In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. Ulrich Kohlenbach introduced abstract types for certain structures such as metric spaces, normed spaces, Hilbert spaces, etc. With these types, the elements of the spaces are given directly, not through the mediation of a representation. However, these abstract spaces presuppose the real numbers. In this paper, we show how to set up an abstract type for the real numbers. The appropriateness of (...) our construction works in tandem with the bounded functional interpretation. (shrink)

It has been known for long time that most of the existing cosmology models have singularity problem. Cosmological singularity has been a consequence of excessive symmetry of flow, such as “Hubble’s law”. More realistic one is suggested, based on Newtonian cosmology model but here we include the vertical-rotational effect of the whole Universe. We review a Riccati-type equation obtained by Nurgaliev, and solve the equation numerically with Mathematica. It is our hope that the new proposed method can be verified (...) with observation data. (shrink)

This is a book about sensory states and their apparent characteristics. It confronts a whole series of metaphysical and epistemological questions and presents an argument for type materialism: the view that sensory states are identical with the neural states with which they are correlated. According to type materialism, sensations are only possessed by human beings and members of related biological species; silicon-based androids cannot have sensations. The author rebuts several other rival theories, and explores a number of important (...) issues: the forms and limits of introspective awareness of sensations, the semantic properties of sensory concepts, knowledge of other minds, and unity of consciousness. The book is a significant contribution to the philosophy of mind, and has much to say to psychologists and cognitive scientists. (shrink)

Many philosophers believe in things, propositions, which are the things that we believe, assert etc., and which are the contents of sentences. The act-type theory of propositions is an attempt to say what propositions are, to explain how we stand in relations to them, and to explain why they are true or false. The core idea of the act-type theory is that propositions are types of acts of predication. The theory is developed in various ways to offer explanations (...) of the important properties of propositions. I present the core idea of the theory, and some developments of it. I discuss the relationship between the theory and the content–force distinction. I also present an important type of objection that has been raised to the explanations offered by the act-type theory. (shrink)

This paper discusses some of the ways in which Martin-Löf type theory differs from set theory. The discussion concentrates on conceptual, rather than technical, differences. It revolves around four topics: sets versus types; syntax; functions; and identity. The difference between sets and types is spelt out as the difference between unified pluralities and kinds, or sorts. A detailed comparison is then offered of the syntax of the two languages. Emphasis is put on the distinction between proposition and judgement, drawn (...) by type theory, but not by set theory. Unlike set theory, type theory treats the notion of function as primitive. It is shown that certain inconveniences pertaining to function application that afflicts the set-theoretical account of functions are thus avoided. Finally, the distinction, drawn in type theory, between judgemental and propositional identity is discussed. It is argued that the criterion of identity for a domain cannot be formulated in terms of propositional identity. It follows that the axiom of extensionality cannot be taken as a statement of the criterion of identity for sets. (shrink)

In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as predicates of entities as in Montague’s semantics. This brings about important advantages in linguistic interpretations but also leads to a limitation of expressive power because there are fewer operations on types as compared with those on predicates. The theory of coercive subtyping adequately extends the modern type theories and, as shown in this paper, plays a very useful role in making (...) type theories more expressive for formal semantics. It not only gives a satisfactory solution to the basic problem of ‘multiple categorisation’ caused by interpreting common nouns as types, but provides a powerful formal framework to model interesting linguistic phenomena such as copredication, whose formal treatment has been found difficult in a Montagovian setting. In particular, we show how to formally introduce dot-types in a type theory with coercive subtyping and study some type-theoretic constructs that provide useful representational tools for reference transfers and multiple word meanings in formal lexical semantics. (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 logic of identity contains riches not seen through the coarse lens of predicate logic. This is one of several lessons to draw from the subtle treatment of identity in Martin‐Löf type theory, to which the reader will be introduced in this article. After a brief general introduction we shall mainly be concerned with the distinction between identity propositions and identity judgements. These differ from each other both in logical form and in logical strength. Along the way, connections to (...) philosophical debates concerning identity are noted. Some use of logical symbolism is inevitable in any serious discussion of type theory, but the emphasis here is on basic ideas rather than technicalities. (shrink)

