The expression "artificial animal" denotes a range of different objects from teddy bears to the results of genetic engineering. As a basis for further investigation, this article first of all presents the main interpretations and traces their systematic interconnections. The subsequent sections concentrate on artificial animals in the context of play. The development of material toys is fueled by robotics. It gives toys artificial sense organs, limbs, and cognitive abilities, thus enabling them to act in the real world. The (...) second line of development, closely related to research into Artificial Life, creates virtual beings "living" on computer screens. Themost essential difference between these variants are the sense modalities involved in interaction. Virtual beings can only be seen and heard, whereas material toys can be touched as well. Therefore, the simulation of haptic qualities plays an important role. In order to complete the proposed typology, two further areas are outlined, namely artificial animals outside play and "artificial animals in the medium of flesh" which are alive but designed and created by man. Research on artificial animals belongs to an extended notion of ecosemiotics, as they are part of ecosystems which may themselves be virtual such as the Internet. (shrink)
Teddy Seidenfeld – CMU An old, wise, and widely held attitude in Statistics is that modest intervention in the design of an experiment followed by simple statistical analysis may yield much more of value than using very sophisticated statistical analysis on a poorly designed existing data set.
Seidenfeld (Seidenfeld, T. [1988a], Decision theory without 'Independence' or without 'Ordering', Economics and Philosophy 4: 267-290) gave an argument for Independence based on a supposition that admissibility of a sequential option is preserved under substitution of indifferents at choice nodes (S). To avoid a natural complaint that (S) begs the question against a critic of Independence, he provided an independent proof of (S) in his (Seidenfeld, T. [1988b], Rejoinder [to Hammond and McClennen], Economics and Philosophy 4: 309-315). In reply to (...) my (Rabinowicz, W. [1995], To have one's cake and eat it too: Sequential choice and expected-utility violations, The Journal of Philosophy 92: 586-620), in which I argue that the proof is invalid, Seidenfeld (Seidenfeld, T. [2000], Substitution of indifferent options at choice nodes and admissibility: A reply to Rabinowicz, Theory and Decision 48: 305â310 this issue) submits that I fail to give due consideration to one of the underlying assumptions of his derivation: it is meant to apply only to those cases in which the agent's preferences are stable throughout the sequential decision process. The purpose of this note is to clarify the notion of preference stability so as meet this objection. (shrink)
An agent who violates independence can avoid dynamic inconsistency in sequential choice if he is sophisticated enough to make use of backward induction in planning. However, Seidenfeld has demonstrated that such a sophisticated agent with dependent preferences is bound to violate the principle of dynamic substitution, according to which admissibility of a plan is preserved under substitution of indifferent options at various choice nodes in the decision tree. Since Seidenfeld considers dynamic substitution to be a coherence condition on dynamic choice, (...) he concludes that sophistication cannot save a violator of independence from incoherence. In response to McClennenâs objection that relying on dynamic substitution when independence is at stake must be question-begging, Seidenfeld undertakes to prove that dynamic substitution follows from the principle of backward induction alone, provided we assume that the agentâs admissible choices from different sets of feasible plans are all based on a fixed underlying preference ordering of plans. This paper shows that Seidenfeld's proof fails: depending on the interpretation, it is either invalid or based on an unacceptable assumption. (shrink)
John Muir, who had seen enough natural beauty for ten life times, simply fumbles his words when it comes to describing Prince William Sound: one of the richest, most glorious mountain landscapes I ever beheld— peak over peak lying deep in the sky, a thousand of them, icy and shining…. and great breadth of sun-spangled, ice-dotted waters in front…. grandeur and beauty in a thousand forms awaiting us at every turn in this bright and spacious wonderland. Prince William Sound, which (...) sits at the northernmost part of the Gulf of Alaska, defines the water border of the Chugach National Forest. Established by Teddy Roosevelt in 1907, it is next in size only to the Tongass National Forest—the largest in the .. (shrink)
