Whereas many others have scrutinized the Allais paradox from a theoretical angle, we study the paradox from an historical perspective and link our findings to a suggestion as to how decision theory could make use of it today. We emphasize that Allais proposed the paradox as a normative argument, concerned with ‘the rational man’ and not the ‘real man’, to use his words. Moreover, and more subtly, we argue that Allais had an unusual sense (...) of the normative, being concerned not so much with the rationality of choices as with the rationality of the agent as a person. These two claims are buttressed by a detailed investigation – the first of its kind – of the 1952 Paris conference on risk, which set the context for the invention of the paradox, and a detailed reconstruction – also the first of its kind – of Allais’s specific normative argument from his numerous but allusive writings. The paper contrasts these interpretations of what the paradox historically represented, with how it generally came to function within decision theory from the late 1970s onwards: that is, as an empirical refutation of the expected utility hypothesis, and more specifically of the condition of von Neumann–Morgenstern independence that underlies that hypothesis. While not denying that this use of the paradox was fruitful in many ways, we propose another use that turns out also to be compatible with an experimental perspective. Following Allais’s hints on ‘the experimental definition of rationality’, this new use consists in letting the experiment itself speak of the rationality or otherwise of the subjects. In the 1970s, a short sequence of papers inspired by Allais implemented original ways of eliciting the reasons guiding the subjects’ choices, and claimed to be able to draw relevant normative consequences from this information. We end by reviewing this forgotten experimental avenue not simply historically, but with a view to recommending it for possible use by decision theorists today. (shrink)

This article documents the history of the Allais paradox, and shows that underneath the many discussions of the various protagonists lay different, irreconcilable epistemological positions. Savage, like his mentor von Neumann and similar to economist Friedman, worked from an epistemology of generalized characterizations. Allais, on the other hand, like economists Samuelson and Baumol, started from an epistemology of exact descriptions in which every axiom was an empirical claim that could be refuted directly by observations. As a result, (...) the two sides failed to find a common ground. Only a few decades later was the now so-called Allais paradox rediscovered as an important precursor when a new behavioural economic subdiscipline started to adopt the epistemology of exact descriptions and its accompanying falsifications of rational choice theory. (shrink)

This paper uses a two-dimensional version of a standard common consequence experiment to test the intransitivity explanation of Allais-paradox-type violations of expected utility theory. We compare the common consequence effect of two choice problems differing only with respect to whether alternatives are statistically correlated or independent. We framed the experiment so that intransitive preferences could explain violating behavior when alternatives are independent, but not when they are correlated. We found the same pattern of violation in the two cases. (...) This is evidence against intransitivity as an explanation of the Allais Paradox. The question whether violations of expected utility are mainly due to intransitivity or to violation of independence is important since it is exactly on this issue the main new decision theories differ. (shrink)

This paper investigates the relation between consequentialism, as conceived of in moral theory, and standard expected utility theory. I argue that there is a close connection between the two. I show furthermore that consequentialism is not neutral with regard to the values of the agent. Consequentialism, as well as standard expected utility theory, is incompatible with the recognition of considerations that depend on what could have been the case, such as regret and disappointment. I conclude that consequentialism should be rejected (...) as a principle of rational choice and that there are reasons to doubt its plausibility in the realm of moral theory. Moreover, this is a reason to doubt whether standard expected utility theory is a plausible theory of rational choice. (shrink)

In The Enterprise of Knowledge, I proposed a general theory of rational choice which I intended as a characterization of a prescriptive theory of ideal rationality. A cardinal tenet of this theory is that assessments of expected value or expected utility in the Bayesian sense may not be representable by a numerical indicator or indeed induce an ordering of feasible options in a context of deliberation. My reasons for taking this position are related to my commitment to the inquiry-oriented approach (...) to human knowledge and valuation favored by the American pragmatists, Charles Peirce and John Dewey. A feature of any acceptable view of inquiry ought to be that during an inquiry points under dispute ought to be kept in suspense pending resolution through inquiry. (shrink)

In Risk and Rationality, Lara Buchak advertised REU theory as able to recover the modal preferences in the Allais paradox. But we pointed out that REU theory only applies in the “grand world” setting, where it actually struggles with the modal Allais preferences. Buchak offers two replies. Here we enumerate technical and philosophical problems they face.

