A note on two theorems by Adams and McGee

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Three-valued accounts of conditionals frequently promise (a) to conform to the probabilistic view that conditionals are evaluated by conditional probabilities, and (b) to yield a plausible account of compounds of conditionals. However, McGee (1981) shows that probabilistic validity, the conception of validity most naturally associated with the probabilistic view, cannot be characterized by a finite matrix. Adams (1995) indicates a further generalization of this result. Nevertheless, Adams (1986) provides a description of probabilistic validity in three-valued terms by going beyond the standard framework. Yet the language Adams considers is severely restricted: it does not contain compounds of conditionals. Thus, a natural question arises: Is there a plausible three-valued account of compounds of conditionals which agrees with probabilistic validity on the restricted language? In this note, I develop a general framework in which to address this question. The answer will be negative.Peer Reviewe

Trivalent Conditionals: Truth Conditions, Probability and Bayesian Inference

This paper develops a trivalent semantics for indicative conditionals and extends it to a probabilistic theory of valid inference and inductive learning with conditionals.} On this account, (i) all complex conditionals can be rephrased as simple conditionals, connecting our account to Adams's theory of p-valid inference; (ii) we obtain Stalnaker's Thesis as a theorem while avoiding the well-known triviality results; (iii) we generalize Bayesian conditionalization to an updating principle for conditional sentences. The final result is a unified semantic and probabilistic theory of conditionals with attractive results and predictions

The Conditional in Three-Valued Logic

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'' or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning

Conditional sentences: truth conditions and probability

This dissertation supports a partial semantics for conditional statements, which wants to be a middle way between the idea conditionals have not truth conditions and that they always are true or false. Adams showed an important intuitive result, called “Equation”, according to which the probability of a conditional is its conditional probability. In a first moment the Equation was satisfied by Stalnaker’s theory, but Lewis 1975 showed—by the Triviality Result—the incompatibility between the assumption that the probability of a proposition is the probability it is true and the Equation. Consequently, supporting Stalnaker’s semantic means to reject Adams’ logic—and vice versa. In front of Lewis’ result, Adams concluded conditionals do not have truth conditions, suggesting a non-propositional view. Contrary, Stalnaker proposed to consider conditional sentences as standard propositions, giving up the Equation such as a general satisfied principle. With the intent to hold the Equation—and Adams’ logic—without denying conditionals have any kind of truth conditions, the dissertation analyzes Alberto Mura’s proposal—the Theory of Hypervaluated Trievents. It concerns a semantic built on de Finetti’s three-valued logic with the intent to avoid Lewis’ result, incorporating Adams’ logic and extending it to every trievent. Demonstrating that every trievent is simple, Mura tried to provide a theory able to deal with both simple and compound conditionals.</br