Multi-Valued Logics and Measurement

A whole new domain of relevant material opened up to me recently as a result of ongoing dialogues with colleagues concerning quantum logic and the implications for measurement and metrology. Given the vague and imprecise ways popular conceptions of “quantum logic” get thrown around, I’m going to take the time to provide some background.

The basic philosophical point concerns the departure from the binary logic of Aristotle’s laws of the excluded middle and of noncontradiction and a move toward alternative ternary, three-valued, multi-valued, and probabilistic logics. This shift has been kind of hanging in the air ever since Godel took down Whitehead and Russel’s Principia Mathematica, but it hasn’t coalesced into a coherent perspective. Multivalued logics have broad implications beyond the theme I’m taking up here; I’m taking those up elsewhere.

The point is pretty simple. Aristotle’s example, concerning whether there will or will not be a sea battle tomorrow, is plainly not an either/or matter; it depends on the weather, the supply lines, the conditions of the ships, the morale of the navies, etc., and so becomes probabilistic and contingent on circumstances (Pykacz, 2015, p. 4374). As might be guessed, one of the recurring problems philosophers have in this area is that they do not distinguish between the probabilistic form of statistical models motivated by sampling problems and the sufficiency of the mean and standard deviation, and the probabilistic form of measurement models motivated by response processes and the sufficiency of the score. This distinction captures the crux of the opportunity and the challenge.

So, in this context, one of the physicist-philosophers who pointed me toward metrology and revealed the solid philosophical basis we have for taking psychological and social measurement in that direction was the late Patrick Heelan. I encountered his work when, after I mentioned Ackermann’s (1985) “Data, Instruments, and Theory” to my AM advisor at Chicago, David Tracy, he alerted me to Heelan’s (1984) talk at a U Chicago conference, where another of my professors, Stephen Toulmin, had given his own perspective in reply.

I stole part of the title of my dissertation (“hermeneutic of instrumentation;” Fisher, 1988; also see Fisher, 1992a, 1994, 2003, 2004) from one of Heelan’s articles (Heelan, 1983b). Heelan worked and conversed extensively with Schrodinger, Wigner, and Heisenberg. As I recount in my dissertation, from Heisenberg, Heelan conceived of all measurement as requiring a simultaneous conjoint structure (Heelan, 1965, pp. 64-65). He furthermore was thoroughly versed in continental philosophy and provides a definitive, to my mind, elaboration of a philosophical foundation for measurement based in the work of Husserl, Heidegger, Gadamer, Merleau-Ponty, Ricoeur, and others (Heelan, 1972a/b, 1977, 1983a/b/c/d, 1991, 1998, 2003a/b/c, etc.; Heelan & Schulkin, 1998). I unfortunately failed to communicate the opportunity to apply all of this in psychological measurement when I met Heelan in 1995 or so at a philosophy of science conference in Chicago.

Getting to the first part of the point, Heelan (1970, 1971, 1974, 1983d) elaborates what he calls a quantum logic of framework transpositions, which looks to me to be akin to common-item test equating. Figure 10.1 in Heelan (1983d, p. 183) illustrates what looks to me to be an obvious conceptual connection with Figure 5.6.1 in Wright and Stone (1979, p. 97).

Figure 10.1: Extensional relationships between context-dependent languages of a particular quantum logic or Q-lattice (Heelan, 1983d)

Figure 5.6.1: Linking easy and hard tests via common items (Wright & Stone, 1979, p. 97)

What Heelan refers to as “context-dependent languages” are known in the context of Rasch’s probabilistic measurement models as scale-dependent counts of correct answers. Heelan’s “logic of framework transpositions” then concerns how these context dependencies can be incorporated into a broader frame of reference, which in the context of Rasch’s measurement models is accomplished via common-sample, common-item, or theory-based equating methods. The focus on the response process as the motivation for making the model probabilistic is key here; the statistical concern for sampling distributions applies at the level of the uncertainty, and that leads to another matter.

This second part of the point concerns the Gaussian shape of the error distributions, which has been the focus of recent work by David Andrich (Andrich & Pedler, 2019). Here, I was surprised just a couple of days ago to come across a whole field making use of what are referred to as quantum metrology and quantum Fisher information.

Giovannetti, et al. (2011) use the central limit theorem and the Gaussian distribution as a basis for uncertainty estimation, suggesting possible simplification and systematization in the Rasch context. For instance, Giovannetti, et al. (2006) say: “The typical quantum precision enhancement is of the order of the square root of the number of times the system is sampled.” On p. 4, in the conclusion, they say: “Schematic: (1) entangle N probes on the basis of eigenstates of H; (2) let the probes interact with the system; (3) measure on a dual basis. Result: a [sq rt]N precision enhancement.”

It seems plain to me that in Rasch’s perspective on measurement (Andrich, 1988, 2019; Fisher & Wright, 1994; Mari, et al., 2021; Pendrill, 2019; Pendrill & Fisher, 2015; Rasch, 1960; Wilson, 2005; Wright, 1977, 1997; Wright & Stone, 1979; etc.), the number of items on the test or survey is the analogue of the entangled N probes, the variation in their scale locations is the analogue of the eigenstates of H, and, as they are based in counts of the numbers of items administered, the Rasch standard error, mean square error, and separation reliability are analogues of the sq rt N precision enhancement. That is, N replications defines the precision of the unit as 1/N, and the uncertainty is the inverse of the information (Andrich & Pedler, 2019; Linacre, 2006; Wright & Stone, 1979, pp. 16, 26-27, 135-136). The stochastic treatment of the information function here is likely closely related to, if not identical with, quantum Fisher information.

Tóth and Appellaniz (2014, p. 3) similarly say: “multipartite entanglement is a prerequisite for maximal metrological precision in many very general metrological tasks. … In section 6, we review some of the very exciting recent findings showing that uncorrelated noise can change the scaling of the precision with the particle number under very general assumptions.”

Of course, as is also the case with connections concerning the phenomenon of stochastic resonance (Fisher, 1992b, 2011; Fisher & Wilson, 2015, pp. 69-71) the specific details of the overlaps here with probabilistic models for measurement will have to be spelled out, but it seems clear that these connections open up onto some even more exciting possibilities for generalizing measurement across the sciences. There’s a lot more that needs to be said on these topics, but it can all wait for another day.

References

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Andrich, D. (1988). Sage University Paper Series on Quantitative Applications in the Social Sciences. Vol. series no. 07-068: Rasch models for measurement. Sage Publications.

Andrich, D. (2019, November). Exemplifying natural science measurement in the social sciences with Rasch measurement theory. Journal of Physics: Conference Series, 1379(012006).

Andrich, D., & Pedler, P. (2019). A law of ordinal random error: The Rasch measurement model and random error distributions of ordinal assessments. Measurement, 131, 771-781.

Fisher, W. P., Jr. (1988). Truth, method, and measurement: The hermeneutic of instrumentation and the Rasch model [Diss]. Dissertation Abstracts International (Dept. of Education, Division of the Social Sciences), 49, 0778A. (376 pages, 23 figures, 31 tables)

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Tags: measurement, physics, psychology, Quantum logic, Rasch, science