Similarity of rectangles: An analysis of subjective dimensions

Two defining properties of psychological dimensions (intradimensional subtractivity and interdimensional additivity) are introduced and their consequences, formulated in terms of an ordinal dissimilarity scale, are derived. These consequences are investigated using dissimilarity judgments between rectangles to determine which of two alternative dimensional structures area (A) and shape (S), or width (W) and height (H), satisfies additivity and/or subtractivity. The results show that neither dimensional structure is acceptable, although A × S provides a better account for the data of most Ss than does W × H. Tests of relative straightness show that A is the least “curved” of the four attributes. Methodological and substantive implications of the study are discussed.     

The mean-integral representation of rectangles

To assess perceptual interaction between the height and width of rectangles, we used an accuracy variant of the Garner paradigm. We measured the discriminability of height and width (baseline tasks) and size and shape (correlated tasks). From thed′ values in these conditions, we estimated perceptual distances and inferred amean-integral representation in which height and width corresponded to nonindependent dimensions in a perceptual space. This model accounted well for performance in these two-stimulus conditions, and it also explained 70% –80% of the decline in performance in selective and divided attention. In a second experiment, conducted for purposes of comparison with the rectangle discrimination Experiment, we studied the discrimination of horizontal and vertical line segments connected in an L-shape. In size discrimination, observers were equally good with line pairs and rectangles, suggesting holistic perception; but in shape discrimination, they appeared to combine information from the two line-pair components of the rectangle independently. The mean-integral model was again successful in relating performance in the Garner tasks quantitatively.     

Mapping the mental space of rectangles.

The cognitive structure of a shape space--the space of rectangles--is explored by a nonmetric scaling technique. Our experiment was designed to extract the major transformational paths or 'modes' that characterize the mental shape space. Earlier studies of rectangle similarities using multidimensional scaling have provided conflicting evidence about whether the coordinate system of the mental rectangle space is based on height and width or on area and shape (i.e. aspect ratio). Our study reveals shape to be the single dominant factor. We suspected that earlier evidence for a height-width parameterization might have been due to the presentation of rectangles upright in a pseudo-gravitational coordinate system (whereas our rectangles are randomly rotated). In a control experiment with upright (vertical or horizontal) rectangles, the heavy bias towards shape preservation was still the dominant mode. In addition, however, a secondary bias towards change of height or width emerged, exactly following the pattern expected from the biasing change in context. This finding established a concrete path by which context and frame can influence the way shape is represented. The relevance of these findings to the cognitive organization of more complex shape spaces is discussed.     

Saliency metric for subadditive dissimilarity judgments of rectangles

Previous findings that dissimilarity judgments for rectangles are strongly subadditive, that qualitative individual differences are always present, and that four-parameter psychophysical maps can reproduce the average ratings (Sch?nemann & Lazarte, 1987) are replicated. However, the present study in addition suggests that the metric for bounded response transformation, previously used to restore additivity of responses, has a modest effect on the fits of the psychophysical maps. The differential weighing of the coordinates already incorporates segmental subadditivity predictions into the maps. In fact, the psychophysical maps define a subadditive saliency metric that depends on assigning larger weights to larger coordinates. This constraint on the weights of these maps, together with the response times, allows us to identify a strategy shift when subjects respond to two classes of stimulus pairs: For rectangle-rectangle pairs, subjects center on the shape and size of the rectangles; for square-rectangle pairs, the focus is on height and width.     

Mapping the perceptual structure of rectangles through goodness-of-fit ratings

Three experiments were carried out to investigate the internal structure of a rectangular frame to test Arnheim's (1974 Art and Visual Perception, 1988 The Power of the Center) proposals about its 'structural skeleton'. Observers made subjective ratings of how well a small probe circle fit within a rectangle at different interior positions. In experiment 1, ratings of 77 locations were highest in the center, decreased with distance from the center, greatly elevated along vertical and horizontal symmetry axes, and somewhat elevated along the local symmetry axes. A linear regression model with six symmetry-related factors accounted for 95% of the variance. In experiment 2 we measured perceived fit along local symmetry axes versus global diagonals near the corners to determine which factor was relevant. 2AFC probabilities were elevated only along the local symmetry axes and were higher when the probe was closer to the vertex. In experiment 3 we examined the effect of dividing a rectangular frame into two rectangular 'subframes' using an additional line. The results show that the primary determinant of good fit is the position of the target circle within the local subframes. In general, the results are consistent with Arnheim's proposals about the internal structure of a rectangular frame, but an alternative interpretation is offered in terms of the Gestalt concept of figural goodness.     