In Richard Bradley’s book, Decision Theory with a Human Face, we have selected two themes for discussion. The first is the Bolker-Jeffrey theory of decision, which the book uses throughout as a tool to reorganize the whole field of decision theory, and in particular to evaluate the extent to which expected utility theories may be normatively too demanding. The second theme is the redefinition strategy that can be used to defend EU theories against the Allais and Ellsberg paradoxes, a (...) strategy that the book by and large endorses, and even develops in an original way concerning the Ellsberg paradox. We argue that the BJ theory is too specific to fulfil Bradley’s foundational project and that the redefinition strategy fails in both the Allais and Ellsberg cases. Although we share Bradley’s conclusion that EU theories do not state universal rationality requirements, we reach it not by a comparison with BJ theory, but by a comparison with the non-EU theories that the paradoxes have heuristically suggested. (shrink)

Sorensen provides a panoramic view of paradoxes of theoretical and practical rationality. These puzzles are organized as apparent counterexamples to attractive principles such as the principle of charity, the transitivity of preferences, and the principle that we should maximize expected utility. The following paradoxes are discussed: fearing fictions, the surprise test paradox, Pascal’s Wager, Pollock’s Ever Better wine, Newcomb’s problem, the iterated prisoner’s dilemma, Kavka’s paradoxes of deterrence, backward inductions, the bottle imp, the preface paradox, Moore’s problem, Buridan’s (...) ass, Condorcet’s paradox of cyclical majorities, the St. Petersburg paradox, weakness of will, the Ellsberg paradox, Allais’s paradox, and Peter Cave’s puzzle of self-fulfilling beliefs. (shrink)

The desirability of what actually occurs is often influenced by what could have been. Preferences based on such value dependencies between actual and counterfactual outcomes generate a class of problems for orthodox decision theory, the best-known perhaps being the so-called Allais Paradox. In this paper we solve these problems by extending Richard Jeffrey's decision theory to counterfactual prospects, using a multidimensional possible-world semantics for conditionals, and showing that preferences that are sensitive to counterfactual considerations can still be desirability (...) maximising. We end the paper by investigating the conditions necessary and sufficient for a desirability function to be an expected utility. It turns out that the additional conditions imply highly implausible epistemic principles. (shrink)

It is always possible to construct a real function f, given random quantities X and Y with continuous distribution functions F and G, respectively, in such a way that f(X) and f(Y), also random quantities, have both the same distribution function, say H. This result of De Finetti introduces an alternative way to somehow describe the `opinion' of a group of experts about a continuous random quantity by the construction of Fields of coincidence of opinions (FCO). A Field of coincidence (...) of opinions is a finite union of intervals where the opinions of the experts coincide with respect to that quantity of interest. We speculate on (dis)advantages of Fields of Opinion compared to usual `probability' measures of a group and on their relation with a continuous version of the well-known Allais' paradox. (shrink)

The Expected Shortfall or Conditional Value-at-Risk (CVaR) has been playing the role of main risk measure in the recent years and paving the way for an enormous number of applications in risk management due to its very intuitive form and important coherence properties. This work aims to explore this measure as a probability-dependent utility functional, introducing an alternative view point for its Choquet Expected Utility representation. Within this point of view, its main preference properties will be characterized and its utility (...) representation provided through local utilities with an explicit dependence on the assessed revenue’s distribution (quantile) function. Then, an intuitive interpretation for the related probability dependence and the piecewise form of such utility will be introduced on an investment pricing context, in which a CVaR maximizer agent will behave in a relativistic way based on his previous estimates of the probability function. Finally, such functional will be extended to incorporate a larger range of risk-averse attitudes and its main properties and implications will be illustrated through examples, such as the so-called Allais Paradox. (shrink)

An explicitly frame related interpretation of a very general more for less result is used to establish a correspondingly general class of frame related switching results. These are used in turn to show how preference reversals of kinds found by Allais and others may not only be essentially non-paradoxical in character, but can be expected to be frequently observed, even under conditions of certainty and of complete information.

Independence is the condition that, if X is preferred to Y, then a lottery between X and Z is preferred to a lottery between Y and Z given the same probability of Z. Is it rationally required that one’s preferences conform to Independence? The main objection to this requirement is that it would rule out the alleged rationality of Allais and Ellsberg Preferences. In this paper, I put forward a sequential dominance argument with fairly weak assumptions for a variant (...) of Independence (called Independence for Constant Prospects), which shows that Allais and Ellsberg Preferences are irrational. Hence this influential objection (that is, the alleged rationality of Allais and Ellsberg Preferences) can be rebutted. I also put forward a number of sequential dominance arguments that various versions of Independence are requirements of rationality. One of these arguments is based on very minimal assumptions, but the arguments for the versions of Independence which are strong enough to serve in the standard axiomatization of Expected Utility Theory need notably stronger assumptions. (shrink)

