In a previous work [the second and the third author, “On paraconsistent deontic logic”, Philosophia 16, 293-303 (1986)] investigated certain systems of paraconsistent deontic in order to investigate the problem of contradiction in the domain of ethics. This paper continues this line of research, studying some paraconsistent systems containing alethic and deontic modalities. This approach allows us to treat the principles of Kant (OA→ \diamond A) and Hintikka (\square A (...) → OA) from the classical and from the paraconsistent point of view, and to propose systems in which either, both, or neither of these principles is valid. (shrink)

A contrary-to-duty obligation (sometimes called a reparational duty) is a conditional obligation where the condition is forbidden, e.g. “if you have hurt your friend, you should apologise”, “if he is guilty, he should confess”, and “if she will not keep her promise to you, she ought to call you”. It has proven very difficult to find plausible formalisations of such obligations in most deontic systems. In this paper, we will introduce and explore a set of temporal alethic (...) dyadic deontic systems, i.e., systems that include temporal, alethic and dyadic deontic operators. We will then show how it is possible to use our formal apparatus to symbolise contrary-to-duty obligations and to solve the so-called contrary-to-duty (obligation) paradox, a problem well known in deontic logic. We will argue that this response to the puzzle has many attractive features. Semantic tableaux are used to characterise our systems proof theoretically and a kind of possible world semantics, inspired by the so-called T× W semantics, to characterise them semantically. Our models contain several different accessibility relations and a preference relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of them. (shrink)

Impure systems contain Objects and Subjects: Subjects are human beings. We can distinguish a person as an observer (subjectively outside the system) and that by definition is the Subject himself, and part of the system. In this case he acquires the category of object. Objects (relative beings) are significances, which are the consequence of perceptual beliefs on the part of the Subject about material or energetic objects (absolute beings) with certain characteristics.The IS (Impure System) approach is as follows: Objects (...) are perceptual significances (relative beings) of material or energetic objects (absolute beings). The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. Ontic existence of possibility entails that inferential relations have Deontic modality: obligation, permission, prohibition, faculty and analogy. We distinguished between theorems (natural laws) and norms (ethical, legislative and customary rules of conduct). (shrink)

The difficulty to interpret the iteration of modalities, already ontic and still more deontic, incites to pay attention to the system B of basic modal logic that John L. Pollock proposed in 1967. The Pollock’s system brought all the theses which, in the classical ontic modal systems, from Sl to S5, contain no iteration of the modal functors. With this basic ontic system we characterize a basic deontic system, and a basic ontico-deontic system, the former (...) including all the theses of the first two. Each of the three systems is based axiomatically and assorted with a semantics for which the soundness and the completeness hold. (shrink)

Standard deontic logic (SDL) is defined on the basis of possible world semantics and is a logic of alethic-deontic modalities rather than deontic modalities alone. The interpretation of the concepts of obligation and permission comes down exclusively to the logical value that a sentence adopts for the accessible deontic alternatives. Here, we set forth a different approach, this being a logic which additionally takes into consideration whether sentences stand in relation to the normative (...) system or to the system of values under which we predicate the deontic qualifications. By taking this aspect into account, we arrive at a logical system which preserves laws proper to a deontic logic but where the standard paradoxes of deontic logic do not arise. It is a logic of strictly-deontic modalities DR. (shrink)

