Book synopsis: Vagueness is currently the subject of vigorous debate in the philosophy of logic and language. Vague terms-such as "tall", "red", "bald", and "tadpole"—have borderline cases ; and they lack well-defined extensions. The phenomenon of vagueness poses a fundamental challenge to classical logic and semantics, which assumes that propositions are either true or false and that extensions are determinate. Another striking problem to which vagueness gives rise is the sorites paradox. If you remove one grain from a heap of (...) sand, surely you must be left with a heap. Yet apply this principle repeatedly as you remove grains one by one, and you end up, absurdly, with a solitary grain that counts as a heap. This anthology collects papers in the field. After an introduction that surveys the field, the essays form four groups, starting with some historically notable pieces. The 1970s saw an explosion of interest in vagueness, and the second group of essays reprints classic papers from this period. The following group of papers represent current work on the logic and semantics of vagueness. The essays in the final group are contributions to the continuing debate about vague objects and vague identity. (shrink)

This paper finds its origin in a debate between Georges Bataille (1897-1962) and Octavio Paz (1914-1998) on what is central to the idea of eroticism. Bataille posits that violence and transgression are fundamental to eroticism, and without prohibition, eroticism would cease to exist. Paz, however, views violence and transgression as merely intersecting with, rather than being intrinsic to, eroticism. Paz places focus on imagination, and transforms eroticism from a transgressive, to a ritualistic act. Eroticism thus functions as an intermediary, turning (...) animal sexuality into ceremony and rite. In this paper, I argue that Paz and Bataille are not fundamentally at odds in their conceptualisation of eroticism. Instead, Paz redefines the locus of eroticism by subverting its unsettling aspects. His conception of eroticism gives precedence to the imagination, which in itself is an act of violation. Imagination transgresses normal life, but this transgression – the violation of the habitual world – also introduces a glimpse of the numinous that Paz associates with belongingness. The paper further explores eroticism, as this is perceived today, and argues that it ultimately eliminates the threshold marked by transgression. The rise of the contemporary pornographic society is marked by a loss of imagination and transcendence, which ultimately reduces the subject into a mere automaton that demands instant gratification. (shrink)

The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: consistency and deductive closure for belief sets, conformity to the axioms of probability for degrees of belief, and the Lockean Thesis concerning the relationship between belief and degree of belief. Assuming that the beliefs and degrees of belief of both individuals and collectives satisfy the preceding three constraints, I discuss what further constraints may be imposed (...) on the aggregation of beliefs and degrees of belief. Some possibility and impossibility results are presented. The possibility results suggest that the three proposed rationality constraints are compatible with reasonable aggregation procedures for belief and degree of belief. (shrink)

It is often natural to compare two events by describing one as ‘more of a cause’ of some effect than the other. But what do such comparisons amount to, exactly? This paper aims to provide a guided tour of the recent literature on ‘degrees of causation’. Section 2 looks at what I call ‘dependence measures’, which arise from thinking of causes as difference‐makers. Section 3 looks at what I call ‘production measures’, which arise from thinking of causes as jointly (...) sufficient for their effects. Finally, section 4 examines the important question of whether there is any sense in which an agent is more responsible for an outcome in virtue of her action being more of a cause of it. I describe a puzzle that emerges from this question, and explore various strategies for resolving it. (shrink)

The representation theorems of expected utility theory show that having certain types of preferences is both necessary and sufficient for being representable as having subjective probabilities. However, unless the expected utility framework is simply assumed, such preferences are also consistent with being representable as having degrees of belief that do not obey the laws of probability. This fact shows that being representable as having subjective probabilities is not necessarily the same as having subjective probabilities. Probabilism can be defended on (...) the basis of the representation theorems only if attributions of degrees of belief are understood either antirealistically or purely qualitatively, or if the representation theorems are supplemented by arguments based on other considerations (simplicity, consilience, and so on) that single out the representation of a person as having subjective probabilities as the only true representation of the mental state of any person whose preferences conform to the axioms of expected utility theory. (shrink)

