Recent debates in metaphysics have highlighted the significance of type theories, such as Simple Type Theory (STT), for our philosophical analysis. In this chapter, I present the salient features of a constructive type theory in the style of Martin-Löf, termed CTT. My principal aim is to convey the flavour of this rich, flexible and sophisticated theory and compare it with STT. I especially focus on the forms of quantification which are available in CTT. A further (...) aim is to argue that a comparison between a plurality of theories is beneficial to the philosophical analysis. We may, for example, discover helpful features of one theory that we may want to import into another context, thus enriching our repertoire of formal tools. Or, through comparison with a less well-known theory, we may gain a better understanding of the characteristics of the theories we are familiar with. As argued in this chapter, CTT has much to offer in all of these respects. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the simple observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims (...) that the search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher-order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that any ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher-Order Pluralism’ on the theory of content. One theme of the discussion is that, if Higher-Order Pluralism is correct, then there is no fixed metatheory from which to characterize higher-order reality. (shrink)

So-called 'dynamic' semantic theories such as Kamp's discourse representation theory and Heim's file change semantics account for such phenomena as cross-sentential anaphora, donkey anaphora, and the novelty condition on indefinites, but compare unfavorably with Montague semantics in some important respects (clarity and simplicity of mathematical foundations, compositionality, handling of quantification and coordination). Preliminary efforts have been made by Muskens and by de Groote to revise and extend Montague semantics to cover dynamic phenomena. We present a new higher-order theory (...) of discourse semantics which improves on their accounts by incorporating a more articulated notion of context inspired by ideas due to David Lewis and to Craige Roberts. On our account, a context consists of a common ground of mutually accepted propositions together with a set of discourse referents preordered by relative salience. Employing a richer notion of contexts enables us to extend our coverage beyond pronominal anaphora to a wider range of presuppositional phenomena, such as the factivity of certain sentential-complement verbs, resolution of anaphora associated with arbitrarily complex definite descriptions, presupposition 'holes' such as negation, and the independence condition on the antecedents of conditionals. Formally, our theory is expressed within a higher-order logic with natural number type, separation-style subtyping, and dependent coproducts parameterized by the natural numbers. The system of semantic types builds on proposals due to Thomason and to Pollard in which the type of propositions (static meanings of sentential utterances) is taken as basic and worlds are constructed from propositions (rather than the other way around as in standard Montague semantics). (shrink)

This chapter provides an introduction to higher-order metaphysics as well as to the contributions to this volume. We discuss five topics, corresponding to the five parts of this volume, and summarize the contributions to each part. First, we motivate the usefulness of higher-order quantification in metaphysics using a number of examples, and discuss the question of how such quantifiers should be interpreted. We provide a brief introduction to the most common forms of higher-order logics (...) used in metaphysics, and indicate a number of questions which can be raised in such systems using logical vocabulary alone. Using a further example, we return to applications of higher-order logics in metaphysics. We also mention key developments in the history of higher-order logic as it pertains to metaphysics. Finally, we mention certain arguments which have been raised against the use of higher-order logic, and some ways of responding to them. (shrink)

Recent work in the ontology of music suggests that we will avoid confusion if we distinguish between two kinds of question that are typically posed in music ontology. Thus, a distinction has been made between fundamental ontology and higher-order ontology. The former addresses questions about the basic metaphysical options from which ontologists choose. For instance, are musical works types, indicated types, classes of particulars, or some other kind of entity? Higher-order ontology addresses the question of what (...) lies ‘at the centre’ of a specific form of music, such as rock or jazz—or perhaps classical music. The argument of this essay is, first, that a close examination of the best efforts in two of these territories shows that they have the effect of pressing the music in each sphere into implausible Procrustean beds. Second, it is argued that the general question that higher-order ontologies pose, that is, ‘What work-kind is it that lies at the centre of a given kind of music, F?’ is a question based on a mistaken but seductive assumption, namely that the concept of the work of F has actual application. In fact, these concepts—upon which higher-order ontology depends—are mere artefacts of philosophy. The question is also addressed why the assumption is so seductive. Finally, the question finally is posed about what, if anything, is implied from the foregoing about the traditional ontology of classical music. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in (...) which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, (...) propositions, and identity, as well as on the metaphysics of modality. (shrink)

