Cholak, Groszek and Slaman proved in J Symb Log 66:881–901, 2001 that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low2 c.e. degree to a nonlow2 degree. In Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree (...) by a low2 c.e. degree b. It is natural to ask in which subclass of low2 c.e. degrees can b in Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) be located. Wu proved in Math Log Quart 50:189–201, 2004 that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results. (shrink)

We construct a ${\Sigma^0_2}$ e-degree which is both high and noncuppable. Thus demonstrating the existence of a high e-degree whose predecessors are all properly ${\Sigma^0_2}$.

We prove that for every ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degree b there exists a noncuppable ${\mathrm{\Sigma }}_{2}^{0}$ degree a > 0 e such that b′ ≤ e a′ and a″ ≤ e b″. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding l: D T → D e , that there exist ${\mathrm{\Sigma }}_{2}^{0}$ noncuppable enumeration degrees at every possible—i.e., above (...) low₁—level of the high/low jump hierarchy in the context of D e. (shrink)

We construct a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^0_2}$$\end{document}e-degree which is both high and noncuppable. Thus demonstrating the existence of a high e-degree whose predecessors are all properly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^0_2}$$\end{document}.

Let [NB]₁ denote the ideal generated by nonbounding c.e. degrees and NCup the ideal of noncuppable c.e. degrees. We show that both [NB]₁ ∪ NCup and the ideal generated by nonbounding and noncuppable degrees are new, in the sense that they are different from M, [NB]₁ and NCup—the only three known definable ideals so far.

We show that there exist downwards properly $\Sigma _{2}^{0}$ (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

We prove that each Σ 0 2 set which is hypersimple relative to $\emptyset$ ' is noncuppable in the structure of the Σ 0 2 enumeration degrees. This gives a connection between properties of Σ 0 2 sets under inclusion and and the Σ 0 2 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to $\emptyset$ '.

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

We regularly make graded normative judgements in the epistemic domain. Recent work in the literature examines degrees of justification, degrees of rationality, and degrees of assertability. This paper addresses a different dimension of the gradeability of epistemic normativity, one that has been given little attention. How should we understand degrees of epistemic criticizability? In virtue of what sorts of factors can one epistemic failing be worse than another? The paper develops a dual-factor view of degrees (...) of epistemic criticizability. According to the view, degrees of epistemic criticizability are (i) an inverse function of degrees of doxastic justification and (ii) a function of degrees of agent culpability. The paper develops an account of each factor, and explains how they should be weighted. The paper also addresses the importance of modelling degrees of epistemic criticizability in a broader context. I focus on the role that such a model can play in the ethics of epistemic criticism. (shrink)

What propositions are rational for one to believe? With what confidence is it rational for one to believe these propositions? Answering the first of these questions requires an epistemology of beliefs, answering the second an epistemology of degrees of belief.

This paper combines various structures representing degrees of belief, degrees of disbelief, and degrees of non-belief (degrees of expectations) into a unified whole. The representation uses relations of comparative necessity and possibility, as well as non-probabilistic functions assigning numerical values of necessity and possibility. We define all-encompassing necessity structures which have weak expectations (mere hypotheses, guesses, conjectures, etc.) occupying the lowest ranks and very strong, ineradicable ('a priori') beliefs occupying the highest ranks. Structurally, there are no (...) differences from the top to the bottom. I argue that belief is a vague notion, and that thresholds for belief, if there are any, are context-dependent. (shrink)

Mandarin Chinese, along with Japanese, Yorùbá, Mòoré, and Samoan, has been argued to lack ‘degree abstraction’, a configuration at LF involving lambda abstraction over a degree variable. These languages are claimed to have a negative setting for a hypothesized ‘Degree Abstraction Parameter’. Recent work, however, has argued for degree abstraction in Japanese and Yorùbá, and degree abstraction has been detected in a number of additional languages. Could it in fact be universal? Here, we focus on the case of Mandarin, and (...) argue that Mandarin has degree abstraction too. We offer three arguments in favor of degree abstraction in Mandarin, based on attributive comparatives, comparatives with embedded predicates, and scope interactions with modals. We also rebut prior arguments for the lack of degree abstraction in Mandarin, considering degree questions, measure phrases, and negative island effects. Taken together, these results show that degree abstraction is not a parameter along which Mandarin and English vary, and suggest rather that degree abstraction may be universally available. (shrink)