Can there be good reasons for judging one set of probabilistic assertions more reliable than a second? There are many candidates for measuring "goodness" of probabilistic forecasts. Here, I focus on one such aspirant: calibration. Calibration requires an alignment of announced probabilities and observed relative frequency, e.g., 50 percent of forecasts made with the announced probability of.5 occur, 70 percent of forecasts made with probability.7 occur, etc. To summarize the conclusions: (i) Surveys designed to display calibration curves, from which a (...) recalibration is to be calculated, are useless without due consideration for the interconnections between questions (forecasts) in the survey. (ii) Subject to feedback, calibration in the long run is otiose. It gives no ground for validating one coherent opinion over another as each coherent forecaster is (almost) sure of his own long-run calibration. (iii) Calibration in the short run is an inducement to hedge forecasts. A calibration score, in the short run, is improper. It gives the forecaster reason to feign violation of total evidence by enticing him to use the more predictable frequencies in a larger finite reference class than that directly relevant. (shrink)
De Finetti introduced the concept of coherent previsions and conditional previsions through a gambling argument and through a parallel argument based on a quadratic scoring rule. He shows that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules which might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while (...) preserving the equivalence of de Finetti’s two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti’s fundamental theorem of prevision to deal with conditional previsions. (shrink)
This paper (based on joint work with M.J.Schervish and J.B.Kadane) discusses some differences between the received theory of regular conditional distributions, which is the countably additive theory of conditional probability, and a rival theory of conditional probability using the theory of finitely additive probability. The focus of the paper is maximally "improper" conditional probability distributions, where the received theory requires, in effect, that P{a: P(a|a) = 0} = 1. This work builds upon the results of Blackwell and Dubins (1975).
Experimenters sometimes insist that it is unwise to examine data before determining how to analyze them, as it creates the potential for biased results. I explore the rationale behind this methodological guideline from the standpoint of an error statistical theory of evidence, and I discuss a method of evaluating evidence in some contexts when this predesignation rule has been violated. I illustrate the problem of potential bias, and the method by which it may be addressed, with an example from the (...) search for the top quark. A point in favor of the error statistical theory is its ability, demonstrated here, to explicate such methodological problems and suggest solutions, within the framework of an objective theory of evidence. (shrink)
This essay is, primarily, a discussion of four results about the principle of maximizing entropy (MAXENT) and its connections with Bayesian theory. Result 1 provides a restricted equivalence between the two: where the Bayesian model for MAXENT inference uses an "a priori" probability that is uniform, and where all MAXENT constraints are limited to 0-1 expectations for simple indicator-variables. The other three results report on an inability to extend the equivalence beyond these specialized constraints. Result 2 established a sensitivity of (...) MAXENT inference to the choice of the algebra of possibilities even though all empirical constraints imposed on the MAXENT solution are satisfied in each measure space considered. The resulting MAXENT distribution is not invariant over the choice of measure space. Thus, old and familiar problems with the Laplacian principle of Insufficient Reason also plague MAXENT theory. Result 3 builds upon the findings of Friedman and Shimony (1971; 1973) and demonstrates the absence of an exchangeable, Bayesian model for predictive MAXENT distributions when the MAXENT constraints are interpreted according to Jaynes's (1978) prescription for his (1963) Brandeis Dice problem. Lastly, Result 4 generalizes the Friedman and Shimony objection to cross-entropy (Kullback-information) shifts subject to a constraint of a new odds-ratio for two disjoint events. (shrink)
When real-valued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes – equivalent variables – occupy the same place in a preference ordering. However, if strict preference respects uniform, strict dominance in outcomes between variables, and if indifference between two variables entails indifference between their difference and the status quo, (...) then preferences over rich sets of unbounded variables, such as variables used in the St. Petersburg paradox, cannot preserve indifference between all pairs of equivalent variables. In such circumstances, preference is not a function only of probability and utility for outcomes. Then the preference ordering is not defined in terms of lotteries. (shrink)
• Coherence1 for previsions of random variables with generalized betting; • Coherence2 for probability forecasts of events with Brier score penalty; • Coherence3 probability forecasts of events with various proper scoring rules.
