Fabien Schang

The concept of legal gap is tackled from a number of logical perspectives and semantic methods. After presenting our own goal (Section 1), a first introduction into legal logic refers to Bobbio’s works and his formalization of legal statements (Sections 2 and 3). Then Woleński’s contribution to the area is taken into account through his reference to the distinction between two juridical systems (viz. Common Law vs Civil Law) and the notion of conditional norms (Section 4). The notion of reason is also highlighted in the case of Raz’s legal logic, thereby leading to a future connection with von Wright’s logic of truth and an analogy made with an anti-realist reading of truth-values and norms (Section 5). Our personal contribution is introduced through a reflection on how logic should deal with the logical form of norms (Section 6), before entering a number of crucial definitions and distinctions for the concepts of norm, legal statement, and promulgation (Section 7). The final point is a proposed semantics for legal statements, which is both many-valued and gap-friendly (Section 8). A distinction between a number of requirements for permission and forbiddance leads to a set of non-classical juridical systems in which non-permission and forbiddance are not equivalent with each other any more; this does justice to Woleński’s former distinction between Common Law and Civil Law, also leading ultimately to a non-classical square of legal oppositions in which several legal operators may collapse into other ones (Section 9).

This special issue on Logic and Law consists of four research papers and one interview focusing on epistemological reflections on relationships between logic and law, whether in a reductionist or complementary approach. Logic aims to elucidate through formal frameworks, yet it often grapples with the intricate nuances of everyday legal discourse. While law endeavors to delineate permissible conduct within defined jurisdictions, it often encounters challenges stemming from the ambiguity of terms, leading to frequent judicial interpretations and the perception that proliferating exceptions undermines the efficacy of the rule itself.

Wittgenstein took the _Tractatus Logico-Philosophicus_ to be eventually invalidated by logical atomism. Our main thesis is that it can be revalidated, provided that we subtract the thesis 2.02 (“The object is simple.”) from it: atoms are not simple objects but, rather, bits of information the objects are made of. Starting from an introductory discussion about what is meant by a ‘logic of colors’, an explanatory framework is then proposed in the form of a partition semantics. The philosophical problem of Wittgenstein’s color-statements is reformulated accordingly, thus overcoming the trouble with logical atomism. A Boolean treatment is advanced to solve the initial difficulty and leads to two final results: Wittgenstein’s logic of colors requires a mereological understanding of the relations between colors in order to make sense; the analysis of psychophysical concepts like colors casts some additional doubt about the analytic–synthetic distinction, to the point of treating relations between colors as analytic a posteriori judgments.

The aim of this chapter is to develop a semantics for Calculus CL. CL is a diagrammatic calculus based on a logic machine presented by Johann Christian Lange in 1714, which combines features of Euler-, Venn-type, tree diagrams, squares of oppositions etc. In this chapter, it is argued that a Boolean account of formal ontology in CL helps to deal with logical oppositions and inferences of extended syllogistics. The result is a combination of Lange’s diagrams with an algebraic semantics of terms: Bit-CL, in which any ordered objects are identified by characteristic bitstrings. Then, a number of objections to Bit-CL are answered to, and the process of inference is explained in this new logical framework.

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.

Roman Suszko said that “Obviously, any multiplication of logical values is a mad idea and, in fact, Łukasiewicz did not actualize it.” The aim of the present paper is to qualify this ‘obvious’ statement through a number of logical and philosophical writings by Professor Jan Woleński, all focusing on the nature of truth-values and their multiple uses in philosophy. It results in a reconstruction of such an abstract object, doing justice to what Suszko held a ‘mad’ project within a generalized logic of judgments. Four main issues raised by Woleński will be considered to test the insightfulness of such generalized truth-values, namely: the principle of bivalence, the logic of scepticism, the coherence theory of truth, and nothingness.

