‘In fairy tales fairies can disappear’: Children's reasoning about the characteristics of humans and fantasy figures

Two studies assessed whether children share with adults a concept of fantasy figures as entities that violate causal principles. Inferences about the characteristics of humans and fantasy figures were elicited with a forced‐choice questionnaire. Items from the biological, psychological, and physical domains pitted possible against impossible abilities; social domain items pitted conventional against unconventional behaviours. Older children (6–9 years) and adults attributed few non‐human characteristics from any domain to humans and attributed more impossible than unconventional characteristics to fantasy figures. Younger children (3–5 years) attributed fewer non‐human characteristics to humans than to fantasy figures, but attributed similar patterns of impossible and unconventional characteristics to humans and fantasy figures. Results are discussed in terms of cognitive changes between 3–5 and 6–9 years, and between 6–9 years and adulthood, that promote awareness that impossible abilities are uniquely associated with fantasy.     

Turning a penrose triangle inside out

Why do impossible figures, which cannot exist in three dimensions, appear to make threedimensional sense? In order to shed some light on this question the limits may be tested to which three-dimensional operations on these figures can be performed. In this paper a particularly difficult operation, viz., torus eversion is attempted. Not only is an eversion found to be possible but an unfamiliar impossibility develops. The regular form of the eversion is shown to be unique.     

Increased visual interest and affective responses to impossible figures in early infancy

This research evaluated infants’ facial expressions as they viewed pictures of possible and impossible objects on a TV screen. Previous studies in our lab demonstrated that four-month-old infants looked longer at the impossible figures and fixated to a greater extent within the problematic region of the impossible shape, suggesting they were sensitive to novel or unusual object geometry. Our work takes studies of looking time data a step further, determining if increased looking co-occurs with facial expressions associated with increased visual interest and curiosity, or even puzzlement and surprise. We predicted that infants would display more facial expressions consistent with either “interest” or “surprise” when viewing the impossible objects relative to possible ones, which would provide further evidence of increased perceptual processing due to incompatible spatial information. Our results showed that the impossible cubes evoked both longer looking times and more reactive expressions in the majority of infants. Specifically, the data revealed significantly greater frequency of raised eyebrows, widened eyes and returns to looking when viewing impossible figures with the most robust effects occurring after a period of habituation. The pattern of facial expressions were consistent with the “interest” family of facial expressions and appears to reflect infants’ ability to perceive systematic differences between matched pairs of possible and impossible objects as well as recognize novel geometry found in impossible objects. Therefore, as young infants are beginning to register perceptual discrepancies in visual displays, their facial expressions may reflect heightened attention and increased information processing associated with identifying irreconcilable contours in line drawings of objects. This work further clarifies the ongoing formation and development of early mental representations of coherent 3D objects.     

Putting order in the impossible

The class of visual illusions called 'impossible figures' (illusory spatial interpretations of pictures) is analyzed in order to introduce an ordering into the great variety of such figures. Such an ordering facilitates reference, unifies terminology, and establishes a conceptual framework for further investigations of the subject, making easier the choice and systematic generation of various types of figures (for example, in systematic psychological experiments). First, the notion of 'impossible figure' is defined and certain other related classes of figures (so-called 'likely' and 'unlikely' figures) are distinguished. Second, the fundamental 'impossibility sources' are identified as elementary 'building blocks' of all impossible figures. Finally, two broad classes of impossible figures, multibars (or 'impossible polygons') and striped figures, are briefly described.     

Representation of anatomical constraints in motor imagery: mental rotation of a body segment

Classically, the mental rotation paradigm has shown that when subjects are asked to judge whether objects that differ in orientation are spatially congruent, reaction times increase with angular discrepancy, although some reports have shown that this is not always the case. Would similar results be obtained with realistic figures of body segments? In this work, the mental rotation of a hand attached to its forearm and arm in anatomically possible and impossible starting positions is compared with the mental rotation of a hammer. The main results show that reaction times increase monotonically with the angle of discrepancy for both stimuli and that the speed of rotation is higher for anatomically possible orientations in the case of the hand. Thus, mental rotation of body segments follows the same empirical rules as objects of another nature, and biomechanical constraints imposed to the motility of these segments can be considered as attributes of the mental representation.     

