We claim that Kant's doctrine of the "transcendental ideal of pure reason" contains, in an anticipatory sense, a second-order theory of reality (as a second-order property) and of the highest being. Such a theory, as reconstructed in this paper, is a transformation of Kant's metatheoretical regulative and heuristic presuppositions of empirical theories into a hypothetical ontotheology. We show that this metaphysical theory, in distinction to Descartes' and Leibniz's ontotheology, in many aspects resembles Gödel's theoretical conception of the possibility of a (...) supreme being. The proposed second-order modal formalization of the Kantian doctrine of the "transcendental ideal" of the supreme being includes some specific features of Kantian general logic like "categorical", "hypothetical", and exclusive disjunctive propositions as basic forms. (shrink)

This paper introduces an interesting class of predicates that come in pairs, so-called total and partial predicates. It will be shown that such predicates contribute to an explanation for the weak and strong interpretations of donkey sentences. This paper proposes that the phenomenon of weak and strong interpretations is real, and that whether a sentence receives the weak or the strong interpretation depends on the predicate in the nuclear scope of the sentence. It also proposes that sum (...) individuals are calculated at some level before the nuclear scope of the sentence is processed. Once the sum individuals are calculated, it will be decided whether the nuclear scope is true of at least one element of the sum individual (weak interpretation) or true of all elements of the sum individual (strong interpretation). (shrink)

The topic of our analysis is the argument for the existence of substances given by Bernard Bolzano in Athanasia, where he essentially employs two ontological categories: substance and adherence. Bolzano considers the real and conditioned Inbegriff of all adherences, which are wirklich and nicht selbst bestehen. He claims that the formed collection is dependent on something external and non-adherential, which therefore is a substance. Bolzano’s argumentation turns out to be structurally similar to his argument for the existence of God from (...) Lehrbuch der Religionswissenschaft, but in each of these reasonings, we find different plausible interpretations of the key concept “Inbegriff”. The latter argumentation refers to the mereological totality of existentially conditioned objects. We propose the explication of the Bolzanian Inbegriff of all adherences using two types of predication: we consider its extension as composed of certain intensional counterparts of adherences. In our approach, we use a fragment of the theory of abstract objects formulated by E. Zalta, describing two different relations between individuals and properties: extensional exemplification and intensional encoding. We put our reconstruction in a wider context of Bolzano’s ontology, formulating the needed axioms with two primitive predicates of second order... is an adherence,... is conditioned by something real as well as the conditionally introduced first order predicate constant \ for Inbegriff of all adherential ideas. Finally, we sketch a model for our theory. (shrink)

A subset A $\subseteq$ M of a totally ordered structure M is said to be convex, if for any a, b $\in A: [a . A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some $\emptyset$ -definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory (...) T, any expansion M + of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63), that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories. (shrink)

Inspired by Pohlers' proof-theoretic analysis of KPω we give a straightforward non-metamathematical proof of the (well-known) classification of the provably total functions of $PA, PA + TI(\prec\lceil)$ (where it is assumed that the well-ordering $\prec$ has some reasonable closure properties) and KPω. Our method relies on a new approach to subrecursion due to Buchholz, Cichon and the author.

We show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PT tot ) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PT tot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.

It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that recursive functions are restricted to primitive recursives.

I dare say, a set is contranatural if some pair of its elements has a nonempty intersection. So, we consider only collections of disjoint nonempty elements and call them totalities. We propose the propositional logicTT, where a proposition letters some totality. The proposition is true if it letters the greatest totality. There are five connectives inTT: , , , , # and the last is called plexus. The truth of # means that any element of the totality (...) has a nonempty intersection with any element of the totality . An imbeddingG of the classical predicate logicCPL inTT is defined. A formulaf ofCPL is a classical tautology if and only ifG(f) is always true inTT. So, mathematics may be expounded inTT, without quantifiers. (shrink)

