed in systems of logic) and those that are nonlogical (and are non-fixed).

É em 1789, durante a queda do império romano que se deu a maior evolução do direito, era a fase onde o Estado mandava e o individuo obedecia, os princípios da Revolução Francesa, como liberdade, igualdade e fraternidade, serviram de parâmetro para a criação do Código Civil de 1916, este por sua vez protegia quem possuía uma relação no mundo civil.

This article is focused on answering the question to what extent one can be a sceptic. Sextus Empiricus’s Outlines of Scepticism serves as a guide. In section 1, it is investigated whether three logical laws have a certain foundation or are subject to doubt. In section 2, Sextus’s way to deal with these laws is examined; the question arises how dogmatic his approach is. After that, a possible ‘reply’ by Sextus to the criticism receives attention. Section 3 is concentrated on a possible alternative to Sextus’s approach. Besides logical laws, some important methods are concerned.

Resumen
Las paradojas muestran algo irracional pero de un modo perfectamente racional, mostrando que sólo contra el telón de fondo de una cierta definición de racionalidad algo resulta irracional. La paradoja no es pues un argumento a dilucidar o un problema a esclarecer, sino una perplejidad frente a la recursividad del lenguaje y al principio de información faltante, un agujero en el centro de nuestra lógica. Por ello la emancipación de la razón posmoderna requiere nuevas lógicas, polivalentes, de la vaguedad o paraconsistentes, para habérselas con los sistemas complejos y los desafíos a la que nos enfrentan la física cuántica, retos que exigen nuevas discursividades, así como cartografías lo suficientemente plásticas para desplazarse en medio de un universo de espejos, cuerdas y agujeros negros.  Así las paradojas pueden ser pensadas como un inevitable "nudo gordiano cognitivo", imposible de desatar en las condiciones y con los modos de pensar con que lo creamos, pero que se desvanece en otro espacio conceptual.
Dr. Adolfo Vásquez Rocca

It is usually accepted that deductions are non-informative and monotonic, inductions are informative and nonmonotonic, abductions create hypotheses but are epistemically irrelevant, and both deductions and inductions can’t provide new insights. In this article, I attempt to provide a more cohesive view of the subject with the following hypotheses: (1) the paradigmatic examples of deductions, such as modus ponens and hypothetical syllogism, are not inferential forms, but coherence requirements for inferences; (2) since any reasoner aims to be coherent, any inference must be deductive; (3) a coherent inference is an intuitive process where the premises should be taken as sufficient evidence for the conclusion, which on its turn should be viewed as necessary evidence for the premises in some modal range; (4) inductions, properly understood, are abductions, but there are no abductions beyond the fact that in any inference the conclusion should be regarded as necessary evidence for the premises; (5) monotonicity is not only compatible with the retraction of past inferences given new information, but it is a requirement for it; (6) this explanation of inferences holds true for discovery processes, predictions and trivial inferences.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. MSC: 11-XX (Number Theory)

ARTÍCULOS

DIEGO TAJER
En defensa del argumento finitista

MAXIMILIANO ESCOBAR VIRE
La necesidad moral en Leibniz: su contenido alético y su significación específica

RAMIRO CASO
Una defensa de las aserciones suboracionales

LUIS MARIANO DE LA MAZA
Voluntad, razón práctica y reconocimiento en Hegel y Ricoeur


ESTUDIOS CRÍTICOS

RICARDO IBARLUCIA
La autonomía del arte en Benjamin y Heidegger: a propósito de la interpretación de Burkhardt Lindner


NOTAS

LEANDRO INDAVERA
Rousseau, Smith y las rudas selvas de la naturaleza


COMENTARIOS BIBLIOGRÁFICOS

LORENZINO DE’ MEDICI
Apología (F. Andrade Marambio)

LETICIA CABANAS y OSCAR M. ESQUISABEL (eds.)
Leibniz frente a Spinoza. Una interpretación panorámica (F. Raffo Quintana)

SERGIO SANCHEZ
La insensata fabrica de la vigilia. Nietzsche y el fenómeno del sueño (P. Hunziker)

MARIA CRISTINA DI GREGORI y FEDERICO E. LOPEZ (coords.)
Regreso a la experiencia: lecturas de Peirce, James, Dewey y Lewis (L. Mattarollo)