This paper aims to pay tribute to Figal’s comprehensive and innovative analysis of the artwork and beauty, while challenging both his realist position on the immediacy of meaning and his monist stance that reduces sublimity to beauty. To enquire into the origin of aesthetic feelings and sense, and thus, to break the hermeneutic circle, we first trace the origin of this reduction to the reception of Burke’s concept of the sublime by Mendelssohn and Kant. We then recur to Husserl and (...) Dufrenne to trace the origin of the sense of the feeling of sublimity in the realm of affectivity or sensuousness. This genesis takes place in passive experience, where we identify an excess of sensuous sense over categorial meaning that is constitutive of the affective sense that our bodily sensations give to hyletic material. It is argued that this sensuous excess evokes a sublime experience in the phenomenological sense, involving an affective and disruptive bodily experience and an endless process of sense formation. Affective or sensuous sense is thus an event that irreversibly affects the bodily surface. To elucidate this phenomenon, we draw on the theoretical framework of Breton and Bataille, positing it as a form-transgressive ‘convulsive’ sublime, which we then illustrate in the architectural works of Tschumi and Eisenman. In short, we have retrieved the origin of aesthetic experience and redefined the feeling of the sublime as a phenomenological ‘convulsive’ sublimity. (shrink)

This paper addresses the issue of the moral status of non-human animals, or the question whether sentient animals are morally considerable. The arguments for and against the moral status of animals are discussed, above all the argument from marginal cases. It is argued that sentient animals have moral status based on their having interests in their experiential well-being, but that there are degrees of moral status. Two interest-based approaches are presented and discussed: DeGrazia’s view that sentient animals have interests (...) in continuing to live, and that their interests should be granted moral weight; and McMahan’s TRIA which similarly postulates that animals have interests and that in a given situation we should compare the human and animal interests at stake. Finally, the paper concludes that the anthropocentric approach to animal ethics should be abandoned in favour of the biocentric ethics. (shrink)

In this paper we argue that awareness comes in degrees, and we propose a novel multi-factor account that spans both subjective experiences and perceptual representations. At the subjective level, we argue that conscious experiences can be degraded by being fragmented, less salient, too generic, or flash-like. At the representational level, we identify corresponding features of perceptual representations—their availability for working memory, intensity, precision, and stability—and argue that the mechanisms that affect these features are what ultimately modulate the degree of (...) awareness. We conclude the paper by demonstrating why the original interpretations of certain empirical findings that apparently pose problems for our account are, in fact, flawed. (shrink)

This article argues in defence of human–robot friendship. I begin by outlining the standard Aristotelian view of friendship, according to which there are certain necessary conditions which x must meet in order to ‘be a friend’. I explain how the current literature typically uses this Aristotelian view to object to human–robot friendships on theoretical and ethical grounds. Theoretically, a robot cannot be our friend because it cannot meet the requisite necessary conditions for friendship. Ethically, human–robot friendships are wrong because they (...) are deceptive, and could also make it more likely that we will favour ‘perfect’ robots, and disrespect, exploit, or exclude other human beings. To argue against the above position, I begin by outlining and assessing current attempts to reject the theoretical argument—that we cannot befriend robots. I argue that the current attempts are problematic, and do little to support the claim that we can be friends with robots now. I then use the standard Aristotelian view as a touchstone to develop a new degrees-of-friendship view. On my view, it is theoretically possible for humans to have some degree of friendship with social robots now. I explain how my view avoids ethical concerns about human–robot friendships being deceptive, and/or leading to the disrespect, exploitation, or exclusion of other human beings. (shrink)

Many widely divergent accounts of luck have been offered or employed in discussing an equally wide range of philosophical topics. We should, then, expect to find some unified philosophical conception of luck of which moral luck, epistemic luck, and luck egalitarianism are species. One of the attempts to provide such an account is that offered by Duncan Pritchard, which he refers to as the modal account. This view commits us to calling an event lucky when it obtains in this world, (...) but fails to obtain in a wide class of nearby possible worlds. In support of this account, Pritchard argues that a theory of luck ought to capture the fact that luck comes in degrees and that luck is closely associated with risk. I argue against this claim by suggesting that an understanding of luck grounded in considerations of probability is better able to satisfy these demands, and that the probability theory better explains exemplary cases of luck like those brought up by Pritchard. (shrink)