A analysis of some concepts of logic is proposed, around the work of Edelcio de Souza. Two of his related issues will be emphasized, namely: opposition, and quasi-truth. After a review of opposition between logical systems [2], its extension to many-valuedness is considered following a special semantics including partial operators [13]. Following this semantic framework, the concepts of antilogic and counterlogic are translated into opposition-forming operators [15] and specified as special cases of contradictoriness and contrariety. Then quasi-truth [5] is introduced and equally translated as a product of two partial operators. Finally, the reflections proposed around opposition and quasi-truth lead to a third new logical concept: quasi-opposition, borrowing the central feature of partiality and opening the way to a potential field of new investigations into philosophical logic.

Il s’agit de comprendre dans cet article l’opposition formulée par Gilles-Gaston Granger entre deux types de négation : la négation « radicale », d’un côté, et les négations « appliquées » de l’autre. Nous examinerons les propriétés de cette opposition, ainsi que les enseignements à en tirer sur la philosophie de la logique de Granger. Puis nous proposerons une théorie constructive des valeurs logiques considérées comme des objets structurés, consolidant à la fois l’unité de la théorie logique de Granger et son pluralisme philosophique.

It has been recently argued that the well-known square of opposition is a gathering that can be reduced to a one-dimensional figure, an ordered line segment of positive and negative integers [3]. However, one-dimensionality leads to some difficulties once the structure of opposed terms extends to more complex sets. An alternative algebraic semantics is proposed to solve the problem of dimensionality in a systematic way, namely: partition (or bitstring) semantics. Finally, an alternative geometry yields a new and unique pattern of oppositions that proceeds with colored diagrams and an increasing set of bitstrings.

English summary: Jan Lukasiewicz (1878-1956) was one of the most important members of the Lwow-Warsaw school of logic. The thirteen translated articles in this volume demonstrate the protean form of Lukasiewiczs work, from his texts on Aristotle and the principle of non-contradiction and syllogistics to modal logic, intuitionism, and multivalent logics. The articles show in particular his preoccupations with logical precision and the problem of human liberty. French description: Avec Kazimierz Twardowski, Stanislaw Lesniewski et Alfred Tarski, le logicien et philosophe polonais Jan Lukasiewicz (1878-1956) est l'un des membres les plus importants de l'Ecole de Lvov-Varsovie. Celebre pour ses ouvrages consacres a Aristote, notamment sur le principe de non-contradiction et sur la syllogistique, il a egalement publie des articles majeurs sur la logique propositionnelle, la logique modale, l'intuitionnisme et les logiques multivalentes, dont il fut l'un des fondateurs. Enfin, il a contribue a faire reconnaitre l'importance des travaux en logique de l'ecole megarico-stoicienne, trop longtemps eclipses par ceux d'Aristote sur la syllogistique. Mais Lukasiewicz ne fut pas seulement un logicien et un historien brillant, il s'engagea aussi dans les polemiques philosophiques de son temps. Il defendit ainsi la methode logique en philosophie. Proche sur ce point des membres du Cercle de Vienne, il ne rejetait pas pour autant les questions metaphysiques, commecelle du determinisme qui l'occupa toute sa vie. Les treize articles de ce recueil, fruit d'un travail collectif de traduction, ont pour ambition de montrer le caractere proteiforme de l'oeuvre de Lukasiewicz, preoccupe autant par la precision logique que par le probleme de la liberte humaine.

Un paralogisme semble commis dans la démonstration par Aristote du principe psychologique de non-contradiction : à partir d’un principe performatif d’assertion (dire quelque chose, c’est le croire), une approche moderne nous incline à prétendre qu’Aristote présuppose une transparence référentielle des contextes opaques de croyance afin de corréler les versions psychologique et logique. Nous tenterons de restituer la preuve du principe (I). Au moyen de la formalisation moderne, nous appliquerons cette explication à quelques paradoxes (II). Nous en conclurons la nature de la non-contradiction (III), avant de proposer une "dissolution" syntaxique du problème d'opacité des contextes d'attitudes propositionnelles. Cette dissolution exprimera un certain scepticisme face aux approches formelles de l'intentionnalité.

