On the Logic of β-pregroups

In this paper we concentrate mainly on the notion of β-pregroups, which are pregroups (first introduced by Lambek [18] in 1999) enriched with modality operators. β-pregroups were first proposed by Fadda [11] in 2001. The motivation to introduce them was to limit (locally) the associativity in the calculus considered. In this paper we present this new calculus in the form of a rewriting system, prove the very important feature of this system - that in a given derivation the non-expanding rules must always proceed non-contracting ones in order the derivation to be minimal (normalization theorem). We also propose a sequent system for this calculus and prove the cut elimination theorem for it. As an illustration we show how to use β-pregroups for linguistical applications. Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller     

Dynamic doxastic logic: why,how, and where to?

We investigate the research programme of dynamic doxastic logic (DDL) and analyze its underlying methodology. The Ramsey test for conditionals is used to characterize the logical and philosophical differences between two paradigmatic systems, AGM and KGM, which we develop and compare axiomatically and semantically. The importance of Gärdenfors’s impossibility result on the Ramsey test is highlighted by a comparison with Arrow’s impossibility result on social choice. We end with an outlook on the prospects and the future of DDL.     

Covenants and reputations

In their classic analyses, Hobbes and Hume argue that offensively violating a covenant is irrational because the offense ruins one’s reputation. This paper explores conditions under which reputation alone can enforce covenants. The members of a community are modeled as interacting in a Covenant Game repeated over time. Folk theorems are presented that give conditions under which the Humean strategy of performing in covenants only with those who have never offensively violated or performed with an offensive violator characterizes an equilibrium of the repeated Covenant Game. These folk theorems establish that for certain ideal settings Hobbes’ and Hume’s arguments against offensively violating covenants are compelling. However, these ideal settings presuppose that the community has certain mechanisms that generate common knowledge of the identities of those with whom one should perform. I analyze the results of computer simulations of the interactions in a community whose members must rely upon private communication alone. The computer simulation data show that in this community, reputation effects cannot effectively deter members from offensively violating covenants. I conclude that Hobbes’ and Hume’s warnings against offensive violation are compelling only on condition that the community is sufficiently structured to generate common knowledge among its members. I also conclude that even in such structured communities, the Humean strategy is not the uniquely “correct” policy.     

More on Putnam’s models: a reply to Belloti

In an earlier paper, I claimed that one version of Putnam's model-theoretic argument against realism turned on a subtle, but philosophically significant, mathematical mistake. Recently, Luca Bellotti has criticized my argument for this claim. This paper responds to Bellotti's criticisms.     

How are base‐rates used,when they are used: a comparison of additive and Bayesian models of base‐rate use

Previous research has uncovered many conditions that encourage base‐rate use. The present research investigates how base‐rates are used when conditions are manipulated to encourage their use in the lawyer/engineer paradigm. To examine the functional form of the response to base‐rate, a factorial design was employed in which both base‐rate and the individuating information were varied within‐subject. We compared the performance of several models of base‐rate use, including a model that allows base‐rate and individuating information to be combined in a strictly additive fashion, and a model which presumes that respondents use Bayes' Rule in forming their judgments. Results from 1493 respondents showed that the additive model is a stronger predictor of base‐rate use than any other model considered, suggesting that the base‐rate and individuating information are processed independently in the lawyer/engineer paradigm. A possible mechanism for this finding is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Age and probabilistic reasoning: Biases in conjunctive,disjunctive and Bayesian judgements in early and late adulthood

Probabilistic reasoning plays an essential part in many aspects of our daily routine and it has been argued that as we grow older, the need to make judgements under uncertainty becomes increasingly important. Two studies were conducted to establish whether the propensity to commit probabilistic reasoning errors increased with age. Young (aged 16–24), middle aged (25–54), and older persons (55 years and above) were included. Study 1 revealed systematic biases and errors across a range of judgement tasks. However, no evidence of any age effect in Bayesian inference, the incidence of the conjunction fallacy, or in the number of disjunction errors was found. The results obtained in Study 1 were replicated in Study 2, where the potential mediating role of working memory processes and intellectual capacity were explicitly assessed. While some aspects of probabilistic reasoning performance were correlated with measures of intelligence and working memory functioning among young adults, this was much less evident in older persons. The present findings are discussed in relation to the evolution of the dualistic heuristic–analytical system over the adult lifespan. Copyright © 2005 John Wiley & Sons, Ltd.     

Vagueness-Adaptive Logic: A Pragmatical Approach to Sorites Paradoxes

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.     

Rule Separation and Embedding Theorems for Logics Without Weakening

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR ⁺ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.     

Fitts' Theorem and Movement Time Dissociation for Amplitude and Width Manipulations: Replying to Hoffmann

The commentary by Errol Hoffmann in this issue asserts that previous work by our group provides the spurious conclusion that amplitude and width manipulations to a movement environment elicit dissociable relations between movement time (MT) and P. M. Fitts' (1954) index of difficulty (ID). Hoffmann concludes that any such dissociation is the result of actions evoked entirely as ballistic. In this reply, we demonstrate that Hoffmann's commentary is a clear misrepresentation of the study goals and conclusions stated by our group. Additionally, we provide kinematic evidence that actions involving online trajectory amendments are associated with dissociable MT–ID relations for amplitude versus width manipulations. Finally, we contend that the kinematic analyses of movement trajectories, and Hoffmann's failure to acknowledge its importance, is an important step in further understanding speed–accuracy relations in human movement.     

The road to two theorems of logic

Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.