It is quite plausible to say that you may read or write implies that you may read and you may write (though possibly not both at once). This so-called free choice principle is well-known in deontic logic. Sadly, despite being so intuitive and seemingly innocent, this principle causes a lot of worries. The paper briefly but critically examines leading accounts of free choice permission present in the literature. Subsequently, the paper suggests to accept the free (...) choice principle, but only as a default (or defeasible) rule, issuing to it a ticket-of-leave, granting it some freedom, until it commits an undesired inference. (shrink)

Free choice permission, a crucial test case concerning the semantics/ pragmatics boundary, usually receives a pragmatic treatment. But its pragmatic features follow from its semantics. We observe that free choice inferences are defeasible, and defend a semantics of free choice permission as strong permission expressed in terms of a modal conditional in a nonmonotonic logic.

In this paper, it is argued that Ferguson’s (2003, Argumentation 17, 335–346) recent proposal to reconcile monotonic logic with defeasibility has three counterintuitive consequences. First, the conclusions that can be derived from his new rule of inference are vacuous, a point that as already made against default logics when there are conflicting defaults. Second, his proposal requires a procedural “hack” to the break the symmetry between the disjuncts of the tautological conclusions to which his proposal leads. Third, Ferguson’s (...) proposal amounts to arguing that all everyday inferences are sound by definition. It is concluded that the informal logic response to defeasibility, that an account of the context in which inferences are sound or unsound is required, still stands. It is also observed that another possible response is given by Bayesian probability theory (Oaksford and Chater, in press, Bayesian Rationality: The Probabilistic Approach to Human Reasoning, Oxford University Press, Oxford, UK; Hahn and Oaksford, in press, Synthese). (shrink)

In this paper, we apply relating semantics to the widely discussed problem of free choice between permitted actions or situations in normative systems. Leaving aside contexts in which the free choice principle is obviously unacceptable or uncontroversially valid, we concentrate on free choice for explicit permissions. In order to construct a formal representation of explicit permissions, we introduce a special constant, $\texttt {permit}$, which is analogous to the constant $\texttt {violation}$ used in the Andersonian–Kangerian (...) approach to deontic logic with respect to prohibition and obligation. Consequently, we define a permission operator on the basis of the $\texttt {permit}$ constant and a relation of legitimization. The general idea is that $P \varphi $ is true if and only if $\texttt {permit}$ is true, which means that a permission is actually issued and $\varphi $ is legitimized by $\texttt {permit}$. The intuitive notion of legitimization is formally represented by an operator of relating implication: a non-classical implication that is semantically defined by adding a constraint to the classical meaning of implication to the effect that arguments are related by a special relation. The properties of this relation are based on an informal, intuitive meaning of legitimization and determine the properties of the permission operator. We show that the resulting permission operator possesses the free choice property and avoids certain unwanted consequences that follow from alternative approaches. (shrink)

Sentences such as Olivia can take Logic or Algebra (‘♢∨-sentences’) are typically interpreted as entailing that Olivia can take Logic and can take Algebra. Given a standard semantics for modals and disjunction, those ‘Free choice’ (FC) readings are not predicted from the surface form of ♢∨-sentences. Yet the standard semantics is appropriate for the ‘double prohibition’ reading typically assigned to ¬♢∨-sentences like Olivia can’t take Logic or Algebra. Several extant approaches to FC can account for those two cases, (...) but face challenges when ♢∨, ¬♢∨ and related sentences appear embedded in certain environments. In this paper, we present a novel account of FC that builds on a ‘grammatical’ theory of scalar implicatures — proposed by Bassi et al. (2021) and Del Pinal (2021) — according to which covert exhaustification is a presupposition trigger such that the prejacent forms the assertive content while any excludable or includable alternatives are incorporated at the non-at issue, presuppositional level. Applied to ♢∨, ¬♢∨, and similar sentences, ‘presuppositional exhaustification’ predicts that their default interpretations have an assertive component (roughly, the classical interpretation of the prejacent) and a homogeneity presupposition which projects in standard ways. Those predictions, we then show, support a uniform account of the puzzling behavior of ♢∨, ¬♢∨, and related sentences when embedded under (negative) factives (Marty & Romoli 2020), disjunctions (Romoli & Santorio 2019), and in the scope of universal, existential (Bar-Lev & Fox 2020) and non-monotonic quantifiers (Gotzner et al. 2020). (shrink)

