# Mario Zanotti's research while affiliated with Stanford University and other places

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## Publications (17)

This is an important collection of essays dealing with the foundations of probability that will be of value to philosophers of science, mathematicians, statisticians, psychologists and educationalists. The collection falls into three parts. Part I comprises five essays on the axiomatic foundations of probability. Part II contains seven articles on...

The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that forN > 4 many new elementary inequalities beyond those of Bell must be satisfied by any hidden variable theory.

We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correl...

Suppes: I met de Finetti in May of 1960 at a symposium in Paris on decision theory. We had several lively informal conversations about the role of the axiom of choice in the foundations of probability, especially in relation to the existence of countably additive measures. Over the years I had the opportunity to meet de Finetti on various occasions...

This paper is concerned with inferences from phenomenological variables to hidden causes or hidden variables. A number of theorems of a general sort are stated. The paper concludes with a treatment of Bell’s inequalities and their generalization to more than four observables.

The primary criterion of adequacy of a probabilistic causal analysis is that the causal variable should render the simultaneous phenomenological data conditionally independent. The intuition back of this idea is that the common cause of the phenomena should factor out the observed correlations. So we label the principle the common cause criterion....

Probability kinematics is studied in detail within the framework of elementary probability theory. The merits and demerits of Jeffrey's and Field's models are discussed. In particular, the principle of maximum relative entropy and other principles are used in an epistemic justification of generalized conditionals. A representation of conditionals i...

Proposes 10 models for predicting a student's final grade placement in a computer-assisted instruction curriculum from the time the student spends taking lessons. Using data from approximately 2,000 elementary students, the models are tested both for ability to predict final grade placement and for ability to describe all points observed throughout...

For a variety of reasons there has been considerable interest in upper and lower probabilities as a generalization of ordinary probability. Perhaps the most evident way to motivate this generalization is to think of the upper and lower probabilities of an event as expressing bounds on the probability of the event. The most interesting case conceptu...

Describes a quantitative theory of student trajectories in a computer-assisted instruction course. The theory rests upon qualitative assumptions about information processing, from which a stochastic differential equation can be derived. The differential equation is characteristic of the course, but the constants of integration are estimated separat...

The main purpose of this note is to prove a lemma about random variables, and then to apply this lemma to the characterization of local theories of hidden variables by Bell (1964, 1966) and Wigner (1970), which are focused around Bell’s inequality. We use the results of the lemma in two different ways. The first is to show that the assumptions of B...

This investigation applied experimentally the use of predictive-control models integrated into computer-assisted instruction (CAI) as discussed earlier by Suppes, Fletcher, and Zanotti (1973). Many of those who are engaged in curriculum reform efforts have been dissatisfied with classical evaluations that simply compare the pre- and post-treatment...

The purpose of this article is to bring out in as conceptually clear terms as possible what seems to be a major incompleteness in the probability theory of particles offered by classical quantum mechanics. The exact nature of this incompleteness is illustrated by consideration of some simple quantum-mechanical examples. In addition, these examples...

In the standard theory of fundamental extensive measurement, qualitative axioms are formulated that lead to a numerical assignment unique up to a positive similarity transformation. The central idea of the theory of random quantities is to replace the numerical assignment by a random-variable assignment. This means that each object is assigned a ra...

## Citations

... First, there is no general consensus on the answer. For example, while Wigner wrote that Bell postulated both deterministic hidden variables and locality [3] (see also [4,5]), Bell himself later claimed that determinism was in fact inferred rather than assumed in his 1964 paper [6], and the debate has only intensified since then. Second, an appeal by Bell to the Einstein-Podolsky-Rosen (EPR) incompleteness argument [27] to support his claim puts the latter argument itself into question. ...

... 32 Given this, the reader will not find here the proposal of a clear-cut criterion to distinguish between quantities and nonquantities. At least since Hölder's (1901) paper, several axiomatizations of quantities have been proposed (e.g., Suppes, 1951;Mundy, 1987;Suppes & Zanotti, 1992), and choosing among them is not relevant here. On this matter a general issue is whether order is sufficient for a property to be considered a quantity. ...

... 3. Several of the studies correlate time spent at computer terminals with grade placement gains in the CAI curriculums. These studies reproduce the positive linear relationship that has been found in previous work of the same sort, for example, that reported in Suppes, Fletcher, Zanotti, Lorton, and Searle (1973). We would not expect to be able to find linear gains with indefinite increases in the amount of time spent at computer terminals, but it is clear, from the studies reported here and from other studies, that for a fairly wide range of time measurements an approximate linear relation holds very well. ...

... Each toss produces "head" with probability 1 2 p = that derives from the consideration of an abstract idealized 1 On the philosophical foundations of probability theory, see e.g. Suppes and Zanotti (1996), Gillies (2000), Jeffrey (2004), Galavotti (2005), and Mellor (2005). 2 As stated by Brakel (1976, p. 123), "(…) given arithmetic as a purely doctrine, the theory of probability can be developed without ever consulting reality". 3 coin. ...

... ABILITY In order to determine how much variation and dimensionality in performance are present among our pilot group, we have conducted qualitative tests for achievement and ability. We have included the most significant measures of programming performance as found in the study as well as other individual parameters of students referred to in the literature [4]. Nine parameters were looked at, including, the 4 found as the most significant for program evaluation [operations, length, functions, mem ], the calendar time [ of student B in all measures, and better on at least one. ...

... This is when probabilities come into play. For example, for a three random-variable system like the one above, Suppes and Zanotti (1981) showed that a necessary and sufficient condition for the existence of a joint probability distribution is the satisfaction of the following inequalities ...

Reference: Constraining Meanings With Contextuality

... See de Finetti (1937), Savage (1954, andScott (1964) for notable representation theorems connecting comparative unconditional confidence to unconditional probability. SeeKoopman (1940a) andSuppes and Zanotti (1982) for notable representation theorems connecting comparative conditional confidence to conditional probability. ...

... Cf. the other conditions proposed in(Kraft et al. 1959;Luce 1967;Savage 1972;Suppes and Zanotti 1976); for discussion seeFine (1973, IIC and IIID),Krantz et al. (2006, Chap.5) andSuppes (1994).17 The use of infinitesimals has been disputed byPruss (2012),Pruss (2013),Pruss (2014) andWilliamson (2007).Benci et al. (2016) andWeintraub (2008) attempt to reply to such objections. ...

... Thus one possible way to avoid these no-go results is to relax one or more of the classical probability axioms, leading to a non-classical probability space in which it is possible to come up with a non-contextual model. A number of such proposals have been made, including extended probability spaces which allow negative probabilities [2] or complex probabilities [53][54][55], generalized probability spaces [56] which relax the requirement that we should be able to assign probabilities to all conjunctions of events, upper probability spaces which are subadditive rather than additive on disjoint measurable sets [57], and quantum measures which are not additive on pairs of events but which are additive on triples of events [58]. ...

... 12 In particular, a more general proposal than simple conditionalization; if PðÁjeÞ is not well defined, or if e only loosely constrains posterior probability, minimizing D across distributions that satisfy these constraints determines a unique P e . 13 For explicit arguments to this effect see, for instance, (Jaynes [1957]) or (Shore and Johnson [1980]); for relevant surveys see (Domotor et al. [1980]) and (Csiszár [2008]). choose one view, saying that such-and-such is the content'. ...

Reference: The Semantics Latent in Shannon Information