On qualitative axiomatizations for probability theory

Studia Logica, 1975

The main aim of this paper is to study the logic of a binary sentential operator 'z=', with the intended meaning 'is at least as probable as'. The object language will be simple; to an ordinary language for truth-functional connectives we add '&' as the only intensional operator. Our choice of axioms is heavily dependent of a theorem due to Kraft et al. [8], which states necessary and sufficient conditions for an ordering of the elements of a finite subset algebra to be compatible with some probability measure. Following a construction due to Segerberg [ 121, we show that these conditions can be translated into our language.

Journal of Philosophical Logic, 1975

The main aim of this paper is to study the logic of a binary sentential operator 'z=', with the intended meaning 'is at least as probable as'. The object language will be simple; to an ordinary language for truth-functional connectives we add '&' as the only intensional operator. Our choice of axioms is heavily dependent of a theorem due to Kraft et al. [8], which states necessary and sufficient conditions for an ordering of the elements of a finite subset algebra to be compatible with some probability measure. Following a construction due to Segerberg [ 121, we show that these conditions can be translated into our language.

Most languages for the Semantic Web have their logical basis in some fragment of first-order logic. Thus, integrating first-order logic with probability is fundamental for representing and reasoning with uncertainty in the semantic web. Defining semantics for probability logics presents a dilemma: a logic that assigns a real-valued probability to any first-order sentence cannot be axiomatized and lacks a complete proof theory. This paper develops a first-order axiomatic theory of probability in which probability is formalized as a function mapping Gödel numbers to elements of a real closed field. The resulting logic is fully first-order and recursively axiomatizable, and therefore has a complete proof theory. This gives rise to a plausible reasoning logic with a number of desirable properties: the logic can represent arbitrarily fine-grained degrees of plausibility intermediate between proof and disproof; all mathematical and logical assumptions can be explicitly represented as finite computational structures accessible to automated reasoners; contradictions can be discovered in finite time; and the logic supports learning from observation.

IEEE Transactions on Systems, Man, and Cybernetics, 1991

There are important theoretical and practical reasons to study belief structures. Similar to qualitative probability, qualitative belief can also be described in terms of a preference relation. One of the objectives in this study is to specify the precise conditions that a preference relation must satisfy such that it can be faithfully represented by a belief function. Two special classes of preference relations identified are weak and strict belief relations. It is shown that only strict belief relations are consistent (compatible) with belief functions. More importantly, the axiomatization of qualitative belief provides a foundation to develop a utility theory for decision making based on belief functions. The established relationship between qualitative probability and qualitative belief may also lead to a better understanding and useful applications of belief structures in approximate reasoning.

Theory and Decision, 1978

Probability theory is measure theory specialized by assumptions having to do with stochastic independence. Delete from probability and statistics those theorems that explicitly or implicitly (e.g., by postulating a random sample) invoke independence, and relatively little remains. Or attempt to estimate probabilities from data without assuming that at least certain observations are independent, and little results. Everyone who has worked with or applied probability is keenly aware of the importance of stochastic independence; experimenters go to some effort to ensure, and to check, that repeated observations are independent. Kolmogorov (1933, 1950) wrote: "The concept of mutual independence of two or more experiments holds, in a certain sense, a central position in the theory of probability" (p. 8). "In consequence, one of the most important problems in the philosophy of the natural sciences is-in addition to the well known one regarding the essence of the concept of probability itseff-to make precise the premises which would make it poss~le to regard any given real events as independent" (p. 9). Despite these views, his classical axiomatization of numerical probability brings in stochastic independence not as a primitive, but as a defined quantity. ff (X,~,P) is a finitely or countably additive probability space, then he defines two events A, B in ~ to be (stochastically) independent if and only if ~A n B) = e(A)e(B). The same is true of most presentations of qualitativa probability, such as Savage (1954), in which sufficient axiomatic structure is introduced on (X, ~,~.), where N. is a binary relation of qualitative probability on ~, so as to be able to construct a finitely additive probability representation in terms of which independence is defined in the usual way. An exception is the work of Demeter (1970) who combines axioms involving qualitative probability and independence to construct a finitely additive probability representation. Since it is easy to give examples of qualitative structures for which the representation P is not unique, it is clear that stochastic independence cannot

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.

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.

1998

Numerical probabilities (associated with propositions) are eliminated in favor of qualitative notions, with an eye to isolating what it is about probabilities that is essential to judgments of acceptability. A basic choice point is whether the conjunction of two propositions, each (separately) acceptable, must be deemed acceptable. Concepts of acceptability closed under conjunction are analyzed within Keisler's weak logic for generalized quantifiers --- or more specifically, filter quantifiers. In a different direction, the notion of a filter is generalized so as to allow sets with probability non-infinitesimally below 1 to be acceptable.

Journal of Logic and Computation, 2020

This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.