Categoricity

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a system.     

Why Numbers Are Sets

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.     

Giving the boot to the bootstrap: how not to learn the natural numbers

According to one theory about how children learn the concept of natural numbers, they first determine that "one", "two", and "three" denote the size of sets containing the relevant number of items. They then make the following inductive inference (the Bootstrap): The next number word in the counting series denotes the size of the sets you get by adding one more object to the sets denoted by the previous number word. For example, if "three" refers to the size of sets containing three items, then "four" (the next word after "three") must refer to the size of sets containing three plus one items. We argue, however, that the Bootstrap cannot pick out the natural number sequence from other nonequivalent sequences and thus cannot convey to children the concept of the natural numbers. This is not just a result of the usual difficulties with induction but is specific to the Bootstrap. In order to work properly, the Bootstrap must somehow restrict the concept of "next number" in a way that conforms to the structure of the natural numbers. But with these restrictions, the Bootstrap is unnecessary.     

Existence,presupposition and anaphoric space

The following paper deals with the notion of existence, especially as concerns natural languages. In Section 1, starting from some quite obvious examples drawn from logic, I sketch the problem of the existential presupposition usually ascribed to noun phrases. My opinion is that the point of view frequently adopted in this case is unduly restrictive, for the existence which is believed to be presupposed here is actual existence. Accordingly, I emphasize the need for having a weaker notion of existential presupposition, such that the existence (if this word can still be used) here referred to is relevant only to linguistic goals. Section 2 sketches this notion, by assimilating existence (in the weak sense) to identification in a linguistic space. (I deal here only with intuitive considerations: a more formal account will be given, I hope, in another paper.) Finally, in Section 3, the notion of actual existence is examined by contrast with the linguistic (or weak) notion of existence: and this is a question which of course can't be tackled in terms of a purely linguistic analysis, for it needs a general, epistemo-logical approach.     

弗雷格《概念文字》理解的两点注记

文章旨在简要地讨论弗雷格《概念文字》,指出其中的两个重要但被一些国内学者误解或忽略的贡献：首先我们指出,根据Boolos等人的论证,弗雷格《概念文字》中的逻辑本质上是带完整二阶存在概括规则的二阶逻辑,这点在国内一些学者的著作与文章中存在误解;其次,我们讨论弗雷格如何用遗传性概念来定义祖先关系,进而定义自然数或有穷数,并使得数学归纳法仅根据自然数的定义就得以成立,这也为弗雷格把算术还原为逻辑奠定了基础。     

Syntactic features and synonymy relations: a unified treatment of some proofs of the compactness and interpolation theorems

This paper introduces the notion of syntactic feature to provide a unified treatment of earlier model theoretic proofs of both the compactness and interpolation theorems for a variety of two valued logics including sentential logic, first order logic, and a family of modal sentential logic includingM,B,S ₄ andS ₅. The compactness papers focused on providing a proof of the consequence formulation which exhibited the appropriate finite subset. A unified presentation of these proofs is given by isolating their essential feature and presenting it as an abstract principle about syntactic features. The interpolation papers focused on exhibiting the interpolant. A unified presentation of these proofs is given by isolating their essential feature and presenting it as a second abstract principle about syntactic features. This second principle reduces the problem of exhibiting the interpolant to that of establishing the existence of a family of syntactic features satisfying certain conditions. The existence of such features is established for a variety of logics (including those mentioned above) by purely combinatorial arguments.Presented byMelvin Fitting     

Categoricity Theorems and Conceptions of Set

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to the pure sets of the other.This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine.     

Categoricity Spectra for Polymodal Algebras

We investigate effective categoricity for polymodal algebras (i.e., Boolean algebras with distinguished modalities). We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.     

The First-Order Theories of Dedekind Algebras

A Dedekind Algebra is an ordered pair (B,h) where B is a non-empty set and h is an injective unary function on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called configurations of the Dedekind algebra. There are N₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on omega called its configuration signature. The configuration signature of a Dedekind algebra counts the number of configurations in the decomposition of the algebra in each isomorphism type.The configuration signature of a Dedekind algebra encodes the structure of that algebra in the sense that two Dedekind algebras are isomorphic iff their configuration signatures are identical. Configuration signatures are used to establish various results in the first-order model theory of Dedekind algebras. These include categoricity results for the first-order theories of Dedekind algebras and existence and uniqueness results for homogeneous, universal and saturated Dedekind algebras. Fundamental to these results is a condition on configuration signatures that is necessary and sufficient for elementary equivalence.     