ABSTRACT Backwards-looking operators Saarinen, E. [1979. “Backwards-Looking Operators in Tense Logic and in Natural Language.” In Essays on Mathematical and Philosophical Logic, edited by J. Hintikka, I. Niiniluoto, and E. Saarinen, 341–367. Dordrecht: Reidel] that have the material in their scope depend on higher intensional operators, are known to increase the expressivity of some intensional languages and have thus played a central role in debates about approaches to intensionality in terms of implicit parameters vs. variables explicitly quantifying over them. The (...) current contribution takes a look at these operators from a type-logical perspective. It is shown that extending Gallin's translation from intensional type logic to two-sorted type theory so as to include a version of Yanovich's backwards-looking operators, does not increase the expressive power of formulae with exclusively intensional parameters. The result, which makes use of a theorem by Zimmermann : 65–77]), is illustrated by pertinent examples from the literature. The paper closes by indicating alternative strategies of incorporating backwards-looking operators. (shrink)

Type specimens are used to designate species. What is the nature of the relation between a type specimen and the species it designates? If species names are rigid designators, and type specimens ostensively define species, then that relation is, at the very least, a close one. Levine :325–338, 2001) argues that the relationship of type specimen to a named species is one of necessity—and that this presents problems for the individuality thesis. Namely, it seems odd that (...) a contingently selected specimen should belong to a species of necessity. In considering Levine’s argument, LaPorte suggests that recognizing the distinction between de re and de dicto necessity resolves Levine’s worries. I reconsider the motivating question: does a type specimen belong of necessity to the species that it designates? In light of taxonomic cases and practice the answer is clear: definitively not. This is particularly clear in the case of re-designation of types by taxonomic decree. I explain how this helps reveal how taxonomists prioritize competing theoretical commitments, and offer a defense of the individuality thesis as applied to these particular cases. In short, I demonstrate how to misidentify a type specimen. (shrink)

Homotopy Type Theory is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path induction, (...) motivated from pre-mathematical considerations, without recourse to homotopy theory. (shrink)

In recent years, several formalisms for program construction have appeared. One such formalism is the type theory developed by Per Martin-L f. Well suited as a theory for program construction, it makes possible the expression of both specifications and programs within the same formalism. Furthermore, the proof rules can be used to derive a correct program from a specification as well as to verify that a given program has a certain property. This book contains a thorough introduction to (...) class='Hi'>type theory, with information on polymorphic sets, subsets, monomorphic sets, and a full set of helpful examples. (shrink)

Biological taxonomists rely on the so-called ‘type method’ to regulate taxonomic nomenclature. For each newfound taxon, they lay down a ‘type specimen’ that carries with it the name of the taxon it belongs to. Even if a taxon’s circumscription is unknown and/or subject to change, it remains a necessary truth that the taxon’s type specimen falls within its boundaries. Philosophers have noted some time ago that this naming practice is in line with the causal theory of reference (...) and its central notion of rigid designation: a type specimen fixes the reference of a taxon name without defining it. Recently, however, this consensus has come under pressure in the pages of this journal. In a series of articles by Alex Levine, Joseph LaPorte, and Matthew Haber, it has been argued that type specimens belong only contingently to their species, and that this may bode problems for the relation between type method and causal theory. I will argue that this ‘contingency debate’ is a debate gone wrong, and that none of the arguments in defense of contingency withstand scrutiny. Taxonomic naming is not out of step with the causal theory, but conforms to it. However, I will also argue that this observation is itself in need of further explanation, since application of the type method in taxonomic practice is plagued by errors and ambiguities that threaten it with breaking down. Thus, the real question becomes why taxonomic naming conforms to the causal theory in the first place. I will show that the answer lies in the embedding of the type method into elaborate codes of nomenclature. (shrink)