We consider several puzzles of bounded rationality. These include the Allais- and Ellsberg paradox, the disjunction effect, and related puzzles. We argue that the present account of quantum cognition—taking quantum probabilities rather than classical probabilities—can give a more systematic description of these puzzles than the alternate treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches (...) introduce additional parameters which possibly can be fitted to empirical data but which do not necessarily explain them. Hence, the phenomenological research has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras. They are combined into a uniform system via a mechanism preventing the simultaneous realization of perspectives. Gleason’s theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic macro-systems using the framework of symbolic dynamics. Finally, we discuss several ideas that are useful for justifying complementarity in cognitive systems. (shrink)

Risk-weighted expected utility theory is motivated by small-world problems like the Allais paradox, but it is a grand-world theory by nature. And, at the grand-world level, its ability to handle the Allais paradox is dubious. The REU model described in Risk and Rationality turns out to be risk-seeking rather than risk-averse on one natural way of formulating the Allais gambles in the grand-world context. This result illustrates a general problem with the case for REU theory, (...) we argue. There is a tension between the small-world thinking marshaled against standard expected utility theory, and the grand-world thinking inherent to the risk-weighted alternative. (shrink)

The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or (...) conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity. To give the strategy more structure, philosophers have developed “principles of individuation”; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion. We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory. (shrink)

It is widely maintained that self-interested rationality is a matter of maximizing one's own good or well-being. Rationality more generally is also frequently characterized in maximizing terms: the rational thing to do in any decision context is whatever is best in terms of one's interests or will lead to the greatest preference-satisfaction, My dissertation consists of three independent papers that challenge this orthodoxy by lending support to "satisficing," the idea that it is rational to prefer what is good enough. In (...) the first I argue that the preferences frequently displayed in the Allais Paradox can not be reconciled with expected utility maximizing and therefore must be construed in satisficing terms. To the extent that common sense is an important touchstone for theory choice, this lends support to satisficing. In the second, I argue more directly for satisficing by claiming that satisficing is the best way to deal with the different temporal perspectives available to human beings. In the third, I argue that supererogation, the idea that options which are not optimal can nonetheless be morally or rationally acceptable, depends on "optional reasons," I claim, further, that the criticisms of the very idea of optional reasons are not compelling. I also include an appendix which addresses Derek Parfit's challenge to prudential rationality. (shrink)

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies (Nash, Proceedings of (...) the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players. (shrink)

Neuroeconomics is examined critically using data on the response times of subjects who were asked to express their preferences in the context of the Allais Paradox. Different patterns of choice are found among the fast and slow responders. This suggests that we try to identify types of economic agents by the time they take to make their choices. Nevertheless, it is argued that it is far from clear if and how neuroeconomics will change economics.

This paper argues that any specific utility or disutility for gambling must be excluded from expected utility because such a theory is consequential while a pleasure or displeasure for gambling is a matter of process, not of consequences. A (dis)utility for gambling is modeled as a process utility which monotonically combines with expected utility restricted to consequences. This allows for a process (dis)utility for gambling to be revealed. As an illustration, the model shows how empirical observations in the Allais (...) paradox can reveal a process disutility of gambling. A more general model of rational behavior combining processes and consequences is then proposed and discussed. (shrink)

Recently proposed models of risky choice imply systematic violations of transitivity of preference. This study explored whether people show the predicted intransitivity of the two models proposed to account for the certainty effect in Allais paradoxes. In order to distinguish “true” violations from those produced by “error,” a model was fit in which each choice can have a different error rate and each person can have a different pattern of preferences that need not be transitive. Error rate for a (...) choice is estimated from preference reversals between repeated presentations of the same choice. Results showed that few people repeated intransitive patterns. We can retain the hypothesis that all participants were transitive. (shrink)

It is always possible to construct a real function φ, given random quantities X and Y with continuous distribution functions F and G, respectively, in such a way that φ(X) and φ(Y), also random quantities, have both the same distribution function, say H. This result of De Finetti introduces an alternative way to somehow describe the `opinion' of a group of experts about a continuous random quantity by the construction of Fields of coincidence of opinions (FCO). A Field of coincidence (...) of opinions is a finite union of intervals where the opinions of the experts coincide with respect to that quantity of interest. We speculate on (dis)advantages of Fields of Opinion compared to usual `probability' measures of a group and on their relation with a continuous version of the well-known Allais' paradox. (shrink)

