Refined Verisimilitude pp 1-34 | Cite as

# Introduction and Terminology

## Abstract

The approach-to-the-truth project started with the publication of Popper’s *Conjectures and Refutations* in 1963, which contains the first formal explication of the “verisimilitude” notion. Eleven years after this publication, the twenty-fifth volume of the *British Journal for the Philosophy of Science* gave the project a significant incentive. In their contributions, Miller, Tichý, and Harris, proved the inadequacy of Popper’s definition. In this chapter the rise and fall of Popper’s proposal is sketched, and a technical framework is developed to compare the alternatives to Popper’s proposal. I deal with the general philosophical background of the verisimilitude notion, Popper’s definition and its failure in Sections 1.1–1.2. An explanation of how I shall compare the various proposals presented in Chapters 2–3 is given in Section 1.3. I shall base this comparison on the most elementary mathematical applications of the definitions: propositional languages. The two different ways to paraphrase theories and data in the algebra foreshadow the paramount distinction between two kinds of approach-to-the-truth proposals: the content and likeness definitions. This distinction is reconstructed in Section 1.4 in terms of two different strategies to revise Popper’s original explication. It leads to a formal definition of the contrast between content and likeness definitions. Finally, in the fifth section, I introduce further metatheoretical properties used in later chapters.

## Keywords

Atomic Proposition Atomic Sentence False Proposition Propositional Language False Theory## Preview

