According to current scholarly consensus, the pre- and post-exilic strata of Biblical Hebrew differ sufficiently to allow for the relative dating of biblical texts on linguistic grounds. Challengers to this view have objected that the received orthography of the Hebrew Bible, which is fuller than that of any pre-exilic epigraphic source, shows that no pre-exilic biblical text escaped post-exilic spelling revision. Moreover, so it is claimed, susceptibility to scribal modification on the level of orthography implies susceptibility to scribal modification on (...) the higher linguistic levels as well, which would effectively obscure a text’s original linguistic profile. In the present article it is argued that the degree of scribal modification involved in the process of orthographical revision was not so great as to distort irremediably the picture of linguistic development seen in biblical literature, so that distinctively late linguistic features can still be distinguished from their classical counterparts. Significantly, as has been argued elsewhere, diachronic development is still discernible even in the domain of orthography, a fact demonstrated on the basis of several examples, including the spelling of the qṭol-pattern qal infinitive construct. (shrink)

This article gives insight into the world of 3rd century BCE Alexandrian Judaism by analysing one aspect of the Greek translation of Exodus and provides a detailed evaluation of the way the translator managed to express the essence of the Hebrew text of Exodus while reflecting to some degree the form of the Hebrew text. No previous study only analyses this translator’s treatment of Hebrew paronymous infinitive absolute constructions in Greek Exodus. This research contributes to the preparation of (...) a commentary on Greek Exodus in the Society of Biblical Literature Septuagint Commentary Series. Using the Göttingen text of Greek Exodus prepared by John William Wevers, it evaluates how this translator rendered each occurrence of a paronymous infinitive absolute construction in Hebrew Exodus, defining the primary modes employed and also seeking to explain variations from these norms. The primary method incorporates a close exegetical reading of both the Hebrew and Greek texts and comparing their texts, in order to discern how the translator treated this Hebrew idiom and illustrating his translation approach.Contribution: The results contribute to our understanding of how the translator of Greek Exodus approached his task. He was more concerned to express the sense of the Hebrew text, than producing a literal translation. He generally follows the Hebrew text’s word order, but is creative in his choice of Greek equivalents, often showing sensitivity to context and Greek idiom. This translation approach differentiates this translator’s approach somewhat from those responsible for other portions of the Pentateuch. (shrink)

No doubt my earlier paper has struck a sensitive nerve among existing and prospective constructive empiricists – hence their united reply.1 I shall, for brevity, introduce an imaginary single author of their critique and call him CE. In this rejoinder, I try to show, first, that CE’s counter-arguments do not refute my original arguments; and second, that a claim of CE’s paper is very close to the conclusion of my original paper. A central point of my original piece was that (...) there is a symmetry between scientific realism and constructive empiricism vis à vis van Fraassen’s arguments from the bad lot and from indifference. Scholastic charges of an ‘apparent misunderstanding’ of ‘empirical adequacy’ do not cast any light on the issues at stake. (However, the notion of empirical adequacy I employed, p. 41, is the standard one: ‘a theory is empirically adequate if and only if it saves all phenomena, past, present and future, and squares with all actual and possible observations’.) The issue between CE and myself is more substantive: CE relies on the thesis that for any theory there are ‘indefinitely many empirically equivalent rivals’ (p. 307), in order to infer that there are infinitely many empirically adequate theories, and then to argue that if realists.. (shrink)

Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only a finite amount (...) of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point but “in” the point. We are interested in our research in the diagrams (...) which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role, and an unveiling role. In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

This paper examines the suggestion that infinitivaltoomission errors in English-speaking children can result from competition between two constructions (Kirjavainen et al., First Language 29: 313–339, 2009a). Kirjavainen et al. suggested that the acquisition of two (or more) constructions (e.g., WANT-X and WANT-to) for verbs takingto-infinitival complement clauses can lead to infinitivaltoomissions, reflecting the relative frequencies of the constructions in the input. In the present study we analysed 13 English children's corpora to determine whether the presence of a (...) variety of utterance types in the immediate discourse context preceding WANT-to-VP (e.g.,I want to eat it) and erroneous *WANT-zero-VP (e.g., *I want__drink it) constructions was associated with infinitivaltoproduction/omission. This was done separately for the children's own and their interlocutors' discourse utterances. The data show that the occurrence of WANT-toand WANT-X constructions in the prior discourse was associated with differing proportions of infinitivaltoprovision in the WANT-to/zero-VP construction. This together with Kirjavainen et al.'s (First Language 29: 313–339, 2009a) data suggests that children are learning at least two constructions for the verb WANT, that competition contributes to infinitivaltoomissions, and that the strength of competing representations is affected by overall input frequencies and the preceding discourse. (shrink)