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1984, Volume Two: Symposia and Invited Papers. (1984), pp. 201-212.
We contrast three decision rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: Γ-Maximin, Maximality, and E-admissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that carries greater expected utility for each probability in a (closed) convex set. If the convex set is a singleton, then each rule agrees with maximizing expected utility. We show that, even when the option set is (...) convex, this pairwise comparison between acts may fail to identify those acts which are Bayes for some probability in a convex set that is not closed. This limitation affects two of the decision rules but not E-admissibility, which is not a pairwise decision rule. E-admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes. (shrink)
Part 1 Background on de Finetti’s twin criteria of coherence: Coherence1: 2-sided previsions free from dominance through a Book. Coherence2: Forecasts free from dominance under Brier (squared error) score. Part 2 IP theory based on a scoring rule.
This paper examines definitions of independence for events and variables in the context of full conditional measures; that is, when conditional probability is a primitive notion and conditioning is allowed on null events. Several independence concepts are evaluated with respect to graphoid properties; we show that properties of weak union, contraction and intersection may fail when null events are present. We propose a concept of “full” independence, characterize the form of a full conditional measure under full independence, and suggest how (...) to build a theory of Bayesian networks that accommodates null events. (shrink)
A transformative decision rule alters the representation of a decision problem, either by changing the set of alternative acts or the set of states of the world taken into consideration, or by modifying the probability or value assignments. A set of transformative decision rules is order-independent in case the order in which the rules are applied is irrelevant. The main result of this paper is an axiomatic characterization of order-independent transformative decision rules, based on a single axiom. It is shown (...) that the proposed axiomatization resolves a problem observed by Teddy Seidenfeld in a previous axiomatization by Peterson. (shrink)
This paper offers a comparison between two decision rules for use when uncertainty is depicted by a non-trivial, convex2 set of probability functions Γ. This setting for uncertainty is different from the canonical Bayesian decision theory of expected utility, which uses a singleton set, just one probability function to represent a decision maker’s uncertainty. Justifications for using a non-trivial set of probabilities to depict uncertainty date back at least a half century (Good, 1952) and a foreshadowing of that idea can (...) be found even in Keynes’ (1921), where he allows that not all hypotheses may be comparable by qualitative probability – in accord with, e.g., the situation where the respective intervals of probabilities for two events merely overlap with no further (joint) constraints, so that neither of the two events is more, or less, or equally probable compared with the other. (shrink)
Will the proliferation of devices that provide the continuous archival and retrieval of personal experiences (CARPE) improve control over, access to and the record of collective knowledge as Vannevar Bush once predicted with his futuristic memex? Or is it possible that their increasing ubiquity might pose fundamental risks to humanity, as Donald Norman contemplated in his investigation of an imaginary CARPE device he called the “Teddy”? Through an examination of the webcam experiment of Jenni Ringley and the EyeTap experiments (...) of Steve Mann, this article explores some of the social implications of CARPE. The authors’ central claim is that focussing on notions of individual consent and control in assessing the privacy implications of CARPE while reflective of the individualistic conception of privacy that predominates western thinking, is nevertheless inadequate in terms of recognizing the effect of individual uptake of these kinds of technologies on the level of privacy we are all collectively entitled to expect. The authors urge that future analysis ought to take a broader approach that considers contextual factors affecting user groups and the possible limitations on our collective ability to control the social meanings associated with the subsequent distribution and use of personal images and experiences after they are captured and archived. The authors ultimately recommend an approach that takes into account the collective impact that CARPE technologies will have on privacy and identity formation and highlight aspects of that approach. (shrink)
For Savage (1954) as for de Finetti (1974), the existence of subjective (personal) probability is a consequence of the normative theory of preference. (De Finetti achieves the reduction of belief to desire with his generalized Dutch-Book argument for Previsions.) Both Savage and de Finetti rebel against legislating countable additivity for subjective probability. They require merely that probability be finitely additive. Simultaneously, they insist that their theories of preference are weak, accommodating all but self-defeating desires. In this paper we dispute these (...) claims by showing that the following three cannot simultaneously hold: (i) Coherent belief is reducible to rational preference, i.e. the generalized Dutch-Book argument fixes standards of coherence. (ii) Finitely additive probability is coherent. (iii) Admissible preference structures may be free of consequences, i.e. they may lack prizes whose values are robust against all contingencies. (shrink)