The history of contemporary modal logic dates back to the writings of C. S. Lewis in the early part of this century. Since then, a growing body of literature has attested to professional interest in the area, and in a number of related issues in philosophical logic which have received wide attention. The recent development of powerful formal techniques for modal system building, together with an increasing interest in modal logic as a tool for philosophical analysis, have created a need (...) for an up-to-date text to introduce students to this material. Snyder's book attempts to answer this need. Assuming an understanding of elementary propositional logic, Snyder introduces a reductive technique called cancellation for detecting the theorems of a system of logic. This technique, analogous to the construction of Smullyan trees but more compact, is extended to modal and quantified contexts. Cancellation versions of the systems M and Mn of G. H. von Wright, and S4 and S5 of Lewis are developed. Snyder uses a Hintikka type of semantics, showing how the various formal systems are differentiated at the semantic level by differing conditions which must be imposed on the model systems used as interpretations for them. These model systems, and the conditions imposed upon them, are in turn described by means of a metalanguage which is essentially first-order quantificational logic. The power of this technique stems from the fact that for every object language formula of a system, there is a corresponding formula in the metalanguage which describes the conditions an interpretation must meet in order to satisfy the original formula. The conditions that the interpretations of a given system of logic must meet are reflected in this metalanguage as "antecedent assumptions," and these in turn correspond to cancellation rules for the object language. This correspondence is constructed in such a way that the metalinguistic counterpart of each theorem of the system will be a theorem of first-order logic. Snyder's primary interest is in showing how modal logic can be used, with the help of these techniques, to build a large variety of formal systems and to "tailor" such systems to meet a variety of analytical tasks. Simple modal systems are developed for the articulation of a number of modal concepts, in addition to the classic alethic ones. Examples are taken from temporal, deontic, and epistemic logic to show how specific interpretations of the meanings of the relevant operators lead to the incorporation of appropriate conditions in the formal systems used to articulate them. An entire chapter is devoted to a discussion of the paradoxes of material and strict implication, and to an attempt to articulate the notion of entailment through the development of formal systems which include a dyadic modal operator that is free of these paradoxes. The final chapter discusses such classical issues as proper names, reference, fictional entities, definite descriptions, and existence presuppositions. Appendices deal with such matters as the equivalence of the cancellation systems to more classical axiomatic-deduction systems, and the sketch of a proof of soundness and completeness for all the modal systems presented. While designed as a text, this volume should be of interest to both logicians and those working in metaphysics and language analysis, since the primary concern of the author is to develop techniques that will facilitate the usefulness of modal logic as a tool for philosophical analysis. The instructor's manual contains many suggestions derived from the author's experience in teaching this non-standard treatment of the subject.--K. T. (shrink)

This chapter contains sections titled: Modal Concepts The Semantics of Modalities and Systems of Modal Logic Modality and Quantification Deontic, Epistemic, and Temporal Modalities Epistemic Logic Deontic Logic Temporal Frames Conditional Obligations and Rules of Detachment.

The deontic status of tautologies and contradictions is one of the major puzzles for authors of early works on deontic logic. It is well-known that von Wright addresses this problem by adopting a Principle of Deontic Contingency, which says that tautologies are not necessarily obligatory and contradictions are not necessarily forbidden. A more radical solution is proposed by Anderson within a reductionist approach to deontic logic and consists in restricting the range of application of deontic (...) modalities to contingent propositions. Anderson’s solution has not received much attention in the literature, despite reflecting a typical feature of ordinary deontic reasoning, where non-contingent propositions are rarely, if ever, taken into account. In the present article we explore some of its formal consequences, providing a taxonomy of the properties of the Andersonian operators of obligation and permission for contingent propositions, O′ and P′, in the class of normal alethic systems. (shrink)

A suggestion is made for representing iterated deontic modalities in stit theory, the “seeing-to-it-that” theory of agency. The formalization is such that normative sentences are represented as agentive sentences and therefore have history dependent truth conditions. In contrast to investigations in alethic modal logic, in the construction of systems of deontic logic little attention has been paid to the iteration... of the deontic modalities.

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a major logical theory (...) and offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses the history of paraconsistent logic; language, negation, contradiction, consistency and inconsistency; logics of formal inconsistency and the main paraconsistent propositional systems; many-valued companions, possible-translations semantics and non-deterministic semantics; paraconsistent modal logics; first-order paraconsistent logics; applications to information processing, databases and quantum computation; and applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. (shrink)

Deontic logic handles not only deontic modalities, but also alethic and temporal ones. In addition, individuals like authorities and addressees play an important role. R5-D5 is a system handling alethic, deontic and temporal modalities, whose adequacy has been proved in an earlier paper. Similarly for KD*UXY with sets of individuals . The present article is an attempt to construct a general system mixing R5-D5 and KD*UXY.