In this paper, a pragmatic approach to the phenomenon of free choice permission is proposed. Free choice permission is explained as due to taking the speaker (i) to obey certain Gricean maxims of conversation and (ii) to be competent on the deontic options, i.e. to know the valid obligations and permissions. The approach differs from other pragmatic approaches to free choice permission in giving a formally precise description of the class of inferences that can (...) be derived based on these two assumptions. This formalization builds on work of Halpern and Moses (1984) on the concept of ‘only knowing’, generalized by Hoek et al., (1999, 2000), and Zimmermann’s (2000) approach to competence. (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)

This dissertation develops, justifies, and examines the jurisprudential implications of a non-monotonic theory of common law legal reasoning. Legal rules seem to have exceptions but identifying all of them is difficult. This hinders attempts to formalize legal rules using classical logics. Non-monotonic logics allow defeasible inference, permitting rules that hold generally but can be defeated in the presence of exceptions. This ameliorates the problem of characterizing all exceptions to a rule, because exceptions can be added piecemeal while the (...) rule remains. The first portion of the dissertation rebuts a prominent criticism leveled at a large class of theories of legal reasoning that includes my theory. The charge is that no coherent theory can recognize both the difference between distinguishing and overruling, and the constraint of precedent. The critics argue that is more important than and that can only be explained by monotonic legal rules. Drawing on cognitive psychology as well as legal theory, I show that coherent theories, such as my own, can accommodate both and. The second chapter provides motivation for understanding precedential constraint in terms of non-monotonic default rules and introduces my positive theory, which elaborates on John Horty's work in treating legal rules as prioritized defaults involving reasons. I motivate and implement a relaxation of Horty's restrictions on the form of rules to allow a more fine-grained characterization of precedent. Finally, I explore the relationship between these relaxations and the concept of precedent. The final section explains how my theory fits into the traditional jurisprudential ecosystem. I demonstrate that, contrary to assertions in the legal theory literature, this non-monotonic approach is entirely compatible with positivism's commitment to extracting rules from authoritative legal sources, namely court opinions. I also suggest how it might be attractive to law and economics theorists, pragmatists, and followers of Dworkin. (shrink)

We present a solution to the paradox of free choice permission by introducing strong and weak permission in a deontic logic of action. It is shown how counterintuitive consequences of strong permission can be avoided by limiting the contexts in which an action can be performed. This is done by introducing the only operator, which allows us to say that only is performed (and nothing else), and by introducing contextual interpretation of action terms.

The term "non-monotonic logic" covers a family of formal frameworks devised to capture and represent defeasible inference , i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called "non-monotonic" because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base itself. (...) This is in contrast to classical (first-order) logic, whose inferences, being deductively valid, can never be "undone" by new information. (shrink)

In this paper we address the modelling of reasoning processes in natural agents. We focus on a very basic kind of non-monotonic inference for which we identify a simple and plausible underlying process, and we develop a family of logical models that allow to match this process. Partial worlds models, as we call them, are a variant of Kraus, Lehmann and Magidor’s cumulative models. We show that the inference relations they induce form a strict subclass of cumulative relations and (...) tackle the issue of providing sound and complete sets of rules to characterise them. Taking inspiration from the work of Gabbay and Schlechta, we analyse the question in terms of definable sets of partial worlds and conclude that completeness is probably unreachable using a standard propositional language. This brings us to enrich our language with an additional connective, which makes it possible to distinguish between two kinds of disjunction in the partial worlds context. Within this renewed framework, we provide two representation theorems: one for inference relations induced by precisification-free smooth models, and the other for inference relations induced by precisification-free ranked models. Finally we give an aperçu of how partial worlds models lend themselves to the modelling of learning. (shrink)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be found in human reasoning. (...) Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

