Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's (...) axiomatisation being thus complete in the probabilistic sense. As there are theorems of classical (propositional) logic that are not intuitionistic ones, there are unary probability functions for intuitionistic logic that are not classical ones. Provided here because of this is a means of singling out the classical probability functions from among the intuitionistic ones. (shrink)

This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain (...) point on, to wit: from the probabilities relative to that very B of certain atomic components and conjunctions of atomic components of A, but again to no further extent. These and other results are extended to the less studied case where A and B are compounded from atomic statements by means of `` ∀ '' as well as `` ∼ '' and "&". The absolute probability functions considered are those of Kolmogorov and Carnap, and the relative ones are those of Kolmogorov, Carnap, Renyi, and Popper. (shrink)

In a recent paper Ronald Meester and Timber Kerkvliet argue by example that infinite epistemic regresses have different solutions depending on whether they are analyzed with probability functions or with belief functions. Meester and Kerkvliet give two examples, each of which aims to show that an analysis based on belief functions yields a different numerical outcome for the agent’s degree of rational belief than one based on probability functions. In the present paper we however (...) show that the outcomes are the same. The only way in which probability functions and belief functions can yield different solutions for the agent’s degree of belief is if they are applied to different examples, i.e. to different situations in which the agent finds himself. (shrink)

Consider a language SL having as its primitive signs one or more atomic statements, the two connectives ‘∼’ and ‘&,’ and the two parentheses ‘’; and presume the extra connectives ‘V’ and ‘≡’ defined in the customary manner. With the statements of SL substituting for sets, and the three connectives ‘∼,’ ‘&,’and ‘V’ substituting for the complementation, intersection, and union signs, the constraints that Kolmogorov places in [1] on probability functions come to read:K1. 0 ≤ P,K2. P) = (...) 1,K3. If ⊦ ∼, then P = P + P,K4. If ⊦ A ≡ B, then P = P.2. (shrink)

Using probability functions defined over a simple language as models of states of belief, my goal in this article has been to analyse contractions and revisions of beliefs. My first strategy was to formulate postulates for these processes. Close parallels between the postulates for contractions and the postulates for revisions have been established - the results in Section 5 show that contractions and revisions are interchangeable. As a second strategy, some suggestions for more or less explicit constructive definitions (...) of the revision process (and indirectly also of the contraction process) were then presented. However, the results in Section 6 are less conclusive than in the earlier ones. This problem area still awaits further development. (shrink)

RésuméOn défend ici l'idée que la définition des notions sémantiques à l'aide des fonctions de probabilité devrait être vue non pas comme une généralisation de la sémantique standard en termes d'assignations de valeurs de vérité, mais plutôt comme une généralisation aux degrés de conséquence logique, de la caractérisation de la relation de conséquence que l'on retrouve dans le calcul des séquents de Gentzen.

Professor Roeper adresses a large question, whether probabilistic semantics is a kind of semantics at all. Happily, he does this via an exploration of a specific issue on which he and Professor Leblanc have done important work. That is the issue of how the relationship of logical consequence can be characterized as a relation denned in terms of probability. Let us follow him in calling a relevant relationship of the latter sort the degree of implication, and follow Professor Roeper (...) on his quest for a satisfactory one. (shrink)

Let Ln be a sentential language with n atomic sentences {A1, . . . , An}. Let Sn = {s1, . . . , s2n} be the set of 2n state descriptions of Ln, in the following, canonical lexicographical truth-table order: State Description A1 A2 · · · An−1 An T T T T T s1 = A1 & A2 & · · · &An−1 & An T T T T F s1 = A1 & A2 & · · · (...) &An−1 & ¬An T T T F T s3 = A1 & A2 & · · · & ¬An−1 & An T T T F F s4 = A1 & A2 & · · · & ¬An−1 & ¬An.. (shrink)

Shown here is that a constraint used by Popper in The Logic of Scientific Discovery (1959) for calculating the absolute probability of a universal quantification, and one introduced by Stalnaker in "Probability and Conditionals" (1970, 70) for calculating the relative probability of a negation, are too weak for the job. The constraint wanted in the first case is in Bendall (1979) and that wanted in the second case is in Popper (1959).

The received model of degrees of belief represents them as probabilities. Over the last half century, many philosophers have been convinced that this model fails because it cannot make room for the idea that an agent’s degrees of belief should respect the available evidence. In its place they have advocated a model that represents degrees of belief using imprecise probabilities (sets of probability functions). This paper presents a model of degrees of belief based on Dempster–Shafer belief functions (...) and then presents arguments for belief functions over imprecise probabilities as a model of evidence-respecting degrees of belief. The arguments cover three kinds of issue: theoretical virtues (simplicity, interpretability and flexibility); motivations; and problem cases (dilation and belief inertia). (shrink)

