Using a Non-universal Logic as a Foundation for Statistical Inference and Induction (original) (raw)

On the logic of nonmonotonic conditionals and conditional probabilities

Journal of Philosophical Logic, 1996

In a previous paper I described a range of nonmonotonic conditionals that behave like conditional probability functions at various levels of probabilistic support. These conditionals were defined as semantic relations on an object language for sentential logic. In this paper I extend the most prominent family of these conditionals to a language for predicate logic. My approach to quantifiers is closely related to Hartry Field's probabilistic semantics. Along the way I will show how Field's semantics differs from a substitutional interpretation of quantifiers in crucial ways, and show that Field's approach is closely related to the usual objectual semantics. One of Field's quantifier rules, however, must be significantly modified to be adapted to nonmonotonic conditional semantics. And this modification suggests, in turn, an alternative quantifier rule for probabilistic semantics.

On the Logic of Nonmonotonic Conditionals and Conditional Probabilities: Predicate Logic

Journal of Philosophical Logic, 1998

Conditional probability on a quantum logic

International Journal of Theoretical Physics, 1986

We analyze two approaches to conditional probability, The first approach follows Gudder and Marchand, M~czyfisky, Cassinelli and Beltrametti, Cassinelli and Truini. The second approach follows R6nyi and Kalmfir. The main result is a characterization of the first approach with the help of a function, similarly as in the second approach.

A logic for reasoning about probabilities

Information and Computation, 1990

ABOUT PROBABILITIES 81 pendently of ous) can be extended in a straightforward way to the language of our first logic. The measurable case of our richer logic bears some similarities to the first-order logic of probabilities considered by Bacchus [Bac88]. There are also some significant technical differences; we compare our work with that of Bacchus and the more recent results on first-order logics of probability in [AH89, Ha1891 in more detail in Section 6.

A logic for reasoning about probabilities* 1

Information and computation, 1990

If-Clauses and Probability operators1

2010

Adams' thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams' thesis from the Lewis-Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis's triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifi er theory. Some implications of the Lewis-Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.

The temporal calculus of conditional objects and conditional events

We consider the problem of defining conditional objects (a|b), which would allow one to regard the conditional probability Pr(a|b) as a probability of a well-defined event rather than as a shorthand for Pr(ab)/ Pr(b). The next issue is to define boolean combinations of conditional objects, and possibly also the operator of further conditioning. These questions have been investigated at least since the times of George Boole, leading to a number of formalisms proposed for conditional objects, mostly of syntactical, proof-theoretic vein.

On first-order conditional logics

Artificial Intelligence, 1998

Conditional logics have been developed as a basis from which to investigate logical properties of "weak" conditionals representing, for example, counterfactual and default assertions. This work has largely centred on propositional approaches. However, it is clear that for a full account a first-order logic is required. Existing or obvious approaches to first-order conditional logics are inadequate; in particular, various representational issues in default reasoning are not addressed by extant approaches. Further, these problems are not unique to conditional logic, but arise in other nonmonotonic reasoning formalisms. I argue that an adequate first-order approach to conditional logic must admit domains that vary across possible worlds; as well the most natural expression of the conditional operator binds variables (although this binding may he eliminated by definition). A possible worlds approach based on Kripke structures is developed, and it is shown that this approach resolves various problems that arise in a first-order setting, including specificity arising from nested quantifiers in a formula and an analogue of the lottery paradox that arises in reasoning about default properties. 0 1998 Published by Elsevier Science B.V. All rights reserved.

Towards a New Logic of Indicative Conditionals

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.

Using a Non-universal Logic as a Foundation for Statistical Inference and Induction (original) (raw)

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