A Kripke-styyle Modal Semantics for a Refined Version of de-Finetti' Theory of Tri-events

Partial logic has been recently used to elucidate de Finetti’s notion of the event with respect to Quantum Mechanics and other contexts (Flaminio, Godo, Hosni, 2014). This paper, while resorting to partial logic, takes another direction. The theory of tri-events goes back to de Finetti (1933,1935). De Finetti's aim was to find a logical theory for conditional probability that plays the same role that sentence logic plays with absolute probability. For de Finetti, conditional probability must be viewed as the probability of a conditional event, called by him tri-event. A tri-event is a conditional sentence, which may be true, false, or 'null' (that is devoid of truth value). This allowed de Finetti to represent conditionals sentences by sentences of the form (C | A), where the symbol '|' is a three-valued truth-function. It is easy to show that, in de Finetti’s Logic, a sentence of the form (φ|φ) is not, in general, a tautology. This shows that de Finetti's Logic is at odds with almost all conditional logics, including Adams' p-entailment. This last Logic aims at satisfying the equation Pr(if A then B) = Pr(B | A), which is implicit in de Finetti's Logic. Moreover, while de Finetti did not define any logical consequence relation in terms of his truth-tables, a general result, due to McGee (1981), shows that no such a relation in terms of any standard many valued logic may fit Adams' p-entailment. The fact that (φ|φ) is not valid, shows also that de Finetti's Logic does not fit conditional probability in the way that Boolean logic fits absolute probability. In my previous papers (2009 and 2011), to deal with the difficulties of the original de Finetti’ theory, I introduced a modification of the truth-table algorithm. This algorithm provides a new semantics for tri-events. According to this semantics, every sentence that can be true but cannot be false is considered as valid (dually every sentence that can be false but cannot be true is considered as inconsistent). Due to the way in which the semantics is defined, this theory is called the Theory of the Hypervaluated Tri-Events (THT). THT is in accordance with probability theory, inasmuch two tri-events have the same probability value for every probability function if and only if they have the same truth-conditions according to THT. The purpose of this paper is (a) to reformulate THT adopting a Kripke-style semantics, which is more general than the previous theory inasmuch it allows atomic sentences take the null truth-value, (b) to compare the new theory to its rival theories with respect to compound of conditionals and counterfactuals, and (c) to give details about a new epistemic theory of both indicative and counterfactual conditionals.