The existence of dysfunctions precludes the possibility of identifying the function to do F with the capacity to do F. Nevertheless, we continuously infer capacities from functions. For this and other reasons stated in the first part of this article, I propose a new theory of functions (of the etiological sort), applying to organisms as well as to artefacts, in which to have some determinate probability P to do F (i.e. a probabilistic capacity to do F) is (...) a necessary condition for having the function to do F. The main objective of this paper is to justify the legitimacy of this condition when considering artefacts. I begin by distinguishing “perspectival probabilities”, which reflect a pragmatic interest or an arbitrary state of knowledge, from “objective probabilities”, which depend on some objective feature of the envisageditems. I show that objective probabilities are not necessarily based on physical constitution. I then explain why we should distinguish between considering an object as a physical body and considering it as an artefact, and why the probability of dysfunction to be taken into account is one relative to the object as member of an artefact category. After clarifying how an artefact category can be defined if it is not defined in physical terms, I establish the objectivity of the probability of dysfunction under consideration by showing how it is causally determined by objective factors regulating the production of items of a definite artefact type. Ifocus on the case of industrially produced artefacts where the objective factors determining the probability of dysfunction can be best seen. (shrink)

A conception of probability that can be traced back to Johannes von Kries is introduced: the “Spielraum” or range conception. Its close connection to the so-called method of arbitrary functions is highlighted. Possible interpretations of it are discussed, and likewise its scope and its relation to certain current interpretations of probability. Taken together, these approaches form a class of interpretations of probability in its own right, but also with its own problems. These, too, are introduced, discussed, (...) and proposals in response to them are surveyed, some of which also go back to von Kries. (shrink)

Proteins with nearly the same structure and function (homologous proteins) are found in increasing numbers in phylogenetically different, even very distant taxa (e.g. hemoglobins in vertebrates, in some invertebrates, and even in certain plants). In discussing the origin of those proteins biologists hardly at all consider convergent evolution because the origin of proteins is held to be a random process, at least ultimately, since selection can work only what the random process delivers as having a minimum adaptive value. The repetition (...) of a random process with the same result is considered to be extremely unlikely. The supposed (un)likelihood, however, is almost never determined quantitatively. This paper attempts such a quantitative determination. It appears that the probability for the random origin of a definite protein is greater than what one would expect in view of the enormous number of equally possible nucleotide sequences in the corresponding gene since what is equally possible is not always equally likely. The probability, however, of the convergent evolution of two proteins with approximately the same structure and function is too low to be plausible, even when all possible circumstances are present which seem to heighten the likelihood of such a convergence. If this is so, then the plausibility of a random evolution of two or more different but functionally related proteins seems hardly greater. (shrink)

This paper presents a theory of probability based on the paraconsistent logic D4. The resulting probability functions are then used to define two sorts of Bayesian updating. One sort of updating merely uses the simple rule of conditionalisation. The other sort adds a wrinkle to the simple rule so that agents' beliefs become more consistent as well as more complete through updating.

The Expected Shortfall or Conditional Value-at-Risk (CVaR) has been playing the role of main risk measure in the recent years and paving the way for an enormous number of applications in risk management due to its very intuitive form and important coherence properties. This work aims to explore this measure as a probability-dependent utility functional, introducing an alternative view point for its Choquet Expected Utility representation. Within this point of view, its main preference properties will be characterized and its (...) utility representation provided through local utilities with an explicit dependence on the assessed revenue’s distribution (quantile) function. Then, an intuitive interpretation for the related probability dependence and the piecewise form of such utility will be introduced on an investment pricing context, in which a CVaR maximizer agent will behave in a relativistic way based on his previous estimates of the probability function. Finally, such functional will be extended to incorporate a larger range of risk-averse attitudes and its main properties and implications will be illustrated through examples, such as the so-called Allais Paradox. (shrink)

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability (...) functions, in particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels. (shrink)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ First four axioms of NAP _3.2_ Continuity and conditional (...) class='Hi'>probability _3.3_ The final axiom of NAP _3.4_ Infinite sums _3.5_ Definition of NAP functions via infinite sums _3.6_ Relation to numerosity theory _4_ Objections and Replies _4.1_ Cantor and the Archimedean property _4.2_ Ticket missing from an infinite lottery _4.3_ Williamson’s infinite sequence of coin tosses _4.4_ Point sets on a circle _4.5_ Easwaran and Pruss _5_ Dividends _5.1_ Measure and utility _5.2_ Regularity and uniformity _5.3_ Credence and chance _5.4_ Conditional probability _6_ General Considerations _6.1_ Non-uniqueness _6.2_ Invariance Appendix. (shrink)

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in (...) Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity. (shrink)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable (...) as Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-questionbegging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

This chapter is a philosophical survey of some leading approaches in formal epistemology in the so-called ‘Bayesian’ tradition. According to them, a rational agent’s degrees of belief—credences—at a time are representable with probability functions. We also canvas various further putative ‘synchronic’ rationality norms on credences. We then consider ‘diachronic’ norms that are thought to constrain how credences should respond to evidence. We discuss some of the main lines of recent debate, and conclude with some prospects for future research.

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims (...) to be borne out by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment. (shrink)

We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap’s revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability (...) function. One such property is Leitgeb’s Probabilistic Convention T, which says that the probability of φ equals the probability that φ is true. (shrink)

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how (...) recent work on the qualitative part of probabilistic inference leads to a very broad class of 'proto-probability' functions. (shrink)

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities (...) are sufficient to trivialize the semantics. (shrink)