An experimental study was conducted on children aged 2;6–3;0 and 3;6–4;0 investigating the priming effect of two WANT-constructions to establish whether constructional competition contributes to English-speaking children's infinitival to omission errors. In two between-participant groups, children either just heard or heard and repeated WANT-to, WANT-X, and control prime sentences after which to-infinitival constructions were elicited. We found that both age groups were primed, but in different ways. In the 2;6–3;0 year olds, WANT-to primes facilitated the provision of to (...) in target utterances relative to the control contexts, but no significant effect was found for WANT-X primes. In the 3;6–4;0 year olds, both WANT-to and WANT-X primes showed a priming effect, namely WANT-to primes facilitated and WANT-X primes inhibited provision of to. We argue that these effects reflect developmental differences in the level of proficiency in and preference for the two constructions, and they are broadly consistent with “priming as implicit learning” accounts. The current study shows that children as young as 2;6–3;0 years of age can be primed when they have only heard particular constructions, children are acquiring at least two constructions for the matrix verb WANT, and that these two WANT-constructions compete for production. (shrink)

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon and Rabin and Rabern.

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the Ellentuck (...) space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

We describe in some detail how to build an infinite computing machine within a continuous Newtonian universe. The relevance of our construction to the Church-Turing thesis and the Platonist-Intuitionist debate about the nature of mathematics is also discussed.

Infinite idealizations appear in our best scientific explanations of phase transitions. This is thought by some to be paradoxical. In this paper I connect the existing literature on the phase transition paradox to work on the concept of indispensability, which arises in discussions of realism and anti-realism within the philosophy of science and the philosophy of mathematics. I formulate a version of the phase transition paradox based on the idea that infinite idealizations are explanatorily indispensable to our best science, and (...) so ought to attract a realist attitude. I go on to offer a solution to the paradox by drawing a distinction between two types of indispensability: constructive and substantive indispensability. I argue that infinite idealizations are constructively indispensable to explanations of phase transitions, but not substantively indispensable. This helps to resolve the paradox, I maintain, since realist commitment tracks substantive, and not constructive, indispensability. (shrink)

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I (...) argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. -/- (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, is unjustified, in general. This means my position is stronger against this objection than the paper suggests, since omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences.). (shrink)

According to a standard interpretation of Hume’s argument against infinite divisibility, Hume is raising a purely formal problem for mathematical constructions of infinite divisibility, divorced from all thought of the stuffing or filling of actual physical continua. I resist this. Hume’s argument must be understood in the context of a popular early modern account of the metaphysical status of the parts of physical quantities. This interpretation disarms the standard mathematical objections to Hume’s reasoning; I also defend it on textual (...) and contextual grounds. (shrink)

This article presents an analysis of the structure and use of the Finnish independent infinitives. Although typological studies have shown that syntactically independent non-finite constructions are widespread in many languages, the understanding of their semantic and intersubjective motivation is still in its early stages. The current paper aims to enrich the understanding of independent non-finite constructions by closely looking at free-standing infinitive constructions in spoken and written Finnish: it combines theoretical concepts of Cognitive Grammar with the (...) methodological tools of Interactional Linguistics to explore the nature of independent infinitives as a resource for conceptualization and the intersubjective functions that it affords. The paper suggests that the fact that independent infinitives are grammatically ungrounded makes them useful in interactional and textual sequences involving affect display. As the indexical functions of infinitives can be explained from their own morphosyntactic and semantic characteristics, the paper makes the more general claim that there is no synchronic evidence that would support the assumption that such constructions ever evolved, via ellipsis, from finite constructions. Methodologically and theoretically, the paper advocates an approach that takes into account both the social and cognitive nature of language, and promotes the view that Cognitive Grammar offers a flexible, semantically rich starting point for the description of intersubjective meanings conveyed by grammar, when combined with the context-sensitive and microanalytical methodology of Interactional Linguistics. (shrink)

We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and (...) resetting infinite-time register machines, and α-Turing machines for countable admissible ordinals α. (shrink)

One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large-small filters requires (...) a nonconstructive axiom like AC. (shrink)

Infinitely many intermediate propositional logics with the disjunction property are defined, each logic being characterized both in terms of a finite axiomatization and in terms of a Kripke semantics with the finite model property. The completeness theorems are used to prove that any two logics are constructively incompatible. As a consequence, one deduces that there are infinitely many maximal intermediate propositional logics with the disjunction property.

