Deontic Logic

This is the story of how the noble squire Sancho Panza, while governing what he thought to be an insula, ingeniously solved a paradox not unlike those of modern logic.

I identify and develop a solution to the puzzles of anankastic conditionals that is novel in the sense that it has gone largely unnoticed, but also well-worn in that the materials for it have long been available. The solution involves an integration of the classical Kratzerian premise semantics and a default theory of reasons (such as the one presented in Horty, 2012, leveraging several decades of research on default logic). To stress-test the proposal I also investigate how it might be (...) applied to the variety of anankastic-adjacent data discussed by Condoravdi and Lauer (2016). The resulting approach is not just an exceptional fit for the problem of anankastic conditionals, but also an independently promising upgrade on the classical account of modals. (shrink)

Niko Kolodny and John MacFarlane claim that a situation involving some trapped miners involves a deontic paradox the resolution of which requires rejecting the logical law of modus ponens. I show that the appearance of paradox results from confusion and that the miners case supplies no cogent reason for impugning modus ponens.

Mind, Vol. LXXXVI (April, 1977) pp. 242-248.

There has been a recent surge of work on deontic modality within philosophy of language. This work has put the deontic logic tradition in contact with natural language semantics, resulting in significant increase in sophistication on both ends. This chapter surveys the main motivations, achievements, and prospects of this work.

This paper generalizes the preface paradox beyond the conjunctive aggregation of beliefs and constructs an analogous paradox for deontic reasoning. The analysis of the deontic case suggests a systematic restriction of intuitive rules for reasoning with obligations. This proposal can be transferred to the epistemic case: it avoids the preface and the lottery paradox and saves one of the two directions of the Lockean Thesis (i.e., high credence is sufficient, but not necessary for rational belief). The resulting account compares favorably (...) to competing proposals; in particular, we can formulate the rules of correct doxastic reasoning without reference to probabilistic features of the involved propositions. (shrink)

In the recent epistemological literature much has been written about the nature of suspending judgement or agnosticism. There has also been a surge of recent interest in the nature of ignorance. But what is the relationship between these two epistemically significant states? Prima facie, both suspension and ignorance seem to involve the lack of a correct answer to a question. And, again prima facie, there may be some intuitive attraction to the idea that when one is ignorant whether p, one (...) ought to suspend judgement whether p. In this paper we will explore a range of different possible normative connections between ignorance and suspension of judgement. We will also discuss how both ignorance and suspension are related to having inconclusive evidence. As we will see, the issues can become quite complex due to the fact that there are different, competing views about the nature of ignorance and also about the nature of suspension. Moreover, there are importantly different kinds of epistemic norms. Our overall aim then will be to systematically chart how these different choices will impact the possible normative connections between ignorance and suspension. (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)

G3-style sequent calculi for the logics in the cube of non-normal modal logics and for their deontic extensions are studied. For each calculus we prove that weakening and contraction are height-preserving admissible, and we give a syntactic proof of the admissibility of cut. This implies that the subformula property holds and that derivability can be decided by a terminating proof search whose complexity is in Pspace. These calculi are shown to be equivalent to the axiomatic ones and, therefore, they are (...) sound and complete with respect to neighbourhood semantics. Finally, a Maehara-style proof of Craig’s interpolation theorem for most of the logics considered is given. (shrink)

In her recent book, Russell (2023) examines various so-called “barriers to entailment,” including Hume’s law, roughly the thesis that an ‘ought’ cannot be derived from an ‘is.’ Hume’s law bears an obvious resemblance to the proscription on fallacies of modality in relevance logic, which has traditionally formally been captured by the so-called Ackermann property. In the context of relevant modal logic, this property might be articulated thus: no conditional whose antecedent is box-free and whose consequent is box-prefixed is valid (for (...) the connection, interpret box deontically). While the deontic significance of Ackermann-like properties has been observed before, Russell’s new book suggests a more broad-scoped formal investigation of the relationship between barrier theses of various kinds and corresponding Ackermann-like properties. In this paper, I undertake such an investigation by elaborating a general relevant bimodal logical framework in which several of the barriers Russell examines can be given formal expression. I then consider various Ackermann-like properties corresponding to these barriers and prove that certain systems satisfy them. Finally, I respond to some objections Russell makes against the use of relevance logic to formulate Hume’s law and related barriers. (shrink)