We discuss several features of coherent choice functions —where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of (...) probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility. (shrink)
We extend de Finetti’s (1974) theory of coherence to apply also to unbounded random variables. We show that for random variables with mandated infinite prevision, such as for the St. Petersburg gamble, coherence precludes indifference between equivalent random quantities. That is, we demonstrate when the prevision of the difference between two such equivalent random variables must be positive. This result conflicts with the usual approach to theories of Subjective Expected Utility, where preference is defined over lotteries. In addition, we explore (...) similar results for unbounded variables when their previsions, though finite, exceed their expected values, as is permitted within de Finetti’s theory. In such cases, the decision maker’s coherent preferences over random quantities is not even a function of probability and utility. One upshot of these findings is to explain further the differences between Savage’s theory (1954), which requires bounded utility for non-simple acts, and de Finetti’s theory, which does not. And it raises a question whether there is a theory that fits between these two. (shrink)
The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org.
In 1936 R.A.Fisher asked the pointed question, "Has Mendel's Work Been Rediscovered?" The query was intended to open for discussion whether someone altered the data in Gregor Mendel's classic 1866 research report on the garden pea, "Experiments in Plant-Hybridization." Fisher concluded, reluctantly, that the statistical counts in Mendel's paper were doctored in order to create a better intuitive fit between Mendelian expected values and observed frequencies. That verdict remains the received view among statisticians, so I believe. Fisher's analysis is a (...) tour de force of so-called "Goodness of Fit" statistical tests using c2 to calculate significance levels, i.e., P-values. In this presentation I attempt a defense of Mendel's report, based on several themes. (1) Mendel's experiments include some important sequential design features that Fisher ignores. (2) Fisher uses particular statistical techniques of Meta-analysis for pooling outcomes from different experiments. These methods are subject to critical debate. and (3) I speculate on a small modification to Mendelian theory that offers some relief from Fisher's harsh conclusion that Mendel's data are too good to be true. (shrink)
DAVID BARSAMIAN: REGIME CHANGE is a new term in the lexicon. Kind of like change of address. It sounds somewhat innocuous. It certainly sounds a lot better than invasion, overthrow and occupation. The U.S. is an old hand at regime change. We’re in a year that marks a couple of anniversaries. Today is the 30th anniversary of the U.S.-backed coup in Chile. October 25 marks the 20th anniversary of the U.S. invasion of Grenada. But I’m particularly thinking of regime change (...) in Iran. 50 years ago, in August 1953, Operation Ajax, carried out by a CIA agent who was incidentally Teddy Roosevelt’s grandson, overthrew the conservative parliamentary democracy led by Mohammed Mossadeq and restored the Shah to the Peacock Throne, where he ruled for the next 25 years. (shrink)
DeFinetti took the concept of random variables as gambles very seriously, and used the concept to motivate the familiar concepts of probability and expectation. For each gamble X, he assumed that “You” would assign a value P (X), called the prevision of X so that you would be willing to accept the gamble β[X − P (X)] as fair for all positive and negative values β. The only constraint that deFinetti envisioned for you and your previsions is that you insisted (...) that there be no positive amount that you had to lose for sure. For example, you would not be allowed to call a gamble fair if its supremum were negative. On the other hand, the criterion is weak enough to allow you call a gamble fair if its supremum is 0, even if all of its possible values are negative. (shrink)
Statistical decision theory, whether based on Bayesian principles or other concepts such as minimax or admissibility, relies on the idea of minimizing expected loss or maximizing expected utility. Loss and utility functions are generally treated as unitless numerical measures of how costly or valuable are the various consequences of potential decisions. In this paper, we address directly the issue of the units in which loss and utility are settled and the implications that those units have on the rankings of potential (...) decisions. The simplest example is to imagine that the loss will be paid in units of some currency. If there are multiple currencies available for paying the loss, one must take explicit account of which currency is used as well as the exchange rates between the various available currencies. (shrink)
Conditioning can make imprecise probabilities uniformly more imprecise. We call this effect "dilation". In a previous paper (1993), Seidenfeld and Wasserman established some basic results about dilation. In this paper we further investigate dilation on several models. In particular, we consider conditions under which dilation persists under marginalization and we quantify the degree of dilation. We also show that dilation manifests itself asymptotically in certain robust Bayesian models and we characterize the rate at which dilation occurs.