In section 24 of The Antichrist, Nietzsche notes a problem namely “the origin of Christianity.” He offers two propositions toward its solution: the first is that “Christianity can only be understood on the soil where it grew:” and the second is that “the psychological type of the Galilean is still recognizable, but it had to assume a completely degenerate form (simultaneously mutilated and full of alien features) before it came to be used as a redeemer of humanity” (A 24). (...) Significantly this passage suggests that the origins of Christianity rest on a reinterpretation of the type of the redeemer. This paper seeks to clarify the nature of such a modification and to identify some of its key ramifications. After clarifying the type, the paper argues that the type, thus understood, serves as a link between the texts On the Genealogy of Morality and The Antichrist and, as such, reveals the connection between Nietzsche's genealogical methods and the wider project of reevaluation. Though this reading is not the standard interpretive strategy, the paper argues that it is the strategy that Nietzsche himself recommends. (shrink)

We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.

Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics (...) might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

‘Type’ in biology is a polysemous term. In a landmark article, Paul Farber (Journal of the History of Biology 9(1): 93–119, 1976) argued that this deceptively plain term had acquired three different meanings in early nineteenth century natural history alone. ‘Type’ was used in relation to three distinct type concepts, each of them associated with a different set of practices. Important as Farber’s analysis has been for the historiography of natural history, his account conceals an important dimension (...) of early nineteenth century ‘type talk.’ Farber’s taxonomy of type concepts passes over the fact that certain uses of ‘type’ began to take on a new meaning in this period. At the closing of the eighteenth century, terms like ‘type specimen,’ ‘type species,’ and ‘type genus’ were universally recognized as referring to typical, model members of their encompassing taxa. But in the course of the nineteenth century, the same terms were co-opted for a different purpose. As part of an effort to drive out nomenclatural synonymy – the confusing state of a taxon being known to different people by different names – these terms started to signify the fixed and potentially atypical name-bearing elements of taxa. A new type concept was born: the nomenclatural type. In this article, I retrace this perplexing nineteenth century shift in meaning of ‘type.’ I uncover the nomenclatural disorder that the new nomenclatural type concept dissolved, and expose the conceptual confusion it left in its tracks. What emerges is an account of how synonymy was suppressed through the coinage of a homonym. (shrink)

Mucopolysaccharidosis type I (MPS I) is a rare lysosomal storage disorder treated with bone marrow transplantation or enzyme replacement therapy with laronidase, a high-cost orphan drug. Laronidase was approved by the US Food and Drug Administration and the European Medicines Agency in 2003 and by the Brazilian National Health Surveillance Agency in 2005. Many Brazilian MPS I patients have been receiving laronidase despite the absence of a governmental policy regulating access to the drug. Epidemiological and treatment data concerning MPS (...) I are scarce. This study aims to present a demographic profile of Brazilian patients with MPS I, describe the routes of access to laronidase in Brazil, and discuss associated ethical issues relating to public funding of orphan drugs. (shrink)

A great deal of attention has been paid recently to the claim that traditional Frankfurt-type counterexamples to the Principle of Alternative Possibilities (PAP), which depend for their success on the presence of a perfectly reliable indicator (or prior sign ) of what an agent will freely do if left to act on his own, are guilty of begging the question against incompatibilists, since such indicators seem to presuppose a deterministic relation between an agent’s free action and its causal antecedents. (...) Objections of this sort have given rise to considerable efforts to construct alternative Frankfurt-type counterexamples that do not rely on prior signs of this kind and so do not presuppose determinism in a way that incompatibilists should find objectionable. One consequence of this shift in the way Frankfurt-type counterexamples are formulated is that it provides an opportunity for the forceful resurgence of certain versions of the so-called flicker defense of PAP. In this paper I develop two versions of the flicker defense, indicate their advantages over other versions of this strategy, and defend them against objections. Insofar as either of these is successful, it will show not only that PAP has yet to be falsified by any of the modified Frankfurt-type counterexamples currently on offer but that cases of this sort are in principle incapable of falsifying PAP. (shrink)