The expressibility of fragments of Hybrid Graph Logic on finite digraphs

Hybrid Graph Logic is a logic designed for reasoning about graphs and is built from a basic modal logic, augmented with the use of nominals and a facility to verify the existence of paths in graphs. We study the finite model theory of Hybrid Graph Logic. In particular, we develop pebble games for Hybrid Graph Logic and use these games to exhibit strict infinite hierarchies involving fragments of Hybrid Graph Logic when the logic is used to define problems involving finite digraphs. These fragments are parameterized by the quantifier-rank of formulae along with the numbers of propositional symbols and nominals that are available. We ascertain exactly the relative definability of these parameterized fragments of the logic.     

Finite mathematics

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.I would like to thank Jeff Barrett, Akeel Bilgrami, Leigh Cauman, John Collins, William Craig, Gary Feinberg, Haim Gaifman, Yair Guttmann, Hidé Ishiguro, Isaac Levi, James Lewis, Vann McGee, Sidney Morgenbesser, George Shiber, Sarah Stebbins, Mark Steiner, and an anonymous referee for encouragement and various useful suggestions. The research described in this article and the preparation of the article were supported in part by the Columbia University Council for Research in the Humanities.     

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become “cardinal principle” or “CP” knowers), they have a conceptual insight about how the count list implements the successor function – i.e., that every natural number n has a successor defined as n + 1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children’s ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6 years of age – roughly 2 years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.     

Logical Operations and Invariance

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski–Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.     

First Order Common Knowledge Logics

In this paper we investigate first order common knowledge logics; i.e., modal epistemic logics based on first order logic with common knowledge operators. It is shown that even rather weak fragments of first order common knowledge logics are not recursively axiomatizable. This applies, for example, to fragments which allow to reason about names only; that is to say, fragments the first order part of which is based on constant symbols and the equality symbol only. Then formal properties of "quantifying into" epistemic contexts are investigated. The results are illustrated by means of epistemic representations of Nash Equilibria for finite games with mixed strategies.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

Finite Contractions on Infinite Belief Sets

Contractions on belief sets that have no finite representation cannot be finite in the sense that only a finite number of sentences is removed. However, such contractions can be delimited so that the actual change takes place in a logically isolated, finite-based part of the belief set. A construction that answers to this principle is introduced, and is axiomatically characterized. It turns out to coincide with specified meet contraction.     

When good evidence goes bad: the weak evidence effect in judgment and decision-making

An indispensable principle of rational thought is that positive evidence should increase belief. In this paper, we demonstrate that people routinely violate this principle when predicting an outcome from a weak cause. In Experiment 1 participants given weak positive evidence judged outcomes of public policy initiatives to be less likely than participants given no evidence, even though the evidence was separately judged to be supportive. Experiment 2 ruled out a pragmatic explanation of the result, that the weak evidence implies the absence of stronger evidence. In Experiment 3, weak positive evidence made people less likely to gamble on the outcome of the 2010 United States mid-term Congressional election. Experiments 4 and 5 replicated these findings with everyday causal scenarios. We argue that this “weak evidence effect” arises because people focus disproportionately on the mentioned weak cause and fail to think about alternative causes.     

Relationships and the social brain: integrating psychological and evolutionary perspectives

Psychological studies of relationships tend to focus on specific types of close personal relationships (romantic, parent-offspring, friendship) and examine characteristics of both the individuals and the dyad. This paper looks more broadly at the wider range of relationships that constitute an individual's personal social world. Recent work on the composition of personal social networks suggests that they consist of a series of layers that differ in the quality and quantity of relationships involved. Each layer increases relationship numbers by an approximate multiple of 3 (5-15-50-150) but decreasing levels of intimacy (strong, medium, and weak ties) and frequency of interaction. To account for these regularities, we draw on both social and evolutionary psychology to argue that relationships at different layers serve different functions and have different cost-benefit profiles. At each layer, the benefits are asymptotic but the costs of maintaining a relationship at that level (most obviously, the time that has to be invested in servicing it) are roughly linear with the number of relationships. The trade-off between costs and benefits at a given level, and across the different types of demands and resources typical of different levels, gives rise to a distribution of social effort that generates and maintains a hierarchy of layered sets of relationships within social networks. We suggest that, psychologically, these trade-offs are related to the level of trust in a relationship, and that this is itself a function of the time invested in the relationship.     

Concepts and aims of functional interpretations: towards a functional interpretation of constructive set theory

The aim of this article is to give an introduction to functional interpretations of set theory given by the authorin Burr (2000a). The first part starts with some general remarks on Gödel's functional interpretation with a focus on aspects related to problems that arise in the context of set theory. The second part gives an insight in the techniques needed to perform a functional interpretation of systems of set theory. However, the first part of this article is not intended to be a complete survey of functional interpretations and here we recommend, for example, Avigad and Feferman (1998),Troelstra (1990) and Troelstra (1973).