When can a Bayesian investigator select an hypothesis H and design an experiment (or a sequence of experiments) to make certain that, given the experimental outcome(s), the posterior probability of H will be lower than its prior probability? We report an elementary result which establishes sufficient conditions under which this reasoning to a foregone conclusion cannot occur. Through an example, we discuss how this result extends to the perspective of an onlooker who agrees with the investigator about the statistical model (...) for the data but who holds a different prior probability for the statistical parameters of that model. We consider, specifically, one-sided and two-sided statistical hypotheses involving i.i.d. Normal data with conjugate priors. In a concluding section, using an "improper" prior, we illustrate how the preceding results depend upon the assumption that probability is countably additive. (shrink)
The effects of research ethics training on medical students' attitudes about clinical research are examined. A preliminary randomized controlled trial evaluated 2 didactic approaches to ethics training compared to a no-intervention control. The participant-oriented intervention emphasized subjective experiences of research participants (empathy focused). The criteria-oriented intervention emphasized specific ethical criteria for analyzing protocols (analytic focused). Compared to controls, those in the participant-oriented intervention group exhibited greater attunement to research participants' attitudes related to altruism, trust, quality of relationships with researchers, desire (...) for information, hopes about participation and possible therapeutic misconception, importance of consent forms, and deciding quickly about participation. The participant-oriented group also agreed more strongly that seriously ill people are capable of making their own research participation decisions. The criteria-oriented intervention did not affect learners' attitudes about clinical research, ethical duties of investigators, or research participants' decision making. An empathy-focused approach affected medical students' attunement to research volunteer perspectives, preferences, and attributes, but an analytically oriented approach had no influence. These findings underscore the need to further examine the differential effects of empathy-versus analytic-focused approaches to the teaching of ethics. (shrink)
The degree of incoherence, when previsions are not made in accordance with a probability measure, is measured by either of two rates at which an incoherent bookie can be made a sure loser. Each bet is considered as an investment from the points of view of both the bookie and a gambler who takes the bet. From each viewpoint, we define an amount invested (or escrowed) for each bet, and the sure loss of incoherent previsions is divided by the escrow (...) to determine the rate of incoherence. Potential applications include the treatment of arbitrage opportunities in financial markets and the degree of incoherence of classical statistical procedures. We illustrate the latter with the example of hypothesis testing at a fixed size. (shrink)
Consider two SEU Bayesian decision makers, Dick and Jane, who wish to form a cooperative partnership that will make decisions, constrained by the following two principles governing coherence and compromise.
We extend a result of Dubins [3] from bounded to unbounded random variables. Dubins [3] showed that a finitely additive expectation over the collection of bounded random variables can be written as an integral of conditional expectations (disintegrability) if and only if the marginal expectation is always within the smallest closed interval containing the conditional expectations (conglomerability). We give a sufficient condition to extend this result to the collection Z of all random variables that have finite expected value and whose (...) conditional expectations are finite and have finite expected value. The sufficient condition also allows the result to extend some, but not all, subcollections of Z. We give an example where the equivalence of disintegrability and conglomerability fails for a subcollection of Z that still contains all bounded random variables. (shrink)
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1978, Volume Two: Symposia and Invited Papers. (1978), pp. 478-489.