Let’s twist again! Embodiment effects in spatial judgments on human figures rotated along a vertical axis

We investigated whether individuals used mental rotation and embodiment for arm laterality judgments of human figures that were stepwise rotated from back view to front view along a vertical axis. In Experiment 1, figures’ heads were always shown in profile, while only the bodies were rotated. Judgments were faster and more correct when figures were presented in back view compared to front view, but the relation between reaction times (RTs) and rotation angles was not strictly linear. In addition, judgments on figures in anatomically possible postures were better than on figures in impossible postures. In Experiment 2, figures’ heads were turned together with their body. RTs and rotation angles were linearly related. Results suggest that individuals use both mental rotation and a more direct matching between their own body and that of the figures, when making arm laterality judgments of human figures that are rotated along a vertical axis.     

The Eight Trigrams of the Chinese I Ching and the Eight Primary Emotions

Durkheim and Mauss, in Primitive Classification , concluded that the emotions play a causal role in the history of dual symbolic classification systems, but could not test this intuitive speculation because they saw a classification of the emotions as impossible. In this paper a portion of Plutchik's psychoevolutionary model of the primary emotions are assumed to be valid and are then investigated through analysis of one of their three case studies of primitive classification, that of classification in ancient China, where their emphasis was on the eight "trigrams" or "powers" that they saw arranged in a "divinatory compass." The trigrams are three-line components of the "hexagrams," the six-line figures that are interpreted as master signs in I Ching divination rituals. Using Plutchik's psychoevolutionary classification of the emotions as a basis of comparison, especially his model of the primary emotions as adaptive reactions to the positive and negative experiences of four existential problems – identity, temporality/reproduction, hierarchy, and territoriality – it is found that both trigrams and primary emotions exist as four pairs of opposites. The eight trigrams and eight primary emotions similarly can be seen as adaptive reactions to the four basic problems of life. Through structural analysis, correspondences between the trigrams and the primary emotions are developed, the result being that the primary emotions are structurally isomorphic and very close in first meanings to the primary attributes of the trigrams. Implications of this isomorphism of structure for the development of a social psychology of the emotions are discussed.     

The theory of braids and the analysis of impossible figures

Impossible figures were regarded neither as new perceptual phenomena, nor as examples of known phenomena such as illusions. They were seen as geometric anomalies, not psychological ones, which serve a heuristic function in the study of perception. With this in mind, a topological analysis was undertaken with the anticipation of the following goals: (a) A systematic way (algorithm) of generating these figures should be evident; (b) simple properties of impossible figures which may not have been previously understood should be discovered. Both of these conditions were met.     

Effort and demand logic in medical decision making

Medical decisions, including diagnosis, prognosis, and disease classification, must often be made on the basis of incomplete or unsatisfactory information. Data which are essential to the care of one patient may be unobtainable for technical or ethical reasons in another patient. For this reason the principles of controlled experimentation may be impossible to satisfy in human studies. In this paper, some formal aspects of medical decision making are discussed. Special operators for the intuitive concepts of ‘certainty’, ‘demand’, and ‘effort’, akin to the operators of modal logic, are used to accommodate the technical and ethical limitations on human studies. Theorems are stated and proved which show how this system handles incomplete information. The embryogenesis of the human heart is presented as a sample problem in classification.     