Let us agree that everything that there is exists, and that to be, to be real, and to exist are one and the same. Does everything that there is exist to the same degree? Or do some things exist more than others? Are there gradations of being? I argue that some entities exist more than others. Moreover, many of the notions in play in contemporary metaphysical discourse, such as fundamentality, perfect naturalness, and grounding ought to be cashed out in terms (...) of degree of existence. (shrink)

Is a human more conscious than an octopus? In the science of consciousness, it’s oftentimes assumed that some creatures (or mental states) are more conscious than others. But in recent years, a number of philosophers have argued that the notion of degrees of consciousness is conceptually confused. This paper (1) argues that the most prominent objections to degrees of consciousness are unsustainable, (2) examines the semantics of ‘more conscious than’ expressions, (3) develops an analysis of what it is (...) for a degreed property to count as degrees of consciousness, and (4) applies the analysis to various theories of consciousness. I argue that whether consciousness comes in degrees ultimately depends on which theory of consciousness turns out to be correct. But I also argue that most theories of consciousness entail that consciousness comes in degrees. (shrink)

Various theories try to give accounts of how measures of this confidence do or ought to behave, both as far as the internal mental consistency of the agent as ...

We define a concept of truthmaker function, and prove the functional completeness, w.r.t. truthmaker functions in this sense, of a set of four-valued functions corresponding to standard connectives of the system of relevance logic known as First-Degree Entailment or Belnap–Dunn logic.

It is shown that for every Turing degree d there is an ω-categorical group G such that the isomorphism type of G is of degree d. We also find an ω-categorical group G such that the isomorphism type of G has no degree.

Degrees of belief are familiar to all of us. Our confidence in the truth of some propositions is higher than our confidence in the truth of other propositions. We are pretty confident that our computers will boot when we push their power button, but we are much more confident that the sun will rise tomorrow. Degrees of belief formally represent the strength with which we believe the truth of various propositions. The higher an agent’s degree of belief for (...) a particular proposition, the higher her confidence in the truth of that proposition. For instance, Sophia’s degree of belief that it will be sunny in Vienna tomorrow might be .52, whereas her degree of belief that the train will leave on time might be .23. The precise meaning of these statements depends, of course, on the underlying theory of degrees of belief. These theories offer a formal tool to measure degrees of belief, to investigate the relations between various degrees of belief in different propositions, and to normatively evaluate degrees of belief. (shrink)

Why are conditional degrees of belief in an observation E, given a statistical hypothesis H, aligned with the objective probabilities expressed by H? After showing that standard replies are not satisfactory, I develop a suppositional analysis of conditional degree of belief, transferring Ramsey’s classical proposal to statistical inference. The analysis saves the alignment, explains the role of chance-credence coordination, and rebuts the charge of arbitrary assessment of evidence in Bayesian inference. Finally, I explore the implications of this analysis for (...) Bayesian reasoning with idealized models in science. (shrink)

This paper describes a formal measure of epistemic justification motivated by the dual goal of cognition, which is to increase true beliefs and reduce false beliefs. From this perspective the degree of epistemic justification should not be the conditional probability of the proposition given the evidence, as it is commonly thought. It should be determined instead by the combination of the conditional probability and the prior probability. This is also true of the degree of incremental confirmation, and I argue that (...) any measure of epistemic justification is also a measure of incremental confirmation. However, the degree of epistemic justification must meet an additional condition, and all known measures of incremental confirmation fail to meet it. I describe this additional condition as well as a measure that meets it. The paper then applies the measure to the conjunction fallacy and proposes an explanation of the fallacy. (shrink)