Our aim is to propose a non-referential semantics for the principle of logical charity: neither logical universalism (one logic, one way of thinking), nor logical relativism (several logics, several ways of thinking) afford an adequate conceptual framework to interpret the meaning of any speech act. But neither of them is totally wrong, either. The point is to know to which extent each of these views is partly right, thus leading to a more consensual but paradoxical-sounding "relative principle of charity". After recalling the theoretical background of logical charity, we suggest a four-valued logic of acceptance and rejection (hereafter: AR4); then we explain how such a non-referential semantics does justice both to the champions of logical charity and its opponents. While endorsing coherence as a precondition for rationality, we argue that such a criterion does not entail that classical logic is a necessary conceptual scheme to interpret the others' beliefs. A better application of charity should take account of the questions implicitly asked by a statement, and we bring these questions out in replacing Quine's truth-functions by Quine’s verdict functions while emphasizing upon their varying degrees of strength.

What's wrong with modalities in (Wittgenstein 1922)? In (Suszko 1968), the writer argued that "Wittgenstein was somewhat confused and wrong in certain points. For example, he did not see the clear-cut distinction between language (theory) and metalanguage (metatheory): a confusion between use and mention of expressions". Furthermore, a modal logic was proposed in (von Wright 1986) as depicting Wittgenstein's bipolarity thesis in a S5 frame. The aim of the present paper is to deal with the specific case of epistemic modal logic: such a logic of propositional attitudes assumes a philosophy of language that would violate Wittgenstein's two main assumptions.

An argument for the rationality of religious belief in the existence of God is defended. After reviewing three preconditions for rational belief, I show reasons to privilege the criterion of consistency. Taking the inconsistency of the religious belief in God and the belief in the scientific world picture as the impediment to a rational belief in God, I propose that we can overcome this objection by assuming, firstly, that God is a universal class. This allows us to put the problem of God in set-theoretic terms, such that the antinomy that follows from such an assumption can be overcome by assuming that God is not a subject but a strict class that cannot be individuated. I conclude that that the self-contradictory nature of God does not prevent the believer from making a rational, ethical assessment that the contradiction resides in the possibility of using language to explain his existence, but that this does not make belief in the existence of God unjustifiable – on the contrary. In this way, we can say statements that claim God exists are justifiable.

The relevance of any logical analysis lies in its ability to solve paradoxes and trace conceptual troubles back; with this respect, the task of epistemic logic is to handle paradoxes in connection with the concept of knowledge. Epistemic logic is currently introduced as the logical analysis of crucial concepts within epistemology, namely: knowledge, belief, truth, and justification. An alternative approach will be advanced here in order to enlighten such a discourse, as centred upon the word assertion and displayed in terms of utterance. Insofar as epistemic modalities express some attitudes, the intentionality of discourse will be emphasized within an illocutionary modal logic. Two large views will range over the whole study: declarative and epistemic sentences have one and the same logic (assertion logic); the plurality of languages games doesn’t entail any logical pluralism. La valeur d'une analyse logique réside dans sa capacité à résoudre des paradoxes et à comprendre l'origine de nos problèmes conceptuels ; à ce titre, le rôle d'une logique épistémique est de traiter des paradoxes liés au concept de connaissance. On présente généralement la logique épistémique comme une analyse logique de concepts centraux en épistémologie : connaissance, croyance, vérité, justification. Une autre approche sera proposée ici en vue de clarifier ce genre de discours, centrée sur la notion d'assertion et décrite en termes d'énonciation. Parce que les modalités épistémiques expriment des attitudes, c'est l'intentionnalité du discours qui sera mise en valeur dans le cadre d'une logique modale illocutoire. Deux thèses transversales parcourront l'ensemble du travail : les énoncés déclaratifs et épistémiques partagent la même logique (logique assertorique) ; la pluralité des jeux de langages n'implique pas un pluralisme logique.