The “dynamic turn” in the field of second language acquisition catalyzed scholarly devotion to the complex dynamic relationships between learner and teacher variables and various academic emotions. As such, the present study examined the varying effects of the aforementioned variables on the constructs of positive and negative flow, and determined their strongest predictors, respectively. This study used a mixed-method approach to collect data from 607 Chinese English-as-a-Foreign-Language learners. In stage one of the research, the researcher first assessed the participants’ levels (...) of positive flow and negative flow in a blended learning context, then performed Pearson correlation analysis to confirm a significant, but weak positive relationship between positive flow and negative flow. Then, significant difference tests were run to determine the varied effects of those variables on flow. Finally, two multiple regression analyses were conducted to identify five predictors of positive flow, with the biggest contribution coming from learners’ attitudes toward the foreign language, and three predictors of negative flow, with the learners’ major accounting for the majority of variance. In the second stage of the research, a qualitative corpus was constructed, based on accounts of classroom experiences from 71 participants of the total sample, and further illustrated the quantitative findings. Pedagogical implications for educational psychologists and teachers of second and/or foreign languages are addressed. (shrink)

I argue that Plato held a kinds of reality theory, not a degrees of reality theory, and that this position solves otherwise intractable problems about the Forms, notably the Third Man critique. These problems stem from the fact that Plato applied the same predicate bothto a Form and to its participants. Section I shows that this creates serious difficulties for the Forms, whether the predicate is taken in the same sense or in totally different senses. Section II presents the evidence (...) that Plato had a third way of applying that predicate which obviates those problems. Finally, section III explains how predication by analogy requires a kinds of reality theory, but is incompatible with a degrees of reality theory. Thus, Plato’s kinds of reality theory validates the third way of predication discussed in section II, which in turn solves the problems enumerated in section I. (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)

This book presents a rigorous foundation for defining Boolean categories, in which the relationship between specification and behavior is explored. Boolean categories provide a rich interface between program constructs and techniques familiar from algebra, for instance matrix- or ideal-theoretic methods. The book's distinction is that the approach relies on a single program construct (the first-order theory of categories); others are derived mathematically from four axioms. Development of these axioms (which are obeyed by an abundance of program paradigms) yields Boolean algebras (...) of "predicates" loop-free constructs, and a calculus of partial and total correctness, which is shown to be the standard one of Hoare, Dijkstra, Pratt, and Kozen. (shrink)

We estimate the derivation lengths of functionals in Gödel's system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a (...) collapsing f unction\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal D}:T\to \omega$\end{document} so that for (closed) terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a,b$\end{document} of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} we have: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a$\end{document} reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $b$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega>{\cal D}(a)>{\cal D}(b).$\end{document} By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence new proof of strongly uniform termination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of Troelstra's result on strong normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. Additionally, a slow growing analysis of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document}-calculus. (shrink)

This paper explores the relationship between language, subjectivity and teaching in Emmanuel Levinas’s Totality and Infinity. It aims to elucidate Levinas’s presentation of language as always already predicated on a relationship of responsibility towards that which is beyond the self and the idea that it is only in this condition of being responsible that we are subjects. Levinas suggests that the relation with the Other through which I am a subject as one uniquely responsible is also the scene of (...) teaching. Through examining these ethical conditions of subjectivity, I suggest that this notion of the self as oriented towards the Other in a relation of passivity presents a challenge to many of the standard topoi of teaching and learning and invites us to consider the nature of teaching in a provocative new manner. (shrink)

Interesting observations and problems emerge as the author pursues a complete calculus of an adverbial logic. Author Bartsch establishes a total of 18 adverbial subcategories and proposes an equal number of logico-semantic basic constructions in a predicate logical notation to explain them. The logico-semantic basic constructions bring out certain aspects of adverbial semantics, insofar as they can be specified by predicate logical means. However, for a logical and linguistic analysis of adverbials to be complete, sentence semantic analysis - provided it (...) is carried out in a clear-cut manner with clear paraphrasing criteria, unambiguous notation and complete interpretation rules - must be supplemented by and combined with 'word semantics' and syntax in a more elaborate way. The topic of adverbial semantics is a complicated and rewarding field, and this book contains enough suggestions, challenging hypotheses and stimulating observations, so that anybody directly concerned with the study of adverbials will want to have a look. (shrink)

