Probabilities of conditionals and previsions of iterated conditionals

Abstract

We analyze selected iterated conditionals in the framework of conditional random quantities. We point out that it is instructive to examine Lewis's triviality result, which shows the conditions a conditional must satisfy for its probability to be the conditional probability. In our approach, however, we avoid triviality because the import-export principle is invalid. We then analyze an example of reasoning under partial knowledge where, given a conditional if A then C as information, the probability of A should intuitively increase. In our approach, we explain this intuition by making some implicit background information explicit. We consider several (generalized) iterated conditionals, which allow us to formalize different kinds of latent information. We verify that for these iterated conditionals the prevision is greater than or equal to the probability of A. We also investigate the lower and upper bounds of the Affirmation of the Consequent inference. We conclude our study with some remarks on the supposed "independence" of two conditionals, and we interpret this property as uncorrelation between two random quantities.

The probability of conditionals: A review

“…An ingenious solution that yields the Equation without treating the cases of ¬ A as void is Jeffrey’s ( 1991 ) probability table. He argued that confidence in a conditional is defined in a table of probabilities (see Table 1 ), not truth-values (see also Sanfilippo et al, 2020 ). It blocks the “paradoxes” of the partial truth table, but does not state the cases in which conditionals are true or are false.…”

Probability Propagation in Selected Aristotelian Syllogisms

“…While, in the present paper, we connected the syllogistic terms S and P in the basic syllogistic sentence types by conditional events P |S, this theory of compounds of conditionals allows for obtaining generalized syllogistic sentence types like If S 1 are P 1 , then S 2 are P 2 (i.e., pP 2 |S 2 q|pP 1 |S 1 q) by suitable nestings of conditional events. Interestingly, in the context of conditional syllogisms, the resulting uncertainty propagation rules coincide with the respective non-nested versions (see, e.g., Sanfilippo et al, 2017Sanfilippo et al, , 2018Sanfilippo et al, , 2020. Future research is needed to investigate whether similar results can be obtained in the context of such generalized Aristotelian syllogisms.…”

Interpreting connexive principles in coherence-based probability logic

“…In particular, when X is (the indicator of) an event A, then PpX|Hq " P pA|Hq. The notion of coherence can be generalized to the case of prevision assessments on a family of conditional random quantities (see, e.g., [25,51]). Given a random quantity X and an event H ‰ H, with prevision PpX|Hq " µ, likewise formula (2) for the indicator of a conditional event, an extended notion of a conditional random quantity, denoted by the same symbol X|H, is defined as follows X|H " XH `µ s H. We recall now the notion of conjunction of two (or more) conditional events within the framework of conditional random quantities in the setting of coherence ( [19,22,24,26], for alternative approaches see also, e.g., [31,37]).…”

“…We gave two notions of validity, namely for non-iterated and iterated connexive principles, respectively. Approach 2 allows for dealing with logical operations on conditional events and avoids (see, e.g., [51]) the well known Lewis' triviality results (see, e.g., [35]). It therefore offers a more unified approach to connexive principles, which is reflected by a unique definition of validity for both, iterated and non-iterated connexive principles.…”