Linear axiomatics of commutative product-free Lambek calculus

Axiomatics which do not employ rules of inference other than the cut rule are given for commutative product-free Lambek calculus in two variants: with and without the empty string. Unlike the former variant, the latter one turns out not to be finitely axiomatizable in that way.     

A schema of deduction theorems for the propositional calculus

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A schema of deduction theorems for the propositional calculus

     

On Finite Models of the Lambek Calculus

We study a class of finite models for the Lambek Calculus with additive conjunction and with and without empty antecedents. The class of models enables us to prove the finite model property for each of the above systems, and for some axiomatic extensions of them. This work strengthens the results of [3] where only product-free fragments of these systems are considered. A characteristic feature of this approach is that we do not rely on cut elimination in opposition to e.g. [5], [9].     

Cross-cultural second order factor structures of the 16PF

Cross-cultural personality research has generated a great amount of data on individual difference patterns in diverse cultures. One of the major instruments used in this research has been Cattell's 16PF. A major question in this research is whether the underlying personality structure is equivalent for different cultures. The present study evaluated the second order factor structure of the 16PF in 101 subjects of European ancestry and 117 subjects of Japanese ancestry. The factor structure for the Japanese was significantly different from that of the caucasian group. The caucasian results did not differ from those reported by Cattell and his associates. The implications of these results for personality theory and for cross-cultural evaluation were briefly discussed.     

A probabilistic theory of second order causation

Larry Wright and others have advanced causal accounts of functional explanation, designed to alleviate fears about the legitimacy of such explanations. These analyses take functional explanations to describe second order causal relations. These second order relations are conceptually puzzling. I present an account of second order causation from within the framework of Eells' probabilistic theory of causation; the account makes use of the population-relativity of causation that is built into this theory.     

A Rule-Extension of the Non-Associative Lambek Calculus

An extension L ⁺ of the non-associative Lambek calculus Lis defined. In L ⁺ the restriction to formula-conclusion sequents is given up, and additional left introduction rules for the directional implications are introduced. The system L ⁺ is sound and complete with respect to a modification of the ternary frame semantics for L.     

On the Complexity of Nonassociative Lambek Calculus with Unit

Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek Calculus with unit (NL1). Moreover, we show that any Lambek grammar based on NL1 (with assumptions) can be transformed into an equivalent context-free grammar in polynomial time.     

Kreisel,the continuum hypothesis and second order set theory

Summary The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion is correct, his arguments are unsatisfactory. Later sections of the paper advance a different argument that the independence results do not show lack of truth values.     

The role of reversal frequency in learning noisy second order conditional sequences

The hallmark of implicit learning is that complex knowledge can be acquired unconsciously. The second order conditionals (SOCs) of Reed and Johnson (1994) were developed to be complex, and they are popular materials for implicit learning research. Recently, it was demonstrated that in a sequence made noisy (by combining two SOCs), shared features of the SOCs may be learned explicitly (Fu, Fu, & Dienes, 2008). What are these shared features? We hypothesized that low reversal frequency may play a significant role. We have varied reversal frequency, and discovered that reversal frequency affected response times, inclusion exclusion behavior, and recognition ratings. Not only does it appear to be important to distinguish implicit and explicit knowledge, but also to distinguish what the knowledge is of.     

The second order approximation to sample influence curve in canonical correlation analysis

A second order approximation to the sample influence curve (SIC) in canonical correlation analysis has been derived in the literature. However, it does not seem satisfactory for some cases. In this paper, we present a more accurate second order approximation. As a particular case, the proposed method is exact for the SIC of the squared multiple correlation coefficient. An example is given. The authors are most grateful to the associate editor and three reviewers for valuable comments and suggestions which improved the presentation of the paper considerably. The first author was partly supported by a RGC earmarked research grant of Hong Kong.     

Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic

We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.