Abstract
For the purpose of defining truthlikeness some simplifying asumptions will be made about the nature of a factual investigation. As we have already noted, the aim of an inquiry is to find out the truth of some matter. The matter to be investigated is circumscribed by two items: a collection of objects (which is usually called the domain of the inquiry) and a collection of traits (properties the objects might have, relations that might hold between them). Following Tichy, this may be called the intensional basis (or just the basis) of the inquiry.1The domain and basis together make up what will be called a conceptual framework. It is the investigator’s aim to discover just how the traits in the basis are distributed through the domain. At the outset he faces a range of different possibilities, for typically thereare different possible ways in which the traits may be distributed. Any such complete distribution is called a possible world, and the collection of all possible worlds generated by a framework, the logical space of the framework. That distribution which obtains is the actual world, but it is no part of the specification of the framework which world that is. That is left to the facts to decide. Nor is the actual world of a different ontological type from the other possible worlds. It is no more or less ‘concrete’. It is just that distribution of traits which happens to obtain.
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References
See Tichy [1978b], pp. 176-7. Note that what Tichy calls anepistemicframework we have called (below) aconceptualframework.
The basic tenet of possible-world semantics, that a proposition is a dichotomy of a logical space, has its origins in Wittgenstein [1921]. More recent works in which the theory has been developed and extended include Carnap [1947], Kripke [1963], Montague [1974], Lewis [1973] and Tichy [1971]. See Lewis [1973], p. 46, footnote.
Miller [1977a], p. 15.
See f.n. 10, chapter 2.
See the discussion in section 1.2.
Miller [1978a], p. 429.
Some of the results of section 3.1were published in Oddie [1979a] and Oddie [1981].
See, for example, Popper [1959], appendix *7, p. 363 ff.
See again Oddie [1979a] and Oddie [1981]. The basic idea of the Miller-Popper programme is to define truthlikeness in terms of the relation of implication, (or, what this amounts to, the relation of set-theoretic inclusion between classes of worlds) without discriminating between different kinds of consequences of a proposition (or, what this amounts to, failing to take note of the structure of the worlds a proposition contains). Other proposals which can be regarded as close to the spirit of this programme are those in Kuipers [1982] and Newton-Smith [1981]. For another criticism of the general approach, and of Newton-Smith’s definition in particular, see Oddie [1986]. See f.n. 15 of chapter 7 (J). 188) for a summary of the latter article.
See Tichy [1974]. The same definition was put forward independently by Vetter in his [1977], along with many of the very criticisms of Popper’s definitions which Tichy included in his [1974]. (It is suggested in Miller [1976] that essentially the same idea was published independently in Haack [1974], pp. 62 ff. As a purported solution to the problem of truthlikeness it is not quite clear what Haack’s proposal is. One obvious way in which the two theories differ is that according to Haack’s it seems that all true propositions have the same degree of closeness to the truth. Moreover, it seems that a proposition may have different degrees of truthlikeness depending on the way it is formulated.) Note that the order of exposition in the text does not follow the historical order. Miller’s and Popper’s symmetric difference theories were published after Tichy published his.
This example first appeared in Tichy [1978b], pp. 189–90.
Lewis [1973].
Our worlds are just distributions of traits, and can therefore be thought of as mathematical entities. The actual world is no more ‘concrete’ than the others, nor are its inhabitants more ‘real’. Possible worlds share a common domain of objects or individuals—worlds are the different ways those individuals might be. (This point was clearly grasped by Wittgenstein, in his [1921] (1.1), but unfortunately the wisdom of theTractatushas not become common.) The actual world is just one such way that happens to have the idiosyncratic feature of being the way things are in fact. Lewis, who against Wittgenstein espouses the thesis that the world is the collection of things, not facts, would call these worlds ‘ersatz’. 14ffilpinen [ 1976 ], p. 29.
Hilpinen [1976], p.29.
Niiniluoto mentions this defect in his [1978a], p. 440.
ibid.
Lewis [1973], p. 95.
ibid.
It is possible to entertain systems of nested spheres which are not isomorphic to any subset of the real numbers under the less-than relation) but for simplicity we will consider those that are. See Lewis [1973], p. 51.
Formally,Ni(A) = r(Ds(max(A)))+(1-r)(Ds(min(A))), for somerbetween 0 and 1. See Niiniluoto [1977], p. 132.
Let P be a measure of the logicalprobability of propositions. IfD(U)is the distance ofUfromW,Ais aP-measurable set, and D is measurable on A, then the appropriate extension of the Tichy measure would be:D(A) = (? A D dP)/P(A). However, so long as A and B are independent there are measures of probability which disagree as to their relative weights. There is no widespread agreement on how to go about pinning down the ‘correct’ measure of logical probability.
This is a problem Leibniz would have taken to be beyond human ken involving as it does something like the comparison of infinitely complex worlds for degrees of perfection. Only God can perform the infinitely complex calculation necessary.
Lete p (U,V,n)be 0 ifUandVagree on Bnand 1 otherwise. The distance betweenUandVby thenth state, dp(U,V,n) is the quantity:\(\left[ {\sum\limits_{i = 0}^n {{\varepsilon _\pi }\left( {U,V,n} \right)} } \right]/n.\)And the distance betweenUandV,Dp(U,V), is the limit ofdp(U,V,n)asntends to infinity:\(\mathop {\lim }\limits_{n \to \infty } {\delta _\pi }\left( {U,V,n} \right).\)
That is,D min (U,V)=minr:r =Dp(U,V) for some p.
For example, if the measure were to be σ-additive, and there were just denumerably many basic states, then the basic states could not be given equal weights. See any introductory textbook on probability or measure theory, for example, Halmos [ 1950 ].
See Hintikka [1965].
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© 1986 D. Reidel Publishing Company, Dordrecht, Holland
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Oddie, G. (1986). Distance in Logical Space. In: Likeness to Truth. The University of Western Ontario Series in Philosophy of Science, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4658-3_3
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