This paper introduces a generalization of Reiter’s notion of “extension” for default logic. The main difference from the original version mainly lies in the way conflicts among defaults are handled: in particular, this notion of “general extension” allows defaults not explicitly triggered to pre-empt other defaults. A consequence of the adoption of such a notion of extension is that the collection of all the general extensions of a default theory turns out to have a nontrivial algebraic structure. This fact has (...) two major technical fall-outs: first, it turns out that every default theory has a general extension; second, general extensions allow one to define a well-behaved, skeptical relation of defeasible consequence for default theories, satisfying the principles of Reflexivity, Cut, and Cautious Monotonicity formulated by D. Gabbay. (shrink)

This note is on cautious cut elimination for one of Veitman's might logics. Syntactically, the logic is presented as an extension of a sequent system for classical proposition logic . I show that this extension preserves the completeness and decidability of CPL. The proof has cautious cut elimination as a corollary. I also give a rather general syntactic proof of cautious cut elimination. It states that any ‘base’ logic which has a reflexive, monotone consequence relation that allows (...) cautious cut to be eliminated preserves cautious cut elimination when extended to a might logic. (shrink)

We generalize Ripley’s results on the transitivity of consequence relation, without assuming a logic to be monotonic. Following Gabbay, we assume nonmonotonic consequence relation to be inclusive and cautious monotonic, and figure out the implications between different forms of transitivity of logical consequence. Weaker frameworks without inclusiveness or cautious monotonicity are also discussed. The paper may provide basis for the study of both non-transitive logics and nonmonotonic ones.

This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ (...) ( p ∧ ¬ q ) ∧ ◊ p ∧ ◊ ¬ q. We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut. (shrink)

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for (...) P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach. (shrink)

Nonmonotonic logics allow—contrary to classical (monotone) logics— for withdrawing conclusions in the light of new evidence. Nonmonotonic reasoning is often claimed to mimic human common sense reasoning. Only a few studies, though, have investigated this claim empirically. system p is a central, broadly accepted nonmonotonic reasoning system that proposes basic rationality postulates. We previously investigated empirically a probabilistic interpretation of three selected rules of system p. We found a relatively good agreement of human reasoning and principles of nonmonotonic reasoning according (...) to the coherence interpretation of system p. This study reports an experiment on the cautious monotonicity Rule and its “incautious” counterpart that is not contained in system p, namely the monotonicity Rule. In accordance with our previous results, the data suggest that people reason nonmonotonically: the subjects in the cautious monotonicity condition infer significantly tighter intervals close to the coherence interpretation of system p compared with the subjects in the incautious monotonicity condition where rather wide (and hence non-informative) intervals are inferred. (shrink)

In this paper we exploit the notions of conjoined and iterated conditionals, which are defined in the setting of coherence by means of suitable conditional random quantities with values in the interval [0,1]. We examine the iterated conditional (B|K)|(A|H), by showing that A|H p-entails B|K if and only if (B|K)|(A|H) = 1. Then, we show that a p-consistent family F={E1|H1, E2|H2} p-entails a conditional event E3|H3 if and only if E3|H3= 1, or (E3|H3)|QC(S) = 1 for some nonempty subset S (...) of F, where QC(S) is the quasi conjunction of the conditional events in S. Then, we examine the inference rules And, Cut, Cautious Monotonicity, and Or of System P and other well known inference rules (Modus Ponens, Modus Tollens, Bayes). We also show that QC(F)|C(F) = 1, where C(F) is the conjunction of the conditional events in F. We characterize p-entailment by showing that F p-entails E3|H3 if and only if (E3|H3)|C(F) = 1. Finally, we examine Denial of the antecedent and Affirmation of the consequent, where the p-entailment of (E3|H3) from F does not hold, by showing that (E3|H3)|C(F) is not equal to 1. (shrink)

The paper presents an exhaustive menu of nonmonotonic logics. The options are individuated in terms of the principles they reject. I locate, e.g., cumulative logics and relevance logics on this menu. I highlight some frequently neglected options, and I argue that these neglected options are particularly attractive for inferentialists.