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The Independence postulate links current preferences between called-off acts with current preferences between constant acts. Under the assumption that the chance-events used in compound von Neumann-Morgenstern lotteries are value-neutral, current preferences between these constant acts are linked to current preferences between hypothetical acts, conditioned by those chance events. Under an assumption of stability of preferences over time, current preferences between these hypothetical acts are linked to future preferences between what are then and there constant acts. Here, I show that a (...) failure of Independence with respect to current preferences leads to an inconsistency in sequential decisions. Two called-off acts are constructed such that each is admissible in the same sequential decision and yet one is strictly preferred to the other. This responds to a question regarding admissibility posed by Rabinowicz ([2000] Preference stability and substitution of indifferents: A rejoinder to Seidenfeld, Theory and Decision 48: 311â318 [this issue]). (shrink)
We report two issues concerning diverging sets of Bayesian (conditional) probabilities-divergence of "posteriors"-that can result with increasing evidence. Consider a set P of probabilities typically, but not always, based on a set of Bayesian "priors." Fix E, an event of interest, and X, a random variable to be observed. With respect to P, when the set of conditional probabilities for E, given X, strictly contains the set of unconditional probabilities for E, for each possible outcome X = x, call this (...) phenomenon dilation of the set of probabilities (Seidenfeld and Wasserman 1993). Thus, dilation contrasts with the asymptotic merging of posterior probabilities reported by Savage (1954) and by Blackwell and Dubins (1962). (1) In a wide variety of models for Robust Bayesian inference the extent to which X dilates E is related to a model specific index of how far key elements of P are from a distribution that makes X and E independent. (2) At a fixed confidence level, (1-α), Classical interval estimates A n for, e.g., a Normal mean θ have length O(n -1/2 ) (for sample size n). Of course, the confidence level correctly reports the (prior) probability that θ ∈ A n ,P(A n )=1-α , independent of the prior for θ . However, as shown by Pericchi and Walley (1991), if an ε -contamination class is used for the prior on the parameter θ , there is asymptotic (posterior) dilation for the A n , given the data. If, however, the intervals A ′ n are chosen with length $O(\sqrt{\log (\text{n})/\text{n})}$ , then there is no asymptotic dilation. We discuss the asymptotic rates of dilation for ClassClassical and Bayesian interval estimates and relate these to Bayesian hypothesis testing. (shrink)
uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability..
The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org.
Tiebreak rules are necessary for revealing indifference in non- sequential decisions. I focus on a preference relation that satisfies Ordering and fails Independence in the following way. Lotteries a and b are indifferent but the compound lottery f, 0.5b> is strictly preferred to the compound lottery f, 0.5a>. Using tiebreak rules the following is shown here: In sequential decisions when backward induction is applied, a preference like the one just described must alter the preference relation between a and b at (...) certain choice nodes, i.e., indifference between a and b is not stable. Using this result, I answer a question posed by Rabinowicz (1997) concerning admissibility in sequential decisions when indifferent options are substituted at choice nodes. (shrink)
The "traditional" view of normative decision theory, as reported (for example) in chapter 2 of Luce and RaiÃa's [1957] classic work, Games and Decisions, proposes a reduction of sequential decisions problems to non-sequential decisions: a reduction of extensive forms to normal forms. Nonetheless, this reduction is not without its critics, both from inside and outside expected utility theory, It islay purpose in this essay to join with those critics by advocating the following thesis.
Teddy Seidenfeld recently claimed that Kolmogorov's probability theory transgresses the Substitutivity Law. Underscoring the seriousness of Seidenfeld's charge, the author shows that (Popper's version of) the law, to wit: If (∀ D)(Pr(B,D)=Pr(C,D)), then Pr(A,B)=Pr(A,C), follows from just C1. 0≤ Pr(A,B)≤ 1 C2. Pr(A,A)=1 C3. Pr(A & B,C)=Pr(A,B & C)× Pr(B,C) C4. Pr(A & B,C)=Pr(B & A,C) C5. Pr(A,B & C)=Pr(A,C & B), five constraints on Pr of the most elementary and most basic sort.