The aim of this paper is to present a strongly complete first order functional predicate calculus generalized to models containing not only ordinary classical total functions but also arbitrary partial functions. The completeness proof follows Henkin’s approach, but instead of using maximally consistent sets, we define saturated deductively closed consistent sets . This provides not only a completeness theorem but a representation theorem: any SDCCS defines a canonical model which determine a unique partial value for every predicate symbol and any (...) function symbol. Any SDCCS can thus be interpreted as an epistemic state. (shrink)

In this work Ockham proposes a theory of simple predication, which he uses in explicating the truth conditions of progressively more complicated kinds of propositions. His discussion includes what he takes to be the correct semantic treatment of quantified propositions, past tense and future tense propositions, and modal propositions, all of which are receiving much attention from contemporary philosophers. He also illustrates the use of exponential analysis to deal with propositions that prove troublesome in both semantic theory and other disciplines, (...) such as metaphysics, physics, and theology. This type of analysis plays an essential role in his substantive philosophical and theological works, and in many cases then can hardly be understood without a prior acquaintance with this section of the Summa. An introduction by Alfred J. Freddoso clarifies and summarizes Ockham's theory, discusses certain philosophical problems which it engenders, and proposes new interpretations for parts of it. The essay will be helpful both to those interested in Ockham as a historical figure and to those interested in his substantive systematic contributions to logical theory. (shrink)

This chapter, which provides a discussion on the phenomenological foundations of predicative structure, first introduces the different steps in Edmund Husserl's argument. It is noted that not all judgements can be equated with a nexus between a conceptual function and an argument. The predicative structure shows the orientation of consciousness towards one or more objects taken as a theme of interest and utterance. The genetic phenomenological perspective posits the question of the origin of predicative judgement within a much larger context. (...) Husserl's account amounts to granting a paradigmatic character to the theory of experience of ultimate individual substrates and to the theory of perceptual judgement. Levinasian phenomenology implies a radical rethinking of the primacy of the predicative apophansis. It is noted that language can never be taken as an independent and autonomous dimension. (shrink)

This book claims that Aristotle followed an aspect theory of predication.

Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the settheoretic universe is as an indefinite (or “potential”) totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (or “completed”) totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, (...) in the case of predicative analysis, the natural numbers are considered to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. Various such semiconstructive systems of analysis and set theory are formulated here and their proof-theoretic strength is characterized. Interestingly, though the logic is weakened, one can in compensation strengthen certain principles in a way that could be advantageous for mathematical applications. (shrink)

Our topic is the theory of topics. My goal is to clarify and evaluate three competing traditions: what I call the way-based approach, the atom-based approach, and the subject-predicate approach. I develop criteria for adequacy using robust linguistic intuitions that feature prominently in the literature. Then I evaluate the extent to which various existing theories satisfy these constraints. I conclude that recent theories due to Parry, Perry, Lewis, and Yablo do not meet the constraints in total. I then introduce the (...) issue-based theory—a novel and natural entry in the atom-based tradition that meets our constraints. In a coda, I categorize a recent theory from Fine as atom-based, and contrast it to the issue-based theory, concluding that they are evenly matched, relative to our main criteria of adequacy. I offer tentative reasons to nevertheless favour the issue-based theory. (shrink)