Distinguer pour unir, ou Les degres du savoir was first published in 1932 by Jacques Maritain. In this new translation of The Degrees of Knowledge, Ralph McInerny attempts a more careful expression of Maritain's original masterpiece than previous translations. Maritain proposes a hierarchy of the forms of knowledge by discussing the degrees of rational and suprarational understanding. Nine appendices, some longer than the chapters of the book, advance Maritain's thought, often by taking on criticism of earlier editions of (...) the work. Rightly called Maritain's cardinal work, The Degrees of Knowledge is a magnificent and sagacious achievement. Jacques Maritain's masterpiece proposes a hierarchy of forms of knowledge that culminate in mystical experience and wisdom, which is a gift of the Holy Ghost. Maritain argues that the intellectual life is meant to be complemented by the spiritual life and should culminate in sanctity. (shrink)

This research explores the relationship of variant degrees of a country’s economic freedom to the ethical profiles of millennial business students, specifically an individual’s personal value orientation and post-conventional reasoning. Grounded in Social Identity, Personal Values, and Cognitive Moral Development theories, we construct an ethical profile to compare responses provided by millennial business students from eight countries. Our results suggest that a country’s degree of economic freedom has some association with an individual’s ethical profile, yet we also discuss other (...) national influences on an ethical profile. These results and their implications are discussed in the article. (shrink)

The impossibility of an indeterministic evolution for standard relativistic quantum field theories, that is, theories in which all fields satisfy the condition that the generators of space-time translation have spectra in the forward light-cone, is demonstrated. The demonstration proceeds by arguing that a relativistically invariant theory must have a stable vacuum and then showing that stability of the vacuum, together with the requirements imposed by relativistic causality, entails deterministic evolution, if all degrees of freedom are standard degrees of (...) freedom. (shrink)

If the phenomenality of consciousness admits of degrees and can be partial and indeterminate, then Block's inference to the best explanation may need to be revaluated both in terms of the supposed data on phenomenal overflow and the range of alternatives against which his view is compared.

The paper’s target is the historically influential betting interpretation of subjective probabilities due to Ramsey and de Finetti. While there are several classical and well-known objections to this interpretation, the paper focuses on just one fundamental problem: There is a sense in which degrees of belief cannot be interpreted as betting rates. The reasons differ in different cases, but there’s one crucial feature that all these cases have in common: The agent’s degree of belief in a proposition A does (...) not coincide with her degree of belief in a conditional that A would be the case if she were to bet on A, where the belief in this conditional itself is conditioned on the supposition that the agent will have an opportunity to make such a bet. Even though the two degrees of belief sometimes can coincide (they will coincide in those cases when the bet has no expected causal bearings on the proposition A and the opportunity to bet have no evidential bearings on that proposition), it is the latter belief rather than the former that guides the agent’s rational betting behaviour. The reason is that this latter belief takes into consideration potential interferences that bet opportunities and betting itself might create with regard to the proposition to be betted on. It is because of this interference problem that the agent’s degree of belief in A cannot be interpreted as her betting rate for A. (shrink)

In the paper it will be shown that Reichenbach’s Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ , respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ and commuting with the (...) both A and B. Since noncommuting common cause solutions are presented in these states the abandonment of commutativity can modulate this result: noncommutative Common Cause Principles might survive in these models. (shrink)