The knowledge representation and reasoning of both humans and artificial systems often involves conditionals. A conditional connects a consequence which holds given a precondition. It can be easily recognized in natural languages with certain key words, like “if” in English. A vast amount of literature in both fields, both artificial intelligence and psychology, deals with the questions of how such conditionals can be best represented and how these conditionals can model human reasoning. On the other hand, findings in the psychology (...) of reasoning, such as those in the Suppression Task, have led to a paradigm shift from the monotonicity assumptions in human inferences towards nonmonotonic reasoning. Nonmonotonic reasoning is sensitive for information change, that is, inferences are drawn cautiously such that retraction of previous information is not required with the addition of new information. While many formalisms of nonmonotonic reasoning have been proposed in the field of Artificial Intelligence, their capability to model properties of human reasoning has not yet been extensively investigated. In this paper, we analyzed systematically from both a formal and an empirical perspective the power of formal nonmonotonic systems to model possible explicit defeaters, as in the Suppression Task, and more implicit conditional rules that trigger nonmonotonic reasoning by the keywords in such rules. The results indicated that the classical evaluation for the correctness of inferences has to be extended in the three major aspects regarding the inference system, the knowledge base, and possible assumed exceptions for the rule. (shrink)

The distribution of the focus particle even is constrained: if it is adjoined at surface structure to an expression that is entailed by its focus alternatives, as in even once, it must be appropriately embedded to be acceptable. This paper focuses on the context-dependent distribution of such occurrences of even in the scope of non-monotone quantifiers. We show that it is explained on the assumption that even can move at LF Syntax and semantics, 1979). The analysis is subsequently extended to (...) occurrences of negative polarity items in these environments, which mirror the abovementioned distribution of even and which invalidate standard characterizations of NPI licensing conditions in terms of downward-entailingness. The idea behind the extension is that NPIs denote weak elements that are associates of covert even. The paper concludes by discussing two comprehensive theories of NPI licensing and how our proposal relates to them. (shrink)

Let h : ℕ → ℚ be a computable function. A real number x is called h-monotonically computable if there is a computable sequence of rational numbers which converges to x h-monotonically in the sense that h|x – xn| ≥ |x – xm| for all n andm > n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h : ℕ → ℚ, we show that the class h-MC coincides (...) with the classes of computable and semi-computable real numbers if and only if Σi∈ℕ) = ∞and the sum Σi∈ℕ) is a computable real number, respectively. On the other hand, if h ≥ 1 and h converges to 1, then h-MC = SC no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h-MC = c-MC. Finally, if h is monotone and unbounded, then h-MC contains all ω-mc real numbers which are g-mc for some computable function g. (shrink)

The Cautious Jealous Virtue is an illuminating meditation that will interest not only Hume scholars but also those interested in the issues of justice and in ...

A consequence relation \ is monotonic iff for premise sets \ and conclusion \, if \, \, then \; and non-monotonic if this fails in some instance. More plainly, a consequence relation is monotonic when whatever is entailed by a premise set remains entailed by any of its supersets. From the High Middle Ages through the Early Modern period, consequence in theology is assumed to be monotonic. Concomitantly, to the degree the argument formulated by Anselm at Proslogion 2–4 is taken (...) up by later commentators, it is accepted or rejected in accordance with a monotonic notion of consequence. Examining Anselm’s use of parallelism in the Proslogion, I show Anselm embeds his famous argument in Proslogion 2–4 in a non-monotonic context. The results here presented challenge some deeply ingrained ideas governing the historiography of the long twelfth century, particularly concerning how the theology of the later eleventh through the twelfth century relates to the scholasticism of the thirteenth. (shrink)

Area number x is called k-monotonically computable , for constant k > 0, if there is a computable sequence n ∈ ℕ of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k · |x — xn| ≥ |x — xm| for any m > n and x is monotonically computable if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence s ∈ ℕ of (...) rational numbers converging to x such that the sum equation image|xs — xs + 1| is finite. In this paper we show that a mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers. (shrink)

We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and the class of support increasing limitwise monotonic sets on certain domains.