This paper shows that expected utility belongs to the intersection of models satisfying betweenness and a homotheticity condition for risky choice. Betweenness models can accommodate variable risk attitudes, originally highlighted by the Allais paradox, by restricting indifference curves to be linear while allowing non-parallelism. Homotheticity, in contrast, restricts indifference curves to be parallel while permitting non-linearities, such as those highlighted by inverse-S probability weighting. Data from an experiment indicate that approximately 2/3s of participants satisfied homotheticity. Of this group, (...) about half satisfied the expected utility hypothesis. The other half violated expected utility and therefore betweenness. These results indicate that homothetic preferences, and in particular those violating betweenness, can be empirically relevant for decision making under risk. (shrink)

This critical notice explores the distinction central to analytic philosophy between the logical study of the normative principles governing rational thought and the psychological study of the processes of thinking. Thomas Nagel maintains (1) that the fundamental principles of reasoning have normative force and make claims to universal validity; (2) that the fundamental principles of reasoning cannot be construed as the expression of contingent forms of life; and (3) that the identification of fundamental principles of reasoning should be completely independent (...) of psychological facts about how people reason. The notice draws upon intuitionist critiques of classical logic and the Allais paradox in expected utility theory to argue that (1) can be consistently combined with the denial of (2) and (3). (shrink)

This paper presents a new decision theory for modelling choice under risk. The new theory is a two-parameter generalization of expected utility theory. The proposed theory assumes that a decision maker: behaves as if maximizing expected utility; but may experience disappointment when the utility of a lottery’s outcome falls short of the expected utility of the lottery; and may have a preference for gambling. The proposed theory can rationalize the fourfold pattern of risk attitudes; the common ratio effect and the (...) reverse thereof ; the Allais paradox in classical common consequence problems and the reverse Allais paradox—in common consequence problems with an even split of a probability mass; violations of the betweenness axiom; switching behavior in the Samuelson’s example; violations of ordinal, upper and lower cumulative independence ; and preference reversals between valuations and choice. In application to insurance, the theory can rationalize full insurance with an actuarially unfair premium and aversion to probabilistic insurance. In application to optimal portfolio investment, the theory can rationalize the equity premium puzzle. (shrink)

I have claimed that risk-weighted expected utility maximizers are rational, and that their preferences cannot be captured by expected utility theory. Richard Pettigrew and Rachael Briggs have recently challenged these claims. Both authors argue that only EU-maximizers are rational. In addition, Pettigrew argues that the preferences of REU-maximizers can indeed be captured by EU theory, and Briggs argues that REU-maximizers lose a valuable tool for simplifying their decision problems. I hold that their arguments do not succeed and that my original (...) claims still stand. However, their arguments do highlight some costs of REU theory. (shrink)

The orthodox theory of instrumental rationality, expected utility (EU) theory, severely restricts the way in which risk-considerations can figure into a rational individual's preferences. It is argued here that this is because EU theory neglects an important component of instrumental rationality. This paper presents a more general theory of decision-making, risk-weighted expected utility (REU) theory, of which expected utility maximization is a special case. According to REU theory, the weight that each outcome gets in decision-making is not the subjective probability (...) of that outcome; rather, the weight each outcome gets depends on both its subjective probability and its position in the gamble. Furthermore, the individual's utility function, her subjective probability function, and a function that measures her attitude towards risk can be separately derived from her preferences via a Representation Theorem. This theorem illuminates the role that each of these entities plays in preferences, and shows how REU theory explicates the components of instrumental rationality. (shrink)

It is a familiar argument that advocates accommodating the so-called paradoxes of decision theory by abandoning the “independence” postulate. After all, if we grant that choice reveals preference, the anomalous choice patterns of the Allais and Ellsberg problems violate postulate P2 of Savage's system. The strategy of making room for new preference patterns by relaxing independence is adopted in each of the following works: Samuelson, Kahneman and Tversky's “Prospect Theory”, Allais and Hagen, Fishburn, Chew and MacCrimmon, McClennen, and (...) in closely argued essays by Machina. (shrink)

The rule to maximize expected utility is intended for decisions where options involve risk. In those decisions the decision maker's attitude toward risk is important, and the rule ought to take it into account. Allais's and Ellsberg's paradoxes, however, suggest that the rule ignores attitudes toward risk. This suggestion is supported by recent psychological studies of decisions. These studies present a great variety of cases where apparently rational people violate the rule because of aversion or attraction to risk. Here (...) I attempt to resolve the issue concerning expected utility and risk. I distinguish two versions of the rule to maximize expected utility. One adopts a broad interpretation of the consequences of an option and has great intuitive appeal. The other adopts a narrow interpretation of the consequences of an option and seems to have certain technical and practical advantages. I contend that the version of the rule that interprets consequences narrowly does indeed neglect attitudes toward risk. That version of the rule excludes the risk involved in an option from the consequences of the option and, contrary to what is usually claimed, cannot make up for this exclusion through adjustments in probability and utility assignments. I construct a new, general argument that establishes this in a rigorous way. On the other hand, I contend that the version of the rule that interprets consequences broadly takes account of attitudes toward risk by counting the risk involved in an option among the consequences of the option. I rebut some objections to this version of the rules, in particular, the objection that the rule lacks practical interest. Drawing upon the literature on 'mean-risk' decision rules, I show that this version of the rule can be used to solve some realistic decision problems. (shrink)