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

Kant took up the issue of origin in the Metaphysical Deduction of the Categories. He sought to demonstrate that the concepts of metaphysics, considered in themselves, are mere logical functions, that is, ways of synthesizing concepts to form judgments Accordingly, the metaphysical concept of substance/accident contains nothing more than the logical form of subject/predicate, whereby any arbitrary pair of concepts may be united in a judgment; cause and effect merely the hypothetical form of judgment, whereby any arbitrary pair of judgments (...) may be united as condition and conditioned; totality the form of subordination of one concept to another whereby any species of the one has the other as its genus; and so forth. Logical functions are what remain after we abstract from all content of the concepts in a judgment; they are merely the various ways in which any concept, or judgment, may be combined with any other, and so are utterly devoid of objective sense or signification. More importantly, they comprise the totality of that which may be attributed to the understanding taken by itself, in isolation from the other faculties of the mind: sensibility, desire, and feeling. The understanding, for Kant, is the capacity to judge ; it is not, when considered without relation to other faculties, a faculty of concepts, rules, principles, or even categories. Indeed, if categories may be said to express any content at all, it is simply and solely the relation of the understanding to another faculty. In the case of sensibility--a faculty comprising the manifold of the senses and its synthesis in imagination--categories are obtained when this synthesis is represented universally and a priori, and so represent the "pure synthetic unity of the manifold in general." Similarly, the categories tabulated in the Critique of Practical Reason express the relation of the capacity to judge to the capacity of desire. Thus, aside from "their possibility as a priori cognitions of objects of an intuition in general," the categories of the first Critique are nothing over and above sheer logical forms, through which no objective necessary connection whatsoever can be thought; indeed, so entirely void of all content and objective import are they that even Hume almost certainly would have countenanced them. (shrink)

of monadic or relational predicate calculus (Fa, Gb, Rab, Hcd, etc.). • The Boolean Algebra BL set-up by such a language will be such that: – BL will have 2 n states (corresponding to the state descriptions of L) – BL will contain 2 2n propositions, in total. ∗ This is because each proposition p in BL is equivalent to a disjunction of state descriptions. Thus, each subset of the set of..

Ethics is a foundational competency in healthcare inherent in everyday nursing practice. Therefore, the promotion of qualified nurses’ and nursing students’ moral competence is essential to ensure ethically high-quality and sustainable healthcare. The aim of this integrative literature review is to identify the factors contributing to the promotion of qualified nurses’ and nursing students’ moral competence. The review has been registered in PROSPERO (CRD42023386947) and reported according to the PRISMA guideline. Focusing on qualified nurses’ and nursing students’ moral competence, a (...) literature search was undertaken in January 2023 in six scientific databases: CINAHL, Cochrane Library, PsycINFO, PubMed Medline, Scopus and Web of Science. Empirical studies written in English without time limitation were eligible for inclusion. A total of 29 full texts were retrieved and included out of 5233 citations. Quality appraisal was employed using Joanna Briggs Institute checklists and the Mixed Method Appraisal Tool. Data were analysed using inductive content analysis. Research about the factors contributing to the promotion of qualified nurses’ and nursing students’ moral competence is limited and mainly explored using descriptive research designs. The contributing factors were identified as comprising two main categories: (1) human factors, consisting of four categories: individual, social, managerial and professional factors, and ten sub-categories; and (2) structural factors, consisting of four categories: educational, environmental, organisational and societal factors, and eight sub-categories. This review provides knowledge about the factors contributing to the promotion of qualified nurses’ and nursing students’ moral competence for the use of researchers, nurse educators, managers, organisations and policymakers. More research about the contributing factors is needed using complex intervention, implementation and multiple methods designs to ensure ethically sustainable healthcare. (shrink)

