Credence for conclusions: a brief for Jeffrey’s rule

Abstract

Some arguments are good; others are not. How can we tell the difference? This article advances three proposals as a partial answer to this question. The proposals are keyed to arguments conditioned by different degrees of uncertainty: mild, where the argument’s premises are hedged with point-valued probabilities; moderate, where the premises are hedged with interval probabilities; and severe, where the premises are hedged with non-numeric plausibilities such as ‘very likely’ or ‘unconfirmed’. For mild uncertainty, the article proposes to apply a principle referred to as ‘Jeffrey’s rule’, for the principle is a generalization of Jeffrey conditionalization. For moderate uncertainty, the proposal is to extend Jeffrey’s rule for use with probability intervals. For severe uncertainty, the article proposes that even when lack of probabilistic information prevents the application of Jeffrey’s rule, the rule can be adapted to these conditions with the aid of a suitable plausibility measure. Together, the three proposals introduce an approach to argument evaluation that complements established frameworks for evaluating arguments: deductive soundness, informal logic, argumentation schemes, pragma-dialectics, and Bayesian inference. Nevertheless, this approach can be looked at as a generalization of the truth and validity conditions of the classical criterion for sound argumentation

Appendix

This appendix includes the details of three probabilistic calculations referred to in the body of the paper. The first and third calculations rely on Jeffrey’s rule; the second uses Bayes’ theorem. For easy reference, the relevant theorem is restated for each calculation.

• 1. point-valued $p(A \to B)$ in Sect. 2

Consider the argument A, therefore $A \to B$ . Truth-table analyses show that there are logical grounds for the following probabilities:

$$ \begin{array}{l} p\left(E \right) \, = p\left(A \right) \, = \, 1/2 \\ p\left({H|E} \right) \, = p(A \to B|A) \, = \, 1/2 \\ p(H|\overline{E} ) = p(A \to B|\overline{A} ) \, = \, 1. \end{array} $$

To determine the epistemic probability of the conclusion $A \to B$, we can apply Jeffrey’s rule:

$$\begin{aligned} p\left(H \right) &= p\left(E \right)p\left({H|E} \right) + p(\overline{E} )p(H|\overline{E}) \\ p(A \to B) &= p\left(A \right)p(A \to B|A) + p(\overline{A} )p(A \to B|\overline{A} )\\ &= \left( {1/2 \, \times \, 1/2} \right) + \left( {1/2 \, \times \, 1} \right)\\ &= 3/4 \end{aligned}$$

• 2. point-valued $p(A \to B|A)$ in Sect. 2

Once again, the argument is A, therefore $A \to B$. Truth-table analyses can verify that there are logical grounds for these probabilities:

$$ \begin{array}{l} p\left(H \right) \, = p(A \to B) \, = \, 3/4 \\ p\left({E|H} \right) \, = p(A|A \to B) \, = \, 1/3 \\ p(E|\overline{H} )\, = p(A|{-}(A \to B)) \, = \, 1. \end{array} $$

Then Bayes’ theorem can be used to calculate the inductive probability of the conclusion $A \to B$ given the premise A:

$$ p\left({H |E} \right) = \frac{p\left(H \right)\,p(E|H) }{{p\left(H \right)p(E|H) + p\left({\overline{H} } \right)p(E|\overline{H} )}} $$

$$ \begin{aligned} p\left({A \to B|A} \right) &= \frac{{p\left({A \to B} \right)\,p(A|A \to B) }}{{p\left({A \to B} \right)p\left({A |A \to B} \right) + p\left({{-}\left( {A \to B} \right)} \right)p(A|{-}\left( {A \to B} \right))}} \\ &= \frac{3/4 \times 1/3 }{{\left( {3/4 \times 1/3} \right) + \left( {1/4 \times 1} \right)}} \\ &= 1/2 \end{aligned} $$

• 3. interval-valued p(A) in Sect. 3

Here the argument is $A \to B$, B, therefore A. Suppose that there are empirical grounds for the following probabilistic intervals:

$$ \begin{array}{l} p\left(E \right) \, = p((A \to B) \wedge B) \, = \, .6 \, {-} \, .7 \\ p\left({H|E} \right) \, = p(A|(A \to B) \wedge B) \, = \, .5 \, {-} \, .6 \\ p(H|\overline{E} ) \, = p(A|{-}((A \to B) \wedge B)) \, = \, .4 \, {-} \, .5. \end{array} $$

We want to determine the epistemic probability of the conclusion A. Hence we consider the possible combinations of minimum (min) and maximum (MAX) values of these probabilities in the light of Jeffrey’s rule: $p\left(H \right) = p\left(E \right)p\left({H|E} \right) + p(\overline{E} )p(H|\overline{E} )$:

$$ \begin{array}{l} \hbox{min}\,p\left(E \right), \, \hbox{min}\, p\left({H|E} \right), \, \hbox{min}\, p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.6 \, \times \, .5} \right) \, + \, \left( {.4 \, \times \, .4} \right) \, = \, .46 \\ \hbox{min}\,p\left(E \right), \, \hbox{min}\,p\left({H|E} \right),{\text{ MAX}}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.6 \, \times \, .5} \right) \, + \, \left( {.4 \, \times .5} \right) \, = \, .50 \\ \hbox{min}\,p\left(E \right),{\text{ MAX}}\,p\left({H|E} \right), \, \hbox{min}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.6 \, \times \, .6} \right) \, + \, \left( {.4 \, \times \, .4} \right) \, = \, .52 \\ \hbox{min}\,p\left(E \right),{\text{ MAX}}\,p\left({H|E} \right),{\text{ MAX}}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.6 \, \times \, .6} \right) \, + \, \left( {.4 \, \times \, .5} \right) \, = \, .56 \\ {\text{MAX}}p\left(E \right), \, \hbox{min}\,p\left({H|E} \right), \, \hbox{min}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.7 \, \times \, .5} \right) \, + \, \left( {.3 \, \times \, .4} \right) \, = \, .47 \\ {\text{MAX}}p\left(E \right), \, \hbox{min}\,p\left({H|E} \right),{\text{ MAX}}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.7 \, \times \, .5} \right) \, + \, \left( {.3 \, \times \, .5} \right) \, = \, .50 \\ {\text{MAX}}p\left(E \right),{\text{ MAX}}\,p\left({H|E} \right),\hbox{min}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.7 \, \times \, .6} \right) \, + \, \left( {.3 \, \times \, .4} \right) \, = \, .54 \\ {\text{MAX}}p\left(E \right),{\text{ MAX}}\,p\left({H|E} \right),{\text{ MAX}}\,p(H|\overline{E} ):p\left(H \right) \, = p\left(A \right) \, = \, \left( {.7 \, \times \, .6} \right) \, + \, \left( {.3 \, \times \, .5} \right) \, = \, .57. \end{array} $$

Since the lowest of these values is .46 and the highest is .57, the epistemic probability of A is the interval .46–.57.