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with (...) the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)

The purpose of this paper is to describe a set of quantified temporal alethic-deontic systems, i.e., systems that combine temporal alethicdeontic logic with predicate logic. We consider three basic kinds of systems: constant, variable and constant and variable domain systems. These systems can be augmented by either necessary or contingent identity, and every system that includes identity can be combined with descriptors. All logics are described both semantically and proof theoretically. We use a (...) kind of possible world semantics, inspired by the so-called T × W semantics, to characterize them semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics. (shrink)

Computational systems are traditionally approached from control-oriented perspectives; however, as soon as we move from centralized to decentralized computational infrastructures, direct control needs to be replaced by distributed coordination mechanisms that are on par with institutional constructs observable in human societies (contracts, agreements, enforcement mechanisms, etc.). This paper presents a formalization of Hohfeld's framework building upon a logic whose language includes primitive operators of ability and parametric deontic operators. The proposal is meant to highlight the fundamental interaction between (...) deontic and potestative concepts and contains proofs of soundness and completeness with respect to a class of relational models. (shrink)

This paper will introduce and explore a set of quantified counterfactual temporal alethic-deontic systems, that is, systems that combine counterfactual temporal alethic-deontic logic with predicate logic. I will consider three types of systems: constant, variable and constant and variable domain systems. Every system can be combined with either necessary or contingent identity. All logics are described both semantically and proof theoretically. I use a kind of possible world semantics, inspired by the so-called (...) T x W semantics, to characterise them semantically and semantic tableaux to characterise them proof theoretically. Our models contain several different accessibility relations and a similarity relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of these. (shrink)

sBuilding models as a practical aspect of ecological theory has as a principal purpose the determination of relations in formal language. In this paper, the authors provide a formalization of ecological models based on impure systems theory. Impure systems contain objects and subjects: subjects are human beings. We can distinguish a person as an observer that by definition is the subject himself and part of the system. In this case he acquires the category of object. Objects are significances, (...) which are the consequence of perceptual beliefs on the part of the subject about material or energetic objects with certain characteristics. The impure system approach is as follows: objects are perceptual significances of material or energetic objects. The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. In this work we define measures which let us choose the more suitable variables to relate both the model with the ecosystem and with different models. In this way we define different comparison indexes. (shrink)

Though there have been productive interactions between moral philosophers and deontic logicians, there has also been a tradition of neglecting the insights that the fields can offer one another. The most sustained interactions between moral philosophers and deontic logicians have notbeen systematic but instead have been scattered across a number of distinct and often unrelated topics. This chapter primarily focuses on three topics. First, we discuss the “actualism/possibilism” debate which, very roughly, concerns the relevance of what one will (...) do at some future time to what one ought to do at present (§2). This topic is also used to introduce various modal deontic logics. Second we discuss the particularism debate which, very roughly, concerns whether there can be any systematic general theory of what we ought to do (§3). This topic is also used to introduce various non-modal deontic logics. Third, we discuss collective action problems which concern the connection between the obligations of individuals and the behavior and obligations of groups of individuals (§4).This topic is also used to discuss formal systems that allow us to study the relationship between individuals and groups. The chapter also contains a general discussion of the relation between ethical theory and deontic logic (§1) and a brief consideration of other miscellaneous topics (§5). (shrink)

The purpose of this paper is to develop a class of semantic tableau systems for some dyadic deontic logics. We will consider 16 different pure dyadic deontic tableau systems and 32 different alethic dyadic deontic tableau systems. Possible world semantics is used to interpret our formal languages. Some relationships between our systems and well known dyadic deontic logics in the literature are pointed out and soundness results are obtained for every tableau (...) system. Completeness results are obtained for all 16 pure dyadic deontic systems and for 16 alethic dyadic deontic systems. (shrink)

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for (...) defining truth-functional modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics. (shrink)

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on in our field—a (...) book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

We study a modal language for negative operators—an intuitionistic-like negation and its paraconsistent dual—added to (bounded) distributive lattices. For each non-classical negation an extra operator is hereby adjoined in order to allow for standard logical inferences to be opportunely restored. We present abstract characterizations and exhibit the main properties of each kind of negative modality, as well as of the associated connectives that express consistency and determinedness at the object-language level. Appropriate sequent-style proof systems and adequate kripke semantics (...) are also introduced, characterizing the minimal normal logic and a few other basic logics containing such negative modalities and their companions. (shrink)