Cross‐situational word learning (XSWL) tasks present multiple words and candidate referents within a learning trial such that word–referent pairings can be inferred only across trials. Adults encode fine phonological detail when two words and candidate referents are presented in each learning trial (2 × 2 scenario; Escudero, Mulak, & Vlach, ). To test the relationship between XSWL task difficulty and phonological encoding, we examined XSWL of words differing by one vowel or consonant across degrees of within‐learning trial ambiguity (1 (...) × 1 to 4 × 4). Word identification was assessed alongside three distractors. Adults finely encoded words via XSWL: Learning occurred in all conditions, though accuracy decreased across the 1 × 1 to 3 × 3 conditions. Accuracy was highest for the 1 × 1 condition, suggesting fast‐mapping is a stronger learning strategy here. Accuracy was higher for consonant than vowel set targets, and having more distractors from the same set mitigated identification of vowel set targets only, suggesting possible stronger encoding of consonants than vowels. (shrink)

We survey the theory of Mucnik and Medvedev degrees of subsets of $^{\omega}{\omega}$with particular attention to the degrees of $\Pi_{1}^{0}$ subsets of $^{\omega}2$. Sections 1-6 present the major definitions and results in a uniform notation. Sections 7-6 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.

We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps are enumeration (...) degree jumps.) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees. (shrink)

ABSTRACTIn this paper, I argue for an Aristotelian way of accommodating supererogation within virtue ethics by retrieving an account of moral heroism and providing a picture of different degrees of virtue. This, I claim, is the most appropriate virtue-ethical background allowing us to talk about supererogation without falling prey to several dangers. After summarizing the main attempts to deny the compatibility of virtue and supererogation, I will present some recent proposals to accommodate supererogation within virtue ethics. Next, I will (...) argue that these proposals raise substantial worries. I will then offer an analysis of heroism and the degrees of virtue to outline my own alternative definition of supererogation in terms of heroic virtue. (shrink)

It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let $\mbox{\bf a} \cup \mbox{\bf b}$ and $\mbox{\bf a} \cap \mbox{\bf b}$ denote respectively the join and the meet of the degrees $\mbox{\bf a}$ and $\mbox{\bf b}$ . This paper introduces a jump operator ( $\cdot'$ ) on the honest elementary degrees and defines canonical degrees $\mbox{\bf 0},\mbox{\bf 0}', \mbox{\bf 0}^{\prime \prime },\ldots$ and low and high (...) class='Hi'>degrees analogous to the corresponding concepts for the Turing degrees. Among others, the following results about the structure of the honest elementary degrees are shown: There exist low degrees, and there exist degrees which are neither low nor high. Every degree above $\mbox{\bf 0}'$ is the jump of some degree, moreover, for every degree $\mbox{\bf c}$ above $\mbox{\bf 0}'$ there exist degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf c}=\mbox{\bf a} \cup \mbox{\bf b} = \mbox{{\bf a}}'=\mbox{\bf b}'$ . We have $\mbox{\bf a}'\cup \mbox{\bf b}' \leq (\mbox{\bf a}\cup\mbox{\bf b})'$ and $\mbox{\bf a}'\cap \mbox{\bf b}' \geq (\mbox{\bf a}\cap \mbox{\bf b})'$ . The jump operator is of course monotonic, i.e. $\mbox{\bf a}\leq\mbox{{\bf b}}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'$ . We prove that every situation compatible with $\mbox{\bf a}\leq\mbox{\bf b}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'$ is realized in the structure, e.g. we have incomparable degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf a}'<\mbox{\bf b}'$ and incomparable degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf a}' = \mbox{\bf b}'$ etcetera. We are able to prove all these results without the traditional recursion theoretic constructions. Our proof method relies on the fact that the growth of the functions in a degree is bounded. This technique also yields a very simple proof of an old result, namely that the structure is a lattice. (shrink)

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.

Using properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration (...) degree can be capped by degrees of any possible local jump ; every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa. (shrink)

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, (...) and hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

We show that every nontrivial interval in the recursively enumerable degrees contains an incomparable pair which have an infimum in the recursively enumerable degrees.