Kreisel’s conjecture is the statement: if, for all$n\in \mathbb {N}$,$\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$, then$\mathop {\text {PA}} \nolimits \vdash \forall x.\varphi (x)$. For a theory of arithmeticT, given a recursive functionh,$T \vdash _{\leq h} \varphi $holds if there is a proof of$\varphi $inTwhose code is at most$h(\#\varphi )$. This notion depends on the underlying coding.${P}^h_T(x)$is a predicate for$\vdash _{\leq h}$inT. It is shown that there exist a sentence$\varphi $and a total recursive functionhsuch that$T\vdash (...) _{\leq h}\mathop {\text {Pr}} \nolimits _T(\ulcorner \mathop {\text {Pr}} \nolimits _T(\ulcorner \varphi \urcorner )\rightarrow \varphi \urcorner )$, but, where$\mathop {\text {Pr}} \nolimits _T$stands for the standard provability predicate inT. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By the use of reflexion principles, one can obtain a theory$T^h_\Gamma $that extendsTsuch that a version of Kreisel’s conjecture holds: given a recursive functionhand$\varphi (x)$a$\Gamma $-formula (where$\Gamma $is an arbitrarily fixed class of formulas) such that, for all$n\in \mathbb {N}$,$T\vdash _{\leq h} \varphi (\overline {n})$, then$T^h_\Gamma \vdash \forall x.\varphi (x)$. Derivability conditions are studied for a theory to satisfy the following implication: if, then$T\vdash \forall x.\varphi (x)$. This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a functionhsuch that$\vdash _{k \text { steps}}\ \subseteq\ \vdash _{\leq h}$. (shrink)

In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...) maps and show that it provides a sound and complete semantics for constructive set theories like IZF and CZF. The main technical contribution of this paper is that it shows in detail that such categories can always be conservatively embedded in categories that are exact. These exactness properties play a crucial rôle in showing that predicative categories with small maps contain models of set theory and that they are closed under sheaves and realizability. We will prove the former statement in this paper as well, leaving a proof of the closure properties to the papers on realizability and sheaves as mentioned above. (shrink)

For his contribution to the general series of Harper Essays in Philosophy, Hilary Putnam selects only one of several philosophical problems in the interrelated fields of logic and/or mathematics that have interested him, viz. the nominalism-realism issue: Are the "abstract entities" spoken of in these sciences, such as classes, number, functions from various kinds of things to real numbers, things that "really exist" or not? He is concerned to present a detailed argument for his own "qualified realism" rather than a (...) survey of current opinions. He argues persuasively that while a nominalist would like to eliminate all talk of logical classes such as S, M and P in favor of such circumlocutions as "The following schema turns out to be true no matter what words or phrases of the appropriate kind one may substitute for the letters S, M and P," etc., such descriptions avoid references to classes only by replacing them with equally nonphysical entities such as "all possible words, strings of letters, formalized languages," etc., and that the notion of truth or validity itself cannot be applied to physical objects or material inscriptions, but to what sentences mean or say, and the things they say are not physical. He rejects Quine's distinction that first order statements or arguments such as "All crows are black; all back things absorb light, therefore," etc., are logical truths, whereas "For all classes S, M and P, if all S is M and all M is P, then all S is P," are mathematical. This runs counter, he says, to the whole logical tradition. The natural intention of the logician in setting down first-order schemata is to assert implicitly their validity and that itself is a second-order assertion. All logic, therefore, including quantification theory, involves references to classes. This also highlights the difficulty of distinguishing between logic and mathematics in terms of first and second order logic, for if only the former is stipulated to be the proper domain of logic, then we must accept the awkward consequence that validity and implication turn out to be notions that belong to mathematics rather than logic. Since the nominalistic position presents the same problems for mathematics as for logic, Putnam sees no point in trying to draw any sharp distinction between the two, which explains his initial broad interpretation of logic as including all of pure mathematics and also the title of the book. To avoid some of the classical objections to a realistic interpretation of sets, he points up the distinction between a predicative and impredicative conception of set. Whereas the latter speaks of all sets of individuals as a well-defined totality, the former defines "set" only for a particular language N or a hierarchic series of languages, N, N', N", etc. in which it is still nonsensical to speak of "all sets" without qualification, though one can speak of "all" up to any given level in the series. While some form of set theory is necessary for all the sciences, both formal and physical, one can get along with predicative set theory. Putnam's basic argument for realism then comes down to this. Quantification over mathematical entities such as things, real numbers and functions is indispensable for logic, mathematics and the physical sciences and this commits one to accept the existence of such entities. In the final stages of the argument he examines and rejects the alternative options to realism, namely that realistic locutions about such entities represent logically deviant forms of speech, or that mathematical truths are "true by pure convention" or that such "entities" are useful fictions. In the closing paragraphs he indicates two other philosophical problems in logic that merit extensive discussion, namely the problem of equivalent constructions which is relevant to the question of whether mathematics needs "foundations" and the problem of whether we may not have to revise logical principles as we have geometrical ones on the grounds that in some sense logic may have empirical foundations.--A. B. W. (shrink)

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor differ...