Many interesting philosophical principles include two kinds of modalities, e.g. epistemic and doxastic, alethic and epistemic, or alethic and deontic modalities.The purpose of this essay is to describe a set of bimodal systems, i.e. systems that include two kinds of modal operators, in which it is possible to investigate some formalizations of such principles. All in all we will consider 4,194,304 logics. All logics are described semantically and proof theoretically. We use possible world (...) semantics to characterize the logics semantically, and both axiomatic systems and semantic tableaux to characterize them proof theoretically. We show that all systems are sound and complete with respect to their semantics and we consider some relationships between the various systems. (shrink)

The passionate and staunch defence of logic of the controversial thinker Ibn Ḥazm, Abū Muḥammad ʿAlī b. Aḥmad b. Saʿīd of Córdoba (384-456/994-1064), had lasting consequences in the Islamic world. Indeed, his book Facilitating the Understanding of the Rules of Logic and Introduction Thereto, with Common Expressions and Juristic Examples (Kitāb al-Taqrīb li-ḥadd al-manṭiq wa-l-mudkhal ilayhi bi-l-alfāẓ al-ʿāmmiyya wa-l-amthila al-fiqhiyya), composed in 1025-1029, was well known and discussed during and after his time; and it paved the way for the studies (...) of his compatriots Ibn Bājjah (d. 1138), Ibn Ṭufayl (1185), and Ibn Rushd (1198), who each gave demonstrative reasoning a privileged place within the methods of attaining knowledge. Unfortunately, as too often in the history of science, Ibn Ḥazm’s innovative perspectives and contributions in logic have been overlooked or considered with an attitude of contempt. On the one hand, his work has been seen, at best, as promulgating the benefits of studying Aristotle’s logic, so that his contribution is assessed as more didactical than conceptual. And on the other hand, those who do examine his innovations often consider them to be mistaken. As indicated by Chejne (1984, p. 2) contempt towards the logical work of Ibn Ḥazm was also present in its reception by the Eastern philosophers who accused him of deviating from Aristotelian logic and of dabbling in things beyond his capability. However, a reassessment of his work on logic has since begun, by delving into the ways the thinker of Córdoba studied the links between deontic and modal qualification of propositions. In this context Lameer’s (2013) paper on the logical sources of Ibn Ḥazm is worth mentioning; the author (p. 417, footnote 1) observes that, although, strictly speaking, it was al-Fārābī who first drew the parallelism between deontic and modal concepts, it was Ibn Ḥazm who developed it and worked it out in a more precise manner. A primary aim of this paper is to help fill some of these gaps by stressing the role of the work of Ibn Ḥazm in developing a notion of deontic necessity deeply rooted in legal normativity, and in explicitly discussing the transference between deontic and modal concepts. According to our view; the basic units of Islamic deontic logic are what we might call, indulging in terminological anachronism, heteronomous imperatives. As it turns out, the heteronomy of imperatives within Islamic legal systems contrasts with those of the purely moral realm, which seem to be closer to an autonomous conception of moral law. There it concurs again with Leibniz’s proposal to define obligatory as “what is necessary for a good person to do”. In the present paper, we will focus on the heteronomous imperatives of legal systems rather than on the imperatives of the purely moral realm. In this context, the work of Ibn Ḥazm extends the parallelism between the necessity of events and that of human actions stressed by his predecessors. According to our understanding, Ibn Ḥazm’s parallelism can be rendered explicit formally by means of a conditional (or hypothetical) structure shared by both deontic and modal propositions. Moreover, this structure makes apparent that this parallelism displays an underlying system of “degrees”. Indeed, while in the domain of events, given some conditions, it makes sense to distinguish between an event that is more likely to happen than another. Ibn Ḥazm speaks of near and distant possibility (such as the higher degree of likelihood of rain, given the condensation of clouds in December, and its lower degree, when the condensation occurs in summer); and, in the domain of actions, the Islamic notion of weighting actions determines degrees of virtue (or of legal and moral value). So whereas the degree of virtue of performing a forbidden act as determined by the distribution of sanction and reward is 0, we obtain the same value by pondering the likelihood of an impossible event to take place. Furthermore, as discussed in section II.3.2, the system of values at work in the parallelism can be seen as opening the way to a more direct correspondence. Thus, while in the realm of nature, likelihood of occurrence of an event is dependent upon the conditions specific to the occurrence of that event; while, in the realm of jurisprudence, likelihood of performing an action is dependent upon the distribution of reward and sanction specific to that action (being that, given the choice to perform or not perform a given action, those that will be rewarded are more likely to be performed than those that are not). According to this perspective, the point of the parallelism is that both deontic and modal qualifications measure the degree of an action or event to become actual; that is – indulging once more in anachronism (but this time from the Leibnizian background) – the degree measures how feasible (facile) an action or event may be. This suggests that Ibn Ḥazm’s perspectives already herald the links to probability and possibility explored by Leibniz six centuries later – see for example Leibniz’s (1671, A VI, I, p. 424-26) use of probability in the context of conditional right. Nevertheless, it also shows a crucial difference to the approach developed in the Elementa Jura Naturalis: whereas Leibniz’s studies seek to define what is to be just or virtuous, the logical system of deontic imperatives within Islamic jurisprudence presupposes that what is to be virtuous has already been settled. Determining what is to be virtuous is not achieved by logical reasoning within the system of legal jurisprudence, but by delving into the higher objectives of the Sharīʿa. While developing our point, we will delve into the logical structure of the heteronomous imperatives. This distinguishes our contribution from the existing literature, such as the papers of Chejne (1984), Lameer (2014), and Guerrero (1997, 2010, 2014), which do not provide a logical analysis of the deontic concepts put into work by Ibn Ḥazm. The true antecedent to the present paper is the work of Farid Zidani (2007, 2015), who, so far as we know, was the first to undertake such a task. Some of our own developments and general epistemological thoughts go beyond Ibn Ḥazm’s framework and motivations; mainly those contained in sections II.3.2 and IV. However, according to our view, these reflections suggest that Ibn Ḥazm’s approach has the substance for a broader and deeper exploration of the logic of norms. The paper is structured as follows:I.Ibn Ḥazm’s Logic of Heteronomous Imperatives. After presenting some extracts of the relevant text, we proceed by providing a formal reconstruction of the five forms of deontic modalities. II. A Landmark in the History of the Logical Analysis of Norms. Duties and Modalities. In this section, we study the transferences from deontic to modal necessity and possibility. We briefly compare the deontic system of Islamic Jurisprudence with that of Leibniz.III. Leibniz and Hypothetical Imperatives in Law.IV. Beyond Ibn Ḥazm: Conclusions and the Work Ahead. We will conclude the paper by discussing briefly some conceptual points that distinguish the logic of heteronomous imperatives from contemporary deontic logic. Our final words will discuss deontic-modal parallelism in the context of Ibn Ḥazm’s rejection of reasoning by conjecture and what we call “the internalization of nature.”. (shrink)