In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong (...) correlation with predicativity, predicative logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf''s Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russell's orders, (2) the placing of the historical system RTT underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf''s type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS. (shrink)

We consider a class of Lindström extensions of first-order logic which are susceptible to a natural Skolemization procedure. In these logics Ehrenfeucht Mostowski (EM) functors for theories with arbitrarily large models can be obtained under suitable restrictions. Characteristic dependencies between algebraic properties of the quantifiers and the maximal domains of EM functors are investigated.Results are applied to Magidor Malitz logic,L(Q <ω), showing e.g. its Hanf number to be equal to ℶω(ℵ1) in the countably compact case. Using results of Baumgartner, (...) the maximal number of isomorphism types of linearly ordered models of regular cardinality is shown to be achieved for theories that admit an EM functor on a typically restricted domain. (shrink)

Holistic theories of meaning have, at least since Dummett’s Frege: The Philosophy of language, been assumed to be problematic from the perspective of the incremental nature of natural language learning. In this essay I argue that the general relationship between holism and language learning is in fact the opposite of that claimed by Dummett. It is only given a particular form of language learning, and a particular form of holism, that there is a problem at all; in general, for all (...) forms of holism, and irrespective of how language learning is understood, semantic holism is conducive to language learning. The paper has three main parts. In the first, I demonstrate with the use of a simple formal system, that the form of holism that generates the problem that Dummett draws attention to is really decomposable into three distinct components, each of which is necessary for the problem to arise. In the second part, I demonstrate that even Dummett’s strong form of holism is compatible with one natural way in which to understand the incremental nature of language learning. In the third part, I outline the reasons why all forms of holism are conducive to language learning and offer two ways in which this general fact can be spelled out precisely. I end the paper by addressing some possible objections, and in doing so I draw attention to some affinities between semantic holism and the principle of compositionality, a semantic principle which has long been assumed to be conducive to language learning. (shrink)

Martin-Löf's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom - treating the category of propositions as a set and thereby enabling higher order quantification - leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of (...) all well-founded sets). The occurrence of the contradiction is explained in set theoretical terms. Crucial here is the way a proof-object of an existential proposition is understood. It is shown that also Russell's paradox can be translated into type theory. The type theory extended with the axiom mentioned above contains constructive higher order logic, but even if one only adds constructive second order logic to type theory the contradictions arise. (shrink)

In recruitment, promotion, admission, and other forms of wealth and power apportion, an evaluator typically ranks a set of candidates in terms of their competence. If the evaluator is prejudiced, the resulting ranking will misrepresent the candidates’ actual rankings. This constitutes not only a moral and a practical problem, but also an epistemological one, which begs the question of what we should do—epistemologically—to mitigate it. In a recent paper, Jönsson and Sjödahl in [Episteme 14:499–517, 2017], argue that the epistemic problem (...) can be fruitfully addressed by way of a novel statistical method that changes the products of biased behaviour, i.e. the rankings themselves, rather than the biased persons. Jönsson and Sjödahl’s pioneering proposal is a both a welcome addition to the literature on implicit bias, due the problems with existing implicit bias interventions [see e.g. Lai et al. in J Exp Psychol Gen 143:1765–1785; J Exp Psychol Gen 145:1001–1016, 2014; 2016; Forscher et al. in J Person Soc Psychol 117:522–559, 2019] but also to the literature on prejudice more generally, where many proposed prejudice-reduction strategies enjoy less than adequate empirical support [Paluck and Green in Ann Rev Psychol 60:339–367, 2009]. Their proposal, however, needs supplementation in two ways: the circumstances that must hold in order for it to work needs to be refined, and their claim that it works as intended in these circumstances needs to be validated. We argue that four of Jönsson and Sjödahl’s method’s presumed presuppositions can be weakened, but needs to be supplemented by two additional assumptions, overlooked by Jönsson and Sjödahl. Moreover, we demonstrate that the method does work as intended by way of a statistical simulation. (shrink)

Simple type theory is a higher-order predicate logic for reasoning about truth values, individuals, and simply typed total functions. We present in this paper a version of simple type theory, called PF*, in which functions may be partial and types may have subtypes. We define both a Henkin-style general models semantics and an axiomatic system for PF*, and we prove that the axiomatic system is complete with respect to the general models semantics. We also (...) define a notion of an interpretation of one PF* theory in another. PF* is intended as a foundation for mechanized mathematics. It is the basis for the logic of, an Interactive Mathematical Proof System developed at The MITRE Corporation. (shrink)

