Application scenarios for nonstandard log-linear models

In this article, the authors have 2 aims. First, hierarchical, nonhierarchical, and nonstandard log-linear models are defined. Second, application scenarios are presented for nonhierarchical and nonstandard models, with illustrations of where these scenarios can occur. Parameters can be interpreted in regard to their formal meaning and in regard to their magnitude. The interpretation of the meaning of parameters is the main focus of this article. Design matrices are used to describe the hypotheses tested in models and to illustrate cases in which parameters are interpretable. Also, design matrices are used to show where and how nonstandard models differ from standard hierarchical models. Coding schemes are discussed, in particular, dummy coding and effects coding. Data examples are given with data and models discussed in the literature.     

Reference to numbers in natural language

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted.     

Multicomponent models for the retention of numbers

Summary Several multicomponent models of memory (Bower, 1967) are presented and applied to the retention of three-digit numbers. Their application is based on the notion that both the digits making up a number stimulus and the position in which they appear can be interpreted as the components of a memory vector. The models are fitted to the data obtained in a Peterson-type experiment in which a single three-digit number was shown per trial. During the retention interval the subject engaged in an arithmetical task for either 13.6, 19.6, or 25.6 s. Following this, the subject attempted recall. Confidence ratings were also obtained. Twenty-four different types of correctly and incorrectly recalled responses were scored. Their frequency distributions were best predicted at each retention interval by a dual encoding model which relies on the assumption that stimuli are stored both component-wise and by means of a single unit code encoding the entire stimulus number. It is also shown that the confidence ratings may be successfully predicted from estimates of the expected number of components retained. About 80% of the rating variance is predicted by a parameter-free procedure.The original experimental study was funded by Grant Ey 4/3, Deutsche Forschungsgemeinschaft, Bad Godesberg. Further developments were made possible by Grant Sch 350/1 to R.S. from the same agency. Part of the theoretical work was done while D.V. was an NSF postdoctoral fellow at the Department of Psychology of New York University, New York. Free computer time was generously supplied by the Courant Institute of Applied Mathematics. We are grateful to Dietrich Albert, Stephanie Kelter, Micha Razel, and Paul Vitz for stimulating discussions.     

Monoidal categories with natural numbers object

The notion of a natural numbers object in a monoidal category is defined and it is shown that the theory of primitive recursive functions can be developed. This is done by considering the category of cocommutative comonoids which is cartesian, and where the theory of natural numbers objects is well developed. A number of examples illustrate the usefulness of the concept.This work was partially carried out while both authors were guests of McGill University and while the second author was a guest of Dalhousie University. Both authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada and the Québec Department of Education.     

Bayesian inference for psychology,part III: Parameter estimation in nonstandard models

We demonstrate the use of three popular Bayesian software packages that enable researchers to estimate parameters in a broad class of models that are commonly used in psychological research. We focus on WinBUGS, JAGS, and Stan, and show how they can be interfaced from R and MATLAB. We illustrate the use of the packages through two fully worked examples; the examples involve a simple univariate linear regression and fitting a multinomial processing tree model to data from a classic false-memory experiment. We conclude with a comparison of the strengths and weaknesses of the packages. Our example code, data, and this text are available via https://osf.io/ucmaz/.     

The Definability of the set of natural numbers in the 1925 Principia Mathematica

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Gödel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered.     

A note on natural numbers objects in monoidal categories

The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.     

Inducing ordinal and cardinal representations of the first five natural numbers

The prediction that the ordinal property of natural number symbols (using these symbols to represent the terms in an ordered progression) is more easily learned than the cardinal property of natural number symbols (using these symbols to represent the manyness of collections) was examined in this experiment. Preschoolers who evidenced no proficiency with either the ordinal or cardinal properties of natural number symbols were trained to acquire these properties via simple feedback. Both properties proved to be trainable. The most important findings were that the ordinal property was much easier to train than the cardinal property, ordinal training effects were more durable across a 1-week interval than cardinal training effects, and ordinal training appeared to transfer better than cardinal training.     

A coding of the countable linear orderings

Associate to any linear ordering on the integers the mapping whose value on n is the cardinality of {k<n; kn}: a purely combinatorial characterization for the mappings associated to the well-orderings is established.This work was partially supported by a CNRS grant PRC mathématiques and informatique.     

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.     

How to learn the natural numbers: inductive inference and the acquisition of number concepts

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51-B60.] argue that such an inductive inference is consistent with a representational system that clearly does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles.     

Some nonstandard methods in combinatorial number theory

A combinatorial result about internal subsets of ^*N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdös and Graham.The author wishes to thank the logic group at the University of Wisconsin, and especially Professor Keisler, for their generous support.     

Man-Machine interaction in natural language: Computational models of dialogue

This article presents a state-of-the-art review of studies on computational modeling of dialogue. Particular attention is given to the treatment of ill-formed input, the prevention of the hearer’s misconceptions, the inference of the speaker’s plans, the generation of language, and the recognition of dialogue focus—particularly in terms of implications for linguistics. Specifications and initial results of some recent studies carried out by the author are provided. This article is a revised version of a lecture held at the Third Scientific Meeting “Computer Processing of Linguistic Data,” Bled, 1985.     

All roads lead to violations of countable additivity

This paper defends the claim that there is a deep tension between the principle of countable additivity and the one-third solution to the Sleeping Beauty problem. The claim that such a tension exists has recently been challenged by Brian Weatherson, who has attempted to provide a countable additivity-friendly argument for the one-third solution. This attempt is shown to be unsuccessful. And it is argued that the failure of this attempt sheds light on the status of the principle of indifference that underlies the tension between countable additivity and the one-third solution.     

The countable homogeneous universal model of B 2

We give a detailed account of the Algebraically Closed and Existentially Closed members of the second Lee class B ₂ of distributive p-algebras, culminating in an explicit construction of the countable homogeneous universal model of B ₂. The axioms of Schmid [7], [8] for the AC and EC members of B ₂ are reduced to what we prove to be an irredundant set of axioms. The central tools used in this study are the strong duality of Clark and Davey [3] for B ₂ and the method of Clark [2] for constructing AC and EC algebras using a strong duality. Applied to B ₂, this method transfers the entire discussion into an equivalent dual category X ₂ of Boolean spaces which carry a pair of tightly interacting orderings. The doubly ordered spaces of X ₂ prove to be much more readily constructed and analyzed than the corresponding algebras in B ₂.     

Absolutely independent axiomatizations for countable sets in classical logic

The notion of absolute independence, considered in this paper has a clear algebraic meaning and is a strengthening of the usual notion of logical independence. We prove that any consistent and countable set in classical prepositional logic has an absolutely independent axiornatization.