Impossible Worlds

Impossible worlds are representations of impossible things and impossible happenings. They earn their keep in a semantic or metaphysical theory if they do the right theoretical work for us. As it happens, a worlds‐based account provides the best philosophical story about semantic content, knowledge and belief states, cognitive significance and cognitive information, and informative deductive reasoning. A worlds‐based story may also provide the best semantics for counterfactuals. But to function well, all these accounts need use of impossible and as well as possible worlds. So what are impossible worlds? Graham Priest claims that any of the usual stories about possible worlds can be told about impossible worlds, too. But far from it. I'll argue that impossible worlds cannot be genuine worlds, of the kind proposed by Lewis, McDaniel or Yagisawa. Nor can they be ersatz worlds on the model proposed by Melia or Sider. Constructing impossible worlds, it turns out, requires novel metaphysical resources.     

Concept formation and categorization of complex,asymmetric, and impossible figures

Impossible figures are striking examples of inconsistencies between global and local perceptual structures, in which the overall spatial configuration of the depicted image does not yield a coherent three-dimensional object. In order to investigate whether structural “impossibility” is an important perceptual property of depicted objects, we used a category formation task in which subjects were asked to divide pictures of shapes into groups that seemed most natural to them. Category formation is usually unidimensional, such that sorting is dominated by a single perceptual property, so this task can serve as a measure of which dimensions are most salient. In Experiment 1, subjects received sets of 12 line drawings consisting of six possible and six impossible objects. Very few subjects grouped the figures by impossibility on the first try, and only half did so after multiple attempts at sorting. In Experiment 2, we investigated other global properties of figures: symmetry and complexity. Subjects readily sorted objects by complexity, but seldom by symmetry. In Experiment 3, subjects were asked to draw each of the figures before sorting them, which had only a minimal effect on categorization. Finally, in Experiment 4, subjects were explicitly instructed to divide the shapes by symmetry or impossibility. Performance on this task was perfect for symmetry, but not for impossibility. Although global properties of figures seem extremely important to our perception, the results suggest that some of these cues are not immediately obvious or salient for most observers.     

The Different Ways in which Logic is (said to be) Formal

I argue that, in the Prior Analytics, higher and above the well-known ‘reduction through impossibility’ of figures, Aristotle is resorting to a general procedure of demonstrating through impossibility in various contexts. This is shown from the analysis of the role of adunaton in conversions of premises and other demonstrations where modal or truth-value consistency is indirectly shown to be valid through impossibility. Following the meaning of impossible as ‘non-existent’, the system is also completed by rejecting any invalid combinations of terms in deductions or conversions. The notion of impossibility reaches the core of Aristotle's system in the Prior Analytics. On the one hand, the use of adunaton shows that he is following one of the two requisites for demonstrative science formulated in the Posterior Analytics, i.e. to demonstrate that it is impossible for things to be otherwise than stated. On the other hand, that demonstrations through impossibility are rooted in the notion of contradiction supports the claim that Aristotle might have been trained to use this specific procedure in the context of dialectical exercises in the academy. This need not rule out other influences on Aristotle's preferred procedures of proving or counter-proving, but it paves a way to a better understanding of Aristotle's logic under the light of Plato's dialectic.     

“Impossible figures” and interpretations of polyhedral figures

The concept “impossible figure” is analyzed by formalizing the interpretations of a polyhedral figure by an observer, and defining several types of inconsistency in such interpretations. Tests for these inconsistencies are developed using simple graph theory, and a sufficient condition is established for the feasibility of an interpretation. Finally, techniques are derived for the automatic construction of usual or unusual multibar figures.     

Systems of modal logic for impossible worlds

The intuitive notion behind the usual semantics of most systems of modal logic is that of ‘possible worlds’. Loosely speaking, an expression is necessary if and only if it holds in all possible worlds; it is possible if and only if it holds in some possible world. Of course, contradictory expressions turn out to hold in no possible worlds, and logically true expressions turn out to hold in every possible world. A method is presented for transforming standard modal systems into systems of modal logic for impossible worlds. To each possible world there corresponds an impossible world such that an expression holds in the impossible world if and only if it does not hold in the possible world. One can then talk about such worlds quite consistently, and there seems to be no logical reason for excluding them from consideration.