Simple type theory is a higher-order predicate logic for reasoning about truth values, individuals, and simply typed total functions. We present in this paper a version of simple type theory, called PF*, in which functions may be partial and types may have subtypes. We define both a Henkin-style general models semantics and an axiomatic system for PF*, and we prove that the axiomatic system is complete with respect to the general models semantics. We also (...) define a notion of an interpretation of one PF* theory in another. PF* is intended as a foundation for mechanized mathematics. It is the basis for the logic of , an Interactive Mathematical Proof System developed at The MITRE Corporation. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces (...) Local Set Theory, an important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

One may formulate the dependent product types of Martin-Löf type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the other constructors of type theory. It is known that the latter rules are at least as strong as the former: we show that they are in fact strictly stronger. We also show, in the presence of the identity types, that the (...) elimination rule for dependent products–which is a “higher-order” inference rule in the sense of Schroeder-Heister–can be reformulated in a first-order manner. Finally, we consider the principle of function extensionality in type theory, which asserts that two elements of a dependent product type which are pointwise propositionally equal, are themselves propositionally equal. We demonstrate that the usual formulation of this principle fails to verify a number of very natural propositional equalities; and suggest an alternative formulation which rectifies this deficiency. (shrink)

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial (...) time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. (shrink)

Church’s type theory, aka simple type theory, is a formal logical language which includes classical first-order and propositional logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory. It is particularly well suited to the formalization of mathematics and other disciplines and to specifying and verifying hardware and software. It also plays an important role in the study of the formal semantics (...) of natural language. When utilizing it as a meta-logic to semantically embed expressive non-classical logics further topical applications are enabled in artificial intelligence and philosophy. A great wealth of technical knowledge can be expressed very naturally in it. With possible enhancements, Church’s type theory constitutes an excellent formal language for representing the knowledge in automated information systems, sophisticated automated reasoning systems, systems for verifying the correctness of mathematical proofs, and a range of projects involving logic and artificial intelligence. Some examples and further references are given in Sections 1.2.2 and 5 below. Type theories are also called higher-order logics, since they allow quantification not only over individual variables, but also over function, predicate, and even higher order variables. Type theories characteristically assign types to entities, distinguishing, for example, between numbers, sets of numbers, functions from numbers to sets of numbers, and sets of such functions. As illustrated in Section 1.2.2 below, these distinctions allow one to discuss the conceptually rich world of sets and functions without encountering the paradoxes of naive set theory. Church’s type theory is a formulation of type theory that was introduced by Alonzo Church in Church 1940. In certain respects, it is simpler and more general than the type theory introduced by Bertrand Russell in Russell 1908 and Whitehead & Russell 1927a. Since properties and relations can be regarded as functions from entities to truth values, the concept of a function is taken as primitive in Church’s type theory, and the λ-notation which Church introduced in Church 1932 and Church 1941 is incorporated into the formal language. Moreover, quantifiers and description operators are introduced in a way so that additional binding mechanisms can be avoided, λ-notation is reused instead. λ-notation is thus the only binding mechanism employed in Church’s type theory. (shrink)

This chapter offers an opinionated introduction to higher-order formal languages with an eye towards their applications in metaphysics. A simply relationally typed higher-order language is introduced in four stages: starting with first-order logic, adding first-order predicate abstraction, generalizing to higher-order predicate abstraction, and finally adding higher-order quantification. It is argued that both β-conversion and Universal Instantiation are valid on the intended interpretation of this language. Given these two principles, it is (...) then shown how we can use pure higher-order logic to ask, and begin to answer, metaphysical questions with non-trivial implications. In particular, while we must reject the popular idea that structural differences between sentences correspond to parallel distinctions in the logical structure of extra-linguistic reality, it may still be possible to give a purely logical characterization of objectual aboutness and related notions. (shrink)