The article aims to show how the acceptance of non-monotonic logic enables arguments to be held between science and religion in a way that does not exclude either of these two spheres. The starting point of the analyses is the idea of the 13th century Danish philosopher, Boethius of Dacia, who states that it is both acceptable that: a natural scientist negates that the world had a beginning, and a Christian theologian asserts that the world had a beginning, because (...) each of them is basing their statements on the principles of their respective discipline: the first on the principles of nature, and the latter on knowledge supplemented by divine revelation. What is more, analogically: a metaphysician, when limited to his principles, cannot settle the issue, as he takes into account supranatural beings and their powers, but cannot know what God or another powerful supranatural would have decided in a such a case. The paper shows that Boethius’s approach: violates the rule of monotonicity, cannot be finally interpreted in terms of classical logic and assumes certain non-monotonic logic as its inference framework. Other presented examples of arguments between religious beliefs and the statements of natural science are resolved in the same way. Thus, it is shown how non-monotonic thinking allows us to seriously treat both scientific and religious inference as compatible. (shrink)

This paper analyses the logical structure of the balancing of conflicting normative arguments, and asks whether non-monotonic logic is adequate to represent this type of legal or practical reasoning. Norm conflicts are often regarded as a field of application for non-monotonic logics. This paper argues, however, that the balancing of normative arguments consists of an act of judgement, not a logical inference, and that models of deductive as well as of defeasible reasoning do not give an adequate account (...) of its structure. Moreover, it argues that as far as the argumentation consists in logical inferences, deductive logic suffices for reconstructing the argumentation from the internal point of view of someone making normative judgements. (shrink)

We introduce a new understanding of deontic modals that we callobligations as weakest permissions. We argue for its philosophical plausibility, study its expressive power in neighborhood models, provide a complete Hilbert-style axiom system for it and show that it can be extended and applied to practical norms in decision and game theory.

In the 1980s, Pollock’s work on default reasons started the quest in the AI community for a formal system of defeasible argumentation. The main goal of this paper is to provide a logic of structured defeasible arguments using the language of justification logic. In this logic, we introduce defeasible justification assertions of the type t : F that read as “t is a defeasible reason that justifies F”. Such formulas are then interpreted as arguments and their acceptance semantics is (...) given in analogy to Dung’s abstract argumentation framework semantics. We show that a large subclass of Dung’s frameworks that we call “warranted” frameworks is a special case of our logic in the sense that Dung’s frameworks can be obtained from justification logic-based theories by focusing on a single aspect of attacks among justification logic arguments and Dung’s warranted frameworks always have multiple justification logic instantiations called “realizations”. We first define a new justification logic that relies on operational semantics for default logic. One of the key features that is absent in standard justification logics is the possibility to weigh different epistemic reasons or pieces of evidence that might conflict with one another. To amend this, we develop a semantics for “defeaters”: conflicting reasons forming a basis to doubt the original conclusion or to believe an opposite statement. This enables us to formalize non-monotonic justifications that prompt extension revision already for normal default theories. Then we present our logic as a system for abstract argumentation with structured arguments. The format of conflicting reasons overlaps with the idea of attacks between arguments to the extent that it is possible to define all the standard notions of argumentation framework extensions. Using the definitions of extensions, we establish formal correspondence between Dung’s original argumentation semantics and our operational semantics for default theories. One of the results shows that the notorious attack cycles from abstract argumentation cannot always be realized as justification logic default theories. (shrink)

In this paper we present an embedding of abstract argumentation systems into the framework of Barwise and Seligmans logic of information flow. We show that, taking P.M. Dungs characterization of argument systems, a local logic over states of a deliberation may be constructed. In this structure, the key feature of non-monotonicity of commonsense reasoning obtains as the transition from one local logic to another, due to a change in certain background conditions. Each of Dungs extensions of argument systems leads to (...) a corresponding ordering of background conditions. The relations among extensions becomes a relation among partial orderings of background conditions. This introduces a conceptual innovation in Barwise and Seligmans representation of commonsense reasoning. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency.

This paper reports on an approach to model generalized implicatures using nonmonotonic logics. The approach, called compositional, is based on the idea of compositional semantics, where the implicatures carried by a sentence are constructed from the implicatures carried by its constituents, but it also includes some aspects nonmonotonic logics in order to model the defeasibility of generalized implicatures.