Different notions of the consistency of obligations collapse in standard deontic logic. In justification logics, which feature explicit reasons for obligations, the situation is different. Their strength depends on a constant specification and on the available set of operations for combining different reasons. We present different consistency principles in justification logic and compare their logical strength. We propose a novel semantics for which justification logics with the explicit version of axiom D, jd⁠, are complete for arbitrary constant specifications. Consistency is (...) sometimes formulated in terms of permission. We therefore study permission in the context of justification logic, introducing a notion of free-choice permission for the first time. We then discuss the philosophical implications with regard to some deontic paradoxes. (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)

According to Act Consequentialism, an act is right if and only if its outcome is not worse than the outcome of any alternative to that act. This view, however, leads to deontic paradoxes if the alternatives to an act are all other acts that can be done in the situation. A typical response is to only apply this rightness criterion to maximally specific acts and to take the alternatives to a maximally specific act to be the other maximally specific acts (...) that can be done in the situation. (This view can then be supplanted by a separate account for the rightness of acts that are not maximally specific.) This paper defends a rival view, Binary Act Consequentialism, where, for any voluntary act, that act is right if and only if its outcome is not worse than the outcome of not doing that act. Binary Act Consequentialism, which dates back to Jeremy Bentham, has few supporters. A number of seemingly powerful objections have been considered fatal. In this paper, I rebut these objections and put forward a positive argument for the view. (shrink)

Resolving the Ross and Prior paradoxes proved to be a difficult task. Its first two stages, involving the identification of the true natures of the implication and truth values, are described in the articles "Solving the Paradox of Material Implication – 2024" and "Solving Jörgensen's Dilemma – 2024". This article describes the third stage, which involves the discovery of missing modal operators and the Classical Modal Calculus. Finally, procedures for solving both paradoxes are provided.

Dual-role approaches to reasons say, roughly, that reasons can relate to actions in two fundamentally different ways: they can either require conformity, or justify an action without requiring that it be taken. This paper develops a formal dual-role approach, combining ideas from defeasible logic and practical philosophy. It then uses the approach to shed light on the phenomenon of supererogation and resolve a well-known puzzle about supererogation, namely, Horton’s All or Nothing Problem.

Havstad (2022) argues that the argument from inductive risk for the claim that non-epistemic values have a legitimate role to play in the internal stages of science is deductively valid. She also defends its premises and thus soundness. This is, as far as we are aware, the best reconstruction of the argument from inductive risk in the existing literature. However, there is a small flaw in this reconstruction of the argument from inductive risk which appears to render the argument invalid. (...) This flaw is superficial, and a small amendment to it rescues the claim of validity. (shrink)

I attempt to provide a general account of deontic context, equally applicable within and an intensional and a hyperintensional framework; I compare Rothschild's and Yablo's accounts of the semantics for deontic logic, deontic updating and denotic duality with my own accounts; and I conclude with some general remarks on negation.

This paper develops a deontic logic based on dynamic logic for reasoning about permission and prohibition of sequential actions. Our approach is characterized by two main features. First, permission and prohibition of sequential actions are not necessarily reduced to permission and prohibition of the actions’ constituent parts. Second, we incorporate the idea that actions may be permitted or prohibited conditional on another action being performed first. The logic interprets actions in terms of sequences of states, and the deontic component of (...) the logic is introduced by relating sequences of states to their legal and illegal future continuations. We consider various different logics obtained by imposing natural constraints on models, and formulate complete axiom systems for several of these logics. (shrink)

One family of maximizing act consequentialist theories are actualist direct theories. Indeed, historically there are at least three different forms of actualist direct consequentialism (due to Bentham, Moore, and contemporary consequentialists). This paper is about the logical differences between these three actualist direct theories and the differences between actualist direct theories and their competitors. Three main points emerge. First, the sharpest separation between actualist direct theories and their competitors concerns the so-called inheritance principle. Second, there are a myriad of other (...) logical differences among actualist direct theories. Third, one theory (Moore's theory) stands out among actualist direct theories because it entails a variety of logical principles. This fact may count in favor of that theory. (shrink)