Recently quantum probability theory started to be actively used in studies of human decision-making, in particular for the resolution of paradoxes (such as the Allais, Ellsberg, and Machina paradoxes). Previous studies were based on a cognitive metaphor of the quantum double-slit experiment—the basic quantum interference experiment. In this paper, we report on an economics experiment based on a triple-slit experiment design, where the slits are menus of alternatives from which one can choose. The test of nonclassicality is based on (...) the Sorkin equality (which was only recently tested in quantum physics). Each alternative is a voucher to buy products in one or more stores. The alternatives are obtained from all disjunctions including one, two or three stores. The participants have to reveal the amount for which they are willing to sell the chosen voucher. Interference terms are computed by comparing the willingness to sell a voucher built as a disjunction of stores and the willingness to sell the vouchers corresponding to the singleton stores. These willingness to sell amounts are used to estimate probabilities and to test both the law of total probabilities and the Born Rule. Results reject neither classical nor quantum probability. We discuss this initial experiment and our results and provide guidelines for future studies. (shrink)

The main result of this paper is a formal argument for the principle of maximizing expected utility that does not rely on the law of large numbers. Unlike the well-known arguments by Savage and von Neumann & Morgenstern, this argument does not presuppose the sure-thing principle or the independence axiom. The principal idea is to use the concept of transformative decision rules for decomposing the principle of maximizing expected utility into a sequence of normatively reasonable subrules. It is shown that (...) this procedure provides a resolution of Allais's paradox that cannot be obtained by Savage-style or von Neumann & Morgenstern-style arguments. (shrink)

The paradigm for modelling decision-making under uncertainty has undoubtedly been the theory of Expected Utility, which was first developed by von Neumann and Morgenstern (1944) and later extended by Savage (1954) to the case of subjective uncertainty. The inadequacy of the theory of Subjective Expected Utility (SEU) as a descriptive theory was soon pointed out in experiments, most famously by Allais (1953) and Ellsberg (1961). The observed departures from SEU noticed by Allais and Ellsberg became known as “paradoxes”. (...) The Ellsberg paradox gave rise, several years later, to a new literature on decision-making under ambiguity. The theoretical side of this literature was pioneered by Schmeidler (1989). This literature views the departures from SEU in situations similar to those discussed by Ellsberg as rational responses to ambiguity. The rationality is “recovered” by relaxing Savage's Sure-Thing principle and adding an ambiguity-aversion postulate. Thus the ambiguity-aversion literature takes a normative point of view and does consider Ellsberg-type choices as behavioural “anomalies”. (shrink)

Lucy Allais presents an original interpretation of Kant's transcendental idealism. She argues that his distinction between things in themselves and things as they appear to us has both epistemological and metaphysical components. Kant is committed to a genuine idealism about things as they appear to us, but this is not a phenomenalist idealism. He is committed to the claim that there is an aspect of reality that grounds mind-dependent spatio-temporal objects, and which we cannot cognize, but he does not (...) assert the existence of distinct non-spatio-temporal objects. On Allais's account, we cannot understand Kant's idealism without a clear account of his notion of intuition and of the role of intuition in cognition: she understands Kantian intuitions as representations that give us acquaintance with the objects of thought. This enables us to make sense of Kant's central argument for his idealism in the Transcendental Aesthetic, and to see why he takes the complete idealist position to be established there. (shrink)

:Space is not an empirical concept that has been drawn from outer experiences. For in order for certain sensations to be related to something outside me , thus in order for me to represent them as outside and next to one another, thus not merely different but as in different places, the representation of space must already be their ground. Thus the representation of space cannot be obtained from the relations of outer appearance through experience, but this outer experience is (...) itself first possible only through this representation. 1Abstract:Der Raum is kein empirischer Begriff, der von äußeren Erfahrungen abgezogen worden. Denn damit gewisse Empfindungen auf etwas außer mir bezogen werden, imgleichen damit ich sie als außer und neben einander, mithin nicht bloß verschieden, sondern als in verschiedenen Orten vorstellen können, dazu muß die Vorstellung des Raumes schon zum Grunde liegen. (shrink)