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## Notes

- 1.Peirce (1965) fifth volume section 565.Google Scholar
- 2.See for instance Niiniluoto (1984), p.76–77.Google Scholar
- 3.Whether the scientific claims about these different fields are, or should be, reducible is still a question of debate.Google Scholar
- 4.Laudan (1981).Google Scholar
- 5.Kuipers (1987), p.7.Google Scholar
- 6.also relates to the cognitive problem that is to be solved. I elaborate on this in Chapter 5.Google Scholar
- 7.In the sequel, the term “proposition cp of a language 2” (cp E Sent(2)) refers to the set of equivalent 2-sentences, [cp]
_{=}, and has nothing to do with*intentional objects*. Often, we refer to such a set by one representative of it, and occasionally will use proposition, sentence and statements almost as synonyms. In the same vain, we use the logical connectives of propositions. The set of 2 propositions is designated by Prop(2):= [cp]_ I cp E Sent(2)).Google Scholar - 8.Niiniluoto (1987), p.256.Google Scholar
- 9.The empirical sentences are the non-tautological, and non-contradictory sentences of a language.Google Scholar
- 10.Popper (1963), p 234.Google Scholar
- 11.Popper (1963), p.233.Google Scholar
- 12.Rescher (1967, chap. vi) gives a nice introduction to Kleene’s three-valued logic.Google Scholar
- 13.Weston (1992).Google Scholar
- 14.See Kuipers (1987), p. 88. Not surprisingly, the explication of such a general idea of progress has a wider scope than only Popper’s methodology. There are also implications for instrumentalistic and technological progress.Google Scholar
- 15.Popper (1963), p.233.Google Scholar
- 16.
*Ibid*.Google Scholar - 17.Popper (1963), chapter X, section xi.Google Scholar
- 18.Compare with the
*dual consequences*of Jan Wolenski (1990), p. 619.Google Scholar - 19.The Miller-Tichÿ result has also consequence for followers of Popper. For example, Lakatos (1970), p.116 “follows Popper” and defines “sophisticated falsification”. According to Lakatos’s proposal
*T*is falsified if there new theory*T’*such that there is a true novel fact*e*,such that*T’ I- e I-*^{-}*’T*,and*T’ F- Tv*T (T is the truth); consequence is that*T*can only be falsified using a*true*theory T’(else*T’= 1);*which is at least improbable.Google Scholar - 20.Hattiangadi (1983), note 10; Miller (1974).Google Scholar
- 21.See Tichÿ (1974) and Miller (1974).Google Scholar
- 22.
- 23.See Tichÿ (1974), p.157, note 2; Harris (1974), p.165; Miller (1978), p.415 and Niiniluoto (1987), p. 190.Google Scholar
- 24.McCarthy (1980) on circumscription was an important starting point of non-monotonic reasoning, see also Makinson (1994). Brewka, Dix and Kolonige (1997) is a comprehensive and readable introduction on non-monotonic reasoning.Google Scholar
- 25.There is an increasing interest in the relation between verisimilitude and standard concepts in philosophical logic. See e.g. van Benthem (1987), Ryan and Schobbens (1995).Google Scholar
- 26.Schurz and Weingartner’s (1987) revision of Popper’s definition drops the assumption that all consequences of theories are equally important. The definition only takes the “relevant consequences” of theories into account. According to their analysis, however, one consequence cp may be relevant whereas an equivalent reformulation of cp is not. For instance,
*(p*A^{-}*q)*-]^{-}*’p*is a relevant implication, and it is equivalent to the irrelevant*p -+ (p y q)*(see Schurz and Weingartner (1987, p.54)).Google Scholar - 27.Some of the definitions presented in Chapter 3–4 are to be found in Kuipers (1987); among the most recent proposals are Wolenski (1990), Gerla (1992), Zamora Bonilla (1992/6), Volpe (1995), and Kieseppä (1996). Niiniluoto (1998) gives an excellent overview of the third period“ in the approach-to-the-truth research.Google Scholar
- 28.E.g. see van Benthem (1987).Google Scholar
- 29.Although Popper used the content and similarity notions in his original presentation, Hilpinen (1976, p. 38) stressed the difference between the content and likeness approach. Finally, it was Oddie who has claimed that the content and likeness distinction obtains for almost all definitions. See e.g. Oddie (1990).Google Scholar
- 30.Niiniluoto (1987), p.459.Google Scholar
- 31.Popper (1963), p.391–398.Google Scholar
- 32.Popper (1972), p. 56.Google Scholar
- 33.See also footnote 24
^{a}in Popper (1972), p. 56.Google Scholar - 34.Miller (1978).Google Scholar
- 35.Oddie (1986), sect.1.3.Google Scholar
- 36.See Hilpinen (1975).Google Scholar
- 37.From Niiniluoto (1987b), p. 13–15.Google Scholar
- 38.Kuipers (1992), p.317–319.Google Scholar
- 39.Kuipers (1992b) applies the idealization concretization ideas ofNowak (1980).Google Scholar
- 40.Brink and Heidema (1987).Google Scholar
- 41.For a more elaborate introduction see the Section 2.6.Google Scholar
- 42.See e.g. Suppe (1977), the introduction.Google Scholar
- 43.The example is also to be found in Oddie (1987).Google Scholar
- 44.In the next chapter, we shall show that if the language 2 will be extended into a modal language 2
_{s5}, then Popper’s revision can be interpreted as “more correct permissions and prohibitions”.Google Scholar - 45.Of course, weak theories do not need to imply literals at all. Real likeness definitions are all equipped with technical devices that also measure the distance to the truth of weak theories.Google Scholar
- 46.Miller (1974b).Google Scholar
- 47.See Miller (1978) and Kuipers (1992), respectively.Google Scholar
- 48.Popper (1963), p.233, first paragraph.Google Scholar
- 49.Tichÿ (1974), p158, p.159.Google Scholar
- 50.Oddie (1986), p 13.Google Scholar
- 51.Hilpinen (1976), p.38.Google Scholar
- 52.The complete falsehood notion is not restricted to propositional languages. Regarding more sophisticated languages the complete falsehood is the proposition that contradicts the truth in all restrictions of the language.Google Scholar
- 53.Tichÿ (1974, p.157 note 2).Google Scholar
- 54.The reason for this phenomenon will become clear in Chapter 2.Google Scholar
- 55.In Miller (1994) the author has changed his mind and dismisses the argument.Google Scholar
- 56.The Chapter 2 shows that it is the
*descriptive*nature of the content approaches that causes the trouble; our modal content proposal blocks the child’s-play argument.Google Scholar - 57.Our analysis seems to agree with that of Miller (1994).Google Scholar
- 58.Although Miller uses three propositions, two suffice to formulate the argument.Google Scholar
- 59.Neither does Miller’s (1975) argument against quantitative estimation.Google Scholar
- 60.Miller (1978, p. 431, last paragraph) clearly uses this meta-interpretation.Google Scholar
- 61.For numerous other examples see Niiniluoto (1987), chapter 1.Google Scholar
- 62.Niiniluoto (1987), p.2.Google Scholar
- 63.Miller (1994), p.207–208.Google Scholar
- 64.Niiniluoto (1987) sections 6.4 and 6.5 present and assess quite a number.Google Scholar
- 65.See also Niiniluoto (1987), p.233 (M10).Google Scholar
- 66.The term is Roberto Festa’s, see Kuipers (1987a), p.85.Google Scholar
- 67.Miller (1976) and Niiniluoto (1987), section 6.2.Google Scholar
- 68.E.g. Chang and Keisler (1973), p.19.Google Scholar
- 69.See van Benthem (1996).Google Scholar
- 70.Kuipers (1982), p.352–353.Google Scholar
- 71.Niiniluoto (1987), p.380–382.Google Scholar
- 72.Niiniluoto (1987), sec. 6.8.Google Scholar