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 paper we show how defeasible logic could formally account for the non-monotonic properties involved in motivational attitudes like intention and obligation. Usually, normal modal operators are used to represent such attitudes wherein classical logical consequence and the rule of necessitation comes into play, i.e., ⊢A/⊢ □A, that is from ⊢A derive ⊢ □A. This means that such formalisms are affected by the Logical Omniscience problem. We show that policy-based intentions exhibit non-monotonic behaviour which could be captured (...) through a non-monotonic system like defeasible logic. To this end we outline a defeasible logic of intention that specifies how modalities can be introduced and manipulated in a non-monotonic setting without giving rise to the problem of logical omniscience. In a similar way we show how to add deontic modalities defeasibly and how to integrate them with other motivational attitudes like beliefs and goals. Finally we show that the basic aspect of the BOID architecture is captured by this extended framework. (shrink)

We examine the impact of Suzumura’s (Economica 43:381–390, 1976) consistency property when applied in the context of collective choice rules that are independent of irrelevant alternatives, neutral, and monotonic. An earlier contribution by Blau and Deb (Econometrica 45:871–879, 1977) establishes the existence of a vetoer if the collective relation is required to be complete and acyclical. The purpose of this paper is to explore the possibilities that result if completeness and acyclicity are dropped and Suzumura consistency is imposed (...) instead. A conceptually similar but logically independent version of the combined axiom that requires the collective decision mechanism to be independent, neutral, and monotonic is employed. In the case of a finite population, we obtain an alternative impossibility theorem if a collective choice rule is assumed to be non-degenerate and a modified no veto requirement is imposed instead of Blau and Deb’s (1977) condition. If the population is countably infinite, the impossibility can be avoided but it resurfaces if our new no veto property is extended to a coalitional variant. (shrink)

Executive function has become an important concept in explanations of psychiatric disorders, but we currently lack comprehensive models of normal executive function and of its malfunctions. Here we illustrate how defeasible logical analysis can aid progress in this area. We illustrate using autism and attention deficit hyperactivity disorder (ADHD) as example disorders, and show how logical analysis reveals commonalities between linguistic and non-linguistic behaviours within each disorder, and how contrasting sub-components of executive function are involved across disorders. This analysis reveals (...) how logical analysis is as applicable to fast, automatic and unconscious reasoning as it is to slow deliberate cogitation. (shrink)

In the first part I argue that normic laws are the phenomenological laws of evolutionary systems. If this is true, then intuitive human reasoning should be fit in reasoning from normic laws. In the second part I show that system P is a tool for reasoning with normic laws which satisfies two important evolutionary standards: it is probabilistically reliable, and it has rules of low complexity. In the third part I finally report results of an experimental study which demonstrate that (...) intuitive human reasoning is in well accord with basic argument patterns of system P. (shrink)

The first chapter explores the theory developed in Kripke's "Outline of a Theory of Truth." A tension in Kripke's account of the concept of truth is revealed--a conflict between two intuitions. The first intuition, called the "fixed point conception of truth," is that the whole meaning of the truth predicate is given by the formula "we may assert of a sentence that it is true iff we may assert that sentence." The second intuition, called the "thesis of the supervenience of (...) semantics," is that the interpretation of the truth predicate is determined by the interpretations of all the other non-logical constants of the language. I show that these two intuitions cannot be reconciled within the context of Kripke's theory, and I argue for the choice of the fixed point conception of truth over the principle of the supervenience of semantics. Chief among the considerations that I advance is the fact that the fixed point conception of truth yields a natural logic of truth. In the second chapter I present that logic of truth in the form of a cut-free sequent calculus. I then turn, in chapter three, to a discussion of the significance of such cut-free sequent calculi. I argue that we may view the sequent calculus introduction rules for a logical constant as giving the meaning of that constant, under certain conditions. In particular, we can see our logic of truth as providing an explication of our intuitive concept of truth which complements Kripke's original model-theoretic explication. The fourth chapter takes up some criticisms of Kripke's theory which center on the logic of truth. In addition, the general ideas developed in the first three chapters are applied in a discussion of the various schemes of three-valued logic which can be used in the development of Kripke's approach. Finally, in the last chapter, it is shown that Kripke's technical concepts, and the logical ideas developed in the second and third chapters, can be used to provide a unified account of many contexts in which paradox arises. (shrink)