Draft for presentation at the 14th International Kant-Congress, September 2024. -/- Abstract: Kant claims we intuit infinite space. There’s a problem: Kant thinks full awareness of infinite space requires synthesis—the act of putting representations together and comprehending them as one. But our ability to synthesize is finite. Tobias Rosefeldt has argued in a recent paper that Kant’s notion of decomposing synthesis offers a solution. This talk criticizes Rosefeldt’s approach. First, Rosefeldt is committed to nonconceptual yet determinate awareness of (potentially) infinite (...) space, but I argue that such awareness is (1) prima facie implausible, as well as contravened by textual evidence from Kant’s (2a) definitions of ‘infinity’ and (2b) account of geometrical construction. Second, scrutiny of how Kant thinks awareness of potential infinity is afforded by geometrical construction (e.g., extending a line segment ad indefinitum) indicates no essential role for decomposing synthesis. Familiar synthesis from parts to wholes is sufficient. (shrink)

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The (...) technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy ), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \ from the point of view of the research area in question is its game-theoretic character. (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)

In his interesting paper on babeo and aueo published in CQ 66 , 388–98, Dr. A.S. Gratwick raised a number of questions bearing on my own discussion of the origin and development of the babeo+infinitive construction in CQ 65 , 215–31. First the collapse of the earlier future-tense system. As I said, this was ‘the product of a number of different linguistic events’, phonetic, grammatical, and semantic, which were summarized and illustrated on pp. 220–1 of my paper. Even so (...) Dr. Gratwick believes that I assigned too much weight to ‘phonetic attrition of the future simple’. (shrink)

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our (...) main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (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 $\zeta$, the least ordinal not the length of any eventual output of an Infinite Time Turing machine ; 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$_\zeta$-stables. It also implies that the machines devised are "$\Sigma_2$ 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)

Many cosmological arguments for the existence of a first cause or a necessary being rely on a premise which denies the possibility of an infinite regress ofsome particular sort. Adequate and satisfying support for this premise, however, is not always provided. In this paper I attempt to address this gap in the literature. After discussing the notion of a causal explanation (section I), I formulate three principles which govern any successful causal explanation (section II). I then introduce the notions of (...) a caused being, a causal network, and a causal chain, and argue that (roughly) an infinite causal chain cannot be explained merely by reference to the causal activities of the members of that chain (section III). In a sequel to the present paper, I employ this result to construct two closely related arguments for the existence of a necessary being. (shrink)

In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They illustrate (...) this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (shrink)

In 1993, the American logic S. Yablo was proposed an original infinitive formulation of the classical ≪Liar≫ paradox. It questioned the traditional notion of self-reference as the basic structure for semantic paradoxes. The article considers the arguments underlying two different approaches to analysis of proposals of the ≪Infinite Liar≫ and understanding of the genuine sources for semantic paradoxes. The first approach (V. Valpola, G.-H. von Wright, T. Bolander, etc.) imposes responsibility for the emergence of semantic paradoxes on the negation (...) of the truth predicate. It deprives the ≪Infinite Liar≫ sentences of consistent truth values. The second approach is based on a modified version of anaphoric prosententialism (D. Grover, R. Brandom, etc.). The concepts of truth and falsehood are treated as special anaphoric operators. Logical constructs similar to the ≪Infinite Liar≫ do not attribute any definite truth values to sentences from which they are composed, but only state certain types of relations between the semantic content of such sentences. (shrink)

We re-express a previous general result in a way which seems easier to remember, using the terminology of infinite games. We show how this can be applied to construct recursive linear orderings, showing, for example, that if there is a ▵ 0 2β + 1 linear ordering of type τ, then there is a recursive ordering of type ω β · τ.

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent. May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a (...) probability of a probability of a probability, and so on, {\em ad infinitum}? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines. (shrink)

Is it possible to venture beyond daily living and experience heightened states of awareness? Deepak Chopra says that higher consciousness is available here and now. “Metahuman helps us harvest peak experiences so we can see our truth and mold the universe’s chaos into a form that brings light to the world.”—Dr. Mehmet Oz, attending physician, New York–Presbyterian, Columbia University New York Times bestselling author Deepak Chopra unlocks the secrets to moving beyond our present limitations to access a field of infinite (...) possibilities. How does one do this? By becoming metahuman. To be metahuman, however, isn’t science fiction and is certainly not about being a superhero. To be metahuman means to move past the limitation constructed by the mind and enter a new state of awareness where we have deliberate and concrete access to peak experiences that can transform people’s lives from the inside out. Humans do this naturally—to a point. For centuries the great artists, scientists, writers, and many so-called ordinary people have gone beyond the everyday physical world. But if we could channel these often bewildering experiences, what would happen? Chopra argues we would wake up to experiences that would blow open your body, mind, and soul. Metahuman invites the reader to walk the path here and now. Waking up, we learn, isn’t just about mindfulness or meditation. Waking up, to become metahuman, is to expand our consciousness in all that we think, say, and do. By going beyond, we liberate ourselves from old conditioning and all the mental constructs that underlie anxiety, tension, and ego-driven demands. Waking up allows life to make sense as never before. To make this as practical as possible, Chopra ends the book with a 31-day guide to becoming metahuman. Once you wake up, he writes, life becomes transformed, because pure consciousness—which is the field of all possibilities—dawns in your life. Only then does your infinite potential become your personal reality. (shrink)

