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Psychological tests from a (fuzzy-)logical point of view

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Abstract

Psychometric theory relies on two basic assumptions: (a) psychological constructs refer to latent (unobservable) variables and (b) psychological tests serve as a way to measure these constructs. This view is complemented by an alternative interpretation of psychological constructs, which neither relies on latent variables nor on the concept of measurement. Using the formal apparatus of many-valued logic, psychological constructs are re-interpreted as linguistic concepts (rather than latent variables), which can be inferred by means of logical calculus (as opposed to measurement). Thus, test scores do not refer to the values of latent variables, but to the degree to which the necessary and sufficient conditions for the ascription of a construct are fulfilled. Following this rationale, a formal theory of psychological tests is developed, which models the process of testing as logical inference. Applying the derived procedures, a person’s testing behaviour yields the degree to which a construct describes her adequately.

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Notes

  1. The formalism used within this paper is to be distinguished from ‚Fuzzy Logic in the wide sense‘, which covers a wide variety of methods based on Fuzzy Set Theory as introduced by Zadeh (1965). In contrast to these methods, ‘Fuzzy Logic in the narrow sense’ refers to a branch of mathematical logic. See Hajek (2006) for further elaboration of this distinction.

  2. Actually, the calculus introduced is a many-sorted variant of the actual Łukasiewicz logic with generalized quantifiers.

  3. In fact, the basic alphabet of Ł∀ consists of fewer signs from which the remaining ones are defined (Priest 2008). However, the distinction between basic connectives and derived connectives is not necessary for the purposes of this paper and is therefore omitted.

  4. In fact, the definition of Many is not quite as straightforward as presented here. But since this paper is merely concerned with the application of logical calculus rather than mathematical developements, we spare the technical details here (see Hájek 1998b for a complete presentation).

  5. The test behavior is assumed to be a number between 0 and 1—to achieve this, usually a linear transformation of test answers will be applied. However, other transformations may be justifiable depending on the context.

  6. Usually, we will only include attributes which are related to the construct at hand (i.e. have a membership degree greater than zero). However, from a formal point of view, this is not necessarily the case.

  7. The fact that attributes which are not assessed in the test always evaluate to 1 (irrelevant of the answer a person may have given) may seem counterintuitive: the less attributes the test assesses, the more a person satisfies the tested conditions. However, since every attribute not contained in the test reduces its validity, this does not impose any practical limitations.

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Authors and Affiliations

  1. Foundations of Education, University of Bamberg, Markusplatz 3, 96047, Bamberg, Germany

    Matthias Buntins

  2. Psychology and Methods of Educational Research, University of Bamberg, Wilhelmspost 3, 96047, Bamberg, Germany

    Katja Buntins

  3. Research Methods and Biopsychology, University of Braunschweig - Institute of Technology, Spielmannstr. 19, 38106, Braunschweig, Germany

    Frank Eggert

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  1. Matthias Buntins
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  2. Katja Buntins
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  3. Frank Eggert
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Correspondence to Matthias Buntins.

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Buntins, M., Buntins, K. & Eggert, F. Psychological tests from a (fuzzy-)logical point of view. Qual Quant 50, 2395–2416 (2016). https://doi.org/10.1007/s11135-015-0268-z

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