Debates about the semantics and pragmatics of predicates of personal taste have largely centered on contextualist and relativist proposals. In this paper, I argue in favor of an alternative, absolutist analysis of PPT. Theorists such as Max Kölbel and Peter Lasersohn have argued that we should dismiss absolutism due to its inability to accommodate the possibility of faultless disagreement involving PPT. My aim in the paper is to show how the absolutist can in fact accommodate this possibility by drawing (...) on an account of faultless disagreement that improves upon a recent proposal due to Karl Schafer. In amending Schafer’s proposal, I put forward an empirically informed view of our beliefs regarding matters of personal taste, as well as an account of our assertions concerning such matters. I also argue that absolutists should take disagreement about these matters to be conative, rather than doxastic, in nature. The anticipated result is an independently compelling account of faultless disagreement about matters of personal taste that fits naturally with absolutism. (shrink)

I will present and criticise the two theories of truthmaking David Armstrong offers us in Truth and Truthmakers (Armstrong 2004), show to what extent they are incompatible and identify troublemakers for both of them, a notorious – Factualism, the view that the world is a world of states of affairs – and a more recent one – the view that every predication is necessary. Factualism, combined with truthmaker necessitarianism – ‘truthmaking is necessitation’ – leads Armstrong to an all-embracing totality (...) state of affairs that necessitates not only everything that is the case but also everything else – that which is not the case, that which is merely possible or even impossible. All the things so dear to realists – rocks, natural properties, real persons – become mere abstractions from this ontological monster. The view that every predication is necessary does in some sense the opposite: it does away with totality states of affairs and, arguably, also with states of affairs. We have particulars and universals, partially identical and necessarily connected to everything else. Just by the existence of anything, everything is necessitated – the whole world mirrored in every monad. Faced with the choice between these two equally unappealing alternatives, I suggest returning to Armstrong’s more empiricist past: the world is not an all-inclusive One, nor necessitated by every single particular and every single universal, but a plurality of particulars and universals, interconnected by a contingent and internal relation of exemplification. While a close variant, truthmaker essentialism, can perhaps be saved, this means giving up on truthmaker necessitarianism. This, I think, what it takes to steer a clear empiricist course between the Scylla of Spinozist general factness and the Charybdis of a Leibnizian overdose of brute necessities. (shrink)

Wittgenstein, in the Tractatus, conceives the world as ‘the totality of facts’. Type-stratification threatens that conception : the totality of facts is an obvious example of an illegitimate totality. Wittgenstein’s notion of truthoperation evidently has some role to play in avoiding that threat, allowing propositions, and so facts, to constitute a single type. The paper seeks to explain that role in a way that integrates the ‘philosophical’ and ‘technical’ pressures on the notion of an operation.