Prospetto. Il presente saggio è articolato in cinque parti. La prima parte ( §1) introduce le nozioni basilari e motiva l'importanza dell'oggetto di studio del saggio: la relazione di conseguenza logica. La seconda parte ( §2) introduce al problema della natura della relazione. Sono affrontati tre nodi tematici principali. In primo luogo ( §2.1), si tratta la questione metodologica se vi siano una, molte o nessuna relazioni di conseguenza logica, e in che modo i seguenti dibattiti possano esser visti da queste diverse prospettive. In secondo luogo ( §2.2), si passano in rassegna le quattro principali teorie sulla natura della relazione di conseguenza logica. In terzo luogo ( §2.3), si discute se e in quale senso il vigere della relazione sia connesso a questioni di materia o di forma. La terza parte ( §3) analizza alcuni aspetti strutturali della relazione. Vengono discussi i diversi problemi legati alla questione di quali entità occupino i posti della relazione ( §3.1) e le proprietà formali di essa ( §3.2). Questa parte ha come scopo principale quello di approfondire importanti aspetti del dibattito sulla conseguenza logica che, per semplicità espositiva, abbiamo preferito trascurare nella trattazione della seconda parte, di carattere più introduttivo. La seconda e la terza parte sono quindi complementari: mentre la prima offre un affresco d'insieme del dibattito sulla conseguenza logica, la seconda lo illumina da diversi angoli, evidenziandone singolarmente le diverse linee di fuga. Entrambe queste parti sono suddivise in paragrafi. Ogni paragrafo coincide normalmente con la presentazione di un tema particolare, organizzata in due tempi: a una succinta introduzione fa seguito l'enucleazione degli aspetti problematici tramite la struttura a obiezioni e risposte. La quarta parte ( §4) trae alcune conclusioni dalle discussioni precedenti. A essa segue una quinta parte ( §5) con esercizi, problemi e relative soluzioni, che sarà utile al lettore per migliorare e approfondire la comprensione dei temi trattati. Una breve bibliografia conclude il saggio: essa comprende solo i lavori effettivamente citati in esso.

In 1898 C. S. Peirce declares that the medieval doctrine of consequences had been the starting point of his logical investigations in the 1860s. This paper shows that Peirce studied the scholastic theory of consequentiae as early as 1866–67, that he adopted the scholastics’ terminology, and that that theory constituted a source of logical doctrine that sustained Peirce for a lifetime of creative and original work.

I argue against inferentialism about logic. First, I argue against an analogy between logic and chess, before considering a more basic objection to stipulating inference rules as a way of establishing the meaning of logical constants. The objection-the Mushroom Omelette Objection-is that stipulative acts are partly constituted by logical notions, and therefore cannot be used to explain logical thought. I then argue that the same problem also attaches to following existing conventional rules, since either those rules have logical contents, or following those conventional rules is done for logical reasons. Lastly, I compare this argument with other arguments found in Quine's early work, and consider two attempts to reply to Quine.

Classical propositional logic can be characterized, indirectly , by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit "in the negative". More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

The concept of formal consequence is at the heart of logic today, and by extension, plays an important role in such diverse areas as mathematics, computing, philosophy, and linguistics.

In this dissertation, I trace the roots of this concept in medieval logic from Pseudo-Scotus and John Buridan back to the earliest treatises on consequences, and provide translations of the three earliest known treatises on consequences - two anonymous, a third by Walter Burley.

Chapter one introduces the reader to the dominant philosophical approaches to formal consequence from the turn of the twentieth century to today. After this, I introduce the account of formal consequence advanced by John Buridan, the medieval predecessor to the semantic account advocated by Tarski and his followers.

The second chapter provides a detailed contrast of Buridan's account of formal consequence with those of Tarski, on the one hand, and later classical logic, on the other.

Chapter three examines the account of formal consequence in Pseudo-Scotus. I show that Pseudo-Scotus' account is dependent on that of Buridan, and therefore most post-date it.

Chapter four examines the account of divided modal consequence in William of Ockham. I show that Ockham's divided modalities are not fully assimilable to narrow-scope propositions of classical modal logic; formalize Ockham's account in an extension of first-order modal logic with restricted quantification; and provide a complete account of relations between two-term divided modal propositions on Ockham's account.

Chapter five introduces Walter Burley's thinking about consequences, examining: Burley's division and enumeration of consequences; his distinction between principal and derivative rules licensing good consequences; the relation of the division of consequences into formal and material varieties to Burley's preferred division between natural and accidental consequences; the relation Burley's work bears to Buridan, to the Boethian reception of Aristotle's Topics, and to the earliest treatises on consequences.

The final chapter concludes: highlighting the characteristic marks of medieval and modern approaches to consequences relative to each other; summarizing the various developments that led to the adoption of the account of formal consequence epitomized in Buridan's work; and suggesting prospects for recovering the most promising aspects of the medieval treatments of the topic.