Monotonicity is commonly considered an essential requirement for power measures; violation of local monotonicity or related postulates supposedly disqualifies an index as a valid yardstick for measuring power. This paper questions if such claims are really warranted. In the light of features of real-world collective decision making such as coalition formation processes, ideological affinities, a priori unions, and strategic interaction, standard notions of monotonicity are too narrowly defined. A power measure should be able to indicate that power (...) is non-monotonic in a given dimension of players' resources if – given a decision environment and plausible assumptions about behaviour – itis non-monotonic. (shrink)

This article studies the monotonicity behavior of plural determinersthat quantify over collections. Following previous work, we describe thecollective interpretation of determiners such as all, some andmost using generalized quantifiers of a higher type that areobtained systematically by applying a type shifting operator to thestandard meanings of determiners in Generalized Quantifier Theory. Twoprocesses of counting and existential quantification thatappear with plural quantifiers are unified into a single determinerfitting operator, which, unlike previous proposals, both capturesexistential quantification with plural determiners and respects (...) theirmonotonicity properties. However, some previously unnoticed factsindicate that monotonicity of plural determiners is not always preservedwhen they apply to collective predicates. We show that the proposedoperator describes this behavior correctly, and characterize themonotonicity of the collective determiners it derives. It is proved thatdeterminer fitting always preserves monotonicity properties ofdeterminers in their second argument, but monotonicity in the firstargument of a determiner is preserved if and only if it is monotonic inthe same direction in the second argument. We argue that this asymmetryfollows from the conservativity of generalized quantifiers innatural language. (shrink)

Positive monotone modal logic is the negation- and implication-free fragment of monotone modal logic, i.e., the fragment with connectives and. We axiomatise positive monotone modal logic, give monotone neighbourhood semantics based on posets, and prove soundness and completeness. The latter follows from the main result of this paper: a duality between so-called \-spaces and the algebraic semantics of positive monotone modal logic. The main technical tool is the use of coalgebra.

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)

We extend the limitwise monotonicity notion to the case of arbitrary computable linear ordering to get a set which is limitwise monotonic precisely in the non‐computable degrees. Also we get a series of connected non‐uniformity results to obtain new examples of non‐uniformly equivalent families of computable sets with the same enumeration degree spectrum.

Power indices are commonly required to assign at least as much power to a player endowed with some given voting weight as to any player of the same game with smaller weight. This local monotonicity and a related global property however are frequently and for good reasons violated when indices take account of a priori unions amongst subsets of players (reflecting, e.g., ideological proximity). This paper introduces adaptations of the conventional monotonicity notions that are suitable for voting games (...) with an exogenous coalition structure. A taxonomy of old and new monotonicity concepts is provided, and different coalitional versions of the Banzhaf and Shapley–Shubik power indices are compared accordingly. (shrink)

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 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)

Monotone modal logics have emerged in several application areas such as computer science and social choice theory. Since many of the most studied selfextensional logics have a conjunction, in this paper we study some distributive extensions obtained from a semilattice based deductive system with monotonic modal operators, and we give them neighborhood and algebraic semantics. For each logic defined our main objective is to prove completeness with respect to its characteristic class of monotonic frames.

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov's positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov's theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its name suggests; (...) yet, many standard theorems are also theorems of the new theory since Kolmogorov's other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov's theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works. (shrink)

This paper focuses on the concept of collective essence: that some truths are essential to many items taken together. For example, that it is essential to conjunction and negation that they are truth-functionally complete. The concept of collective essence is one of the main innovations of recent work on the theory of essence. In a sense, this innovation is natural, since we make all sorts of plural predications. It stands to reason that there should be a distinction between essential and (...) accidental plural predications if there is a distinction among singular predications. In this paper I defend the view that the concept of collective essence is governed by the principle of Monotonicity: that something is essential to some items only if it is essential to any items to which they belong. (shrink)

Theorists of aesthetic value since Hume have traditionally aimed to justify at least some comparative judgments of aesthetic value and to explain why we thereby have more reason to appreciate some aesthetic objects than others. I argue that three recent theories of aesthetic value—Thi Nguyen’s and Matthew Strohl’s engagement theories, Nick Riggle’s communitarian theory, and Dominic McIver Lopes’ network theory—face a challenge to carry out this explanatory task in a satisfactory way. I defend a monotonicity principle according to which (...) the strength of our aesthetic reasons to appreciate varies monotonically with aesthetic value and claim that these theories, because they do not respect the principle, are non-monotonic. If they cannot find a plausible way to preserve the link between the aesthetic goodness of an object and our aesthetic reasons to appreciate it, these non-monotonic theories should be rejected. (shrink)

