FOIL Axiomatized

In an earlier paper, [5], I gave semantics and tableau rules for a simple firstorder intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.In this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view.This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

Universalism and Junk

Those who accept the necessity of mereological universalism face what has come to be known as the ‘junk argument’ due to Bohn [2009 Bohn, Einar D. 2009a. An Argument Against the Necessity of Unrestricted Composition, Analysis 69/1: 27–31.[Crossref], [Web of Science ®] , [Google Scholar]], which proceeds from (i) the incompatibility of junk with universalism and (ii) the possibility of junk, to conclude that mereological universalism isn't metaphysically necessary. Most attention has focused on (ii); however, recent authors have cast doubt on (i). This paper undertakes a defence of premise (i) against three main objections. The first is a new objection to the effect that Bohn's defence of that premise presupposes far too much. I show that one can defend premise (i) from a much weaker set of assumptions. The second objection, due to Contessa [2012 Contessa, Gabriel 2012. The Junk Argument: Safe Disposal Guidelines for Mereological Universalists, Analysis 72/3: 455–7.[Crossref], [Web of Science ®] , [Google Scholar]], is that those who accept unrestricted composition should only accept the existence of binary sums (which are compatible with junk) rather than infinitary fusions. I argue that this conception of unrestricted composition is problematic: it is in conflict with an intuitive remainder principle. The final objection is due to Spencer [2012 Spencer, Joshua. 2012. All Things Must Pass Away, in Oxford Studies in Metaphysics: Volume 7, ed. Karen Bennett and Dean W. Zimmerman, Oxford: Oxford University Press: 67–92.[Crossref] , [Google Scholar]]. His view is that there is no absolutely unrestricted plural universal quantifier; so any statement of the unrestricted fusion axiom will simply not rule out the existence of junky worlds. I argue that the failure of unrestricted quantification will not be enough by itself to establish the existence of junk. Furthermore, it is not clear whether this view counts as a form of mereological universalism. As a result, I suggest that if one wants to reject the junk argument, premise (ii) is the only viable option.     

Standard Quantification Theory in the Analysis of English

Standard first-order logic plus quantifiers of all finite orders (SFOL) faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOL models. The typed structure of SFOL models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOL type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOL models.     

Necessary versus Probable Cause

One finds in the systems of natural languages some explicit means of elaborating not only upon the directness of the causal relationship believed to exist between two events X and Y (i.e. some means of specifying just how inevitably event X gives or gave rise to event Y), but also some manner of indicating just who or what is understood to be the primary instigator of the caused event. The goal of the present paper is to explore these notions in detail and arrive at a formal, logic-based means of capturing them.     

On quantifying different types of categorical data

In quantifying categorical data, constraints play an important role in characterizing the outcome. In the Guttman-type quantification of contingency tables and multiple-choice data (incidence data), the trivial solution due to the marginal constraints is typically removed before quantification; this removal, however, has the effect of distorting the shape of the total space. Awareness of this is important for the interpretation of the quantified outcome. The present study provides some relevant formulas for those cases that are affected by the trivial solution and those cases that are not. The characterization of the total space used by the Guttman-type quantification and pertinent discussion are presented.This study was supported by a grant from The Natural Sciences and Engineering Research Council of Canada to S. Nishisato.     

Relationships among several methods of linearly constrained correspondence analysis

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.The work reported in this paper has been supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. We wish to thank Carolyn Anderson, Ulf Böckenholt, Henk Kiers, Shizuhiko Nishisato, Jim Ramsay, Tadashi Shibayama, Cajo ter Braak, and Peter van der Heijden for their helpful comments on earlier drafts of this paper.     

Is There a Place for Traditional Values and Virtues in Society Today?

This paper argues that rather than looking to the past for a previously developed set of traditional values and virtues we should instead look to the past for ways of thinking about morality and ethics which may be found in the “tradition” and which may also be relevant for the contemporary situation. It examines the causes of the disconnection between traditional ways of thinking and the contemporary situation at two levels: the marginalization of morality and the disarray in ethics. Both aspects are found to be rooted in the emergence of the empirical scientific differentiation of consciousness. The paper then goes on to ask what resources in the tradition may be found to overcome the gap between ancient and modern, or traditional and contemporary, ways of thinking about morality and ethics. The contemporary relevance of the thought of Bernard Lonergan to this issue is examined.     

Inscrutability and ontological commitment

There are two doctrines for which Quine is particularly well known: the doctrine of ontological commitment and the inscrutability thesis—the thesis that reference and quantification are inscrutable. At first glance, the two doctrines are squarely at odds. If there is no fact of the matter as to what our expressions refer to, then it would appear that no determinate commitments can be read off of our best theories. We argue here that the appearance of a clash between the two doctrines is illusory. The reason that there is no real conflict is not simply that in determining our theories’ ontological commitments we naturally rely on our home language but also (and more importantly) that ontological commitment is not intimately tied to objectual quantification and a reference-first approach to language. Or so we will argue. We conclude with a new inscrutability argument which rests on the observation that the notion of objectual quantification, when properly cashed out, deflates.

The defeasible nature of coherentist justification

The impossibility results of Bovens and Hartmann (2003, Bayesian epistemology. Oxford: Clarendon Press) and Olsson (2005, Against coherence: Truth, probability and justification. Oxford: Oxford University Press.) show that the link between coherence and probability is not as strong as some have supposed. This paper is an attempt to bring out a way in which coherence reasoning nevertheless can be justified, based on the idea that, even if it does not provide an infallible guide to probability, it can give us an indication thereof. It is further shown that this actually is the case, for several of the coherence measures discussed in the literature so far. We also discuss how this affects the possibility to use coherence as a means of epistemic justification.