Abstract
In the previous chapter we agued that every theory provides a collection of physical possibilities, − they are the (basic) elements of its associated magnitude vector space. The elements of that space are the modal physical possibilities provided by the theory. We will now explain why they are genuine modal possibilities, and not merely a loose way of speaking.
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Notes
- 1.
In anticipation: we will not say anything about modal predicates, but we will study in detail a host of modal operators and with the help of them we will define the important case of modal entities such as the possible outcomes in a probability space, the magnitude states of theories and so forth.
- 2.
A. Koslow, “Laws, explanations and the reduction of possibilities” in Real Metaphysics Essays in Honour of D. H. Mellor, Eds. Hallvard Lillehammer and Gonzalo Rodriguez-Peryra, Routledge, London and New York, 2003, pp. 169–183, where they were called Natural modals. in A. Koslow, A Srtucturalist Theory of Logic, Cambridge University Press, 1992, pp. 239–371, where they were called just modals, and A. Koslow, “The Implicational Nature of Logic”, in European Review of Philosophy, The Nature of Logic, volume 4, ed. Achille C. Varzi, CSLI Publications, Stanford, 1999, pp. 111–155.
- 3.
Technically, we will use the Kolmogorov axiomatization of probability, so that the bearers of probability are sets. Though the outcomes in the case of a thrown die are 1, 2, ..., 6, their probabilities are the probabilities of their unit sets {1}, ..., {6} respectively. The reason is that probabilities are defined on Boolean fields, and they are sets of sets. So when we refer to the possibilities in the sample space, they are the unit sets of each of the outcomes.
- 4.
Poignantly described by Robert Frost’s The Road not Taken, “Two roads diverged in a wood, and I – I took the one less traveled by, And that has made all the difference.”
- 5.
‘Untersuchungen über das logische Schliessen” Mathematische Zeitschrift (1933), 39: 176–210, 405–31; reprinted (1969) as “Investigations into Logical Deduction,” in, M. E. Szabo (ed.) The Collected Papers of Gerhard Gentzen, North-Holland Publishing Co., pp. 68–131.
- 6.
We refer here to probability theory in the form given to it by A.N. Kolmogorov, which was heavily influenced by the way that Hilbert thought of axiomatized theories, Foundations of the Theory of Probability, second English edition Tr. edited by N. Morrison,added bibliography by A.T. Batucha-Reid, Chelsea Publishing Company, New York,1956, original German monograph published in Ergebnisse Der Mathematik, 1933. It is clear that Kolmogorov thought he was axiomatizing probability theory in a way that paralleled Hilbert’s axiomatization of Geometry (1899), and he thought he was following Hilbert’s suggestion, in 1900, that one of the outstanding open mathematical problems included the axiomatization (Hilbert-style) of probability and mechanics. We return to a more detailed discussion of Kolmogorov’s theory of probability in Chap. 12.
- 7.
The use of the term “basic” is intended to call attention to the similar case where the set of possibilities is given by the sample space of a probability space, and the singletons of the elements of such spaces are usually called the elementary events of the space.
- 8.
Cf. G. Boolos, The Unprovability of Consistency, Cambridge university Press, 1979, Theorem 11, p. 31.
- 9.
We regard the familiar bread and butter modal operators, and the Gentzen modals in particular, as only modal with respect to some implication relation on a non-empty set. This kind of modal operator was studied extensively in A. Koslow and there is a lot of evidence that all the familiar modals are also Gentzen modals, and all the systematization of modal theory by Kripke can be recaptured by using Gentzen modals. These efforts can be found in A. Koslow, A Structuralist Theory of Logic, Cambridge University Press, Cambridge 1992, Part IV, pp. 239–371, and “The implicational Nature of Logic: A Structuralist Account”, in European Review of Philosophy, The Nature of Logic, volume 4, edited by A. Varzi, 1999, CSLI Publications, Stanford, pp. 111–155, where it is simply called a modal operator. The second clause of the definition is used for the special case when there is a disjunction operator always available in the implication structure. That is not always the case. The general condition for (II) is given by using the dual of the implication relation i.e. ⇒^ which always exists in implication structures in which there is an implication relation ⇒. It is defined this way: A1, A2, ..., An ⇒^B if and only if for any C in S, if all the Ai s imply (⇒) C, then B implies (⇒) C. In structures in which there is always a disjunction operator available, this is equivalent to the result that the Ai s together dually imply B if and only if B implies the disjunction of all the Ai s. In the single premise case, this comes to A ⇒^ B if and only if B ⇒ A. The remarkable discovery of the dual of an implication relation is due to R. Wojcicki, “Dual Counterparts of Consequence Operations”, presented of Seminar of the Section of Logic, Polish Academy of Sciences, December 1972, pp. 54–57. Condition II in full generality requires that the operator does not distribute over the converse implication relation.
- 10.
This has the nice consequence that the only basic necessity in this structure is the set S. In this structure S is the union of all the singletons of S. Consequently, in this structure the union of sets is their disjunction, so that the only necessity in this structure is the disjunction of all the basic possibilities. That conforms to the usual mathematical way of talking about the necessity of S. in such cases. I.e. in the probability case for example, what is the basic necessity in the structure is the disjunction of all the basic outcomes.
- 11.
Cf. the incisive discussion by J.C. Maxwell “On Ohm’s Law”. Reprinted from the British Association Report, 1876, in The Scientific Papers of James Clerk Maxwell, ed. W.D. Niven, MA., FRS. Two volumes bound as one volume. Volume 2, pp. 533–537.
- 12.
Mach claimed that “The laws of nature are equations between the measurable elements α β γ δ . . ., ω of phenomena.”, Ernst Mach, Science of Mechanics, Chicago, The Open Court Publishing Company, 1907, Tr by T. J. McCormack. Of course special care has to be taken in our understanding of that dictum, since what he intended by “elements of Phenomena, as well as his notion of a mathematical function may be non-standard, and not the same as any contemporary notion.
- 13.
Cf. section (2) above.
- 14.
The case for multiple antecedents is defined in the usual way that requires that either one of the antecedents is not H-designated or the consequent is H-designated.
- 15.
The two attempts discussed in Chap. 8, fall short. One, that could be tacitly assumed by R. Braithwaite requires that the explanation of a law be carried out in a deductiive system that positively extends the deductive system that contains that law. It isn’t explained why this would confer more generality on the explaining theory. The other, provided in explicit detail by E. Nagel generalizes the idea that one generalization like “All Americans are mortal” is less general than the generalization that all human beings are mortal. The proposal has counterxamples, and also allows cases where logically equivalent statements can differ in generality. No immmediate correction is evident.
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Koslow, A. (2019). The Possibilities That Theories Provide (Physical Modals) and the Possibilities of Laws (Nomic Modals). In: Laws and Explanations; Theories and Modal Possibilities. Synthese Library, vol 410. Springer, Cham. https://doi.org/10.1007/978-3-030-18846-7_11
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