In relativistic classical and quantum mechanics with Poincaré-invariant parameter, particle worldlines are traced out by the evolution of spacetime events. The formulation of a covariant canonical framework for the evolving events leads to a dynamical theory in which mass conservation is demoted from a priori constraint to the status of conserved Noether current for a certain class of interactions. In pre-Maxwell electrodynamics—the local gauge theory associated with this framework —events induce five local off-shell fields, which mediate interactions between instantaneous events, (...) not between the worldlines which represent entire particle histories. The fifth field, required to compensate for dependence of gauge transformations on the evolution parameter, enables the exchange of mass between particles and fields. In the equilibrium limit, these pre-Maxwell fields are pushed onto the zero-mass shell, but during interactions there is no mechanism regulating the mass that photons may acquire, even when event trajectories evolve far into the spacelike region. This feature of the off-shell formalism requires the application of some ad hoc mechanism for controlling the photon mass in two opposite physical domains: the low energy motion of a charged event in classical Coulomb scattering, and the renormalization of off-shell quantum electrodynamics. In this paper, we discuss a nonlocal, higher derivative correction to the photon kinetic term, which provides regulation of the photon mass in a manner which preserves the gauge invariance and Poincaré covariance of the original theory. We demonstrate that the inclusion of this term is equivalent to an earlier solution to the classical Coulomb problem, and that the resulting quantum field theory is renormalized. (shrink)

Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an active role in the development of computational science and core mathematics. It is worth exploring some of them in depth, particularly predicative Martin-Löf’s intuitionistic type theory and impredicative Coquand’s calculus of constructions. The logical and philosophical differences and similarities (...) between them will be studied, showing the relationship between these type theories and other fields of logic. (shrink)

Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of (...) type theory is assumed. However, the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance. (shrink)

This paper is motivated by the increasing popularity of efficient designs for stated choice experiments. The objective in efficient designs is to create a stated choice experiment that minimizes the standard errors of the estimated parameters. In order to do so, such designs require specifying prior values for the parameters to be estimated. While there is significant literature demonstrating the efficiency improvements of employing efficient designs, the bulk of the literature tests conditions where the priors used to generate the (...) efficient design are assumed to be accurate. However, there is substantially less literature that compares how different design types perform under varying degree of error of the prior. The literature that does exist assumes small fractions are used, which is in contrast to computer-aided surveys that readily allow for large fractions. Further, the results in the literature are abstract in that there is no reference point to provide clear insight on the magnitude of any issue. Our objective is to analyze the robustness of different designs within a typical stated choice experiment context of a trade-off between price and quality. We use as an example transportation mode choice, where the key parameter to estimate is the value of time. Within this context, we test many designs to examine how robust efficient designs are against a misspecification of the prior parameters. The simple mode choice setting allows for insightful visualizations of the designs themselves and also an interpretable reference point for the range in which each design is robust. Not surprisingly, the D-efficient design is most efficient in the region where the true population VOT is near the prior used to generate the design: the prior is $20/h and the efficient range is $10–$30/h. However, the D-efficient design quickly becomes the most inefficient outside of this range, and the estimation significantly degrades above $50/h. The orthogonal and random designs are robust for a much larger range of VOT. The robustness of Bayesian efficient designs varies depending on the variance that the prior assumes. Implementing two-stage designs that first use a small sample to estimate priors are also not robust relative to uninformative designs. Arguably, the random design performs as well as any design, and it will perform even better if data cleaning is done to remove choice tasks where one alternative dominates the other. (shrink)

Anyone interested in Greek philosophy or in contemporary theories of hierarchy will profit from reading this book. It is a model of correct methodology since O’Meara carefully studies the relevant Greek texts in a chronological order, which he constantly illumines through references to secondary literature. As a result, it offers helpful insights into Plotinus, as these examples indicate. Although the Greek word for "hierarchy" does not itself occur in the history of thought prior to Pseudo-Dionysius, the conception of hierarchic (...) structures is found clearly in Plotinus’ view of existents related to one another as higher/lower, prior/posterior, causally independent/dependent, and so on. His presentation of those structures varies, though. At times he utilizes the Platonic contrast between the sensible and intelligible realms, which, however, are linked through participation. On other occasions he speaks of the One, the Intellect and the Soul. The second of these is occasionally linked with intelligibility, being, and number. The third with reference to the visible universe is described as Nature and Logos. But the hierarchical ordering which remains constant from beginning to end of the Enneads is that of the three hypostases: the One, Intellect, and Soul. These Plotinus often presents from different perspectives and with varying emphases, with the result that one realizes his hierarchical arrangements are not an a priori source of his philosophizing but emerge from it. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist (...) reading). While STT, understood as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell’s about the relationship between quantification and (...) the structure of predication, we offer such an account. We use this account to examine the problem in three different type-theoretic settings, which are increasingly permissive with respect to predication. We conclude that unrestricted quantification is available in all but the most permissive kind of type theory. (shrink)