The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind (...) of$$T \times W$$T×Wmodels, and the proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper. (shrink)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...) predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

We consider an infinite hierarchy of systems of Alethic Modal Logic with so-called Levels of Perfection, and add to them suitable definitions of such interesting deontic categories as those of supererogation, offence, conditional obligation and conditional permission. We then state three problems concerning the proper characterization of the resulting logic(s) for our defined notions, and discuss two of these problems in some detail.

This chapter contains sections titled: Introduction Paraconsistent Logic and Latin America Thinking about Logic The Nature of Paraconsistent Logic A History of Paraconsistent Logic Philosophical Aspects of Paraconsistent Logic References Further Reading.

In this paper we investigate nonnormal modal systems in the vicinity of the Lewis system S1. It might be claimed that Lewis's modal systems (S1, S2, S3, S4, and S5) are the starting point of modern modal logics. However, our interests in the Lewis systems and their relatives are not (merely) historical. They possess certain syntactical features and their frames certain structural properties that are of interest to us. Our starting point is not S1, but a weaker (...) logic S1 (S1 without the schema [T]). We extend it to S1D, which can be considered as a deontic counterpart of the alethic S1. Soundness and completeness of these systems are then demonstrated within a prenormal idiom. We conclude with some philosophical remarks on the interpretation of our deontic logic. (shrink)

A number of systems of logic, including formulations of the two-valued propositional logic, alethic modal propositional logics and deontic modal propositional logics, are described and discussed with the aim of utilizing some of these systems for the sociological analysis of normative structures. Evidence is adduced that certain normative concepts are explicated or partially explicated in these systems.--W.T.

