Truthlikeness Confirmation?

As has been established in the literature, given some truthlikeness/verisimilitude measure Tr(), theory T and evidence E, we can measure the estimated truthlikeness of T given E with:

\text{Tr}_{\text{est}}(T | E) = \displaystyle\sum_{i = 1}^s \text{Tr}(T, w_{i}) \text{Pr}(w_{i} | E)

for each state w_{i} in the logical space.

Now, using a Bayesian confirmation measure such as the following:

\text{C}(E, T) = \text{Pr}(T | E) - \text{Pr}(T)

we can combine it with the estimated truthlikeness measure to get a measure of truthlikeness confirmation:

\text{Tr}_{\text{C}}(T, E) = \text{Tr}_{\text{est}}(T | E) - \text{Tr}_{\text{est}}(T)

So what can be done with this measure? In A Verosimilitudinarian Analysis of the Linda Paradox, the authors suggest this measure for what they term a ‘verisimilitudinarian confirmation account’ of the Linda paradox (they do so in response to a problem with an earlier proposal of theirs that gives an account of the paradox based on estimated truthlikeness alone). But it seems that this approach is doing nothing that an account of the Linda paradox in terms of confirmation alone isn’t already doing.

Thus it would be interesting to think about this idea of truthlikeness confirmation some more. For starters, clearly confirmation and truthlikeness confirmation do not increase/decrease together. Take a logical space with three propositions p1, p2 and p3 and a uniform a priori probability distribution amongst the eight possible states:

  • Whilst (p_{1} \wedge p_{2} \wedge p_{3}) \vee (\neg p_{1} \wedge \neg p_{2}) confirms p_{1} \wedge p_{2} \wedge p_{3} it results in a negative truthlikeness confirmation.
  • Whilst p_{1} \wedge p_{2} \wedge \neg p_{3} disconfirms p_{1} \wedge p_{2} \wedge p_{3} it results in a positive truthlikeness confirmation.