Abstract
K. R. Popper distinguished between two main uses of logic, the demonstrational one, in mathematical proofs, and the derivational one, in the empirical sciences. These two uses are governed by the following methodological constraints: in mathematical proofs one ought to use minimal logical means (logical minimalism), while in the empirical sciences one ought to use the strongest available logic (logical maximalism). In this paper I discuss whether Popper’s critical rationalism is compatible with a revision of logic in the empirical sciences, given the condition of logical maximalism. Apparently, if one ought to use the strongest logic in the empirical sciences, logic would remain immune to criticism and, thus, non-revisable. I will show that critical rationalism is theoretically compatible with a revision of logic in the empirical sciences. However, a question that remains to be clarified by the critical rationalists is what kind of evidence would lead them to revise the system of logic that underlies a physical theory, such as quantum mechanics? Popper’s falsificationist methodology will be compared with the recently advocated extension of the abductive methodology from the empirical sciences to logic by T. Williamson, since both of them arrive at the same conclusion concerning the status of classical logic.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For a detailed and extensive analysis of Popper’s account on deductive inference and logical constants see Schroeder-Heister (1984).
- 2.
Tennant (1985) takes the deductive structure of the experimental refutation of scientific hypotheses, present in Popper’s analysis of scientific method, to have the following schema (P): from hypotheses and boundary conditions we obtain predictions which, together with observational reports, lead to contradiction. The role of logic in the Popperian methodology is taken by Tennant to consist in the retransmission of falsity from the contradiction encountered to at least one of the premises. After he proves that every classically inconsistent set of first order sentences (expressed in ~ , v, &, → , and ∃) is intuitionistically inconsistent and (expressed in ~ , v, & and ∃) is minimally inconsistent, Tennant concludes that both intuitionistic and minimal logic are adequate for Popperian science. I think that the reduction of the role of logic only to the downward direction in Schema P is not a good adviser for judging which logic is adequate for Popperian science. Logic is also needed for deriving the predictions from hypotheses and boundary conditions and this is an important reason for which we have to use all the guns we have, i.e., we have to use all the logical rules available in order to deduce consequences from the empirical hypotheses. From this point of view, classical logic has an advantage, namely, it provides us with more logical tools than the other systems of logic.
- 3.
Mortensen and Burgess (1989, 48–49), in reply to Popper et al. (1970), have proposed a sort of logical minimalism in the empirical sciences, i.e., “prefer a weaker logic”. Their main reason for this principle is that a weaker logic has a larger number of theories and a theory criticized from a weaker logical base excludes the option of modifying it to enter in dispute with those of a stronger logic. This idea, however, in a different shape, is still debated nowadays. Williamson (2017, 337) briefly dismisses the idea that weakness is a strength in logic, since almost every logical principles has been subjected to criticism. Thus, a weaker logic is not necessarily in a better position. Bell (2019, 213–217) offers a detailed analysis of this idea, but he concludes that Williamson’s argument for logical strength does not work, unless we assume a sort of Quinean conservatism principle (see also footnote 10 below).
- 4.
This problem will be discussed in the last section of this paper.
- 5.
Critical rationalism is defined by Miller (2012, 93) as “the generalization, from the empirical sciences to the whole of our knowledge, of the methodological falsificationism (or deductivism) proposed by Karl Popper in Logic der Forschung”.
- 6.
The abductivist approach to logic is characterized by Russell (2019, 550) as consisting in two main claims about the justification of logic. The first one is the holism about justification, namely, that it is entire logics that are subject to justification, and not particular claims of logical consequence—logic being taken as a theory of the relation of logical consequence. The second one is that the justification is given by adequacy to the data, and the possession of virtues and absence of vices.
- 7.
Russell (2019, 556) argues that logical strength is theoretically neutral in the selection process of a logical theory, i.e., it is neither a virtue nor a vice. However, Russell discusses the problem outside a specific context and thus she concludes that “logical strength is something that a logic is supposed to get right, rather than something that it is always good to have more of”. Roughly expressed, the idea is that for certain purposes a weaker logic may be better than a stronger one. Certainly, Popper would agree with this idea, since he actually emphasized that, in the demonstrational use of logic, “the weaker the logical means we use, the less is the danger of consistency” (Popper et al. 1970, 20). Nevertheless, it is worth mentioning that the distinction between the demonstrational use of logic and its derivational use probably does not make much sense for the abductivists, who work, so to say, in a Quinean holist framework.