Innovative, accessible, and easily implemented, One Degree Revolution is acclaimed yoga educator and leadership coach Coby Kozlowski's holistic program for self-inquiry and personal transformation. Her philosophy is deeply connected to living yoga-not just doing yoga. In fact, readers don't need to have ever attended a yoga class to dive into this book: her thoughtful teachings are for anybody interested in learning to navigate the waves of life more skillfully and gracefully. Imagine sailing a boat with a course set for a (...) lifetime. If that route changes by just one navigational degree, what would happen to the journey? How far from the original trajectory would we be in one year? Five years? Ten years? Twenty years? Well, we would end up in a totally different place. In much the same way, we can change the course of our life by making a one degree shift. In other words, we don't have to change everything about ourselves or our world to make a difference. Coby inspires readers to dig deep, to ask powerful questions and to dive into the insights, experiments, and inquiries of living yoga: how can I best be with life? How can the teachings of yoga direct us to see the most aligned choices, let go of past hurts, and discover deep and meaningful connections? And what are the most skillful ways we can learn to savor all that life presents? These yoga philosophies are infused with practical strategies for creating the life you truly want and having a positive impact on the world. One Degree Revolution will guide readers to access infinite personal possibilities, celebrate their authentic selves, find meaning and purpose, trust the unfolding of life, and foster transformational change, one degree at a time. (shrink)

We study for each computably bounded $\Pi ^0_1$ class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect $\Pi ^0_1$ class where all c.e. members have degree a. We also show that every $\Pi ^0_1$ set of c.e. indices is realized in some perfect $\Pi ^0_1$ class, and classify the sets of c.e. degrees which can be realized in some $\Pi ^0_1$ class (...) as exactly those with a computable representation. (shrink)

Here is a puzzle, which I learned from Terence Parsons in his “True Contradictions” [8]. First Degree Entailment is a logic which allows for truth value gaps as well as truth value gluts. If you are agnostic between assigning paradoxical sentences gaps and gluts, then this looks no different, in effect, from assigning them a gap value? After all, on both views you end up with a theory that doesn’t commit you to the paradoxical sentence or its negation. How is (...) the fde theory any different from the theory with gaps alone? In this paper, I will present a clear answer to this puzzle – an answer that explains how being agnostic between gaps and gluts is a genuinely different position than admitting gaps alone, by using the formal notion of a bi-theory, and showing that while such positions might agree on what is to be accepted, they differ on what is to be rejected. (shrink)

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with (...) degrees of categoricity is $\Pi_{1}^{1}$-complete. (shrink)

Defining the degree of categoricity of a computable structure ${\mathcal{M}}$ to be the least degree d for which ${\mathcal{M}}$ is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d.c.e. in and above 0 (n) can be so realized, as can the degree 0 (ω).

This paper explores a novel analysis of adjectives in the comparative and the positive based on the notion of a trope, rather than the notion of a degree. Tropes are particularized properties, concrete manifestations of properties in individuals. The point of departure is that a sentence like ‘John is happier than Mary’ is intuitively equivalent to ‘John’s happiness exceeds Mary’s happiness’, a sentence that expresses a simple comparison between two tropes, John’s happiness and Mary’s happiness. The analysis received particular support (...) from various parallels between adjectival constructions and corresponding adjective nominalizations which make reference to tropes. (shrink)

A computable structure $\mathcal {A}$ is $\mathbf {x}$-computably categorical for some Turing degree $\mathbf {x}$ if for every computable structure $\mathcal {B}\cong\mathcal {A}$ there is an isomorphism $f:\mathcal {B}\to\mathcal {A}$ with $f\leq_{T}\mathbf {x}$. A degree $\mathbf {x}$ is a degree of categoricity if there is a computable structure $\mathcal {A}$ such that $\mathcal {A}$ is $\mathbf {x}$-computably categorical, and for all $\mathbf {y}$, if $\mathcal {A}$ is $\mathbf {y}$-computably categorical, then $\mathbf {x}\leq_{T}\mathbf {y}$. We construct a $\Sigma^{0}_{2}$ set whose degree (...) is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity. (shrink)

A number of authors have noted that vagueness engenders degrees of belief, but that these degrees of belief do not behave like subjective probabilities. So should we countenance two different kinds of degree of belief: the kind arising from vagueness, and the familiar kind arising from uncertainty, which obey the laws of probability? I argue that we cannot coherently countenance two different kinds of degree of belief. Instead, I present a framework in which there is a single notion (...) of degree of belief, which in certain circumstances behaves like a subjective probability assignment and in other circumstances does not. The core idea is that one’s degree of belief in a proposition P is one’s expectation of P’s degree of truth. (shrink)