Given the old conception of the relation greater than, the proposition that the whole is greater than the part is an immediate consequence. But being greater in this sense is not incompatible with being equal in the sense of one-one correspondence. Some who failed to recognize this formulated invalid arguments against the possibility of infinite quantities. Others who did realize that the relations of equal and greater when so defined are compatible, concluded that such relations are not appropriately taken as (...) quantitative relations, at least, not in general. If suitable quantitative relations were to be defined, there was a possibility of retaining either the traditional definition of greater and finding another concept of equality, or of retaining the concept of equality as one-one correspondence and defining greater than so that it is incompatible with equality in this sense. The former alternative was implicitly taken by Bolzano, who never succeeded in defining suitable quantitative relations. The latter alternative was taken by Cantor, and is the basis of his great success in constructing a mathematical theory of the transfinite. (shrink)

Constructive (productive) thinking in the critical philosophy of Hermann Cohen differs significantly from the seemingly similar speculative thinking in J. G. Fichte’s Science of Knowledge (Wissenschaftslehre) (1794/95). The fundamental characteristics of scientific thinking in Cohen’s teaching include: purity, focus on the “fact of science”, the origin (Ursprung), the infinitesimal method, continuity, movement, production, correlation, intensive magnitude, interrelation of thinking and being. According to Cohen, scientific thinking can only be pure and generated by the origin. The origin is continuous action (movement) (...) of thinking to separate the united and bind the divided content. In this process, thinking and being are correlative. Infinitely small reality contains thinking and being simultaneously as a union and in a divided form. The infinitesimal method is thinking that continuously carries out a) the operations of opposing itself as pure thinking to the results of its own production; b) the coincidence of itself with the products of its own generation. Infinitesimal thinking seeks to eliminate the difference between self and being. Nevertheless, being constantly retains autonomy. Being does not merge with thinking and is not absorbed by it. In Fichte’s Science of Knowledge pure thinking moves in a logical circle, having no access into real being. In an effort to break out of this circle and find its causality, thinking turns not to the being of the world of phenomena, but to the Absolute I. Such a speculative approach to consciousness, thinking and being has little in common with Cohen’s critical position. (shrink)

The projective plane of Baldwin 695) is model complete in a language with additional constant symbols. The infinite rank bicolored field of Poizat 1339) is not model complete. The finite rank bicolored fields of Baldwin and Holland 371; Notre Dame J. Formal Logic , to appear) are model complete. More generally, the finite rank expansions of a strongly minimal set obtained by adding a ‘random’ unary predicate are almost strongly minimal and model complete provided the strongly minimal set is ‘well-behaved’ (...) and admits ‘exactly rank k formulas’. The last notion is a geometric condition on strongly minimal sets formalized in this paper. (shrink)

This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed ‘niceness’ by the mathematician Michiel Hazewinkel, a concept we compare to the ‘robustness’ described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but (...) rather whether the choice of our concepts is forced upon us. (shrink)

The project of constructing a logic of scientific inference on the basis of mathematical probability theory was first undertaken in a systematic way by the mid-nineteenth-century British logicians Augustus De Morgan, George Boole and William Stanley Jevons. This paper sketches the origins and motivation of that effort, the emergence of the inverse probability (IP) model of theory assessment, and the vicissitudes which that model suffered at the hands of its critics. Particular emphasis is given to the influence which competing interpretations (...) of probability had on the project, and to the role of the 'lottery' or 'ballot box' metaphor in the philosophical imagination of the proponents of the IP model. (shrink)

Infinity, in various guises, has been invoked recently in order to ‘explain’ a number of important questions regarding observable phenomena in science, and in particular in cosmology. Such explanations are by their nature speculative. Here we introduce the notions of relative infinity, closure, and economy of explanation and ask: to what extent explanations involving relative or real constructed infinities can be treated as reasonable?

We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice.

