Tree proofs for syllogistic

This paper presents a tree method for testing the validity of inferences, including syllogisms, in a simple term logic. The method is given in the form of an algorithm and is shown to be sound and complete with respect to the obvious denotational semantics. The primitive logical constants of the system, which is indebted to the logical works of Jevons, Brentano and Lewis Carroll, are term negation, polyadic term conjunction, and functors affirming and denying existence, and use is also made of a metalinguistic concept of formal synonymy. It is indicated briefly how the method may be extended to other systems.To the memory of Jerzy Supecki     

Henkin Quantifiers and the Definability of Truth

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L _* ¹(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L _* ¹(H) with respect to Boolean operations, and obtain the language L ¹(H). At the next level, we consider an extension L _* ²(H) of L ¹(H) in which every sentence is an L ¹(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).     

An abstract dynamic access control architecture

This paper presents an abstract architecture capturing the most essential characteristics of security architectures for the kind of access control required in multi-agent systems. A security architecture is usually component based; it is dynamically extensible, reflective, and should be able to support a variety of policy strategies and enforcement mechanisms. Therefore, in designing and implementing such an architecture, it is important to accurately describe its structure, components, and their interactions; and we should also provide support for verifying properties that the architecture must satisfy. The abstract Dynamic Access Control Architecture, or DACA, we propose provides the flexibility to express several complex characteristics abstractly, ultimately resulting in a more structured foundation for implementing security architectures for access control.     

An optimal period for setting sustained variability levels

In two experiments, we investigated how explicit reinforcement of highly variable behavior at different points in training affected performance after the requirement was eliminated. Two versions of a computer game, differing in the number of possible solution paths, were used. In each, an optimal period of training for producing sustained high variability was found. Exposure to a high lag requirement shortly after acquisition sustained variability. Rewarding variability at other times did not have a sustained effect. The implications for learning and problem solving are discussed.     

Knowledge of proofs

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch's paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett's meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost.I am much indebted for comments to Lars Bergström, Per Martin-Löf, Wlodek Rabinowicz, Fredrik Stjernberg, Dag Westerståhl and Tim Williamson. I owe even more to the many seminars about truth and meaning, led by Dag Prawitz, at the philosophy department of Stockholm University. These were especially intense in the mideighties, when I was a graduate student.     

The theory of abstract partials: An introduction

It is customary in multivariate analysis to search out relational structure by partitioning measures of total variation or overall relatedness into components variously attributable to different sources. Although the multivariate systems most familiar today—notably, linear correlational analysis, analysis of variance, and Information theory—base their decompositions upon very different substantive quantities, they all embody the very same abstract mathematical structure which, moreover, is capable of generating potentially fruitful patterns of data analysis in many other substantive applications as well.     

An investigation of the nature of abstract concepts

Eight abstract concepts were submitted to a procedure designed to test the logical nature of their definition. The hypothesis that they would show a polymorphous prototype structure similar to that found for concrete categories (Hampton, 1979; Rosch & Mervis, 1975) was confirmed for five of the concepts. Reasons for the lack of fit of the prototype model to the remaining concepts and implications for the generality of existing theories of semantic memory are discussed.     

TPS: A hybrid automatic-interactive system for developing proofs

The theorem proving system Tps provides support for constructing proofs using a mix of automation and user interaction, and for manipulating and inspecting proofs. Its library facilities allow the user to store and organize work. Mathematical theorems can be expressed very naturally in Tps using higher-order logic. A number of proof representations are available in Tps, so proofs can be inspected from various perspectives.     

An enactivist account of abstract words: lessons from Merleau-Ponty

Enactivist accounts of language use generally treat concrete words in terms of motor intentionality systems and affordances for action. There is less consensus, though, regarding how abstract words are to be understood in enactivist terms. I draw on Merleau-Ponty’s later philosophy to argue, against the representationalist paradigm that has dominated the cognitive scientific and philosophical traditions, that language is fundamentally a mode of participation in our world. In particular, language orients us within our milieus in a manner that extends into the depth of the idea-endowed world (where “ideas” are construed in a specifically Merleau-Pontian sense). This conceptualization of language allows us to see that abstract words orient us bodily just as surely as concrete words do, albeit in a manner that is more diffuse across the entirety of given situations, as I will show with an example of abstract language use in Don DeLillo’s novel Underworld. These insights are applied to some of the recent enactivist discourse to suggest some ways in which representationalism maintains a latent presence in this discourse. I conclude by pointing to developments in conceptual metaphor theory that can enrich our sense of how abstract language is involved in embodied understanding.     

How to think about informal proofs

It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the field.