表面相似性对数量关系和相对大小关系理解的影响

研究采用数量关系匹配任务和相对大小关系匹配任务,通过高表面相似性、低表面相似性和交叉映射三种实验条件,探讨表面相似性对儿童和成人数量与相对大小关系理解的影响。研究一对37名4岁儿童进行三次每次间隔半年的追踪测查,结果表明4~5岁间儿童理解数量关系和相对大小关系的水平有显著提高;4岁儿童对数量关系的理解受到表面相似性的影响,4~5岁儿童对相对大小关系的理解受到表面相似性的影响。研究二探究44名成人数量和相对大小关系理解的特点,结果发现,成人完成两种关系匹配任务时也受到表面相似性的影响。     

Similarity in context

Similarity comparisons are highly sensitive to judgment context. Three experiments explore context effects that occur within a single comparison rather than across several trials. Experiment 1 shows reliable intransitivities in which a target is judged to be more similar to stimulus A than to stimulus B, more similar to B than to stimulus C, and more similar to C than to A. Experiment 2 explores the locus of Tversky’s (1977) diagnosticity effect in which the relative similarity of two alternatives to a target is influenced by a third alternative. Experiment 3 demonstrates a new violation of choice independence which is explained by object dimensions’ becoming foregrounded or backgrounded, depending upon the set of displayed objects. The observed violations of common assumptions to many models of similarity and choice can be accommodated in terms of a dynamic property-weighting process based on the variability and diagnosticity of dimensions.     

Similarity and cotenability

In this paper I present some difficulties for Lewis’s and similar theories of counterfactuals, and suggest that the problem lies in the notion of absolute similarity. In order to explain the problem, I discuss the relation between Lewis’s and Goodman’s theory, and show that the two theories are not related in the way Lewis thought they were.     

Constitution and Similarity

Whenever an object constitutes, makes up orcomposes another object, the objects inquestion share a striking number of properties. This paper is addressed to the question of whatmight account for the intimate relation andstriking similarity between constitutionallyrelated objects. According to my account, thesimilarities between constitutionally relatedobjects are captured at least in part by meansof a principle akin to that of strongsupervenience. My paper addresses two mainissues. First, I propose independentlyplausible principles by means of which todelineate, in a non-ad-hoc, non-stipulative andnon-circular fashion, those properties whichcan be expected to be shared amongconstitutionally related objects in virtue oftheir being so related from those which ingeneral cannot be expected to be shared, orwhich are shared for other reasons. Secondly,I spell out in detail the nature of thesupervenience-principle at work in thiscontext. My account thus aims at isolating, ina methodologically responsible fashion, theparticular sort of restricted indiscernibilityprinciple which is a component of theconstitution-relation.     

Similarity as transformation

We propose that similarity is determined by the transformation distance between representations: entities which are perceived to be similar have representations which are readily transformed into one another, whereas transforming between dissimilar entities requires many transformations. We present three experiments that indicate that similarity is strongly influenced by transformation distance. These data present a challenge for featural or spatial accounts of similarity. Finally, we introduce a family of transformation-based accounts of similarity, called 'Representational Distortion', as a specific example of a transformational approach to similarity.     

Wordform Similarity Increases With Semantic Similarity: An Analysis of 100 Languages

Although the mapping between form and meaning is often regarded as arbitrary, there are in fact well‐known constraints on words which are the result of functional pressures associated with language use and its acquisition. In particular, languages have been shown to encode meaning distinctions in their sound properties, which may be important for language learning. Here, we investigate the relationship between semantic distance and phonological distance in the large‐scale structure of the lexicon. We show evidence in 100 languages from a diverse array of language families that more semantically similar word pairs are also more phonologically similar. This suggests that there is an important statistical trend for lexicons to have semantically similar words be phonologically similar as well, possibly for functional reasons associated with language learning.     

Spatio-temporal interaction and brightness judgments in sequentially presented,partially overlapping rectangles

Ss made magnitude estimates of brightness to three areas within a display consisting of two sequentially presented partially overlapping rectangles of light. The three areas were the singly stimulated region of the rectangle presented first (the trailing flank), the doubly stimulated overlap region, and the singly stimulated region of the rectangle presented second (the leading flank). The stimuli were presented in right-left or left-right order with stimulus onset asynchronies (SOAs) of 0, 25, 50, 75, 100, and 125 msec. A further condition was flank width. The results indicated that the apparent brightness of the overlap region decreased with increasing SOA. The leading flank was judged to increase in apparent brightness, while the trailing flank remained constant.