Partial explanations are everywhere. That is, explanations citing causes that explain some but not all of an effect are ubiquitous across science, and these in turn rely on the notion of degree of explanation. I argue that current accounts are seriously deficient. In particular, they do not incorporate adequately the way in which a cause’s explanatory importance varies with choice of explanandum. Using influential recent contrastive theories, I develop quantitative definitions that remedy this lacuna, and relate it to existing measures (...) of degree of causation. Among other things, this reveals the precise role here of chance, as well as bearing on the relation between causal explanation and causation itself. (shrink)

While many authors distinguish belief from acceptance, it seems almost universally agreed that no similar distinction can be drawn between degrees of belief, or credences, and degrees of acceptance. I challenge this assumption in this paper. Acceptance comes in degrees and acknowledging this helps to resolve problems in at least two philosophical domains. Degrees of acceptance play vital roles when we simplify our reasoning, and they ground the common ground of a conversation if we assume context (...) probabilism, i.e., that the common ground must be represented with probability spaces rather than possible worlds. (shrink)

We wish to give a precise formulation of the intuitive concept: The degree to which the known facts support a given hypothesis.

Starting from John MacFarlane's recent survey of answers to the question ‘What is assertion?’, I defend an account of assertion that draws on elements of MacFarlane's and Robert Brandom's commitment accounts, Timothy Williamson's knowledge norm account, and my own previous work on the normative status of logic. I defend the knowledge norm from recent attacks. Indicative conditionals, however, pose a problem when read along the lines of Ernest Adams' account, an account supported by much work in the psychology of reasoning. (...) Furthermore, there seems to be no place for degrees of belief in the accounts of belief and assertion given here. Degrees of belief do have a role in decision‐making, but, again, there is much evidence that the orthodox theory of subjective utility maximization is not a good description of what we do in decision‐making and, arguably, neither is it a good normative guide to how we ought to make decisions. (shrink)

This chapter argues that the distinction between empiricism and metaphysics is not as clear as van Fraassen would like to believe. Almost all inquiry is metaphysical to a degree, including van Fraassen's stance empiricism. Van Fraassen does not make a strong case against metaphysics, since the argument against metaphysics has to happen at the level of meta-stances — the level where one decides which stance to endorse. The chapter maintains that utilizing van Fraassen's own conception of rationality, metaphysicians are rational. (...) Empiricists should not reject all metaphysics, but just the sort of metaphysics which goes well beyond the empirical contexts that most interest them. (shrink)

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$ -computable low $_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high $_{n}$ -degrees are required to present the homeomorphism type (...) of $X_n$. Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. (shrink)

The philosophical work of Jean-Luc Marion has opened new ways of speaking about religious convictions and experiences. In this exploration of Marion’s philosophy and theology, Christina M. Gschwandtner presents a comprehensive and critical analysis of the ideas of saturated phenomena and the phenomenology of givenness. She claims that these phenomena do not always appear in the excessive mode that Marion describes and suggests instead that we consider degrees of saturation. Gschwandtner covers major themes in Marion’s work—the historical event, art, (...) nature, love, gift and sacrifice, prayer, and the Eucharist. She works within the phenomenology of givenness, but suggests that Marion himself has not considered important aspects of his philosophy. (shrink)

Two objects of valuation are said to be incommensurable if neither is better than the other, nor are they equally good. This negative, coarse-grained characterization fails to capture the nuanced structure of incommensurability. We argue that our evaluative resources are far richer than orthodoxy recognizes. We model value comparisons with the corresponding class of permissible preference orderings. Then, making use of our model, we introduce a potentially infinite set of degrees of approximation to better, worse, and equally good, which (...) we interpret as degrees of commensurability.One payoff is the solution our approach provides to a paradox in population ethics, generated by Parfit's “Continuum Argument”. Parfit imagines a sequence of populations, starting with one consisting of excellent lives and, by a sequence of apparent improvements, reaching a much larger population of lives barely worth living. What he dubs “the Repugnant Conclusion” is that the final population is better than the first. Developing Parfit's response, we argue that some of the populations in the sequence are merely almost better than their immediate predecessors. Almost better is not transitive (unlike better). We offer analogies to other ‘spectrum arguments’, Condorcet's paradox, and to developments in formal epistemology. (shrink)

I argue that Aristotle believes that virtue comes in degrees. After dispatching with initial concerns for the view, I argue that we should accept it because Aristotle conceives of heroic virtue as the highest degree of virtue. I support this interpretation of heroic virtue by considering and rejecting alternative readings, then showing that heroic virtue characterized as the highest degree of virtue is consistent with the doctrine of the mean.

