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The main two arguments for probabilism are flawed

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Abstract

Probabilism, the view that agents have numerical degrees of beliefs that conform to the axioms of probability, has been defended by the vast majority of its proponents by way of either of two arguments, the Dutch Book Argument and the Representation Theorems Argument. In this paper I argue that both arguments are flawed. The Dutch Book Argument is based on an unwarranted, ad hoc premise that cannot be dispensed with. The Representation Theorems Argument hinges on an invalid implication.

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Notes

  1. This is actually a weaker version of the axiom of additivity originally proposed by Kolmogorov.

  2. To be paid a negative sum of money means to be deprived of that money.

  3. Typically this procedure is considered a definition of subjective/epistemic probabilities (cf. for example De Finetti 1990, p. 62), while its role of computational algorithm is viewed as secondary. However, in my argument this procedure plays merely the role of computational algorithm, and thus I shall ignore its main role of definition.

  4. \(S\) is assumed to be positive. The reader can easily construct a similar argument for \(S\) negative.

  5. One can merge the bet on \(\upalpha \) with the bet on \(\upbeta \) to form the bet on \(\upalpha \vee \upbeta \) (and vice versa).

  6. A detailed proof can be found in Kemeny (1955).

  7. The proof of this statement is almost identical to that of the Complete Dutch Book Theorem (i.e. to that of the Dutch Book Theorem plus that of the Converse Dutch Book Theorem) and it will not be given here. However, that axioms (I.)–(III.) are valid for the probability assignment \(\bar{{P}}\) (on pain of irrationality) is not surprising, since \(p_\chi =\bar{{p}}_\chi ^3 \) (which immediately follows from the conditions \(S_\chi =p_\chi \times S\) and \(S_\chi =\bar{{p}}_\chi ^3 \times S)\). Consider for example axiom (III.). If Kolmogorov’s axiom of additivity, \(p_{\upalpha \vee \upbeta } =p_\upalpha +p_\upbeta \), is valid for \(P \)(on pain of irrationality), then, given that \(p_\upalpha =\bar{{p}}_\upalpha ^3 \), \(p_\upbeta =\bar{{p}}_\upbeta ^3 \) and \(p_{\upalpha \vee \upbeta } =\bar{{p}}_{\upalpha \vee \upbeta }^3 \), it’s no surprise that axiom (III.), \(\bar{{p}}_{\upalpha \vee \upbeta }^3 =\bar{{p}}_\upalpha ^3 +\bar{{p}}_\upbeta ^3 \), is valid for \(\bar{{P}}\)(on pain of irrationality).

  8. Suppose that the price of the bet on \(\upalpha \uplambda \lnot \lambda \) is \({S}'<S\); then an agent is prepared to sell that bet for \({S}'\), thus incurring the sure loss \(S-{S}'\). Suppose, on the other hand, that the price of the bet on \(\upalpha \uplambda \lnot \lambda \) is \({S}''>S\); then an agent is prepared to buy that bet for \({S}''\), thus incurring the sure loss \({S}''-S\).

  9. Cf. footnote 5.

  10. The proof of this statement is almost identical to that of the Complete Dutch Book Theorem. See also footnote 7 (the condition \(S_\chi =\bar{{p}}_\chi ^3 \times S\) is a particular case of the more general condition \(S_\chi =f(p_\chi ^f )\times S)\).

  11. An agent is indifferent between two lotteries \(a\) and \(b\) if, and only if, \(a\subseteq b\wedge b\subseteq a\).

  12. \(u[\cdot ]\) associates to any outcome \(\upalpha _i \) a unique real number \(u[\upalpha _i ]\).

  13. That is any utility function \(u[\cdot ]\) of U satisfies the equation \(u[\cdot ]=C\times \bar{{u}}[\cdot ]\), where \(C\in \mathfrak R \) and \(\bar{{u}}[\cdot ]\in \)U.

  14. Let \(\Omega \) be the disjunction of all the states, i.e. \(\Omega \equiv S_1 \vee S_2 \vee S_3 \vee \ldots \); then \(P(\Omega )=1\) (normalization). For any state \(S_i \), \(P(S_i )\ge 0\) (non-negativity). For any two states \(S_i \) and \(S_j , P(S_i )+P(S_j )=P(S_i \vee S_j )\) (additivity).

  15. Two of the rationality constraints are: for any two lotteries, either the first is not preferred to the second or the second is not preferred to the first, or both (trichotomy); the preference relation ‘\(\subseteq \)’ must be transitive. The other rationality constraints require an extensive presentation, which is outside the scope of this paper. For a complete and accessible discussion see Maher (1993, ch. 8) and Jeffrey (1983, ch. 9). Krantz and Luce (1971) is a more technical, and demanding, exposition.

  16. Some critics of the RTA have argued that some of the rationality constraints of a representation theorem are not genuine conditions of rationality: an agent may violate some of these constraints and at the same time be perfectly rational. For the sake of the argument I shall ignore this criticism.

  17. Proof Let \(P_\theta (\cdot )\equiv f_\theta \left( {P(\cdot )} \right) \) where \(\theta \in \Theta \), an index set, and \(f_\theta \left( \cdot \right) \) is such that for any two states \(S_h \) and \(S_k \), \(P_\theta (S_h )\ge P_\theta (S_k )\) if and only if \(P(S_h )\ge P(S_k )\). Now let EU\(_\theta (x)\equiv \sum _i {u[x(S_i )]} \times f_\theta ^{-1} \left( {P_\theta (S_i )} \right) \). It is immediate that for any lottery \(x\) and for any \(\theta \in \Theta \), EU\((x)=\)EU\(_\theta (x)\). Therefore, since there are infinitely many \(f_\theta \left( \cdot \right) \), the representation \(P(\cdot )\), EU(\(\cdot )\) is equivalent to and compatible with infinitely many representations (i.e. all of the pairs \(P_\theta (\cdot )\), EU\(_\theta (\cdot ))\). \(\square \)

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  1. USC School of Philosophy, Los Angeles, CA, USA

    Luigi Secchi

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Secchi, L. The main two arguments for probabilism are flawed. Synthese 191, 287–295 (2014). https://doi.org/10.1007/s11229-013-0286-0

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