In this study, we conducted a pseudosign repetition task with 22 children acquiring American Sign Language as a first language from deaf parents. Thirty-nine pseudosigns with varying complexity were developed and organized into eight categories depending on number of hands, number of simultaneous movement types, and number of movement sequences. Pseudosigns also varied in handshape complexity. The children’s performance on the ASL pseudosign task improved with age, displaying relatively accurate production of location and orientation, but much less accurate handshape and (...) movement, a finding in line with real sign productions for both L1 and L2 signers. Handshapes with higher complexity were correlated with lower accuracy in the handshape parameter. We found main effects of sequential and simultaneous movement combinations on overall performance. Items with no movement sequence were produced with higher overall accuracy than those with a movement sequence. Items with two simultaneous movement types or a single movement type were produced with higher overall accuracy than those with three simultaneous movement types. Finally, number of hands did not affect the overall accuracy. Remarkably, movement sequences impose processing constraints on signing children whereas complex hands and two simultaneous movement types do not significantly lower accuracy, indicating a capacity for processing multiple simultaneous components in signs. Spoken languages, in contrast, manifest greater complexity in temporal length. Hearing children’s pseudoword repetition still displays high levels of accuracy on disyllabic words, with complexity effects affecting only longer multisyllabic words. We conclude that the pseudosign repetition task is an informative tool for studies of signing children’s phonological development and that sheds light on potential modality effects for phonological development. (shrink)

In Groenendijk & Stokhof [1989] a system of dynamic predicate logic (DPL) was developed, as a compositional alternative for classical discourse representation theory (DRT ). DPL shares with DRT the restriction of being a first-order system. In the present paper, we are mainly concerned with overcoming this limitation. We shall define a dynamic semantics for a typed language with λ-abstraction which is compatible with the semantics DPL specifies for the language of first-order predicate logic. We shall propose to (...) use this new logical system as the semantic component of a Montague-style grammar (referred to as dynamic Montague grammar, DMG), which will enable us to extend the compositionality of DPL to the subsentential level. Furthermore, we shall extend this analysis also in this sense that we shall add new, dynamic interpretations for logical constants which in DPL were treated in a static fashion. This will substantially increase the descriptive coverage of DMG. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations (...) are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that (...) higher-order realism does not in fact resolve the tropes versus universals dispute. In a constructive spirit, I then develop higher-order realism in a way that leads to a dissolution, rather than a resolution, of this dispute too. (shrink)

In this article we define a logical system called Hybrid Partial Type Theory ( $\mathcal {HPTT}$ ). The system is obtained by combining William Farmer’s partial type theory with a strong form of hybrid logic. William Farmer’s system is a version of Church’s theory of types which allows terms to be non-denoting; hybrid logic is a version of modal logic in which it is possible to name worlds and evaluate expressions with respect to particular worlds. We motivate this (...) combination of ideas in the introduction, and devote the rest of the article to defining, axiomatising, and proving a completeness result for $\mathcal {HPTT}$. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both (...) translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$ -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

According to relationism, for Alice to believe that some rabbits can speak is for Alice to stand in a relation to a further entity, some rabbits can speak. But what could this further entity possibly be? Higher-order metaphysics seems to offer a simple, natural answer. On this view (roughly put), expressions in different syntactic categories (for instance: names, predicates, sentences) in general denote entities in correspondingly different ontological categories. Alice's belief can thus be understood to relate her (...) to a sui generis entity denoted by "some rabbits can speak", belonging to a different ontological category than Alice herself. This straightforward account of the attitudes has historically been deemed so attractive that it was seen as providing an important motivation for higher-order metaphysics itself (Prior [1971]). But I argue that it is not as straightforward as it might seem, and in fact that propositional attitudes present a foundational challenge for higher-order metaphysics. (shrink)

