Picking the one that does not fit - A matter of logical proportions

National audienceQuiz or tests about reasoning capabilities often pertain to the perception of similarity and dissimilarity between situations. Thus, one may be asked to complete a series of entities $A$, $B$, $C$ by an appropriate $X$, or to pick the one that does not fit in a list. It has been shown that the first problem can receive a solution by solving analogical proportion equations between the representations of the entities in a logical setting, where we assume that $X$ should be such that $A$ is to $B$ as $C$ is to $X$. In this paper, we focus on the second problem, and we show that it can be properly handled by means of heterogeneous proportions that are the logical dual of the homogeneous proportions involved in the modeling of analogical proportions and related proportions. Thus, the formal setting of logical proportions, to which homogeneous and heterogeneous proportions belong, provides an appropriate framework for handling the two problems in a coherent way. As it already exists for homogeneous proportions, a particular multiple-valued logic extension of heterogeneous proportions is discussed (indeed being an intruder in a group may be a matter of degree)

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This paper responds to the contributions by Alexander Bird, Nathan Wildman, David Yates, Jennifer McKitrick, Giacomo Giannini & Matthew Tugby, and Jennifer Wang. I react to their comments on my 2015 book Potentiality: From Dispositions to Modality, and in doing so expands on some of the arguments and ideas of the book

Homogenous and heterogeneous logical proportions

International audienceCommonsense reasoning often relies on the perception of similarity as well as dis- similarity between objects or situations. Such a perception may be expressed and summarized by means of analogical proportions, i.e., statements of the form “A is to B as C is to D”. Analogy is not a mere question of similarity between two objects (or situations), but rather a matter of proportion or relation between objects. This view dates back to Aristotle and was enforced by Scholastic philosophy. Indeed, an analogical proportion equates a relation between two objects with the relation between two other objects. As such, the analogical proportion “A is to B as C is to D” poses an analogy of proportionality by (implicitly) stating that the way the two objects A and B, otherwise similar, differ is the same way as the two objects C and D, which are similar in some respects, differ

Tool use and related errors in ideational apraxia: The quantitative simulation of patient error profiles

The behaviour of ideational apraxic patients on simple tasks involving multiple objects is typically marked by a variety of errors. While some of these errors concern the sequential organisation of action through time, many relate to the misuse of, or failure to use, necessary or appropriate tools. In this paper we apply the computational model of Cooper & Shallice (2000) to five standard multiple object tasks used in clinical assessment and demonstrate how, when lesioned, the model can account for the error profiles of two ideational apraxic patients discussed by Rumiati et al. (2001). Application of the model to the multiple object tasks demonstrates the generality of the model, while the account of the error profiles extends previous work (Cooper et al., 2005) in which ideational apraxia was argued to arise from a generalised disturbance of object representations that are held to trigger action schemas

Love and Contracts in \u3cem\u3eDon Quixote\u3c/em\u3e

Viewing love as a contract seems, initially, like mistaking windmills for giants, or a peasant girl for a grand lady. This chapter seeks, like Don Quixote, to convince readers to suspend their practiced views of everyday relationships in order to see them in a new light. What seems crazy at first glance may come to look as good, and sometimes better, than the more conventional view. As a law professor, I usually write about love and contracts by focusing on legal opinions and statutes, and recently I have added real-life stories from books and newspapers, as well as my friends, family, colleagues, and students. But if I am right that love and contracts often complement instead of oppose each other, then my argument that contracts shape the beginning, middle, and demise of love relationships ought to hold true in fiction as well, especially for the jump-off-the-page characters and situations in Don Quixote. Applying this analysis to Don Quixote invites new readings, and may even bring yet more readers to this brilliant text

Inter-temporal variations in the value of time

INTRODUCTION The objective of the research reported here is to examine how the value of time varies over time. A key factor in inter-temporal variations in the value of time is expected to be the impact of income growth, although changes in other socio-economic, demographic, attitudinal, employment and travel characteristics could also lead to variations in the value of time over time. The most widely held convention relating to the adjustment of recommended values of time over time is that they should be linked proportionately to growth in some measure of income. No consideration is given to possible changes in the value of time for other reasons. Even disregarding the latter issue, there is no reason from a theoretical standpoint why the income elasticity for private travel should be unity since it is a matter of personal preference how an individual or household allocates additional income to time savings. In contrast, the case for a close link between the value of time and income is much stronger for business travel. Official recommendations in Britain, as elsewhere, increase the value of non-work travel time over time in line with growth in income. DETR’s Transport Economics Note specifies that both work and non-work time values should be increased in line with real GDP per head. Beesley (1971) pointed out various sources of variation in the value of time over time and, on the basis of the uncertainty as to even the direction in which the values might vary, he argued for in favour of a zero trend value. The first British national value of time study (MVA et al., 1987) claimed that a constant real value of time was on theoretical grounds “equally logical and defensible” as the convention of linking the value of time to income growth. However, it was recognised that there did seem to have been an increase in the value of time over time. It was concluded that, “We do not feel able, therefore, in the absence of any specific work on this topic within our programme, and given the existence of plausible arguments in contrary directions, to come to any firm conclusions. The matter must remain on the agenda for further investigation”. A large amount of recent evidence, which we shall cover, is being taken to indicate that the income elasticity for the value of time spent in private travel is far less than unity. It is important that such a challenge to the widely used convention is tested against the widest body of evidence possible before any conclusions are drawn, particularly given the implications of amended recommendations for practical project evaluations. The aim of this paper is to review the existing evidence relevant to inter-temporal variations in the value of time and to present some fresh empirical evidence. The approach adopted here is threefold. Firstly, we examine the cross-sectional variations in the value of time with income apparent from a number of empirical studies, both British and from other countries, and we develop a model to explain cross-sectional income elasticities across British studies. Secondly, the opportunity exists of analysing two data sets obtained from the same SP design conducted in the same area but at different points in time. Finally, variations in values of time over time are analysed by means of ‘meta-analysis’ of a large data set of British empirical evidence. The structure of this report is as follows. Section 2 contains a discussion of various background issues relating to theoretical maters, methodology and previous findings. Section 3 reports analysis of cross-sectional variations in values of time with income whilst section 4 reports on joint analysis of two data sets collected in the first and second national value of time studies commissioned by the Department of Transport. Section 5 reports the findings of our meta-analysis of a large body of British evidence on the value of time. A discussion of the various findings is provided in section 6 and concluding remarks are provided in section 7. The final stage of the study will draw together this evidence to form recommendations concerning the value of time over time

From Analogical Proportion to Logical Proportions

International audienceGiven a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators ( a∧b and a¯∧b¯), or dissimilarity indicators ( a∧b¯ and a¯∧b) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which corresponds to a statement of the form “a is to b as c is to d”. The paper inventories the whole set of logical proportions by dividing it into five subfamilies according to what they express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4-tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for their negations (a¯,b¯,c¯,d¯)). It appears that only four proportions (including analogical proportion) are homogeneous in the sense that they use only one type of indicator (either similarity or dissimilarity) in their definition. Due to their specific patterns, they have a particular cognitive appeal, and as such are studied in greater details. Finally, the paper provides a discussion of the other existing works on analogical proportions