This paper investigates inquisitive extensions of normal modal logic with an existential modal operator taken as primitive. The semantics of the existential modality is generalized to apply to questions, as well as statements. When the generalized existential modality is applied to a question, the result is a statement that roughly expresses that each way of resolving the question is consistent with the available information. I study the resulting logic both from a semantic and from a proof-theoretic point of view. I (...) argue that it can be used for reasoning about a general notion of ignorance, and for reasoning about choice-offering permissions and obligations. The main technical results are sound and complete axiomatizations, both for the class of all Kripke frames, and for any class of frames corresponding to a canonical normal modal logic. (shrink)

In this work, we provide an extensive analysis of Hohfeld’s theory of normative relations, focusing in particular on diagrammatic structures. Our contribution is threefold. First, we specify an extensional formal language to represent the main notions in the two families of normative relations identified by Hohfeld (i.e. the deontic and the potestative family). Our primary focus is on the part of the theory concerning potestative relations. In this regard, we assign a key role to the concept of ability, which is (...) treated as a primitive notion and used to formulate three fine-grained definitions of power (outcome-centered, change-centered and force-centered). Second, on the basis of these definitions we build Aristotelian diagrams of opposition for deontic and potestative relations, improving, extending and systematizing previous proposals formulated in the literature. Third, we present a model-theoretic interpretation and a logic programming (ASP) implementation of the proposed framework, elaborating on the procedural dimension of normative reasoning. (shrink)

We argued that formal objections to the error theory overgeneralize and therefore fail. Christine Tiefensee and Gregory Wheeler deny this. We argue that they are wrong, for two reasons. The first concerns how we should adjudicate conflicts between formal and substantive commitments. The second concerns an overlooked tension between formal objections and non-error-theoretic views. Our discussion shows that the commitments behind formal objections to the error theory, such as the dual schema, should be regarded as much more contentious than is (...) often assumed. (shrink)

In common treatments of deontic logic, the obligatory is what is true in all deontically ideal possible worlds. In this article, I offer a new semantics for Standard Deontic Logic with Leibnizian intensions rather than possible worlds. Even though the new semantics furnishes models that resemble Venn diagrams, the semantics captures the strong soundness and completeness of Standard Deontic Logic. Since, unlike possible worlds, many Leibnizian intensions are not maximally consistent entities, we can amend the semantics to invalidate the inference (...) rule which ensures that all tautologies are obligatory. I sketch this amended semantics to show how it invalidates the rule in a new way. (shrink)

In this paper, I set out some basic elements of a deontic logic with an implementation appropriate for handling conflicting legal obligations for purposes of programming autonomous machine agents. Kantian justice demands that the prescriptive system of enforceable public laws be consistent, yet statutes or case holdings may often describe legal obligations that contradict; moreover, even fundamental constitutional rights may come into conflict. I argue that a deontic logic of the law should not try to work around such conflicts but, (...) instead, identify and expose them so that the rights and duties that generate inconsistencies in public law can be explicitly qualified and the conflicts resolved. I then argue that a credulous, non-monotonic deontic logic can describe inconsistent legal obligations while meeting Kant’s demand for consistency in the prescriptive system of public law. Finally, I propose an implementation of this logic via a modified form of “answer set programming,” which I demonstrate with some simple examples. (shrink)

The concept of hypodox is dual to the concept of paradox. Whereas a paradox is incompatibly overdetermined, a hypodox is underdetermined. Indeed, many particular paradoxes have dual hypodoxes. So, naively the dual of Russell’s Paradox is whether the set of all sets that are members of themselves is self-membered. The dual of the Liar Paradox is the Truth-teller, and a hypodoxical dual of the Heterological paradox is whether ‘autological’ is autological. I provide some analysis of the duality and I search (...) for modal hypodoxes as duals to known paradoxes. I also try to continue this search further by mapping the relations between some paradoxes and hypodoxes in duality squares. This investigation increases the known extension of the concept of hypodox, and the duality relation between paradoxes and hypodoxes assists in finding some of these. (shrink)

In this paper, we axiomatize the deontic logic in Fusco (2015), which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions (...) Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to “lift” a Kripke frame for a one-dimensional modal logic into two dimensions. (shrink)

