Alberto Mario Mura

Philosophical research on conditionals is a complex topic. Since Ramsey, philosophers have noticed that the degree of assertability of an utterance of the form “if p then
q” may be measured by its probability  P(q | p). However,              involves two propositions while natural language suggests that “if p then q” is a single proposition. Can we regard ‘|’ as
a genuine connective? Clearly, standard material implication is not suitable to represent ‘|’, since from the laws of probability it follows that except in very special cases P(q | p) =/ P(p -> q).  Since there is no other decent Boolean candidate for conditional except material implication, it follows that ‘|’ cannot be seen as a Boolean connective. David Lewis (1976) argued that it cannot be an iterable connective at all (let alone a Boolean one).  Lewis’s  reasoning convinced  many philosophers,  like  Adams (1975), that conditionals are not propositions (capable of being either true or false) and that asserting a conditional amounts to maintaining
that has a very high value. In more recent times, researchers in the field of AI tried to challenge Lewis’s result, by merging the Boolean algebra of ordinary two-valued propositions in a larger non-Boolean lattice (where they are represented by elements) and adopting a three-valued semantics. In these settings ‘|’ is dealt with as a three-valued truth- functional connective. Probability functions defined on the Boolean structure may be extended to the larger lattice. Since, in this setting, condition (ii) holds in general only for two-valued propositions, Lewis’s problem is, at least from a formal viewpoint, solved.
Recent research (developed mainly in the ‘90s) on three-valued approach to ‘|’ was powerfully forerun by de Finetti in a paper presented at the Congress of Scientific Philosophy held in Paris in 1935 and entitled "The Logic of Probability" ([1936] 1995).
In this paper I shall first explain de Finetti’s original contribution and shall try to show the principal algebraic properties of de Finetti’s logic of trievents. Second, I shall show the limits of de Finetti’s contribution (as well as recent equivalent approaches) in dealing with conditionals events
and shall advance some proposals to overcome them. In particular, I shall introduce a new two-stages semantics (based on the new notion of hypervaluation) and which is truth-functional in a generalized sense and allows us to solve some puzzling features of de Finetti’s original approach.

In this paper I will propose a refinement of the semantics of hypervaluations (Mura 2009), one in which a hypervaluation is built up on the basis of a set of valuations, instead of a single val-uation. I shall define validity with respect to all the subsets of valua-tions. Focusing our attention on the set of valid sentences, it may easily shown that the rule substitution is restored and we may use valid schemas to represent classes of valid sentences sharing the same logical form. However, the resulting semantical theory TH turns out to be throughout a modal three-valued theory (modal sym-bols being definable in terms of the non modal connectives) and a fragment of it may be considered as a three-valued version of S5 system. Moreover, TH may be embedded in S5, in the sense that for every formula ϕ of TH there is a corresponding formula ϕ' of S5 such that ϕ' is S5-valid iff ϕ is TH-valid. The fundamental property of this system is that it allows the definition of a purely semantical relation of logical consequence which is coextensive to Adams’ p-entailment with respect to simple conditional sentences, without be-ing defined in probabilistic terms. However, probability may be well be defined on the lattice of hypervaluated tri-events, and it may be proved that Adam’s p-entailment, once extended to all tri-events, coincides with our notion of logical consequence as defined in purely semantical terms.

Under the ltalian legai system, a crirninal defendant's prior convictions generally cannot be adrnitted as evidence on the issue of his guilt or innocence. (We shall refer to this as the mie of exclusion). There is, however, a notable exception to this mIe in some types of prosecutions of alleged members of the Mafia. This paper concems the generai mie of exclusion and its exception. SpecificalIy, we shall discuss the rationale for them both. Our inquiry, however, will extend beyond consideration of the practical and juridical aspects of mles regarding the admissibility of prior convictions. We shall also consider a more theoretical issue: what is the formai, logical basi s, if any, for the generaI prohibition excluding this form of evidence? We will necessarily utilize an interdisciplinary approach to this inquiry, in particular the principles of probabilistic inductive logic.

A new account of partial entailment is developed. Two
meanings of the term ‘partial entailment’ are distinguished, which generalise two distinct aspects of deductive entailment. By one meaning, a set of propositions A entails a proposition q if, supposed that all the elements of A are true, q must necessarily be true as well. By the other meaning, A entails q inasmuch the informative content of q is encapsulated in the informative content of A: q repeats a part of what the elements of A, taken together, convey. It is shown that while the two ideas are coextensive with respect to deductive inferences, they have not a common proper explicatum with respect to the notion of partial entailment. It is argued that epistemic inductive probability is adequate as an explicatum of partial entailment with respect to the first meaning while it is at odds with the second one. A new explicatum of the latter is proposed and developed in detail. It is shown that it does not satisfy the axioms of probability.