The goal of this paper is an account of the semantics and pragmatics of exclamation. I focus on two key observations: first, that sentence exclamations like Wow, John bakes delicious desserts! and exclamatives like What delicious desserts John bakes! express that a particular proposition has violated the speaker’s expectations; and second, that exclamatives are semantically restricted in a way that sentence exclamations are not. In my account of these facts, I propose a characterization of illocutionary force of exclamation, a function (...) from propositions to speech acts of exclamation. The difference in meaning between sentence exclamations and exclamatives has consequences for the type of violated expectation. I end with a comparison to some previous approaches and a tentative extension of parts of the analysis to other constructions. (shrink)

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure A as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of A; the degree of bi-embeddable categoricity of A is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, (...) and we show that every degree d.c.e above 0(α) for α a computable successor ordinal and 0(λ) for λ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra. (shrink)

I first seriously contemplated writing a book on degree theory in 1976 while I was visiting the University of Illinois at Chicago Circle. There was, at that time, some interest in ann-series book about degree theory, and through the encouragement of Bob Soare, I decided to make a proposal to write such a book. Degree theory had, at that time, matured to the point where the local structure results which had been the mainstay of the earlier papers in the area (...) were finding a steadily increasing number of applications to global degree theory. Michael Yates was the first to realize that the time had come for a systematic study of the interaction between local and global degree theory, and his papers had a considerable influence on the content of this book. During the time that the book was being written and rewritten, there was an explosion in the number of global theorems about the degrees which were proved as applications of local theorems. The global results, in turn, pointed the way to new local theorems which were needed in order to make further progress. I have tried to update the book continuously, in order to be able to present some of the more recent results. It is my hope to introduce the reader to some of the fascinat ing combinatorial methods of Recursion Theory while simultaneously showing how to use these methods to prove some beautiful global theorems about the degrees. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

This paper studies degrees of doxastic justification. Dependency relations among different beliefs are represented in terms of causal models. Doxastic justification, on this picture, is taken to run causally downstream along appropriate causal chains. A theory is offered which accounts for the strength of a derivative belief in terms of (i) the strength of the beliefs on which it is based, and (ii) the epistemic quality of the belief-forming mechanisms involved. It is shown that the structure of degrees (...) of justification converges to ranking theory under ideal conditions. (shrink)

We analyze the degree spectra of structures in which different types of immunity conditions are encoded. In particular, we give an example of a structure whose degree spectrum coincides with the hyperimmune degrees. As a corollary, this shows the existence of an almost computable structure of which the complement of the degree spectrum is uncountable.

This paper provides a new account of positive versus negative antonyms. The data includes well-known linguistic generalizations regarding negative adjectives, such as their incompatibility with measure phrases (cf. two meters tall/ *short) and ratio phrases (twice as tall/ #short) as well as the impossibility of truly crosspolar comparisons (*Dan is taller than Sam is short). These generalizations admit a variety of exceptions, e.g., positive adjectives that do not license measure phrases (cf. #two degrees warm/cold) and rarely also negative adjectives (...) that do (cf. two hours late/early). Furthermore, new corpus data is presented regarding the use of twice with positive and negative adjectives. The analysis the paper presents supposes that grammar associates gradable adjectives with measure functions—mapping of entities to a set of degrees isomorphic to the real numbers (Kennedy, Projecting the adjective: The syntax and semantics of gradability and comparison, 1999). On this analysis, negative adjectives map entities to values that are linearly reversed and linearly transformed in comparison with their values in the positive antonyms. As shown, the generalizations, as well as their exceptions, directly follow. Negative polarity is explained in terms of function reversal, and non-licensing of measure phrases is explained in terms of transformation by an unspecified value. (shrink)