Great thinkers have devoted to explaining morality and ethics in human beings. The major reflections have resulted in a well-known dichotomy, that of matters of fact and matters of value, or what is known as the theoretical world, which is objective, and the practical world, that of affections. With the birth of analytic philosophy, the emphasis is placed on language allowing to explain philosophical problems, such as validity. This study proposes the following thesis: it is possible to derive “ought” from (...) “is”, as it is evidenced in the Spanish verb “tocar” and several of its verbal periphrasis. To achieve this, a naturalization of morality is proposed from the postulates of two theories, namely, cognitive semantics (Talmy 2000) and the Modular Theory of Modalities (Gosselin 2010). The methodology applied consists of both a semantic and a cognitive analysis of “tocar”, for which a corpus was collected in literary texts and presidential speeches. The main result shows that “tocar” has a prototypical descriptive semantic value, alethic value, and new prescriptive semantic values. It is concluded that the new semantic values for this verb and its periphrasis are the deontic and the axiological ones, which demonstrates once again the evolutionary character of language and the need to provide empirical information on philosophical issues, such as morality. (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics (...) for the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

In this paper we investigate nonnormal modal systems in the vicinity of the Lewis system S1. It might be claimed that Lewis's modal systems are the starting point of modern modal logics. However, our interests in the Lewis systems and their relatives are not historical. They possess certain syntactical features and their frames certain structural properties that are of interest to us. Our starting point is not S1, but a weaker logic S1$^0$. We extend it to S1$^0$D, (...) which can be considered as a deontic counterpart of the alethic S1. Soundness and completeness of these systems are then demonstrated within a prenormal idiom. We conclude with some philosophical remarks on the interpretation of our deontic logic. (shrink)

In this paper I develop a theory of 'function' and function as a deontic modal word and phenomenon. Kratzer’s account of the semantics for the deontic modals is invoked and using her approach a formal schema for the semantics of 'function'-sentences is proposed. My account of function is a modalized and extended version of Cummins’ systems-type account of function. In the biological and physical sciences, on this account, function is a complex empirical deontic modal property. It (...) is built on the property of X’s doing Y well enough to enable Z, which is implicitly deontic because of the evaluative but nonetheless empirical in its biological and physical applications. This account of function resolves many of the traditional puzzles about biological and physical function, and extends naturally to include the other types of function. With a variant treatment of the semantics, this account is argued also to apply to mathematical function, where it is shown to be interestingly related to Frege’s account of function. (shrink)

According to one view of alethic modality, to say that something is necessary is to say that we must take that thing to be true according to rules of thinking or linguistic rules. In other words, alethic modality is reduced to deontic modality with respect to thoughts or language. This view has been argued to have many philosophical advantages over the traditional view that takes alethic modality to describe something in the world. In this paper, I (...) argue that the deontic view also enjoys a wide range of empirical supports from linguistics and psychology. (shrink)

In this paper we present an overview, with historical and critical remarks, of two articles by S. Jaśkowski ([20, 21] 1948 and [22, 23] 1949), which contain the oldest known formulation of a paraconsistent logic. Jaśkowski has built the logic – he termed discussive (D2) – by defining two new connectives and by introducing a modal translation map from D2 systems into Lewis’ modal logic S5. Discussive systems, for their formal details and their original philosophical justification, have (...) attracted discrete attention among experts. Indeed, in what follows, after having introduced Jaśkowski methodology of building D2 and his main philosophical motivations for providing such a system, we will explore some of the main contributions to the development of D2. (shrink)

The Moral Law is fulfilled iff everything that ought to be the case is the case, and The Good is realised in a possible world w at a time t iff w is deontically accessible from w at t. In this paper, I will introduce a set of temporal modal deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of (...) class='Hi'>systems without any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete. (shrink)

The purpose of this paper is to describe a set of temporal alethic–deontic systems, i.e. systems that include temporal, alethic and deontic operators. All in all we will consider 2,147,483,648 systems. All systems are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by the so-called T x W semantics, to characterize our systems semantically and semantic tableaux to characterize them proof theoretically. We also show (...) that all systems are sound and complete with respect to their semantics. (shrink)

Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, cKN4, scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and cKN4 are connexive variants of the FDE-based modal logics BK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cBK}^-$$\end{document} and cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} are shown to be algebraizable. The algebraizability of cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} is derived from proving cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} to be definitionally equivalent to MBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MBL}$$\end{document}. All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics. (shrink)

Leibniz suggested that deontic modalities can be defined in terms of the alethic modalities; according to him, the permitted (licitum) is what possible for a good man to do and the obligatory (debitum) is what is necessary for a good man to do. The paper starts from specifying a connection of deontic concepts with the moral values. The connection comes down to define an isomorphism of two Boolean algebras: from deontic one onto axiological one. (...) The work presents theories of two algebras of deontic notions: the algebra of sets and the Boolean algebra. The theory of deontic set is based on the two axioms: x∈V (an act x is an element of the set of acts subordinated to some norm or law) and x\'\'=x (an act x is identical with double denial of x). By means of definitions following notions are introduced: Λ (the empty set of acts), N (the set of ordered acts), Z (the set of forbidden acts), P (the set of obligatory acts), F (the set of optional acts), D (the set of permitted acts), I (the set of indifferent acts).The calculus is structured by rules of the Słupecki-Borkowski’s suppositional deduction. Forty five theorems are proven in this calculus. The second theory presented in the paper, is a Boolean algebra of deontic notions. Added to the theory of equality, it takes axioms from the theory of Boolean algebras with addition of a specific axiom for the deontic system i.e., N = N∩D. Sixty four theorems are proven in this calculus. (shrink)

Deontic reasoning is thinking about whether actions are forbidden or allowed, obligatory or not obligatory. It is proposed that social norms, imposing constraints on individual actions, constitute the fundamental concept for the system of these four deontic modalities, and that people reason from such norms flexibly according to deontic core principles. Two experiments are presented, one on deontic conditional reasoning, the other on “pure” deontic reasoning. Both experiments demonstrate people's high deontic competence and (...) confirm the proposed representational and inferential principles. Experiment 1 additionally shows small effects of the conditional formulations. These findings support the dual source approach (Beller & Spada, 2003) that distinguishes between domain-specific and domain-general inferences. Implications for other theories of deontic reasoning are discussed. (shrink)

The aim of this note is to delineate some new systems of logic which govern the interconnections between two kinds of obligation: moral obligation and legal obligation . In consequence, we shall have in such a logic two kinds of forbideness of premission, and of indifference, but obviously definable in terms of the two obligation operators and negation.

This volume deals with approaches to handling contradictory information. These include approaches for actual contradiction - both A and not-A can be proven from the information - and approaches for potential contradiction - where the information may contain arguments for A and arguments for not-A, but the system suppresses the contradiction by, for example, preferring some arguments over others. Approaches covered include paraconsistent logics, modal logics, default logics, conditional logics, defeasible logics and paraconsistent semantics for logic programming. The (...) volume is of interest to students, researchers and practitioners in artificial intelligence, software engineering, logic, language and philosophy. This volume is the first handbook to give a comprehensive coverage of handling contradictory information. (shrink)

This book is an introductory logic text of moderate difficulty which contains added topics not usually found in an introductory book. The book has two parts--basic logic and advanced logic. The basic logic contains propositional logic through conditional proofs, syllogistic logic, the fundamentals of set theory and their application to both syllogistic and non-syllogistic inferences along with the use of Venn and Carroll diagrams, and concludes with predicate logic using the rules for Universal Instantiation, Existential Instantiation, Universal Generalization, and Existential (...) Generalization. The part on advanced logic begins by extending the predicate logic to identity, descriptions, and relations. A presentation of modal logic includes C. I. Lewis' modal systems S3, S4, and S5; modal logic with quantification; epistemic, doxastic, and deontic modalities; and mixed modalities. A chapter on the logic of ordinary language considers analytic statements, empty expressions, and equivalence and implication in ordinary language. The author also discusses "tools of analysis" such as definition, the theory of meaning, probability theory, and scientific inference. Each chapter concludes with a section entitled "Philosophical Applications" or "Philosophical Difficulties." One of these sections contains the author's version of Hartshorne's modal proof of Anselm's ontological argument. Answers are provided for the even-numbered exercises.--T. G. N. (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics (...) for the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