We investigate the application of Courcelle’s theorem and the logspace version of Elberfeld et al. in the context of non-monotonic reasoning. Here we formalize the implication problem for propositional sets of formulas, the extension existence problem for default logic, the expansion existence problem for autoepistemic logic, the circumscriptive inference problem, as well as the abduction problem in monadic second order logic and thereby obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, (...) for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu, resp., XLnu) under standard complexity assumptions. (shrink)

There has been considerable discussion in the past about the assumptions and basis of different ethical rules. For instance, it is commonplace to say that ethical rules are defaults rules, which means that they tolerate exceptions. Some authors argue that morality can only be grounded in particular cases while others defend the existence of general principles related to ethical rules. Our purpose here is not to justify either position, but to try to model general ethical rules with artificial intelligence formalisms (...) and to compute logical consequences of different ethical theories. More precisely, this is an attempt to show that progress in non-monotonic logics, which simulates default reasoning, could provide a way to formalize different ethical conceptions. From a technical point of view, the model developed in this paper makes use of the Answer Set Programming (ASP) formalism. It is applied comparatively to different ethical systems with respect to their attitude towards lying. The advantages of such formalization are two-fold: firstly, to clarify ideas and assumptions, and, secondly, to use solvers to derive consequences of different ethical conceptions automatically, which can help in a rigorous comparison of ethical theories. (shrink)

It is not unusual in real-life that one has to choose among finitely many alternatives when the merit of each alternative is not perfectly known. Instead of observing the actual utilities of the alternatives at hand, one typically observes more or less precise signals that are positively correlated with these utilities. In addition, the decision-maker may, at some cost or disutility of effort, choose to increase the precision of these signals, for example by way of a careful study or the (...) hiring of expertise. We here develop a model of such decision problems. We begin by showing that a version of the monotone likelihood-ratio property is sufficient, and also essentially necessary, for the optimality of the heuristic decision rule to always choose the alternative with the highest signal. Second, we show that it is not always advantageous to face alternatives with higher utilities, a non-monotonicity result that holds even if the decision-maker optimally chooses the signal precision. We finally establish an operational first-order condition for the optimal precision level in a canonical class of decision-problems, and we show that the optimal precision level may be discontinuous in the precision cost. (shrink)

Disjunctions in the scope of possibility modals give rise to a conjunctive inference, generally referred to as ‘free choice.’ For example, Emma can take Spanish or Calculus suggests that Emma can take Spanish and can take Calculus. This inference is not valid on standard semantics for modals in combination with a Boolean semantics for disjunction. Hence free choice has sparked a whole industry of theories in philosophy of language and semantics. This paper investigates free (...) class='Hi'>choice in sentences involving a non-monotonic modified numeral, under which we embed a possibility modal scoping over disjunction. One example is Exactly one student can take Spanish or Calculus. As we point out, the presence of certain readings of these sentences is a key test for a prominent approach, which analyzes free choice as a kind of scalar implicature. We report on two experiments investigating the readings of such sentences, using an inferential task. Our results are challenging for the implicature approach. We sketch two possible solutions within this approach, either adopting a different recent implicature algorithm, or exploring a different meaning for modified numerals with exactly. Both of them suffer from a variety of problems. We then discuss a third solution, which exploits a recent account of free choice based on homogeneity. This approach can account for our results, in combination with plausible assumptions about homogeneity projection, though it too has open issues with related cases. Regardless of which solution is chosen, non-monotonic contexts turn out to be an important test case for theories of free choice, implicature, and modified numerals. (shrink)

Classic deductive logic entails that once a conclusion is sustained by a valid argument, the argument can never be invalidated, no matter how many new premises are added. This derived property of deductive reasoning is known as monotonicity. Monotonicity is thought to conflict with the defeasibility of reasoning in natural language, where the discovery of new information often leads us to reject conclusions that we once accepted. This perceived failure of monotonic reasoning to observe the defeasibility of natural-language arguments (...) has led some philosophers to abandon deduction itself (!), often in favor of new, non-monotonic systems of inference known as `default logics'. But these radical logics (e.g., Ray Reiter's default logic) introduce their desired defeasibility at the expense of other, equally important intuitions about natural-language reasoning. And, as a matter of fact, if we recognize that monotonicity is a property of the form of a deductive argument and not its content (i.e., the claims in the premise(s) and conclusion), we can see how the common-sense notion of defeasibility can actually be captured by a purely deductive system. (shrink)