The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees are isomorphic. In the second (...) part of the paper we study the jumps of the ω-enumeration degrees below . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω-enumeration degrees below which are high n for some n and of the class L of the ω-enumeration degrees below which are low n for some n. (shrink)

We present a necessary and sufficient condition for the embeddability of a principally decomposable finite lattice into the computably enumerable degrees. This improves a previous result which required that, in addition, the lattice be ranked. The same condition is also necessary and sufficient for a finite lattice to be embeddable below every non-zero computably enumerable degree.

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, the first-order theory (...) of the closed Muchnik degrees, the first-order theory of the compact Muchnik degrees, and the second-order theory of true arithmetic. Our coding methods also prove that neither the closed Medvedev degrees nor the compact Medvedev degrees are elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees. (shrink)

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the [Formula: see text] relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the [Formula: see text] relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low 1 is parameter definable, and we provide methods that lead to a new example of a ∅-definable ideal. Moreover, we (...) prove that automorphisms restricted to intervals [d, 1], d ≠ 0, are [Formula: see text]. We also show that, for each c ≠ 0, can be interpreted in [0, c] without parameters. (shrink)

Metaphysical emergence has often been used to help understand the relationship between the entities of physics and the entities of the special sciences. What are the prospects of using metaphysical emergence within physics, to help understand the relationship between three-dimensional physical entities, and the non-three-dimensional entities that have been recently posited in certain interpretations of quantum mechanics and quantum gravity? This paper explores Jessica Wilson’s (2021) analysis of certain cases of metaphysical emergence in terms of degrees of freedom and (...) raises several questions that need to be answered in order to better understand whether this analysis can be used to handle cases of metaphysical emergence within physics. (shrink)

We show that the theory of the local structure of the enumeration degrees is computably isomorphic to the theory of first order arithmetic. We introduce a novel coding method, using the notion of a K-pair, to code a large class of countable relations.

We complete a study of the splitting/non-splitting properties of the enumeration degrees below by proving an analog of Harrington’s non-splitting theorem for the enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.

Plausibility models are Kripke models that agents use to reason about knowledge and belief, both of themselves and of each other. Such models are used to interpret the notions of conditional belief, degrees of belief, and safe belief. The logic of conditional belief contains that modality and also the knowledge modality, and similarly for the logic of degrees of belief and the logic of safe belief. With respect to these logics, plausibility models may contain too much information. A (...) proper notion of bisimulation is required that characterises them. We define that notion of bisimulation and prove the required characterisations: on the class of image-finite and preimage-finite models, two pointed Kripke models are modally equivalent in either of the three logics, if and only if they are bisimilar. As a result, the information content of such a model can be similarly expressed in the logic of conditional belief, or the logic of degrees of belief, or that of safe belief. This, we found a surprising result. Still, that does not mean that the logics are equally expressive: the logics of conditional and degrees of belief are incomparable, the logics of degrees of belief and safe belief are incomparable, while the logic of safe belief is more expressive than the logic of conditional belief. In view of the result on bisimulation characterisation, this is an equally surprising result. We hope our insights may contribute to the growing community of formal epistemology and on the relation between qualitative and quantitative modelling. (shrink)

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals (...) associated with the jump operator satisfy a Spector criterion, and correspond to the L ζ -stables. It also implies that the machines devised are "Σ 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)

The "Dutch Book" argument, tracing back to Ramsey and to deFinetti, offers prudential grounds for action in conformity with personal probability. Under several structural assumptions about combinations of stakes, your betting policy is coherent only if your fair odds are probabilities. The central question posed here is the following one: Besides providing an operational test of coherent betting, does the "Book" argument also provide for adequate measurement of the agents degrees of beliefs? That is, are an agent's fair odds (...) also his/her personal probabilities for those events? We argue the answer is "No!" The problem is caused by the possibility of state dependent utilities. (shrink)

We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half a nontrivial relativized K-pair. This leads to a structural dichotomy in the enumeration degrees.