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the (...) normality of the associated measures. (shrink)

Girard and Reynolds independently invented System F to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking. Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has (...) been settled for restrictions and extensions of F and related systems and complexity lower-bounds have been determined for typability in F, but this report is the first to resolve whether these problems are decidable for System F. This report proves that type checking in F is undecidable, by a reduction from semi-unification, and that typability in F is undecidable, by a reduction from type checking. Because there is an easy reduction from typability to type checking, the two problems are equivalent. The reduction from type checking to typability uses a novel method of constructing lambda terms that simulate arbitrarily chosen type environments. All of the results also hold for the λI-calculus. (shrink)

In this paper I undertake to resolve a main pragmatic puzzle triggered by Bank-type cases. After accepting ‘sanitized’ intuitions about Truth-Values, as reflected in x-phi experiments, the pragmatic puzzle about whether the husband is inconsistent remains, and if he isn’t, which intuitively is the case, how are we to explain it. The context in such cases is pragmatic, with awareness of high risks, and the treatment I propose is pragmatic as well, but not Gricean. I offer a new Pragmatics (...) whose main tools are Steering Thrust and Posting. It focuses on what I call Action-Directed Pragmatics. An overall view of Pragmatics centers on the primary resort to Sayability rather than to Truth-Values, and to the Pragmatic phenomenon of co-present operative norm-types, which can conflict in some cases vis-à-vis particular actions and about what subjects can say. The very satisfactory explanation of the above puzzle that I offer is in such terms. In particular, attention is focused on the phenomenon of Negative-Polarity Pragmatic Functors. (shrink)

The type identity theory, according to which types of mental state are identical to types of physical state, fell out of favour for some years but is now being considered with renewed interest. Many philosophers are critically re-examining the arguments which were marshalled against it, finding in the type identity theory both resources to strengthen a comprehensive, physicalistic metaphysics and a useful tool in understanding the relationship between developments in psychology and new results in neuroscience. This volume brings (...) together leading philosophers of mind, whose essays challenge in new ways the standard objections to type identity theory, such as the multiple realizability objection and the modal argument. Other essays show how cognitive science and neuroscience are lending new support to type identity theory and still others provide, extend and improve traditional arguments concerning the theory's explanatory power. (shrink)

This paper contains a novel and coherent reading of Weberian ideal type construction, based on recent philosophical approaches to conceptual engineering. This reading makes transparent the dialectics of Weber's approach, resulting in a more nuanced interpretation of his methodological work. It will become apparent that Weber, when introducing his notion of an ideal type, did not merely summarize his views on methodology in the social sciences, but, rather, presented a two-step argument in favor of these views. The reconstruction (...) will directly bear on issues in the methodology of the social sciences. Weber not only presented a methodology that is, at some level of abstraction, strikingly similar to Carnapian explication (the canonical point of reference in the contemporary debate on conceptual engineering). He also diagnosed, in a systematic fashion, types of problems we face when dealing with ordinary and scientific concepts that play a key role in the description of social reality, and he foreshadowed what is today sometimes described as “debunking” projects. From a philosophy of science perspective, I propose Weber's view (on the reconstruction proposed here) as a serious candidate for an adequate theory of concept replacement in the social sciences. (shrink)

This study empirically examines whether firms’ environmental capital expenditures impact institutional investors’ investment decisions in the Chinese market. We particularly examine the impact of ownership type on the relationship of environmental capital expenditures and the behavior of different types of institutional investors by classifying institutional investors into two categories, short-term and long-term investors. In addition, this study further investigates whether environmental capital expenditures related to ownership type increase firm value. We find that long-term institutional investors tend to invest (...) in state-owned firms making environmental capital expenditures. Results also indicate that, with governmental backing and encouragement, the market value of SOEs making more environmental capital expenditures is likely to increase. However, no similar results are found for non-SOEs. (shrink)