In this paper I investigate Aristotle’s account of predication in Topics I 9. I argue for the following interpretation. In this chapter Aristotle (i) presents two systems of predication cutting across each other, the system of the so-called four ‘predicables’ and of the ten ‘categories’, in order to distinguish them and explore their mutual relationship. I propose a semantic interpretation of the relationship between them. According to this reading, every proposition formed through a predicable constitutes at the same time a (...) predication according to one of the ten categories, and, consequently, signifies one of them, expressing one of the predicative relationship conveyed by them. Further, Aristotle (ii) explains the predicative connection between these two systems and the ten items signified by the ‘things said without any combination’ enumerated in Chapter 4 of the Categories, whose list is almost identical with that of categories in Top. I 9, with the only exception of their first members. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Diacritics 29.3 (1999) 63-80 [Access article in PDF] The Transversality of Michel de Certeau: Foucault's Panoptic Discourse and the Cartographic Impulse Bryan Reynolds and Joseph Fitzpatrick Above all (and this is a corollary, but an important one), the phenomenological and praxiological analysis of cultural trajectories must allow to be grasped at once a composition of places and the innovation that modifies it by dint of moving and cutting across (...) them. --Michel de Certeau, Culture in the PluralThrough his rigorous examinations of the common, the quotidian, the personal, the plural practices that establish and continually transform societies, Michel de Certeau develops a conception of culture born of the perpetual transgression, however subtle or pronounced, of the boundaries imposed by all totalizing systems, whether theoretical or social. This view of culture as a "cutting across" of boundaries, which can be found in many forms throughout his work, achieves a special significance for the study of subjectification in his discussion of "Spatial Practices" that comprises the third section of The Practice of Everyday Life. In an important essay on Certeau's theory of space, Ian Buchanan argues that "our understanding of culture must commence with an understanding of the formation of the subject," and that the latter is addressed by Certeau through his essays on space [129-30]. Following this assertion, we want to argue that Certeau's analyses of spatial practices in Heterologies, "The Gaze: Nicholas of Cusa," and The Practice of Everyday Life [particularly the chapter "Walking in the City" 91-110], outline a theory that describes subjectivity as what amounts to, as Certeau puts it in Culture in the Plural, "a composition of places and the innovation that modifies it by dint of moving and cutting across them" [146]. Furthermore, by articulating the close relationship between Certeau's spatial theory and our own work on "transversal movement," we will attempt to show that the method of analysis Certeau uses in "Walking in the City" (and describes in several other works) is itself marked by the "transversal" tactics that his theory describes. 1 By focusing in particular on Certeau's methodological critique of Foucault in both The Practice of Everyday Life and Heterologies, we will demonstrate the various ways in which he moves beyond what Bryan Reynolds and James Intriligator have labeled the "dissective-cohesive mode" of analysis, an analytical [End Page 63] [Begin Page 65] approach (characteristic of most dialectical argumentation) that breaks its subject matter into constituent parts and examines those parts with the goal of reassembling them into a unified and accountable whole (such as, say, the city or the human subject in "Walking in the City"). 2 Postmodern Space: Certeau and Jameson In his essay "Heterophenomenology, or Certeau's Theory of Space," Ian Buchanan situates Certeau's writings on the production of space within the postmodern debate over the autonomy of the subject, opposing Certeau's view to Fredric Jameson's "assumption that the subject takes his or her psychic bearings from the built environment and only has certain existence so long as he or she can cognitively 'map' this environment" [Buchanan 114]. Citing the example of the Westin Bonaventure hotel in Los Angeles, Jameson argues that "this latest mutation in space--postmodern hyperspace--has finally succeeded in transcending the capacities of the individual human body to locate itself, to organize its immediate surroundings perceptually, and cognitively to map its position in a mappable external world" [44]. The effect on cultural politics, according to Jameson, is that the subject "submerged" by this postmodern hyperspace is deprived of the "critical distance" that makes possible the "positioning of the cultural act outside of the massive Being of capital" [48]. Buchanan finds in "Walking in the City" the basis for a critique of this view though a conception of space stemming from "heterophe-nomenology," defined as "a phenomenology predicated by a heterogeneously constituted subject which does not take for granted the unity of the body" [112]. Focusing on Certeau's debt to Merleau-Ponty's theories of perception (which can best be seen in... (shrink)

Armstrong appeals to the existence of totalities in order to solve the problem of negative truths. The totality of first-order states of affairs is a truthmaker for all negative truths, but it involves things which are irrelevant to many such truths. To solve this problem, Armstrong claimed that negative truths have minimal truthmakers which usually consist in totalities smaller than the totality of first-order states of affairs. Merricks objects to this claim by arguing that given Armstrong’s definition of (...) minimal truthmakers, the totality of first-order states of affairs is a minimal truthmaker for every negative truth. In this article, I respond to Merricks’s objection. I will first show that the definition of minimal truthmakers that he proposes is not plausible. However, Merricks’s objection, I will argue, also works on the standard definition of minimal truthmakers. I will then show that for independent reasons, the standard definition should be revised, and, given my revised definition, Merricks’s objection fails. (shrink)

