Part II: Suppes and the mature theory. Representation and uniqueness.

When beliefs are quantified as credences, they are related to each other in terms of closeness and accuracy. The “accuracy first” approach in formal epistemology wants to establish a normative account for credences based entirely on the alethic properties of the credence: how close it is to the truth. To pull off this project, there is a need for a scoring rule. There is widespread agreement about some constraints on this scoring rule, but not whether a unique scoring rule stands (...) above the rest. The Brier score equips credences with a structure similar to metric space and induces a “geometry of reason.” It enjoys great popularity in the current debate. I point out many of its weaknesses in this article. An alternative approach is to reject the geometry of reason and accept information theory in its stead. Information theory comes fully equipped with an axiomatic approach which covers probabilism, standard conditioning, and Jeffrey conditioning. It is not based on an underlying topology of a metric space, but uses a non-commutative divergence instead of a symmetric distance measure. I show that information theory, despite initial promise, also fails to accommodate basic epistemic intuitions; and speculate on its remediation. (shrink)

Part II: Suppes and the mature theory. Representation and uniqueness.

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable (...) us to treat formally errors arising in nominal and ordinal measurements. (shrink)

In this paper, we provide a re-interpretation of qualitative and quantitative modeling from a representationalist perspective. In this view, both approaches attempt to construct abstract representations of empirical relational structures. Whereas quantitative research uses variable-based models that abstract from individual cases, qualitative research favors case-based models that abstract from individual characteristics. Variable-based models are usually stated in the form of quantified sentences. This syntactic structure implies that sentences about individual cases are derived using deductive reasoning. In contrast, case-based models are (...) usually stated using context-dependent existential sentences. This syntactic structure implies that sentences about other cases are justifiable by inductive reasoning. We apply this representationalist perspective to the problems of generalization and replication. Using the analytical framework of modal logic, we argue that the modes of reasoning are often not only applied to the context that has been studied empirically, but also on a between-contexts level. Consequently, quantitative researchers mostly adhere to a top-down strategy of generalization, whereas qualitative researchers usually follow a bottom-up strategy of generalization. Depending on which strategy is employed, the role of replication attempts is very different. In deductive reasoning, replication attempts serve as empirical tests of the underlying theory. Therefore, failed replications imply a faulty theory. From an inductive perspective, however, replication attempts serve to explore the scope of the theory. Consequently, failed replications do not question the theory per se, but help to shape its boundary conditions. We conclude that quantitative research may benefit from a bottom-up generalization strategy as it is employed in most qualitative research programs. Inductive reasoning forces us to think about the boundary conditions of our theories and provides a framework for generalization beyond statistical testing. In this perspective, failed replications are just as informative as successful replications, because they help to explore the scope of our theories. (shrink)

Although databases have been well-defined and thoroughly discussed in the computer science literature, the actual users of databases often have varying definitions and expectations of this essential computational infrastructure. Systems administrators and computer science textbooks may expect databases to be instantiated in a small number of technologies, but there are numerous examples of databases in non-conventional or unexpected technologies, such as spreadsheets or other assemblages of files linked through code. Consequently, we ask: How do the materialities of non-conventional databases differ (...) from or align with the materialities of conventional relational systems? What properties of the database do the creators of these artifacts invoke in their rhetoric describing these systems—or in the data models underlying these digital objects? To answer these questions, we conducted a close analysis of four non-conventional scientific databases. By examining the materialities of information representation in each case, we show how scholarly communication regimes shape database materialities—and how information organization paradigms shape scholarly communication. These cases show abandonment of certain constraints of relational database construction alongside maintenance of some key relational data organization strategies. We discuss the implications that these relational data paradigms have for data use, preservation, and sharing, and discuss the need to support a plurality of data practices and paradigms. (shrink)