A new account of partial entailment is developed. Two
meanings of the term ‘partial entailment’ are distinguished, which generalise two distinct aspects of deductive entailment. By one meaning, a set of propositions A entails a proposition q if, supposed that all the elements of A are true, q must necessarily be true as well. By the other meaning, A entails q inasmuch the informative content of q is encapsulated in the informative content of A: q repeats a part of what the elements of A, taken together, convey. It is shown that while the two ideas are coextensive with respect to deductive inferences, they have not a common proper explicatum with respect to the notion of partial entailment. It is argued that epistemic inductive probability is adequate as an explicatum of partial entailment with respect to the first meaning while it is at odds with the second one. A new explicatum of the latter is proposed and developed in detail. It is shown that it does not satisfy the axioms of probability.

A new account of partial entailment is developed. Two
meanings of the term ‘partial entailment’ are distinguished, which generalise two distinct aspects of deductive entailment. By one meaning, a set of propositions A entails a proposition q if, supposed that all the elements of A are true, q must necessarily be true as well. By the other meaning, A entails q inasmuch the informative content of q is encapsulated in the informative content of A: q repeats a part of what the elements of A, taken together, convey. It is shown that while the two ideas are coextensive with respect to deductive inferences, they have not a common proper explicatum with respect to the notion of partial entailment. It is argued that epistemic inductive probability is adequate as an explicatum of partial entailment with respect to the first meaning while it is at odds with the second one. A new explicatum of the latter is proposed and developed in detail. It is shown that it does not satisfy the axioms of probability.

The theory of personal probability as a fair betting quotient goes back to Ramsey [1926/90] and de Finetti [1931]. Both claimed that this theory presupposes the mIe of maximization of expected gain (supposedly linear in utility). Ramsey [p. 79] writes that the theory «is based throughout on the idea of mathematical expectation.» De Finetti uses almost the same words [p. 304] when he writes that the theory «is in substance based on the notion of mathematical expectation.» This claim has never, as far as I know, been questioned, so that it is quite genernlly taken for granted.
On the other hand, there are go od reasons for considering too strong the axioms ofpreference that lead to the linear utility functions. In particular, Allais [1953, 1987/90] convincingly asserts that a violation of Savage's sure-thing principle [1954/72] is implied by very reasonable preferences. If Ramsey's and de Finetti's thesis were correct, any weakening of the sure-thing principle would imply loss of the Dutch Book argument along with loss of any characterisation of personal probability as a fair betting quotient.

This paper presents a new account of Hume’s “probability of causes”. There are two main results attained in this investigation. The first, and perhaps the most significant, is that Hume developed – albeit informally – an essentially sound system of probabilistic inductive logic that turns out to be a powerful forerunner of Carnap’s systems. The Humean set of principles include,

Keywords: Conditionals; Probability; Counterfactuals; Triviality, Tri-events; Trievents, Partial Logic; Contraction; Three-valued Logic, de Finetti; Stalnaker; David Lewis; Ernest Adams; Gärdenfors; Leibniz.

This paper aims at addressing some problems involving the nature of conditional sentences. The general outlines of a new theory (whose formalism has been spelled out by the author elsewhere in a pretty technical manner) are discussed here and exposed from a philosophical point of view, trying to avoid as far as possible all technicalities. In this paper it is argued that the logic of conditionals is throughout epistemic, so that it is relative to a deductively closed stock K of bivalent sentences whose truth is taken for granted. Originally, the new formalism was meant to deal only with indicative conditionals. In the present paper, thanks to the epistemic characterization of conditionals, a further generalization is introduced: it allows a unified treatment of both indicative and counterfactuals conditionals in a single framework, without any limitation in compositionality. It is argued that counterfactual conditionals are relative to a counterfactual stock K´, obtained by contracting K in such a way that, ceteris paribus, the antecedent is dropped. It is stressed that the new theory (a) is able to satisfy, in a general way and without contradicting the so-called "Lewis's Triviality Results", the equation by which the probability of a conditional of the form «if A then C» is equal to the probability of C given A, (b) allows the definition of a new semantic (i.e. based on truth-conditions) notion of logical consequence. This new notion generalizes the classical relation of logical consequence and encompasses as well as extends the so-called "Adams's p-entailment" to compounds of conditionals. These results challenge the popular view according to which conditionals intrinsically lack truth conditions, and their probability is nothing more than a «degree of assertability». Another remarkable tenet of this article is that counterfactuals that are not epistemically necessary, are contingent, meaning that they are true at certain worlds and false at other worlds, while they are neither true nor false at the actual world. In spite of this, they may be probabilistically relevant for the actual world, in the sense that analogical or inductive probabilistic inferences about the actual world may be reasonably drawn on the basis of the high probability of contingent counterfactuals.