Andrew Bacon, John Hawthorne, and Gabriel Uzquiano have recently argued that free logics—logics that reject or restrict Universal Instantiation—are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior. These logics encompass ramified and contextualist approaches to paradoxes, and broadly speaking, there are two kinds of criticism they face. First, they fail to address every version of the Priorean paradoxes. Second, the theoretical considerations behind the logics make absolutely general statements about all propositions, properties (...) of propositions, etc., but because this sort of intensional quantification is always restricted in the logics, they cannot even express those considerations. I present a novel sort of free logic, which rejects the standard Universal Instantiation but validates a restricted form of the rule, and which addresses both kinds of criticism by allowing some propositions to be unrestricted in their quantification. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. Wedefine the logics BK+ and BK′+ by adding to B+ the K rule and to BK+the characteristic S4 axiom, respectively. These logics are endowed witha relatively strong non-constructive negation. We prove that all the logicsdefined lack the K axiom and the standard paradoxes of consistency.

The semantics for a deontic action logic based on Boolean algebra is extended with an interpretation of action expressions in terms of sets of alternative actions, intended as a way to model choice. This results in a non-classical interpretation of action expressions, while sentences not in the scope of deontic operators are kept classical. A deontic structure based on Simons’ supercover semantics is used to interpret permission and obligation. It is argued that these constructions provide ways to handle various (...) problems related to free choice permission. The main result is a sound and complete axiomatization of the semantics. (shrink)

ABSTRACT Inspired by legal reasoning, this paper presents a semantics and proof theory of a system for defeasible argumentation. Arguments are expressed in a logic-programming language with both weak and strong negation, conflicts between arguments are decided with the help of priorities on the rules. An important feature of the system is that these priorities are not fixed, but are themselves defeasibly derived as conclusions within the system. Thus debates on the choice between conflicting arguments can also be modelled. (...) The semantics of the system is given with a fixpoint definition, while its proof theory is stated in dialectical style, where a proof takes the form of a dialogue between a proponent and an opponent of an argument: an argument is shown to be justified if the proponent can make the opponent run out of moves in whatever way the opponent attacks. (shrink)

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the (...) inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations. (shrink)

ABSTRACT The major approaches to belief revision and non monotonic reasoning proposed in the literature differ along a number of dimensions, including whether they are “syntax- based” or “semantic-based”, “foundational” or “coherentist”, “consistence-restoring” or “inconsistency-tolerant”. Our contribution towards clarifying the connections between these various approaches is threefold: •We show that the two main approaches to belief revision, the foundations and coherence theories, are mathematically equivalent, thus answering a question left open in [Gar90, Doy92], The distinction between syntax-based approaches to (...) revision and approaches based on (semantic) preferential structures falls along similar lines, and their expressive equivalence is a consequence of this result. •We formally clarify the connection between belief revision and non monotonic reasoning, in a particularly simple way which also throws light on the connection between consistence- restoring and reasoning-from-inconsistency approaches [BDP95]. •As a direct application of the above, we show that Poole's (syntax-based) system of default reasoning and Shoham's preferential semantic for non monotonic reasoning are also expressively equivalent, in that they can represent the same set of non monotonic consequence relations. (shrink)

Since the earliest formalisation of default logic by Reiter many contributions to this appealing approach to nonmonotonic reasoning have been given. The different formalisations are here presented in a general framework that gathers the basic notions, concepts and constructions underlying default logic. Our view is to interpret defaults as special rules that impose a restriction on the juxtaposition of monotonic Hubert-style proofs of a given logicL. We propose to describe default logic as a logic where the (...) juxtaposition of default proofs is subordinate to a restriction condition . Hence a default logic is a pair (L, ) where properties of the logic , like compactness, can be interpreted through the restriction condition . Different default systems are then given a common characterization through a specific condition on the logicL. We also prove cumulativity for any default logic (L, ) by slightly modifying the notion of default proof. We extend, in fact, the language ofL in a way close to that followed by Brewka in the formulation of his cumulative default system. Finally we show the existence of infinitely many intermediary default logics, depending on and called linear logics, which lie between Reiter's and ukaszewicz' versions of default logic. (shrink)

