Abstract
Suppose you lived in a world of individuals \(a_1,a_2,a_3, \ldots\) (which exhaust the universe) and a finite set of predicates \(P(x), P_1(x),P_2(x), R(x,y), \ldots\) but no other constants or function symbols. You observe that \(P(a_1)\) and \(P(a_2)\) hold, and nothing else. In that case what probability, \(w(P(a_3))\) say, in terms of willingness to bet, should you assign to \(P(a_3)\) also holding?
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Notes
- 1.
We shall put aside the problem of what to do if \(w(\varphi)=0\), in what follows that difficulty will not arise. (For a thorough consideration of this problem see [5].)
- 2.
In [26] the spectrum was defined as the vector \(\langle s_1, s_2, \ldots, s_r \rangle\) where the s i are the sizes of equivalence classes in non-increasing order. Clearly the two versions are equivalent.
- 3.
Again in [26] we adopted the equivalent formulation in terms the vector of the s i in non-decreasing order.
- 4.
The principle applies equally to a formula \(\varphi(x)\) in place of \(\alpha(x)\).
- 5.
Nevertheless some of it’s ‘natural generalizations’ have surprising consequences, see [25]
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Paris, J.B., Vencovská, A. (2011). From Unary to Binary Inductive Logic. In: van Benthem, J., Gupta, A., Pacuit, E. (eds) Games, Norms and Reasons. Synthese Library, vol 353. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0714-6_12
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