We identify a new monotonicity condition (called cover monotonicity) for tournament solutions which allows a discrimination among main tournament solutions: The top-cycle, the iterated uncovered set, the minimal covering set, and the bipartisan set are cover monotonic while the uncovered set, Banks set, the Copeland rule, and the Slater rule fail to be so. As cover monotonic tournament solutions induce social choice rules which are Nash implementable in certain non-standard frameworks (such as those set by Bochet and Maniquet (...) (CORE Discussion Paper No. 2006/84, 2006) or Özkal-Sanver and Sanver (Social Choice and Welfare, 26(3), 607–623, 2006), the discrimination generated by cover monotonicity becomes particularly notable when implementability is a concern. (shrink)

The paper is about the interpretation of opaque verbs like “seek”, “owe”, and “resemble” which allow for unspecific readings of their (indefinite) objects. It is shown that the following two observations create a problem for semantic analysis: (a) The opaque position is upward monotone: “John seeks a unicorn” implies “John seeks an animal”, given that “unicorn” is more specific than “animal”. (b) Indefinite objects of opaque verbs allow for higher-order, or “underspecific”, readings: “Jones is looking for something Smith is looking (...) for” can express that there is something unspecific that both Jones and Smith are looking for. Given (a) and (b), it would seem that the following inference is hard to escape, if the premisses are construed unspecifically and the conclusion is taken on its under- specific reading: Jones is looking for a sweater. Smith is looking for a pen. Smith is looking for something Jones is looking for. (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)

In this paper we will discuss constraints on the number of (non-dummy) players and on the distribution of votes such that local monotonicity is satisfied for the Public Good Index. These results are compared to properties which are related to constraints on the redistribution of votes (such as implied by global monotonicity). The discussion shows that monotonicity is not a straightforward criterion of classification for power measures.

A path scheme for a game is composed of a path, i.e., a sequence of coalitions that is formed during the coalition formation process and a scheme, i.e., a payoff vector for each coalition in the path. A path scheme is called population monotonic if a player’s payoff does not decrease as the path coalition grows. In this study, we focus on Shapley path schemes of simple games in which for every path coalition the Shapley value of the associated subgame (...) provides the allocation at hand. Obviously, each Shapley path scheme of a game is population monotonic if and only if the Shapley allocation scheme of the game is population monotonic in the sense of Sprumont (Games Econ Behav 2:378–394, 1990). We prove that a simple game allows for population monotonic Shapley path schemes if and only if the game is balanced. Moreover, the Shapley path scheme of a specific path is population monotonic if and only if the first winning coalition that is formed along the path contains every minimal winning coalition. We also show that each Shapley path scheme of a simple game is population monotonic if and only if the set of veto players of the game is a winning coalition. Extensions of these results to other efficient probabilistic values are discussed. (shrink)

Felicitas Goodman (1990) observed that naive participants experienced unique trance states, characterized by specific visionary content, when they assumed particular postures and listened to monotonous rattling. Students (n = 284), enrolled in various sections of the course Introduction to Psychology, experienced one of four conditions with their eyes closed: Sitting Quietly with and without Drumming, Standing (Feather Serpent) Posture plus Drumming with and without Suggested Experiences. Participants completed the Phenomenology of Consciousness Inventory (Pekala 1982, 1991c) and wrote narratives following their (...) respective stimulus conditions. Participants reported less imagery, joy, and absorption, but more internal dialogue, self‐awareness, and muscular tension in the posture, compared to the sitting quietly, conditions. Goodman's results of specific posture (content‐related) effects with naive participants were not replicated. (shrink)

A proposal by Ferguson [2003, Argumentation 17, 335–346] for a fully monotonic argument form allowing for the expression of defeasible generalizations is critically examined and rejected as a general solution. It is argued that (i) his proposal reaches less than the default-logician’s solution allows, e.g., the monotonously derived conclusion is one-sided and itself not defeasible. (ii) when applied to a suitable example, his proposal derives the wrong conclusion. Unsuccessful remedies are discussed.