Adams' famous thesis that the probabilities of conditionals are conditional probabilities is incompatible with standard probability theory. Indeed it is incompatible with any system of monotonic conditional probability satisfying the usual multiplication rule for conditional probabilities. This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength. It shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, but that a standard (...) complementation operation cannot be defined within them, on pain of collapsing probability into bivalence. (shrink)

In this paper we explore the thesis that the role of argumentation in practical reasoning in general and legal reasoning in particular is to justify the use of defeasible rules to derive a conclusion in preference to the use of other defeasible rules to derive a conflicting conclusion. The defeasibility of rules is expressed by means of non-provability claims as additional conditions of the rules.We outline an abstract approach to defeasible reasoning and argumentation which includes many existing formalisms, including (...) class='Hi'>default logic, extended logic programming, non-monotonic modal logic and auto-epistemic logic, as special cases. We show, in particular, that the admissibility semantics for all these formalisms has a natural argumentation-theoretic interpretation and proof procedure, which seem to correspond well with informal argumentation. (shrink)

Expected Utility Theory (EUT) has anomalies when interpreted descriptively and tested empirically. Experiments show that the way in which options are formulated is, in most cases, relevant for decision-making. This kind of anomaly is directly related, however, not with a proper axiom of EUT but rather with the logical principle of extensionality and its decision theoretic version: the principle of invariance. This paper focuses on the phenomenon of framing effects (FE) and the associated failures of invariance. FE arise when different (...) descriptions of the same object lead to different preferences, beliefs, or choices. A formal reconstruction of the Gain–Loss (G-L) frame is proposed, expanding the domain of standard decision theory to incorporate not only prospects but also descriptions of prospects. The system is extended by adding non-monotonic logical resources so as to account for the ceteris paribus character of FE. Finally, the same method is applied to sketching a formal representation for an anomalous case that involves subjective probabilities. The logical method is a general one that can be applied to most cases of FE. The philosophical consequences of the strategy pursued are also discussed. (shrink)

Non-monotonicity in logic is a symptom that may have many causes. In the formalisation of defeasible reasoning, an epistemic diagnosis has largely prevailed according to which some inferences are non-monotonic because they are provisionally drawn in the absence of relevant or complete information. The Gabbay-Makinson rules for cumulative consequence relations are a paradigmatic example of this epistemic approach. In this paper a different approach to defeasible reasoning is introduced, based on the idea of inferential perspectives. According to this approach, (...) some inferences are non-monotonic because they are drawn as from another reasoner’s perspective. Rules are introduced and discussed for a Perspectival Calculus, which show both similarities and interesting differences with respect to cumulative systems. (shrink)

ABSTRACT The major approaches to belief revision and non monotonic reasoning proposed in the literature differ along a number of dimensions, including whether they are ?syntax- based? or ?semantic-based?, ?foundational? or ?coherentist?, ?consistence-restoring? or ?inconsistency-tolerant?. Our contribution towards clarifying the connections between these various approaches is threefold: ?We show that the two main approaches to belief revision, the foundations and coherence theories, are mathematically equivalent, thus answering a question left open in [Gar90, Doy92], The distinction between syntax-based approaches to (...) revision and approaches based on (semantic) preferential structures falls along similar lines, and their expressive equivalence is a consequence of this result. ?We formally clarify the connection between belief revision and non monotonic reasoning, in a particularly simple way which also throws light on the connection between consistence- restoring and reasoning-from-inconsistency approaches [BDP95]. ?As a direct application of the above, we show that Poole's (syntax-based) system of default reasoning and Shoham's preferential semantic for non monotonic reasoning are also expressively equivalent, in that they can represent the same set of non monotonic consequence relations. (shrink)

This paper refutes the widespread view that disjunctions of imperatives invariably grant free choice between the actions named by their disjuncts. Like other disjunctions they can also express a correlation with some factual distinction, but as with modalized declaratives used for non-assertive speech acts this needs to be indicated explicitly. A compositional analysis of one such indicator, depending on, constitutes the point of departure for a uniform analysis of disjunctions across clause types. Disjunctions are analyzed as sets of (...) propositional alternatives that correlate with a partition that the speaker may or may not be able to indicate explicitly. Free choice arises as the specific case where the partition is induced by the preferences of the addressee. (shrink)