- 8.
In reply to Williamson, Russell (2019, 557–562) argues that logical strength does not entail scientific strength since, among other reasons, weaker and stronger logics are on a par if informativeness refers to the classification of arguments in valid or invalid. Certainly, this sense of informativeness is internal to the province of a certain logical theory, i.e., the validity of an argument is relative to a certain logical theory. However, as Williamson (2017, 337) exemplifies, if we extend the classical and the intuitionistic propositional calculi to include quantification into sentence position, ˹(∀p)(p v ~ p)˺ will be a theorem in the extended PC, while ˹ ~ (∀p)(p v ~ p)˺ will be assertable in the extended IC. Since a universal generalization is more informative than its negation, there is at least one clear sense in which a logical theory is more informative than another. (For an elaborate discussion on the relation between logical and scientific strength see Incurvati and Nicolai (2020)).
- 9.
- 10.
The severe tests that Williamson refers to are interpreted by Bell (2019, 220) in the sense that the theories which have successfully passed the testing process appear to be closed under the classical relation of logical consequence. However, Bell does not consider this fact as a disadvantage of weaker logical theories, since a theory closed under classical logic is also closed under a sub-classical one. Moreover, the successful testing of classical logic in the history of mathematics appears so since in mathematics the theoretical possibilities are usually reduced in practice to classical models. Certainly, Popper would agree with Bell here, since, in the demonstrational use of logic, using weaker means is a better option.
- 11.
Williamson does not seem to make a clear-cut distinction between the use of logic in the empirical sciences and in the mathematical ones, a thing quite natural if we consider the Quinean background of the abductive methodology. Nevertheless, since he selects classical logic as the best candidate simpliciter, its use in the empirical sciences is also presupposed here.
- 12.
By this I mean both epistemic and psychological attachment. Our ideas are embedded in the overall web of our mind and are important tools for the accommodation in natural and social environment. Since they are ours, we do not have all the time an objective and critical attitude towards them.
- 13.
To have a clear cut understanding of this idea we may think to the decision procedures in a logical system. Think for instance to the method or normal truth tables in classical propositional logic. We can easily decide I this case that there is no permissible possibility to have true premises and a false conclusion for modus ponens once we precisely disambiguate all the premises and the conclusion and accept the principle of bivalence. (See also Miller (2012, 100–103) for a discussion of the status of the logical rules of inference from a critical rationalist perspective).
- 14.
Popper takes a logical calculus to be either a derivational logic or a demonstrational one. The former is a system of logic “intended from the start to be a theory of inference in the sense that it allows us to derive from certain informative (non-logical) statements other informative statements” Popper (1947b, 230). This logic contains rules of inference for drawing consequences from hypotheses. A system of natural deduction fits very well this role. In contrast, “most systems of modern logic are not purely derivational, and some (for example, in the case of Hilbert-Ackerman) are not derivational at all.” Popper (1947b, 230). These systems are demonstrational logics and are serving better in the demonstrational use of logic. The derivations conducted in them usually start from logical axioms, definitions or theorems. It should be noted however that this distinction refers primarily to the deductive format of the system and not to its strength.
- 15.
“As far as the propositions of mathematics refer to reality, they are not certain; and as far as they’ are certain, they do not refer to reality”.
- 16.
We apply the trial and error method whenever we are faced with a problem, we tentatively advance a solution, and then criticize it as much as possible in order to eliminate the possible errors. A detailed analysis between the dialectic and the trial and error methods is offered by Popper (1940). As he emphasized, the dialectic development could be actually explained on the basis of the trial and error method, which has a wider application.
- 17.
Certainly, if the deduction theorem holds for the system of logic we use, this distinction is less relevant.
- 18.
It is important to mention that Birkhoff and von Neumann’s proposal, and likewise Reichenbach’s proposal, of introducing a new logic to deal with quantum phenomena was not for testing the quantum mechanics, but rather for obtaining a better understanding of it. See Putnam (2012) for a captivating story of the philosophical development of quantum logic.
- 19.
In addition, there are also some general considerations which suggest that logic cannot be treated in the same manner in which the physical hypotheses are, although both of them are criticizable. For a philosophical discussion on the status of classical logic, viewed from a metalogical point of view, see Brîncuş (2019).