We study the complexity of proving the Pigeon Hole Principle in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We prove a size-depth trade-off upper bound for monotone proofs of the standard encoding of the PHP as a monotone sequent. At one extreme of the trade-off we get quasipolynomia -size monotone proofs, and at the other extreme we get subexponential-size bounded-depth monotone proofs. This result is a consequence of deriving the basic properties of certain monotone formulas (...) computing the Boolean threshold functions. We also consider the monotone sequent expressing the Clique-Coclique Principle defined by Bonet, Pitassi and Raz [9]. We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one-to-one and onto PHP, and so CLIQUE also has quasipolynomia -size monotone proofs. As a consequence of our results, Resolution, Cutting Planes with polynomially bounded coefficients, and Bounded-Depth Frege are exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudlák. (shrink)

Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

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)

We provide necessary and sufficient conditions determining how monotonicity of some classes of reducible quantifiers depends on the monotonicity of simpler quantifiers of iterations to which they are equivalent.

Natural languages exhibit manysemantic universals, that is, properties of meaning shared across all languages. In this paper, we develop an explanation of one very prominent semantic universal, the monotonicity universal. While the existing work has shown that quantifiers satisfying the monotonicity universal are easier to learn, we provide a more complete explanation by considering the emergence of quantifiers from the perspective of cultural evolution. In particular, we show that quantifiers satisfy the monotonicity universal evolve reliably in an (...) iterated learning paradigm with neural networks as agents. (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)

In this thesis, we study the least fixed point principle in a constructive setting. A constructive theory of functions and sets has been developed by Feferman. This theory deals both with sets and with functions over sets as independent notions. In the language of Feferman's theory, we are able to formulate the least fixed point principle for monotone inductive definitions as: every operation on classes to classes which satisfies the monotonicity condition has a least fixed point. This is called (...) the principle of monotone inductive definition. Furthermore, we may formulate this principle in a uniform way as: there is an operation which maps a monotone operation to its least fixed point. This is called the principle of uniform monotone inductive definition. Feferman raised the question of the strength of the principle of monotone inductive definition when adjoined to his theory . This question is our primary concern in this thesis. ;Our main results are the consistency of both the principle of monotone inductive definition and the principle of uniform monotone inductive definition adjoined to his theory, and the proof theoretical equivalence between the principle of monotone inductive definition with Feferman's system without the inductive generation axiom and the system of the Pi-one-one Comprehension Axiom. Determination of the proof-theoretical strength of the principle of monotone inductive definition adjoined to Feferman's theory still remains open. ;The consistency of both the principle of monotone inductive definition and the principle of uniform monotone inductive definition with Feferman's theory will be achieved by constructing their models in set theoretical sense. The proof theoretical equivalence between the principle of monotone inductive definition with Feferman's system without the inductive generation axiom and the system of the Pi-one-one Comprehension Axiom can be obtained by a careful examination of the model construction for the principle of monotone inductive definition with Feferman's system without the inductive generation axiom, which is parallel to the model construction for the principle of monotone inductive definition with his theory. (shrink)

We present a method for relevance sensitive non-monotonic inference from belief sequences which incorporates insights pertaining to prioritized inference and relevance sensitive, inconsistency tolerant belief revision. Our model uses a finite, logically open sequence of propositional formulas as a representation for beliefs and defines a notion of inference from maxiconsistent subsets of formulas guided by two orderings: a temporal sequencing and an ordering based on relevance relations between the putative conclusion and formulas in the sequence. The relevance relations are ternary (...) (using context as a parameter) as opposed to standard binary axiomatizations. The inference operation thus defined easily handles iterated revision by maintaining a revision history, blocks the derivation of inconsistent answers from a possibly inconsistent sequence and maintains the distinction between explicit and implicit beliefs. In doing so, it provides a finitely presented formalism and a plausible model of reasoning for automated agents. (shrink)

Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already Z.Z(1 — sg) is not MMF. It is proved that extensionality is not Dialectica-realizable by MMF, and a simple example of a MF which is not hereditarily majorizable is given.