Can international relations (IR) be a distinctive discipline? In the present paper I argue that such a discipline would be a social science that could be formulated within the perspective of comparative paradigms. The objections to scientific methods are thus overcome by the logic of international oppositions, in other words a model takes several paradigms into account and considers three kinds of foreign relation (enemy, friend, and rival) in the light of three main questions: what is IR about (ontology); what does relate therein (epistemology); and how to assess such a relation (logic).

Does it make sense to employ modern logical tools for ancient philosophy? This well-known debate2 has been re-launched by the indologist Piotr Balcerowicz, questioning those who want to look at the Eastern school of Jainism with Western glasses. While plainly acknowledging the legitimacy of Balcerowicz's mistrust, the present paper wants to propose a formal reconstruction of one of the well-known parts of the Jaina philosophy, namely: the saptabhangi, i.e. the theory of sevenfold predication. Before arguing for this formalist approach to philosophy, let us return to the reasons to be reluctant at it.

This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question–Answer Semantics. We develop a generalized account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression ‘No’ is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators.

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth.

A logical theory of oppositions deals with the relation between propositions and their truth values. On the basis of a formal semantics that proceeds by means of questions-answers, three theses are claimed in the following: (1) the concept of opposition usually refers to incompatibility, but our logical analysis focusses upon a broader relation of difference; (2) more generally, opposition has to do with negativity; our semantics accounts for it through opposite-forming operators; (3) subalternation is a particular case of non-contradiction expressing an “opposition of an opposition”; it is not a proper opposition, accordingly, but its very meaning depends upon the latter. There is not only one but rather a number of expressions for negativity and the paper presents a detailed analysis of these "floating" expressions in a twofold sense of the word: floating quantification is closely related to the logical operations of sentential negation and denial; to what extent natural languages do not clearly express the meaning of a negative thought will be exemplified. L'objectif du texte qui suit est double : non seulement proposer une analyse logique du concept d'opposition, susceptible d'apporter de la précision à cette notion tout aussi vague que courante ; mais aussi, proposer une explication des différentes façons dont la négation se manifeste dans les langues naturelles, en particulier dans notre modèle local de la langue française. Tandis que la communauté des linguistes propose couramment une distinction entre une négation logique et une négation linguistique, l'article qui suit a la prétention d'une explication unifiée de la négation, c'est-à-dire un cadre théorique commun dans lequel ces deux aspects de l'opérateur de négation apparaissent comme des cas particuliers aux significations distinctes. Il n'y aurait donc pas une négation logique, d'un côté, et une négation linguistique ou "métalinguistique", de l'autre. Une approche structurelle des expressions négatives devrait apporter un éclairage sur ce point, par le biais d'une analyse algébrique de la signification en termes de questions-réponses. Mais revenons pour commencer sur l'origine philosophique du concept d'opposition, avant d'en exposer les propriétés logiques par la suite et les expressions linguistiques pour conclure.

It is claimed hereby that, against a current view of logic as a theory of consequence, opposition is a basic logical concept that can be used to define consequence itself. This requires some substantial changes in the underlying framework, including: a non-Fregean semantics of questions and answers, instead of the usual truth-conditional semantics; an extension of opposition as a relation between any structured objects; a definition of oppositions in terms of basic negation. Objections to this claim will be reviewed.

Misunderstanding occurs between speakers when they disagree about the meaning of words in use. In the case of truth-values, Frege took these to be referents of sentences which consist of classes of accepted (i.e. “true”) or rejected (i.e. “false”) sentences. From this usual depiction of truth and falsity, a general algebraic framework is proposed to systematize the use of truth-values from a dialogical point of view of logic. A special attention will be paid to two radically opposed pseudo-speakers: Heraclites and Nagarjuna, according to whom truth-values respectively refer to everything or nothing. Finally, dialectical (pseudo-Hegelian) negation will be rendered as a very special function: an ontological object-forming operator, similar to the arithmetic successor-forming operator Sn+1(x) on integers and consistent within our generalized theory of truth-values as variable referents.