A unified theory is offered to account for three types of definite descriptions: with singular, plural, & mass predicates, & to provide an account for the word the in descriptions. It is noted that B. Russell's analysis ("On Denoting," Mind, 1905, 14, 479-493) failed to account for plural & mass descriptions. The proposed theory differs from Russell's only by the substitution of the notation (less than or equal to) for Russell's =. It is suggested that for every predicate G (...) there is an appropriate "part of" or "some of" relation on the extension of G. For singular predicates, (less than or equal to) becomes just the identity relation =. The predicate G is formulated as "the G" & analyzed with the formula ((schwa)x) (Gx . (Gy (includes) yy (less than or equal to) x)), which accounts for all cases of definite descriptions. It is suggested that the primary use of the is not to indicate uniqueness, but totality. C. Ornatowski. (shrink)

In the debate between contextualism and relativism about predicates of taste, the challenge from disagreement (the objection that contextualism cannot account for disagreement in ordinary exchanges involving such predicates) has played a central role. This paper investigates one way of answering the challenge consisting on appeal to certain, less focused on, uses of predicates of taste. It argues that the said thread is unsatisfactory, in that it downplays certain exchanges that constitute the core disagreement data. Additionally, several (...) arguments to the effect that the exchanges in question don’t amount to disagreement are considered and rejected. (shrink)

A formal semantic analysis of the _tough_-construction is provided building on the well-known observation that _events_ play a central role. A close look at the semantic characteristics of the class of _tough_-predicates and the syntactic and semantic properties of nonfinite clauses reveals the link between these pieces, expanding on recent advances in the semantics of clauses (Moulton in Natural selection and the syntax of clausal complementation, PhD thesis, University of Massachusetts at Amherst, 2009 ). Building on Salzmann (Reconstruction and (...) resumption in indirect A′-dependencies: On the syntax of prolepsis and relativization in (Swiss) German and beyond, de Gruyter, Berlin 2017b ), a formal semantic and syntactic analysis of prolepsis is provided to explain the antecedent gap chain in the _tough_-construction. In total, this paper offers a description and explanation of (i) the class of _tough_-predicates; (ii) the properties of the nonfinite clause that appears in the _tough_-construction; (iii) why no other predicates or clauses are permitted in the construction; (iv) the many properties of the antecedent gap chain in the _tough_-construction, primarily the fact that the chain is “weakly” unbounded; and (v) the semantic contribution of prolepsis as it applies in the _tough_-construction. (shrink)

The fundamental belief in destiny in Yoruba culture is explained within the tradition that for every individual person who comes to aye (earth), there is a package of destiny containing the totality of all that such person will be. However, the content of this destiny is not known to any person except Orunmila, one of the deities. Therefore, it is believed that a person dies if and when he/she has exhausted the content of his/her ori (package of destiny). Included (...) also in the Yoruba belief system is that a youthful death is a sorrowful death. This is predicated on the premise that a young man could not have completed the content of his earthly mission. His death is therefore sorrowful, and he could therefore not be admitted into Orun(“heaven”) to join the league of the ancestors. This is the explanation for the belief in reincarnation, and, more specifically, the belief in akudaaya or abarameji(reincarnated persons).This paper argues for two main points. First, in spite of the Yoruba belief that no human being is privy to the content of destiny, the Yoruba belief in sorrowful or sudden death (ikuofo or iku-ojiji) presupposes some knowledge of the content of each person’s destiny. Second, the beliefs in destiny on one hand, and in sudden or sorrowful death on the other, are mutually exclusive. This means that if one is true the other will be false and vice-versa, so that holding the two together would be contradictory. Because such contradictions are common in the entire gamut of Yoruba belief, there is need for a critical examination of theYoruba belief system in the quest for a modern culture. (shrink)