Fechner's conception of psychophysics included both “outer psychophysics” the relation between stimulus intensity and the response reflecting sensation strength, and “inner psychophysics” the relation between neurelectric responses and sensation strength. In his own time outer psychophysics focussed on the form of the psychophysical law, with Fechner espousing a logarithmic law, Delboeuf a variant of the logarithmic law incorporating a resting level of neural activity, and Plateau a power law. One of the issues on which the dispute was focussed concerned the (...) appearance of contrasts if the overall illumination was increased or decreased; another issue was the question of whether a sensation of a “just noticeable difference” established for one value of a sensory dimension appeared the same for a value elsewhere on the dimension. The development of “inner psychophysics” led through the works of Delboeuf, Solomons, Jastrow, and Thurstone to modern signal detection theory. A third line of research, devoted to the question of what was meant by the “measurement” of sensation strength, stemmed from the criticism of Fechner's work by von Kries (1882) and others. Although a valid body of science could be built up without the intervening variable called “sensation strength,” such a science might be a cumbersome representation of reality. When an optical contrast is set up, and its overall illumination is increased or decreased, subjective contrasts involving medium levels of lightness vary little as illumination varies (as a power law based on sensation ratios or a logarithmic law based on sensation differences predict), but subjective contrasts involving extreme levels of lightness might be subject to the effects of other variables. (shrink)

In voting theory, monotonicity is the axiom that an improvement in the ranking of a candidate by voters cannot cause a candidate who would otherwise win to lose. The participation axiom states that the sincere report of a voter’s preferences cannot cause an outcome that the voter regards as less attractive than the one that would result from the voter’s non-participation. This article identifies three binary distinctions in the types of circumstances in which failures of monotonicity or participation can occur. (...) Two of the three distinctions apply to monotonicity, while one of those and the third apply to participation. The distinction that is unique to monotonicity is whether the voters whose changed rankings demonstrate non-monotonicity are better off or worse off. The distinction that is unique to participation is whether the marginally participating voter causes his first choice to lose or his last choice to win. The overlapping distinction is whether the profile of voters’ rankings has a Condorcet winner or a cycle at the top. This article traces the occurrence of all of the resulting combination of characteristics in the voting methods that can exhibit failures of monotonicity. (shrink)

In the context of computational social choice, we study voting methods that assign a set of winners to each profile of voter preferences. A voting method satisfies the property of positive involvement (PI) if for any election in which a candidate x would be among the winners, adding another voter to the election who ranks x first does not cause x to lose. Surprisingly, a number of standard voting methods violate this natural property. In this paper, we investigate different ways (...) of measuring the extent to which a voting method violates PI, using computer simulations. We consider the probability (under different probability models for preferences) of PI violations in randomly drawn profiles vs. profile-coalition pairs (involving coalitions of different sizes). We argue that in order to choose between a voting method that satisfies PI and one that does not, we should consider the probability of PI violation conditional on the voting methods choosing different winners. We should also relativize the probability of PI violation to what we call voter potency, the probability that a voter causes a candidate to lose. Although absolute frequencies of PI violations may be low, after this conditioning and relativization, we see that under certain voting methods that violate PI, much of a voter's potency is turned against them - in particular, against their desire to see their favorite candidate elected. (shrink)

A working theory can involve a compromise between pragmatic application and pure theory. This is illustrated by the development of Coombs's theory of data. The original version of the theory purported to guide investigators in the use of scaling techniques, but subsequent revisions including a logico-mathematical statement of the theory do not show whether the practical recommendations made earlier still apply. These practical recommendations, some of which Coombs has discarded to obtain a comprehensive theory, are important in spite of shortcomings (...) as they reveal differences between attitude and mental test data. The different forms of Coombs's theory point to the existence of a general problem, how to find a compromise between practice and pure theory. (shrink)

We reflect upon quantification in biology in two ways. First, from a sensory scientific perspective, we address theories and methods for studying sensation, perception, and cognition. Sensory science concerns action of the human senses, which are not passive receivers but operate in an active and fundamental way for human beings in various social and environmental contexts. In the past one could only handle one-to-one relationships within a univariate framework. Today we have tools to capture complexity closer to real world situations. (...) Thus, the second perspective is rooted in multivariate thinking and modeling tools. Relying mainly on soft modeling-exploratory methods, such as partial least squares regression, we claim that quantification in sensory biology in the future should be met with a dialectic view that simultaneously exploits the two perspectives. (shrink)