Abstract
Popper emphasised both the problem-solving nature of human knowledge, and the need to criticise a scientific theory as strongly as possible. These aims seem to contradict each other, in that the former stresses the problems that motivate scientific theories while the one ignores the character of the problems that led to the formation of the theories against which the criticism is directed. A resolution is proposed in which problems as such are taken as prime in the search for knowledge, and subject to discussion. This approach is then applied to the problem of induction. Popper set great stake to his solution of it, but others doubted its legitimacy, in ways that are clarified by changing the form of the induction problem itself. That change draws upon logic, which is the subject of another application: namely, in contrast to Popper’s adhesion to classical logic as the only welcome form (because of the maximal strength of criticism that it dispenses), can other logics be used without abandoning his philosophy of criticism?
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Notes
Popper stated his position on these four features, and others, in various places; in this essay I shall cite from him either the main source, or an easily available representative one.
Unfortunately I have not been able to identify a source for the quotation in Mac Lane, and I never asked him about it. But it fits in with his contributions, especially to category theory; and in any case it stands on its own.
It has been suggested to me that the principle of the continuity of nature be construed as part of the meta-level description of the test. I see no advantage here; at least as Popper and his critics discuss it, it is an object-level assertion about the physical environment in which the test is being conducted.
Neither do scientists seem to have much skill in classical logic. In particular because of its special relevance to Popper’s philosophy of refutation, their ability to spot the modus tollens rule of inference is very limited (Kern et al. 1983)!
Popper often used the word ‘logic’ in its other sense, concerning the logic of a situation, most prominently in his book ‘The logic of scientific discovery’ (Popper 1959); I do not consider this sense in this essay.
The notion of corroboration itself will need to be modified for the chosen logic. So will verisimilitude, which in Popper’s form using classical logic faces well-known difficulties (see, for example, (Andersson 1978)).
Fuzzy logic, an offshoot of fuzzy set theory, has gained much attention among various communities but not much from logicians. The logic is formulated from these sets as a valuation function onto [0,1], deploying structure similarity such as associating minimisation of truth-values with conjunction and universal quantification and maximisation with disjunction and existential quantification, after the manner of a de Morgan algebra. Classical logic comes out as the ‘crisp’ case where the only values are 0 and 1—or [0,0] and [1,1] in my preferred version of the theory, where the range of the function takes sub-intervals of [0, 1] rather than numbers within it, and where the metalogic is also fuzzy (Grattan-Guinness 1979).
Whether one grants the valuation function the status of a logic is, as mentioned, a matter of convention. But I do not find Haack’s reservations over fuzzy logic (1996, 232–242) very convincing. The purpose of fuzzy logic is not so much to study vague arguments (pp. 229, 233) as to develop arguments using vague predicates and relations (such as ‘healthy’ and ‘the prettiest among’) that are handled in fuzzy set theory. (Dubois and Prade 1980) contains an early survey of the range of applications.
The theory has complications, but they show that even other many-valued logics are too simple, or else not sufficiently relevant to the applications. So the claims of simplicity made by some fuzzy logicians are indeed not happy, especially (for example) regarding linguistics; nor are some of the permitted qualifiers of truth. In general, the literature in fuzzy set theory and logic exhibits an unwelcome quantity of philosophical fuzziness!
For a review of the main positions on these topics, and on the psychology of reasoning see, for example, (Hanna 2006, chs. 4–5). Part of my motive for focussing upon the psychological sciences is a reservation about the practise of most philosophies of science, Popper’s included: an excessive emphasis on the physical sciences.
Darrigol (2005) surveys several parts of their histories, though without focussing much on testing. There must be many comparably complicated testing situations in technology, the life sciences and medicine, but they lie outside my specialist knowledge.
For example, Wittgensteinians explicitly reject or avoid hierarchies and prefer the showing-saying distinction.
Mathematics has long formed another part of this hope for certainty, although it has always stood a long way apart from logic, and still usually does: mathematicians’ logic ≠ logicians’ logic (Grattan-Guinness 2000)! I discussed this curious situation several times with Popper, who was very intrigued by it.
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Grattan-Guinness, I. Levels of Criticism: Handling Popperian Problems in a Popperian Way. Axiomathes 18, 37–48 (2008). https://doi.org/10.1007/s10516-007-9017-9
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DOI: https://doi.org/10.1007/s10516-007-9017-9