References
Bell, Jc.: On Williamson’s new Quinean argument against nonclassical logic. Aust. J. Log. 16(7), 202–230 (2019)
Brîncuş, C.C.: Philosophical accounts of first-order logical truths. Acta Anal. 34(3), 369–383 (2019)
Carnap, R.: Empiricism, semantics and ontology. Rev. Int. Philos. 4(2), 20–40 (1950)
Einstein, A.: Geometry and experience (1921). In: Einstein, A. (ed.) Ideas and Opinions (fifth printing), pp. 232–246, Crown Publishers, New York, (1960)
Haack, S.: Deviant logic, fuzzy logic: beyond the formalism. The University of Chicago Press (1996)
Incurvati, L., Nicolai, C.: On logical and scientific strength. Unpublished manuscript. https://philarchive.org/archive/INCOLA Accessed 27 July 2020
Miller, D.: Critical rationalism: a restatement and defence. Open Court, Chicago (1994)
Miller, D.: Overcoming the justificationist Addiction. Studia Philosophica Wratislaviensia: Supplementary Volume, 93–103 (2012)
Mortensen, C., Burgess, T.: On logical strength and weakness. Hist. Philos. Log. 10, 47–51 (1989)
Popper, K.R.: What is dialectic?. Mind XLIX(194), 403–426 (1940)
Popper, K.R.: Why are the calculuses of logic and arithmetic applicable to reality?. In: Ryle, G., Lewy, C., Popper, K.R. (eds.) Symposium: Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?. Aristotelian Society Supplementary, vol. 20, no. 1, pp. 40–60 (1946)
Popper, K.R.: Logic without assumptions. Proc. Aristot. Soc. 47, 251–292 (1947a)
Popper, K.R.: New foundations for logic. Mind 56 (223), 193–235 (1947b)
Popper, K.R.: Science: conjectures and refutations. In: Popper, K.R. (ed.) Conjectures and Refutations: The Growth of Scientific Knowledge, pp. 32–65, Basic Books, London, Routledge (1962)
Popper, K.R.: A realist view of logic, physics, and history. In: Bondi, H. (ed.) Physics, Logic, and History, pp. 1–37. Plenum Press, New York (1970)
Putnam, H.: The curious story of quantum logic. In Putnam, H. (ed.) Philosophy in an Age of Science: Physics, Mathematics, and Skepticism, ed. M. De Caro and D. Macarthur, pp. 162–177, Cambridge, MA: Harvard University Press (2012)
Quine, W.O.V.: Two dogmas of empiricism. Philosophical Review 60(1), 20–43 (1951)
Quine, W.O.V.: Philosophy of logic. Prentice Hall, Englewood Cliffs, NJ (1970)
Russell, G.: Deviance and vice: strength as a theoretical virtue in the epistemology of logic. Philos. Phenomenol. Res. XCXI(3), 548–563 (2019)
Schroeder-Heister, P.: Popper’s theory of deductive inference and the concept of logical constant. Hist. Philos. Log. 5, 79–110 (1984)
Tarski, A.: Introduction to logic and to the methodology of the deductive sciences (1941). Dover Publications, New York, Translated by Olaf Helmer (1995)
Tennant, N.: Minimal logic is adequate for Popperian science. Br. J. Philos. Sci. 36(3), 325–329 (1985)
Williamson, T.: Semantic paradoxes and abductive methodology. In: Armour-Garb, B. (ed.) Reflections on the Liar, pp. 325–346, Oxford University Press (2017)
Acknowledgements
I would like to thank my audiences in Klagenfurt and Prague, where parts of my work on this paper have been presented, in particular Bernard Burgoyne, David Miller, Jagdish Hattiangadi, Olival Freire, Flavio Del Santo, and Stephen Senn. I also want to thank Mircea Flonta and Iulian Toader for helpful discussions. This work was supported by a grant of the Romanian Ministry of Education and Research, CNCS - UEFISCDI, project number PN-III-P1-1.1-PD-2019-0901, within PNCDI III.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Brîncuş, C.C. (2021). Logical Maximalism in the Empirical Sciences. In: Parusniková, Z., Merritt, D. (eds) Karl Popper's Science and Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-030-67036-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-67036-8_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-67035-1
Online ISBN: 978-3-030-67036-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)