Theories of possible worlds abound. Since the introduction of modal logic, the term of a possible world, and the very nature of an entity denoted by the term, have stood on the top of metaphysical inquiries. A possible world, roughly speaking, is a complete way things could have been. On the face of it, whatever is possible takes place in some possible world, and whatever is not possible, does not. The aim of the present book is to argue that even (...) impossible things happen. By taking David Lewis’s Modal Realism (henceforth as “MR”) seriously, I claim that besides infinitely many concrete ways things could have been, there exist ways things could not have been. I call them concrete impossible worlds. In Chapter I, I outline Lewis’s well-defined conception of philosophical analysis. I present its structure, aims, methodology and criteria for success. Besides the virtues of the Lewisian conception, I point out several limitations the theory has and offer very simple solution: the admission of concrete impossible worlds. In Chapter II, I present a version of an epistemological objection against MR and a positive account in favour of Modal Fictionalism (henceforth as “MF”). I then present the puzzle in this matter and attempt to generalise my point. 9 Introduction Chapter III demonstrates how flexible Lewis’s theory is. I consider a putative impossibility of there being island universes – spatiotemporally disjoint spacetimes – and show what a proponent of modal realism can do in order to account for it. I present three possible moves: a revision of our pre-theoretical opinions, a modification of our definitions and, finally, an ex- tension of our metaphysical commitments. I consider two ersatz theories of impossible worlds in Chapter IV. In particular, I scrutinize Franz Berto’s and Edwin Mares’s (hybrid) theories of MR. Although I admit that the theories radically extend the scope of applications of MR without extending its metaphysical commitments, the approaches do have some limits. Finally, in Chapter V, I present Extended Modal Realism (henceforth EMR), a theory according to which the best way to go in order to account for impossible phenomena is to ex- tend our metaphysical base by genuine impossibilia. I try to (at least) weaken, if not meet, the crucial objections to the proposal. Namely, I challenge the proclaimed universality of classical logic and, subsequently, motivate certain kind of paraconsistent approach to modal reality. Although the consequences are very hard to swallow, I argue that it is because of the fact that we are unsure of our pre-theoretical opinions concerning the impossible. (shrink)

The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. (...) Reversion to the One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction. (shrink)

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as (...) the given totality. (shrink)

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the arithmetic addition of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such (...) as accelerations, are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions. (shrink)

Though acknowledged by scholars, Plato’s identification of the Beautiful and the Good has generated little interest, even in aesthetics where the moral concepts are a current topic. The view is suspect because, e.g., it is easy to find examples of ugly saints and beautiful sinners. In this paper the thesis is defended using ideas from Plato’s ancient commentators, the Neoplatonists. Most interesting is Proclus, who applied to value theory a battery of linguistic tools with fixed semantic properties—comparative adjectives, associated gradable (...) adjectives, mass nouns, and predicate negations—all with a semantics that demand a privative scale of value. It is shown how it is perfectly possible to interpret value terms Platonically over privative Boolean algebras so that beautifuland good diverge while at higher levels other value terms are coextensional. Considerations are offered that this structure conforms to actual usage. (shrink)

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some (...) remarks on standard models are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55. (shrink)

This chapter explores the metaphysical views about higher-order logic held by two individuals responsible for introducing it to philosophy: Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970). Frege understood a function at first as the remainder of the content of a proposition when one component was taken out or seen as replaceable by others, and later as a mapping between objects. His logic employed second-order quantifiers ranging over such functions, and he saw a deep division in nature between (...) objects and functions. Russell understood propositional functions as what is obtained when constituents of propositions are replaced by variables, but eventually denied that they were entities in their own right. Both encountered contradictions when supposing there to exist as many objects as functions, and both adopted views about the meaningfulness of higher-order discourse that were difficult to state from within their own strictures. (shrink)

We investigate Hilbert’s varepsilon -calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed varepsilon -terms. We extend the usual topos semantics for type theories to the varepsilon -operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined‘ varepsilon -term. MSC: 03B15, 03B20, 03G30.