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only use (...) arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction. The results are of interest in their own right, (...) but also, we factor the mathematics so as to maximise the chances that it could be used off-the-shelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semi-automatic theorem to transfer results from classes of infinitely-supported nominal models to the more restricted class of models with finite support. In conclusion, we consider different permissive-nominal syntaxes and nominal models and discuss how they relate to the results proved here. (shrink)

Donald Coxeter died in 2003, at a ripe old age of 96. Though I had regularly seen him at mathematics talks in Toronto for over twenty years, I never felt rushed to seek him out. It seemed he would go on forever. His death left me regretting my missed opportunity and Siobhan Robert's excellent book makes me regret it even more. Like any good biography of an intellectual, King of Infinite Space contains personal details and mathematical achievements in some detail. (...) Thus, we learn of the traumatic effects on Coxeter of his parents' divorce, his search for a spouse, his vegetarianism, and his progressive politics. We also learn a fair bit about the kaleidoscopes he made while in Cambridge during his student days in order to study the symmetry properties of polyhedra. These involved mirrors that Coxeter had specially constructed for this purpose. Along the way we are treated to interesting tidbits, such as G.H. Hardy's detestation of mirrors . There are brief accounts of the brilliant notation Coxeter invented, known as Coxeter diagrams, and of course, the now famous Coxeter groups. Short appendices fill in a bit more detail. It is all very well done and thoroughly engrossing.Wittgenstein befriended Coxeter, who was part of the very small group to whom …. (shrink)

Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...) geometrical spaces is a proper subset of metaphysical space, thus, metaphysical space is infinite. Kant's paradoxical doctrine of metaphysical space is necessary to reconcile his empiricism with his transcendental idealism. (shrink)

Continuing the paper [7], in which the Blum-Shub-Smale approach to computability over the reals has been generalized to arbitrary algebraic structures, this paper deals with computability and recognizability over structures of infinite signature. It begins with discussing related properties of the linear and scalar real structures and of their discrete counterparts over the natural numbers. Then the existence of universal functions is shown to be equivalent to the effective encodability of the underlying structure. Such structures even have universal functions satisfying (...) the s-m-n theorem and related features. The real and discrete examples are discussed with respect to effective encodability. Megiddo structures and computational extensions of effectively encodable structures are encodable, too. As further variants of universality, universal functions with enumerable sets of program codes and such ones with constructible codes are investigated. Finally, the existence of m-complete sets is shown to be independent of the effective encodability of structures, and the linear and scalar structures are discussed once more, under this aspect. (shrink)

Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use the (...) method to 1. give a new proof of a well-known trade-off theorem , which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. construct a continuous normalization operator with an explicit modulus of continuity. (shrink)

We study in a constructive manner some problems of topology related to the set Irr of irrational reals. The constructive approach requires a strong notion of an irrational number; constructively, a real number is irrational if it is clearly different from any rational number. We show that the set Irr is one-to-one with the set Dfc of infinite developments in continued fraction . We define two extensions of Irr, one, called Dfc1, is the set of dfc of rationals and irrationals (...) preserving for each rational one dfc, the other, called Dfc2, is the set of dfc of rationals and irrationals preserving for each rational its two dfc. We introduce six natural distances over Irr wich we denote by dfc0, dfc1, dfc2, d, dmir and dcut. We show that only the four distances dfco, dfc1, d and dmir among the six make Irr a complete metric space. The last distances define in Irr the same topology in a constructive sens. We study further the set Dfc1 in which we show that the irrationals constitute a closed subset. Finally, we make a particular study of the completion Dfc2 of Dfc for the two equivalent metrics dfc2 and dcut. (shrink)

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences. In what follows, (...) we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general. (shrink)

This paper investigates frequency effects in the L2 acquisition of the catenative verb construction by German learners of English from a usage-based perspective by presenting findings from two experimental studies and a complementary corpus study. It was examined if and to what extent the frequency of the verb in the catenative verb construction affects the choice of the target-like complement type and if the catenative verb construction with a to-infinitive complement, which is highly frequent in English, is more accurately (...) acquired and entrenched than the less frequent variant with an -ing complement. In all three studies, the more frequent construction with a to-infinitive yielded higher numbers of target-like complement choices. Furthermore, it was shown that the verb’s faithfulness to the construction made a significant prediction of a target-like complement preference. It is argued that a higher faithfulness promotes a target-like entrenchment of the construction and motivates a taxonomic generalisation across related exemplars. Furthermore, the results provide support for the idea that the mental representation of language is comprised of item-specific as well as more abstract schema knowledge, where frequency determines the specificity with which the construction is entrenched. (shrink)

For any Boolean Algebra A, let cmm be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, and the construction of an (...) interval algebra in which every well-ordered chain has size less than cmm. (shrink)