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Symmetry’s End?

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Abstract

We examine the idea that similar problems should have similar solutions (to paraphrase van Fraassen’s slogan ‘Problems which are essentially the same must receive essentially the same solution’, see van Fraassen in Laws and symmetry, Oxford Univesity Press, Oxford, 1989, p. 236) in the context of symmetries of sentence algebras within Inductive Logic and conclude that by itself this is too generous a notion upon which to found the rational assignment of probabilities. We also argue that within our formulation of symmetry the paradoxes associated with the so called ‘Principle of Indifference’ collapse, but only to be replaced by genuinely irremediable examples of the same phenomenon.

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Notes

  1. Recall that \(w:SL {\to}[0,1]\) is a probability function if it satisfies that for all θ, ϕ, ∃x ψ(x)  ∈ SL:

    1. (P1)

      If ⊧θ then w(θ) = 1.

    2. (P2)

      If \({\models}{\neg}({\theta}{\wedge}\phi)\) then w(θ∨ϕ) = w(θ) + w(ϕ).

    3. (P3)

      \(w(\exists x \psi(x)) = \lim_{m \to \infty} w\left(\bigvee^m_{i=1}\psi(a_i)\right)\).

  2. One might have suggested the alternative of simply the Tarski–Lindenbaum Sentence Algebra L. However this omits the information that the a i exhaust the universe.

  3. See for example Paris (1994, p. 10).

  4. For example Schlipp (1963, pp. 969–979) and Paris (1994).

  5. By adopting a still stronger principle, Johnson’s Sufficientness Principle.

  6. In fact there is no loss in taking this particularization.

  7. Ignoring the ‘pathological case’ when \(w(\forall x P(x))=w(\forall x \neg P(x))=1/2\).

  8. Meaning that μ is invariant under permutations of the coordinates.

  9. Having said that however the support provided by INV for c 0 might cause us to at least reconsider for a moment that possible choice. In our everyday lives it seems (to us) that we do frequently jump to the conclusion that essentially all instances a i will be the same on the basis of a single case, for example the holotype of a new species of insect (though obviously it would be rash of us to claim that this was a response to the sort of symmetry considerations presented in this paper!).

  10. This seems to be precisely what van Fraassen has already claimed of some more complicated examples, see van Fraassen (1989, Chapter 12), though we would dispute that they actually fulfill the strict symmetry requirements set out in this note, see the next section.

  11. In fact a similar argument to show INV consistent (for empty knowledge) works for a general polyadic language with finitely many relations (so now not necessarily unary), constants a 1, a 2, a 3, ... as before and no functions. In this case the finitely many structures M j , j = 1, 2, ..., s in \({\mathcal T}\) for which all the a i look identical viz-a-viz the relations again correspond to atoms in BL and the probability function w defined by

    $$ w(\theta) = s^{-1} \left|\left\{ j | M_j \models \theta \right\}\right| $$

    satisfies INV (in the case of empty knowledge).

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Acknowledgments

Supported by a UK Engineering and Physical Sciences Research Council (EPSRC) Research Associateship.

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    J. B. Paris & A. Vencovská

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  1. J. B. Paris
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  2. A. Vencovská
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Correspondence to J. B. Paris.

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Paris, J.B., Vencovská, A. Symmetry’s End?. Erkenn 74, 53–67 (2011). https://doi.org/10.1007/s10670-010-9251-1

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