We show that the partial order of Σ0 3-sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

§1. Introduction. Natural sets that can be enumerated by a computable function always seem to be either actually computable or of the same complexity as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K?Let be the r.e. degrees, i.e., the r.e. sets (...) modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order with least element 0, the degree of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of.The solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved:Theorem 1.1. Every countable partial ordering or even uppersemilattice can be embedded into.Theorem 1.2. For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatb ∨ c = a.Theorem 1.3. For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b. (shrink)

Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 3–37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results (...) for any ideal of the r.e. degrees. (shrink)

Plausibility models are Kripke models that agents use to reason about knowledge and belief, both of themselves and of each other. Such models are used to interpret the notions of conditional belief, degrees of belief, and safe belief. The logic of conditional belief contains that modality and also the knowledge modality, and similarly for the logic of degrees of belief and the logic of safe belief. With respect to these logics, plausibility models may contain too much information. A (...) proper notion of bisimulation is required that characterises them. We define that notion of bisimulation and prove the required characterisations: on the class of image-finite and preimage-finite models, two pointed Kripke models are modally equivalent in either of the three logics, if and only if they are bisimilar. As a result, the information content of such a model can be similarly expressed in the logic of conditional belief, or the logic of degrees of belief, or that of safe belief. This, we found a surprising result. Still, that does not mean that the logics are equally expressive: the logics of conditional and degrees of belief are incomparable, the logics of degrees of belief and safe belief are incomparable, while the logic of safe belief is more expressive than the logic of conditional belief. In view of the result on bisimulation characterisation, this is an equally surprising result. We hope our insights may contribute to the growing community of formal epistemology and on the relation between qualitative and quantitative modelling. (shrink)

Cognition means different things to different psychologists depending on the position held on the mind-matter problem. Ecological psychologists reject the implied mind-matter dualism as an ill-posed theoretic problem because the assumed mind-matter incommensurability precludes a solution to the degrees of freedom problem. This fundamental problem was posed by both Nicolai Bernstein and James J. Gibson independently. It replaces mind-matter dualism with animal-environment duality -- a better posed scientific problem because commensurability is assured. Furthermore, when properly posed this way, a (...) conservation law is suggested that encompasses a psychology of transactional systems, a biology of self-actional systems, and a physics of interactional systems. For such a solution, a theory of cognition for goal-directed behaviour is needed. A sketch is supplied for how such a theory might be pursued in the spirit of the new physics of evolving complex systems. (shrink)

Pode o conhecimento de uma dada verdade admitir gradações? Sim, de fato, segundo o gradualismo deste artigo. O artigo introduz o conceito do saber-como que p – isto é, o conceito de saber como é que p. Saber-como que p é claramente gradual – admitindo gradações, dado que se pode saber mais ou menos como é que p. E a vinculação que este artigo faz entre sabercomo que p e saber que p revela que este último tipo de conhecimento também (...) é gradual. A teoria dos criadores-deverdade [truthmakers] é, então, invocada para enriquecer a análise, e a análise é aplicada às tarefas de fazer sentido do fundacionismo e de barrar uma certa forma de ceticismo. PALAVRAS-CHAVE – Conhecimento. Gradualismo. Absolutismo. Ceticismo. Fundacionismo, Criadores-de-verdade. BSTRACT Can knowledge of a particular truth admit of degrees? Indeed so, according to this paper’s gradualism. The paper introduces the concept of how-knowledge that p – that is, the concept of knowing how it is that p. Howknowledge that p is clearly gradational – admitting of degrees, as one can know more or less of how it is that p. And this paper’s linking of how-knowledge that p with knowledge that p reveals the latter kind of knowledge, too, to be gradational. Truthmaker theory is then called upon to enrich the analysis, and the analysis is applied to the tasks of understanding foundationalism and of thwarting one form of scepticism. KEY WORDS – Knowledge. Gradualism. Absolutism. Scepticism. Foundationalism. Truthmakers. (shrink)

We show that the elementary theory of the recursively enumerable tt-degrees has the same computational complexity as true first-order arithmetic. As auxiliary results, we prove theorems about exact pairs and initial segments in the tt-degrees.