This paper argues that Malebranche’s semantics sheds light on his metaphysics and epistemology, and is of interest in its own right. By recasting issues linguistically, it shows that Malebranche assumes a Neoplatonic semantic structure within Descartes’ dualism and Augustine’s theory of illumination, and employs linguistic devices from the Neoplatonic tradition. Viewed semantically, mental states of illumination stand to God and his ideas as predicates stand in Neoplatonic semantics to ideas ordered by a privative relation on “being.” The framework sheds light (...) on interpretive puzzles in Malebranche studies such as the way ideas reside in God’s mind, the notion of resemblance by which bodies imitate their exemplar causes, and the issue of direct vs. indirect perception through a mechanism by which agents can see bodies by “seeing” ideas. Malebranche’s semantics is of interest in its own right because it gives a full account of the mediating relations that determine indirect reference; lays out a correspondence theory of truth for necessary judgments; defines contingent truth as based on an indirect reference relation that is both descriptive and causal but that does not appeal to body-mind causation; and within his theory of perception, works out an account of singular reference in which singular terms carry existential import, refer indirectly via causal relations, but describe their referents only in a general way. (shrink)

Dynamic Theodicy (DT) is a broad concept we bring up to designate some modern Philosophical Theology attempts to reconcile the necessary and perfect existence of God with the contingent characteristics of human life. In this paper we analyze such approaches and discuss how they have become incomprehensible because the metaphysical assumptions implicit in these explanations have lost their intrinsic relation to the natural human desire for salvation. In the first part we show Charles Hartshorne's DT-model, arising from the modal logic (...) of perfection, and the modern rational problems of this position in making infinite-necessary Being (God) and finite-contingent being (human) compatible. We note that at the heart of the contradictions in this DT account is a dialectical mode of thinking that makes it difficult to find a correct solution to this dichotomy, and to assume a human desire that could be considered related to lifelong goals. In the second part, supported by the proposal of Hans Urs von Balthasar's DT, we develop the concepts of bodily vulnerability, corporeal intentionality, and natural desire for salvation, which come from an Aristotelian-Thomistic thought. This theory is established in order to build an argument, following Alasdair MacIntyre’s ethical framework, on how to make possible the recovery of a metaphysical and anthropological desire that transcends natural aging and goes beyond death. We conclude that both human dependence and the virtues that arise naturally when human beings decide to seek the good of their transcendent condition, make it possible to recover the natural desire for salvation through divine and human love. (shrink)

University students will be our future business leaders, and will have to address social problems caused by business by implementing solutions such as social entrepreneurship ventures. In order to facilitate the learning process that will foster social entrepreneurship, however, a more holistic pedagogy is needed. Based on learning theory, we propose that students’ social entrepreneurship actions will depend on their learning about CSR and their absorptive capacity. We propose that instructors and higher education institutions can enhance this absorptive (...) capacity by exploiting Web 2.0 technologies. We tested our proposition with a sample of 425 university students using structural equation modeling and found support for the proposed relationships. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Spinoza’s Cosmological Argument in the EthicsMogens Lærke (bio)1. IntroductionIn this paper,1 i discuss Spinoza’s version of the cosmological argument for the existence of God (hereafter CA), specifically as it can be found in EIP11D3.2 By a CA, I broadly understand an argument which infers a posteriori the existence of an independent, necessary being, usually identified as God, from the experience that there exists some other being, often oneself, whose (...) existence is dependent on this independent, necessary being.3 Generally, I will speak of the dependence [End Page 439] relation that any CA relies on in terms of “existential dependence.” Such a dependence relation can be considered as causal, conceptual, or other. Much of the following will be a discussion of what kind of existential dependence relation governs Spinoza’s CA. The CA in EIP11D3, as we shall see, explicitly turns on a relation of “being in” (i.e. the fact that all finite things are in some way in God and that God is in himself). I argue however that the context of the argument makes this “in-relation” inseparable from both Spinoza’s theory of causation and his theory of power, and that it consequently must be understood in the context of what Alexandre Matheron dubbed Spinoza’s “ontology of power.”4The study attempts to fill a lacuna in the study of Spinoza’s arguments for the existence of God. Don Garret, in his classic “Spinoza’s Ontological Arguments,” and Martin Lin, in his “Spinoza’s Arguments for the Existence of God,” discuss Spinoza’s various arguments for the existence of God.5 None of them, however, consider the passage in EIP11D3 which contains Spinoza’s CA, but focus exclusively on other proofs. There is something to be said for not attaching too much importance to the CA in EIP11D3. Spinoza himself, as we shall see, did not consider it his principal argument for the existence of God. Moreover, there is good reason to think that at least part of Spinoza’s argument in EIP11D3 is unsound. One could then reasonably argue for not giving it any consideration on the grounds that it is both unimportant and unsound.Nonetheless, studying the argument helps clarify what Spinoza thought in general about the necessary existence of God6 and about the relations between the infinite being, God, and finite beings such as ourselves. There is thus something to add to the existing accounts. Indeed, I believe that when we do include [End Page 440] considerations regarding this argument, we must considerably revise the general metaphysical framework that especially Lin argues underlies Spinoza’s other arguments for the existence of God. Lin’s approach is strongly informed by Michael Della Rocca’s reading of Spinoza, developed in a multitude of articles over the last decade and recently presented in full in Spinoza from 2008. Della Rocca’s work constitutes the broader context and motivation for this article. His book is an impressive and already hugely influential attempt to account for the systematic unity and fundamental principles of Spinoza’s metaphysics. To my mind, however, some of its basic interpretative choices are problematic. I have chosen to discuss the CA and the form it takes in the Ethics as a good point of departure for arguing this and for suggesting what I take to be a better approach. Following Della Rocca’s account, all dependence relations in Spinoza can be reduced to conceptual ones, including the sort of relations that the CA relies on. I maintain to the contrary that Spinoza’s version of the CA is inextricably linked to his theories of causation and power. Thus, the CA in EIP11D3 provides a good example of how Spinoza’s philosophy is an ontology of power grounded in a theory of causation, and not the sort of conceptualist ontology that Della Rocca attributes to him.72. The Cosmological Argument in Eip11d3Before scrutinizing in detail Spinoza’s CA in EIP11D3, it is important to note the subsidiary role of the argument in relation to Spinoza’s various other demonstrations of the existence of God. Contrary to Thomas Aquinas, for example, who denies that a... (shrink)