This paper offers an explanation of the fact that sentences of the form (1) ‘X may A or B’ may be construed as implying (2) ‘X may A and X may B’, especially if they are used to grant permission. It is suggested that the effect arises because disjunctions are conjunctive lists of epistemic possibilities. Consequently, if the modal may is itself epistemic, (1) comes out as equivalent to (2), due to general laws of epistemic logic. On the other hand, (...) on a deontic reading of may, (2) is only implied under exceptional circumstances – which usually obtain when (1) is used performatively. (shrink)

This paper contributes to the semantic typology of dependent indefinites, by accounting for the distribution and interpretation of the Romanian indefinite vreun. It is shown that its occurrences are restricted to negative polarity and a subset of modal contexts. More specifically, the study of its behavior in intensional environments reveals that vreun is systematically incompatible with non-epistemic operators, a restriction we capture by proposing a novel empirical generalization (‘the epistemic constraint’). To account for the observed pattern, we adopt the unitary (...) approach to polarity in Chierchia (Logic in Grammar: polarity, free choice and intervention, 2013a; in: Fălăuş (ed.) Alternatives in semantics, 2013b) and derive the properties of vreun from its obligatory association with alternatives. Its distributional restrictions are argued to follow from the interplay between the types of alternatives it activates (scalar and subdomain alternatives), the way these alternatives are factored into meaning (via an independently motivated mechanism of exhaustification) and the lexical semantics of the operators in the context of occurrence. We propose that the epistemic constraint arises from the lexicalization of an inference (anti-total variation) that non-epistemic operators can never satisfy. (shrink)

This paper presents a number of apparent anomalies in rational choice scenarios, and their translation into the logic of everyday reasoning. Three classes of examples that have been discussed in the context of probabilistic choice since the 1960s (by Debreu, Tversky and others) are analyzed in a non-probabilistic setting. It is shown how they can at the same time be regarded as logical problems that concern the drawing of defeasible inferences from a given information base. I argue that (...) initial appearances notwithstanding, these cases should not be classed as instances of irrationality in choice or reasoning. One way of explaining away their apparent oddity is to view certain aspects of these examples as making particular options salient. The decision problems in point can then be solved by ‘picking’ these options, although they could not have been ‘chosen’ in a principled way, due to ties or incomparabilities with alternative options. (shrink)

ABSTRACT This paper defines a new modal logic based theory for non-monotonic reasoning. This logic expresses notions about hypotheses and known information. These notions are defined in the framework of the modal system τ. A translation of default logic in terms of hypothesis theory is given with which it is possible to fully characterize default logic by giving a necessary and sufficient criterion for the existence and the non-existence of extensions. Moreover several problems relating to non-monotonic (...) reasoning are discussed (and solutions are presented), such as case analysis, multiple extensions and contraposition. (shrink)

I will try to do three things in this paper. First, I want to situate certain problems in natural language semantics with respect to larger trends in logicism, including: (i) Attempts by positivist philosophers earlier in this century to provide a logical basis for the physical sciences; (ii) Attempts by linguists and logicians to develop a “natural language ontology” (and, presumably, a logical language that is related to this ontology by formally explicit rules) that would serve as a framework for (...) natural language semantics. (shrink)

In An Introduction to Non-Classical Logic: From If to Is Graham Priest presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest's rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest's rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest's branching universal and particular instantiation (...) rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest is grappling with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest. (shrink)

Inventing novel knowledge to solve problems is a crucial, creative, mechanism employed by humans, to extend their range of action. In this paper, we present TCL (typicality-based compositional logic): a probabilistic, non monotonic extension of standard Description Logics of typicality, and will show how this framework is able to endow artificial systems of a human-like, commonsense based, concept composition procedure that allows its employment in a number of applications (ranging from computational creativity to goal-based reasoning to recommender systems and (...) affective computing). -/- The framework relies on 3 main ingredients: a non monotonic extension of standard Description Logics (allowing to reason on exceptions to inheritance in standard knowledge bases) a probabilistic extension coming from the field of logic programming (the DISPONTE semantics) a cognitive heuristics known ad HEAD-Modifier determining preference rules for the inheritance mechanisms of the novel concepts to be generated via commonsense rules. -/- We report the obtained results and the lessons learned of this research path in the context of model-based AI applications. -/- . (shrink)