The aim of this book is to introduce the reader to some new areas of contemporary logic which generally fall under the rubric of philosophical logic. It succeeds in this task to a degree, although the chapters are for the most part adaptations of journal articles published by Rescher over the last ten years and are more self-contained than they might have been. But the book should renew interest in the problems of philosophical logic. It contains many interesting discussions and (...) a great deal of useful information. Rescher begins with chapters on modal logic which include some discussion of intuitionistic logic and the causal modalities as well as the alethic modalities. He then discusses the notion of belief as a representative notion of epistemic logic. A long chapter is devoted to a history and survey of the main systems of many valued logics. Shorter, but still substantial, chapters are devoted to the logic of existence, non-standard quantification theory, chronological or tense logic, topological or positional logic, logic of assertion and logic of preference. Still shorter chapters are given to deontic logic, probability logic, a discussion of random individuals and self-reference. An interesting final chapter provides a new "discourse on method" for the philosophical or applied logician.--R. H. K. (shrink)

The main idea animating the present paper is that the general aim of debates, such as the one involving the notorious case of the Master Argument, is the ponderation of logical principles by confronting them with some set of assertions and other endorsed principles on the meaning explanation of connectives, quantifiers and modality. As suggested by Seel (2017), the point of the specific case of the MA is about examining Aristotle’s notion of possibility – as implemented by the Possibility Principle (...) (i.e., If it is necessary that if A then B, then B is possible if A is) and the Possibility Rule (i.e., If A is possible, and if B follows from the assumption of A, then B is is possible) in the context of a dialectical debate where it has been asserted that a given proposition is possible, even it is neither true now nor it will ever be, and where at the same time the irrevocability of the past has been endorsed. This suggests to study the MA debate in a setting based on Marion’s (2020) reconstruction of Aristotle’s Dialectical Bouts, adapted to propositions affected by modalities. The setting of Dialectical Bouts for the MA proposed includes•Propositions affected by Reichenbach’s (1948, § 5) three-folded temporal reference, namely: the point of speech S, the point of reference R and the point of the event E. This constitutes our take on Vuillemin’s (1977, 1979, 1984, 1996) suggestion of a “double indexation”.•The dialogical meaning explanation for modalities understood as involving hypotheticals. •The dialogical rules that settle the commitments undertaken when endorsing Aristotle’s Possibility Principle. In fact, in relation to the notion of possibility at work in Vuillemin’s (1996) reconstruction we explored different formulations of the Possibility Rule. Still, there remains a dearth of researches confronting the different formulations of the Possibility Rule with the texts studied by Crubellier (2010) and Rosen and Malink (2012). More generally and coming back to my remarks above, one of the most fruitful insights that can be gained from Vuillemin’s writings on the MA, is that such kind of debates provide an instance of a general pattern for reasoning that I shall call Dialectical Bouts of Material Admissibility (BMA). The rationale behind such kind of argumentation-pattern is to examine the incorporation of new (epistemological, ontological or logical) principles against the background of the commitments undertaken by endorsing a specific set of assertions and meaning explanations. They display the kind of abductive procedure stressed by Crubellier’s (2014) study of syllogism. Under this perspective Ebbinghaus’s (1964) casting of the inferential moves that reduce imperfect syllogisms to perfect ones as implementing an admissibility procedure, represents a logical generalization of the material procedures at stake in the Dialectical Bouts of Material Admissibility. Furthermore if, as stressed by Duthil-Novaes (2005) Medieval Obligations represent a development of Aristotelian Bouts aimed at the ludic exercise of consistency maintenance, DMA should be rather linked to the Islamic tradition of dialectical theory [jadal, munāẓara, ādāb al-baḥth], aimed at the epistemological task of what Young (2017) calls the forging of principles, including legal and deontic ones. In a more contemporary setting, DMA can be related to Brandom’s (2000, pp. 66-77) considerations on Non-Conservative Extensions, where the point of dialectical procedures is put under public Socratic scrutiny the discovery and elucidation of concepts. This general point on Dialectical Bouts of Material Admissibility can be seen as underlying Vuillemin’s project of linking the study of the MA with the classification of philosophical systems.The present paper does not contain a fully developed study on such Dialectical Bouts, but it is rather is an invitation to pick up the gauntlet that Jules Vuillemin threw into the ring. (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)