Although most natural law ethical theories recognize moral absolutes, there is not much agreement even among natural law theorists about how to identify them. The author argues that in order to understand and determine the morality (or immorality) of a human action, it must be considered in relation to the organized system of human practices within which it is performed. Such an approach, he argues, is to be found in the natural law theory of Thomas Aquinas, especially once it (...) is recognized that the logical structure of Aquinas's ethical theory is basically that of an Aristotelian science. In order to depict this structure and to explain how it bears upon the analysis of action, the author investigates a number of issues that have attracted the attention of Thomistic and Aristotelian scholarship. He examines the nature of practical reason, its relationship with theoretical reason, the derivation of lower from higher ethical principles, the incommensurability of human goods, the relationship between will and intellect, and the principle of double effect. The book will be useful to students and scholars interested in ethics, especially from an Aristotelian and/or Thomistic perspective. One appendix reproduces the Leonine text of the De malo (question 6), with facing English translation. Another appendix provides facing Latin text and English translation of the Summa Theologiae I-II (question 94, article 2). ABOUT THE AUTHOR: Kevin L. Flannery, S.J., author of many works on the history of logic--particularly Aristotelian logic--is dean of the faculty of philosophy and professor of the history of ancient philosophy at the Pontifical Gregorian University in Rome. PRAISE FOR THE BOOK: ""I know of no more careful and detailed study of Thomas's principle that the realms of theoretical and practical rationality are parallel. This principle is indeed central to an understanding of Thomas Aquinas's ethical thought, and Father Flannery examines its ramifications most perspicaciously. His book constitutes a valuable and important addition to the Thomistic literature. Father Flannery's writing is lucid, if not always extremely engaging, and his arguments are well supported by references to, and quotations from, the Aristotelian and Thomistic texts upon which they are based.""--The Medieval Review ""It would be hard to overstate the importance of this book at the present juncture of Thomistic studies.""--Ralph McInerny, University of Notre Dame ""... of interest and of great value mostly to specialists in Aquinas's ethics or moral theory more generally. Flannery presents complex matters clearly, and his explanations of the logical presuppositions of Aquinas's moral thought are always illuminating.""--Jean Porter, Theological Studies ""This is a significant work that belongs on the bookshelf of any serious student of Aquinas.... An important contribution to Thomistic studies.""--Irish Theological Quarterly ""This scholastic text will particularly appeal to thinkers interested in Aristotelian mathematics and logic and their practical integration into Thomistic natural law ethics. However, all natural law scholars must address the questions of context and method raised by Flannery's insights and his careful, precise textual analyses."" -- Beverly Whelton, Review of Metaphysics. (shrink)

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate (...) simple term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (shrink)

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics in which the identity relation is not definable. The point is that the (...) definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to , where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective. (shrink)