A weak symmetry condition for probabilistic measures of confirmation

Abstract

This paper presents a symmetry condition for probabilistic measures of confirmation which is weaker than commutativity symmetry, disconfirmation commutativity symmetry but also antisymmetry. It is based on the idea that for any value a probabilistic measure of confirmation can assign there is a corresponding case where degrees of confirmation are symmetric. It is shown that a number of prominent confirmation measures such as Carnap’s difference function, Rescher’s measure of confirmation, Gaifman’s confirmation rate and Mortimer’s inverted difference function do not satisfy this condition and instead exhibit a previously unnoticed and rather puzzling behavior in certain cases of disconfirmation. This behavior also carries over to probabilistic measures of information change, causal strength, explanatory power and coherence.

Appendix

Proof of Theorem 1

The measures $C_{KEY}$ and $C_{CAR^{*}}$ are commutative (cf. Eells and Fitelson 2002) and therefore do not satisfy NCS. For all other measures the joint probability distribution shown in Fig. 2 with the corresponding results given in Table 3 suffices as a proof that they satisfy NCS. $\square$

Proof of Theorem 2

Since the measures $C_{KEY}$ and $C_{CAR^{*}}$ are commutative they trivially satisfy DCS. Moreover, since in cases of disconfirmation the measure $C_{CRU}$ is identical to $C_{KEY}$ $-1$ (cf. Crupi et al. 2007) it also satisfies DCS. For all other measures the aforementioned probability distribution in Fig. 2 with corresponding results given in Table 3 suffices as a counterexample against the satisfaction of DCS. $\square$

Proof of Theorem 3

For all measures any probability function P under which two propositions $x_1$ and $x_2$ are independent suffices as a proof that these measures satisfy NAS. $\square$

Proof of Theorem 4

We first prove that the measures $C_{CAR}$, $C_{RES}$, $C_{GAI}$ and $C_{MOR}$ do not satisfy WCS. Observe that in order to guarantee symmetric degrees of confirmation, the marginal probabilities must be identical. Therefore, let $P(x_1)=P(x_2)=p$ where due to definedness $p\in (0,1)$ which puts the usual constraint on $P(x_1\wedge x_2)=q$ such that $q\in [\max (0,2p-1),p]$. We can observe that this reduces the measures as shown in Table 5 making all of them commutative.

Now, let $x_1$ and $x_2$ be disconfirmatory, which is equivalent to $q<p^2$. In order to derive a contradiction for the measure $C_{CAR}$ we assume that $q/p-p<-0.5$ which is equivalent to $q<p^2-(p/2)$. For any admissible choice of p and q this assumption is inconsistent with our assumption of negative dependence. Hence it must hold that $q/p-p\ge -0.5$. The same applies for the measure $C_{MOR}$ since it is just $C_{CAR}$ with interchanged arguments and for the measure $C_{RES}$ since it is identical to $C_{MOR}$ in cases of disconfirmation. In order to derive a contradiction for the measure $C_{GAI}$ we assume that $(1-p)/[(p-q)/(1-p)]<0.5$. This yields $p/[(p-q)/(1-p)]<0.5$ which is equivalent to $q<p^2\times (2-1/p)$. Again, for any admissible choice of p and q this assumption is inconsistent with our assumption of negative dependence. Hence, it must hold that $(1-p)/[(p-q)/(1-p)]\ge 0.5$.

We continue with the proof that the remaining measures do satisfy WCS. For the measures $C_{KEY}$ and $C_{CAR^{*}}$ the proofs are trivial due to their commutativity. For all other measures the basic idea is quite simple. We show how it works for the measure $C_{NOZ}$, for all other measures the readers are invited to do the proofs as an exercise. Let us assume that the reduced measure takes some value from its range by letting $p/q-(p-q)/(1-p)=r$ where $r\in [-1,1]$. We can express $q=pr+p^2(1-r)$ and inspect whether it holds that q falls into the Boole-Fréchet bounds for joint probabilities, i.e. $q\in [\max (0,2p-1),p]$. For the upper bound it has to be shown that $q\le p$ for all $r\in [-1,1]$. Substituting q yields $pr+p^2(1-r)\le p$ which is equivalent to $p\le 1$ and guaranteed by the probability axioms. For the lower bound it has to be shown that $q\ge \max (0,2p-1)$ for all $r\in [-1,1]$. If $\max (0,2p-1)=0$, then the inequality $pr+p^2-p^2r\ge 0$ needs to be satisfied, which is equivalent to satisfying $p\ge r(p-1)$. For $r\in [0,1]$ any p will satisfy the inequality, for $r\in [-1,0)$ it can easily be seen that all $p\ge 0.5$ are suitable. If $\max (0,2p-1)=2p-1$, then the inequality $pr+p^2-p^2r\ge 2p-1$ needs to be satisfied. Since we have just shown that $q\ge p$ we can infer that $p\ge 2p-1$ which is equivalent to $p\le 1$ and guaranteed by the probability axioms. This completes the proof. $\square$

Table 5 Reduced probabilistic measures of confirmation

Proof of Theorem 5

First, in order to prove that the set $\{{\mathbf{NCS}},{\mathbf{DCS}},{\mathbf{WCS}}\}$ is consistent simply observe that the measure $C_{CRU}$ satisfies all three conditions. Second, in order to prove that DCS and WCS are logically independent it needs to be shown that they are consistent and there are cases where one of the two is satisfied while the other is violated. The proof of consistency is already given by the fact that the measure $C_{CRU}$ satisfies both conditions. The proof that it is possible to violate DCS while satisfying WCS is given by the measures $C_{KEM}$, $C_{POP}$, $C_{NOZ}$ and $C_{CHR}$. The proof that it is possible to satisfy DCS while violating WCS is given by the following artificial confirmation measure: the measure is identical to $C_{CRU}$ in cases of disconfirmation and identical to $P(x_1)-P(x_1|x_2)$ in cases of confirmation. In cases of confirmation this artificial measure behaves like the measure $C_{CAR}$ in cases of disconfirmation. Third, in order to prove that neither NCS and DCS nor NCS and WCS are logically independent simply observe that violating NCS entails commutativity which trivially entails satisfying DCS as well as WCS. $\square$