We introduce a multimodal framework of deontic action logic which encodes the interaction between two fundamental procedures in normative reasoning: conceptual classification and deontic classification. The expressive power of the framework is noteworthy, since it combines insights from agency logic and dynamic logic, allowing for a representation of many kinds of normative conflicts. We provide a semantic characterization for three axiomatic systems of increasing strength, showing how our approach can be modularly extended in order to get different levels of analysis (...) of normative reasoning. Finally, we discuss ways in which the framework can be used to capture other formalisms proposed in the literature, as well as to model searching problems in Artificial Intelligence. (shrink)

In this paper, we axiomatize the deontic logic in Fusco 2015, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions (...) Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to ‘lift’ a Kripke frame for a one- dimensional modal logic into two dimensions. (shrink)

In this thesis, I develop and investigate various novel semantic frameworks for deontic logic. Deontic logic concerns the logical aspects of normative reasoning. In particular, it concerns reasoning about what is required, allowed and forbidden. I focus on two main issues: free-choice reasoning and the role of norms in deontic logic. -/- Free-choice reasoning concerns permissions and obligations that offer choices between different actions. Such permissions and obligations are typically expressed by a disjunctive clause in the scope of a deontic (...) operator. For instance, the sentence "Jane may take an apple or a pear" intuitively offers Jane a choice between two permitted courses of action: she may take an apple, and she may take a pear. In the first part of the thesis, I develop semantic frameworks for deontic logic that account for free-choice reasoning. I show that the resulting logics avoid problems that arise for other logical accounts of free-choice reasoning. The main technical contributions are completeness results for axiomatizations of the different logics. -/- Semantic frameworks for deontic logic typically only talk about norms implicitly. In the second part of the thesis, I clarify the role of norms in deontic logic. I develop a formal model of normative systems where norms are represented explicitly. The model takes into account both the hierarchical structure of normative systems, and the function of norms to regulate agent behavior. I show how the model can be used to clarify issues in the study of normative systems. I also develop a norm-based semantics for deontic action logic based on the model. (shrink)

In the present work we provide a logical analysis of normatively determined and non-determined propositions. The normative status of these propositions depends on their relation with another proposition, here named reference proposition. Using a formal language that includes a monadic operator of obligation, we define eight dyadic operators that represent various notions of “being normatively (non-)determined”; then, we group them into two families, each forming an Aristotelian square of opposition. Finally, we show how the two resulting squares can be combined (...) to form an Aristotelian cube of opposition. (shrink)

Consider three cases: -/- Turn: A trolley is about to kill five innocent strangers. You can turn the trolley onto me, saving the five and killing me. -/- Hurl: A trolley is about to kill five innocent strangers. You can hurl me at the trolley, saving the five and paralyzing me. -/- TurnHurl: A trolley is about to kill five innocent strangers. You can turn the trolley onto me, saving the five and killing me. You can instead hurl me at (...) the trolley, saving the five and paralyzing me. -/- Most find the following four claims intuitively plausible: -/- (1) It is permissible to turn the trolley onto me in Turn. (2) It is impermissible to hurl me at the trolley in Hurl. (3) It is impermissible to turn the trolley onto me in TurnHurl. (4) It is permissible to hurl me at the trolley in TurnHurl. -/- But how does turning go from permissible to impermissible, and hurling from impermissible to permissible, when both alternatives are available? I argue that such “secondary permissibility” claims are explained by contrastive consent. Even if I do not consent to being harmed, it is likely I’ll consent to being hurled at the trolley rather than being turned onto. (shrink)

This paper introduces two new paradoxes for standard deontic logic (SDL). They are importantly related to, but distinct from Ross' paradox. These two new paradoxes for SDL are the simple weakening paradox and the complex weakening paradox. Both of these paradoxes arise in virtue of the underlaying logic of SDL and are consequences of the fact that SDL incorporates the principle known as weakening. These two paradoxes then show that SDL has counter-intuitive implications related to disjunctive obligations that arise in (...) virtue of deontic weakening and in virtue of decisions concerning how to discharge such disjunctive obligations. The main result here is then that theorem T1 is a problematic component of SDL that needs to be addressed. (shrink)

This short paper offers a skeptical solution to Åqvist's paradox of epistemic obligation. The solution is based on the contention that in SDL/KDT logics the externalist features of knowledge, about which we cannot have obligations, are obscured.