Corcoran, J. 1989. Argumentations and Logic. Argumentation 3: 17-43

Argumentations are at the heart of the deductive and the hypothetico-deductive methods, which are involved in attempts to reduce currently open problems to problems already solved.  These two methods span the entire spectrum of problem-oriented reasoning from the simplest and most practical to the most complex and most theoretical, thereby uniting all objective thought whether ancient or contemporary, whether humanistic or scientific, whether normative or descriptive, whether concrete or abstract.  Analysis, synthesis, evaluation, and function of argumentations are described.  Perennial philosophic problems, epistemic and ontic, related to argumentations are put in perspective.  So much of what has been regarded as logic is seen to be involved in the study of argumentations that logic may be usefully defined as the systematic study of argumentations, which is virtually identical to the quest of objective understanding of objectivity.

KEY WORDS: hypothesis, theorem, argumentation, proof, deduction, premise-conclusion argument, valid, inference, implication, epistemic, ontic, cogent, fallacious, paradox, formal, validation.

“Argumentations and Logic” has been used in courses at University of Barcelona, Bryn Mawr College, University of Buffalo, Buffalo State College, State University of Campinas, Canisius College, Federal University of Goiás, Fredonia University, Grand Valley State University, University of Lausanne, Niagara University, University of Santiago de Compostela, SUNY Potsdam, and elsewhere.
It has been translated into Spanish, Portuguese, and Persian. Insha’Allah, in the fullness of time there will be translations into Chinese and Czech.

This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of indicative conditionals and probability operators in natural language. Along the way we investigate some of the interactions of these expressions with knows, and we call into question the thesis that all knowledge ascriptions have truthconditions. A probabilistic dynamic semantics for probability operators, conditionals, and acceptance attitudes is developed around the idea of representing the common ground of a conversation as a set of probability spaces.

Quantificational accounts of logical consequence account for it in terms of truth-preservation in all cases – be it admissible substitutional variants or interpretations with respect to non-logical terms. In this second of my three connected studies devoted to the quantificational tradition, I set out to reconstruct the seminal contributions of Russell, Carnap, Tarski and Quine and evaluate them vis-à-vis some of the most pressing objections. This study also prepares the ground for my discussion of the standard model-theoretic account of consequence to be found in the concluding study.

I study here the implications of Suszko-like reductive results for the common wisdom on the internal logic of a topos. My aim here is to question certain slogans about topos logic, and especially on its many-valuedness. The originality of this paper comes not from the observation that the internal logic of a topos is bivalent because it is in the scope of Suszko’s or any other reduction; that is easy. Rather, the originality comes from (1) recognizing the import of applying Suszko’s and similar results in a topos-theoretical setting, (2) the suggestions to give categorial content to the reduction, and (3) the extrapolation of the debate about Suszko’s thesis to the topos-theoretical framework, which gives us some insight about the scope of another theorem, namely that stating the intuitionistic character of the internal logic of a topos.

В статье рассматривается понятие факта, а также другие базовые понятия ситуационной семантики, как они даны в версии А. Кратцер. Выявляются преимущества способа их конструирования, используемого Кратцер, перед способом, который использовался Дж. Барвайсом и Дж. Перри. Особое внимание уделяется функциям понятия факта в версии Кратцер в семантике эпистемических установок, а также в теории особого вида следования («сильного» следования), выделяемого в ситуационной семантике.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

This is a survey paper in German covering a number of central themes in the philosophy of logic, including the debates surrounding the notion of logical consequence, the problem of demarcating logic, logical pluralism vs. logical monism and the question of the normativity of logic.

This special issue collects together nine new essays on logical consequence: the relation obtaining between the premises and the conclusion of a logically valid argument. This paper is a partial, and opinionated, introduction to the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarily preserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field's contention that such objections show consequence to be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We make three main claims: (i) that the accounts in question have the resources to meet the objections standardly thought to herald their demise; (ii) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (iii) that consequence is a paradoxical notion if truth is. for helpful discussion over the years on some of the topics discussed herein. Special thanks to Gil Sagi for detailed and very helpful comments on an earlier draft. Julien Murzi gratefully acknowledges the Analysis Trust, the Alexander von Humboldt Foundation, the University of Kent, and the British Academy for generous financial support during the time this paper or its ancestors were written. Part of §4 is partly drawn from ?, ? and ?.

We give an elementary introduction into the theory of consequence operations. We proof some elementary results concerning basic notions of logic like tautology, consistency, independence and completeness. We show in particular that every nite axiomatizable set is independent axiomatizable and that every consistent set has relative to a nitary consequence operation a maximal consistent extension. Finally we provide an abstract semantics for consequence operations. 1

Developed out of earlier work on Aristotelian topics, syllogistic, and fallacies, by the early fourteenth century the medieval theory of consequence came to provide the first unified framework for the treatment of inference as such. With this development came the task of unifying the various justifications for inferences treated in earlier frameworks.