We describe the foundations and the systematization of natural logic-like monotonic inference using unscoped episodic logical forms (ULFs) that as reported by Kim et al. (Proceedings of the 1st and 2nd Workshops on Natural Logic Meets Machine Learning (NALOMA), Groningen, 2021a, b) introduced and first evaluated. In addition to providing a more detailed explanation of the theory and system, we present results from extending the inference manager to address a few of the limitations that as reported by Kim et al. (...) (Proceedings of the 1st and 2nd Workshops on Natural Logic Meets Machine Learning (NALOMA), Groningen, 2021b) naive system has. Namely, we add mechanisms to incorporate lexical information from the hypothesis (or goal) sentence, enable the inference manager to consider multiple possible scopings for a single sentence, and match against the goal using English rather than the ULF. (shrink)

Sustainable development is a common goal in the public sector but may be difficult to implement due to epistemic uncertainties and the long time frames required. This paper proposes that some of these problems can be solved by formulating cautious utopias, entailing a relationship between means and goals differing from both utopian and realistic goal-setting. Cautiously utopian goals are believed, but not certain, to be achievable and to remain desirable, but are open to future adjustments due to changing desires (...) and/or factual circumstances. Quality criteria for such goals are suggested. (shrink)

This paper presents a generalization of the standard notions of left monotonicity (on the nominal argument of a determiner) and right monotonicity (on the VP argument of a determiner). Determiners such as “more than/at least as many as” or “fewer than/at most as many as”, which occur in so-called propositional comparison, are shown to be monotone with respect to two nominal arguments and two VP-arguments. In addition, it is argued that the standard Generalized Quantifier analysis of numerical determiners (...) such as “more than three/at least three” is a simplification which ignores the fundamental parallellism with the propositional comparatives. Furthermore, the symmetric monotonicity configurations of the existential “some” and “no” are shown to be straightforwardly related to those of numerical comparatives with the limit numeral zero, whereas the asymmetric configurations of universal “all” and “not all” involve the extra complicating mechanism of polarity reversal, which is related to Keenan's notion of co-intersectivity (as opposed to intersectivity). A second aim of the paper is to investigate some of the factors reducing the inferential potential of determiners. Different types of comparative determiners and their modification will be considered in detail. In addition, a systematic interaction will be revealed between the monotonicity properties of the determiners in propositional comparatives and the differ ent types of ellipsis in the “than”-complement. Different degrees of ellipsis are defined in terms of informational dependencies between the “than”-complement and the main clause. A general balancing mechanism is observed by virtue of which an increase in informational dependency is compensated by a reduction of the inferential potential of the comparative determiner. The structure of the paper is as follows. Part two deals with propositional comparison and introduces the two basic monotonicity configurations. Part three distinguishes three types of informational dependencies or ellipsis between the “than”-complement and the main clause. In Section 3.1 the “than”-complement only contains a VP constituent, whereas in Section 3.2. it only consists of a nominal constituent. Section 3.3 considers more complex instances of multiple dependencies. Section 3.4 then looks at the interaction of these three types with the modifier “proportionally”. Part four presents the two basic monotonicity configurations for numerical comparison, whereas part five deals with more complex numerical determiners. Bounding determiners, such as “between five and ten”, and other boolean combinations are discussed in Section 5.1, whereas Section 5.2 goes into approximative determiners involving modifiers such as “only” or “almost”. Section 5.3. is devoted to proportional determiners such as “more than two out of three”. Part six then considers the standard existential and universal quantifiers. Section 6.1 reformulates the standard Generalized Quantifier analysis in comparative terms, whereas Section 6.2 deals with exception determiners of the form “all but five”. (shrink)

In voting theory, monotonicity is the axiom that an improvement in the ranking of a candidate by voters cannot cause a candidate who would otherwise win to lose. The participation axiom states that the sincere report of a voter’s preferences cannot cause an outcome that the voter regards as less attractive than the one that would result from the voter’s non-participation. This article identifies three binary distinctions in the types of circumstances in which failures of monotonicity or participation (...) can occur. Two of the three distinctions apply to monotonicity, while one of those and the third apply to participation. The distinction that is unique to monotonicity is whether the voters whose changed rankings demonstrate non-monotonicity are better off or worse off. The distinction that is unique to participation is whether the marginally participating voter causes his first choice to lose or his last choice to win. The overlapping distinction is whether the profile of voters’ rankings has a Condorcet winner or a cycle at the top. This article traces the occurrence of all of the resulting combination of characteristics in the voting methods that can exhibit failures of monotonicity. (shrink)