Inheritance is the principle that deontic `ought' is closed under entailment. This paper is about a tension that arises in connection with Inheritance. More specifically, it is about two observations that pull in opposite directions. One of them raises questions about the validity of Inheritance, while the other appears to provide strong support for it. We argue that existing approaches to deontic modals fail to provide us with an adequate resolution of this tension. In response, we develop a positive analysis, (...) and show that this proposal provides a satisfying account of our intuitions. (shrink)

The article offers an overview of the deontic theory developed by the philosophical school of Mīmāṃsā, which is, and has been since the last centuries BCE, the main source of normative concepts in Sanskrit thought. Thus, the Mīmāṃsā deontics is interesting for any historian of philosophy and constitutes a thought-provoking occasion to rethink deontic concepts, taking advantage of centuries of systematic reflections on these topics. Some comparison with notions currently used in Euro-American normative theories and metaethical principles is offered in (...) order to show possible points of contact and deep divergences. In more detail, after an introduction explaining the methodology and aims of our work, we discuss how Mīmāṃsā authors distinguished and defined some fundamental deontic concepts, such as different types of prescriptions and prohibitions. We then discuss how Mīmāṃsā authors approached the problem of conflicts among commands without jeopardising the validity of the normative text issuing them. In the second part of the article we introduce our formal apparatus, which is construed around the main taxonomic and conceptual distinctions used in the first part. Our formal rendering captures the most important features of the Mīmāṃsā theory and can thus serve as a concise and rigorous presentation of it for scholars working in deontic logic. (shrink)

The present article provides a taxonomic analysis of bimodal logics of normative ideality and normative awfulness, two notions whose meaning is here explained in terms of the moral values pursued by a given community. Furthermore, the article addresses the traditional problem of a reduction among deontic concepts: we explore the possibility of defining other relevant normative notions, such as obligation, explicit permission and Hohfeldian relations, in terms of ideality and awfulness. Some proposals in this respect, which have been formulated in (...) the literature over the years, are here improved and discussed with reference to the various logics that we will introduce. (shrink)

In normative reasoning one typically refers to intervals of time across which norms are intended to hold, as well as to alternative possibilities representing hypothetical developments of a given scenario. Thus, deontic modalities are naturally intertwined with temporal and metaphysical ones. Furthermore, contemporary debates in philosophy suggest that a proper understanding of fundamental ethical principles, such as the Ought-Implies-Can thesis, requires a simultaneous analysis of these three families of concepts. In the present article we propose a general formal framework which (...) allows for fine-grained multimodal reasoning in the normative domain. We provide an axiomatization for a novel system of propositional logic encoding the way in which possibilities and norms arising from different sources change over intervals of time. The usefulness of our framework is illustrated by analysing an ancient and particularly challenging `cold case', the Paradox of the Court. (shrink)

We provide a fine-grained analysis of notions of regret and responsibility (such as agent-regret and individual responsibility) in terms of a language of multimodal logic. This language undergoes a detailed semantic analysis via two sorts of models: (i) relating models, which are equipped with a relation of propositional pertinence, and (ii) synonymy models, which are equipped with a relation of propositional synonymy. We specify a class of strictly relating models and show that each synonymy model can be transformed into an (...) equivalent strictly relating model. Moreover, we define an axiomatic system that captures the notion of validity in the class of all strictly relating models. (shrink)

Obligations can be affected by knowledge. Several approaches exist to formalize knowledge-based obligations, but no formalism has been developed yet to capture the dynamic interaction between knowledge and obligations. We introduce the dynamic extension of an existing logic for knowledge-based obligations here. We motivate the logic by analyzing several scenarios and by showing how it can capture in an original manner several fundamental deontic notions such as absolute, prima facie and all-things-considered obligations. Finally, in the dynamic epistemic logic tradition, we (...) provide reduction axioms for the dynamic operator of the new logic. (shrink)

Conflicts between legal norms are common in reality. In many legislations, legal conflicts between norms are resolved by applying ordered principles. This work presents a formalization of the conflict resolution mechanism and introduces action legal logic (⁠ALL) to reason about the normative consequences of possibly conflicting legal systems. The semantics of ALL is explicitly based on legal systems consisting of norms and ordered principles. Legal systems specify the legal status of transitions in transition systems and the language of ALL describes (...) the legal status of paths in transition systems. The formalization is used to study abstract revisions of legal systems. The expressivity of ALL is studied and its completeness is proved. (shrink)