Prior to the appearance of theories of consequences, the task of providing a real foundation, or grounding, for good inferences is shared between theories of demonstration, such as those provided in commentaries on Aristotle’s Posterior Analytics, and theories of topical inference, passed on to the medievals via Boethius. But by the time of the earliest consequentiae, most consequences were grounded in the theory of supposition, which began its own development in the twelfth
century.

Secondary literature on supposition has generally held that in the most common form of supposition, personal supposition, a term is taken to stand for individuals falling under it. In this paper, I show that for the earliest consequentiae this is false: prior to William of Ockham’s work, personal supposition could also involve descent to concepts or types falling under a term, previously thought to be the exclusive provision of simple supposition. As such, a greater variety of ways of grounding consequence exists in the period than has hitherto been recognized.

"Substitutional and interpretational accounts of logical consequence explicate it as truth-preservation in all cases, cases being construed either as admissible substitutional variants or admissible semantic interpretations. This series of three connected studies aims to examine the merits and demerits of “quantificational accounts” of this sort, focusing on seminal contributions of Bolzano, Russell, Tarski, Carnap, Quine and the standard model-theoretic approach. Though extremely influential in the tradition, quantificational accounts have received much critical attention, especially in the wake of Etchemendy 1990. I reconstruct the main objections, arguing that the model-theoretic account appears as the most promising of quantificational accounts vis-à-vis them. Finally, by way of responding to one allegedly devastating objection due to Kneale and Etchemendy, I explain that quantificational slogans do not by themselves capture all there is to logical properties, if not backed up by a plausible account of semantic structure and the specific contribution of logical elements to it. But proponents of the interpretational approach in model-theoretic style need not be paralysed, since they have such a semantic story at their disposal.

This, in a nutshell, is my overall agenda. In the present study, I focus on the beginning of systematic theorizing of consequence in Aristotle‘s work, which contains the rudiments of both modal and quantificational accounts of logical consequence. It is pointed out, inter alia, that (1) there is no evidence for the claim that Aristotle would have subscribed to the reductionist spirit of the latter, and that (2) for a full-fledged quantificational approach we need to turn to Bolzano’s substitutional approach, whose motivation, structure and problems are explained in the second part of this study. "

Oppositional geometry gives a mathematical model of oppositional phenomena through “oppositional structures” (logical squares, hexagons, cubes, …). Its so far known formal entities, the backbone of which are the “oppositional bi-simplexes (and poly-simplexes) of dimension m”, are distributed into three families (the alpha-, beta- and gamma-structures). However, some recent studies by different authors exhibit strange structures, notably strange variations of the notion of “oppositional hexagon” (or “logical hexagon”). In this paper we show that inside the oppositional tetrahexahedron, i.e. the beta3-structure (discovered in 1968 and rediscovered in 2008) – a 3D solid made of a logical cube and six logical “strong hexagons”, containing 14 vertices and 36 implication arrows – there are in fact C(6|14) = 30030 strange hexagons, which we call “hybrid hexagons”. In this paper, through a systematic study of those among them which have as invariant property a regular perimeter made of alternated arrows (henceforth “arrow-hexagons”), we show that they divide into a much smaller number of families, nine, each containing several isomorphic instances of the same oppositional pattern. An interesting result seems to be that when seen from the viewpoint of their mutual transformations (i.e. moving from one to another kind of arrow-hexagon, just by exchanging one of its 6 vertices with one among the remaining 14-6=8 vertices of the tetrahexahedron), these arrow-hexagonal patterns taken as points can be displayed into a new kind of 3D structure. The latter, by putting into order these points (each representing a family of arrow-hexagons), gives some kind of morphogenetic cartography of the arrow-hexagons of the beta3-structure. As we will argue, since several arrow-hexagons play the role of “attractors”, there are reasons for thinking that such a cartography could be very meaningful in the future for modelling “oppositional dynamics”, that is the systematic formal study of the situations where a given complex oppositional structure sees its shape change within time.

In standard model-theoretic semantics, the meaning of logical terms is said to be fixed in the system while that of nonlogical terms remains variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing their meaning precisely amounts to. My proposal is that when a term is considered logical in model theory, what gets fixed is its intension rather than its extension. I provide a rigorous way of spelling out this idea, and show that it leads to a graded account of logicality: the less structure a term requires in order for its intension to be fixed, the more logical it is. Finally, I focus on the class of terms that are invariant under isomorphisms, as they render themselves more easily to mathematical treatment. I propose a mathematical measure for the logicality of such terms based on their associated Löwenheim numbers.

In this paper I argue against the view that logical consequence is a truth-preserving relation and, in general, that it is a reflexive and transitive relation. I expound a number of non-reflexive or non-transitive notions of logical consequence and argue that they satisfy the requirements to be considered part of the so-called “core tradition” of logic, and that they thus give rise to logics bona fide.