Probabilistic Models for Propositional and Modal Logics

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 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 $\mathcal{F}\mathcal{P}\mathcal{M}$.

Synthese, 1979

Epistemic modal expressions like possible, likely, and certain routinely give rise to scalar inferences: (i) It’s possible that we will hire Mary or Sue. a. ↝ It’s possible that we will hire Mary b. ↝ It’s possible that we will hire Sue (ii) It’s likely that we will hire Mary or Sue. a. ↝ It’s possible that we will hire Mary b. ↝ It’s possible that we will hire Sue (iii) It’s certain that we will hire Mary or Sue. a. ↝ It’s possible that we will hire Mary b. ↝ It’s possible that we will hire Sue These scalar effects are usually classified and treated differently. The effect in (i) is labeled a ‘free choice inference,’ those in (ii) and (iii) ‘distributive inferences.’ Free choice and distributive inferences receive separate treatments in the literature. Distributive inferences are routine predictions of most standard accounts of im- plicature (Sauerland 2004, Fox 2007, among others), while free choice inferences are notoriously problematic (Kratzer & Shimoyama 2002, Fox 2007, Chierchia 2013 among others). We show that a degree-based semantics for modals can easily predict all these effects via the same mechanism and without making controversial assumptions. Our proposal dovetails well with (though does not require endorsing) recent probability-based semantics for epistemic modals (Yalcin 2010, Lassiter 2011, 2014, Moss in press, Swanson 2015), and hence can be seen as an indirect argument for analyses in this vein. We conclude by showing how the proposal can be extended to deontic modals, given reasonable assumptions about the form of a degree semantics for these modals.

2019

Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions. Instead, they are best thought of as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rat...

Springer eBooks, 2003

The aims of this paper are (i) to summarize the semantics of (the propositional part of) a unified epistemic/doxastic logic as it has been developed at greater length in Lenzen [1980] and (ii) to use some of these principles for the development of a semi-formal pragmatics of epistemic sentences. While a semantic investigation of epistemic attitudes has to elaborate the truth-conditions for, and the analytically true relations between, the fundamental notions of belief, knowledge, and conviction, a pragmatic investigation instead has to analyse the specific conditions of rational utterance or utterability of epistemic sentences. Some people might think that both tasks coincide. According to Wittgenstein, e.g., the meaning of a word or a phrase is nothing else but its use (say, within a certain community of speakers). Therefore the pragmatic conditions of utterance of words or sentences are assumed to determine the meaning of the corresponding expressions. One point I wish to make here, however, is that one may elaborate the meaning of epistemic expressions in a way that is largely independent of-and, indeed, even partly incompatible with-the pragmatic conditions of utterability. Furthermore, the crucial differences between the pragmatics and the semantics of epistemic expressions can satisfactorily be explained by means of some general principles of communication. In the first three sections of this paper the logic (or semantics) of the epistemic attitudes belief, knowledge, and conviction will be sketched. In the fourth section the basic idea of a general pragmatics will be developed which can then be applied to epistemic utterances in particular. 1 The Logic of Conviction Let 'C(a,p)' abbreviate the fact that person a is firmly convinced that p, i.e. that a considers the proposition p (or, equivalently, the state of affairs expressed by that proposition) as absolutely certain; in other words, p has maximal likelihood or probability for a. Using 'Prob' as a symbol for subjective probability functions, this idea can be formalized by the requirement: (PROB-C) C(a,p) ↔ Prob(a,p)=1. Within the framework of standard possible-worlds semantics <I,R,V>, C(a,p) would have to be interpreted by the following condition: (POSS-C) V(i,C(a,p))=t ↔ ∀j(iRj → V(j,p)=t). Here I is a non-empty set of (indices of) possible worlds; R is a binary relation on I such that iRj holds iff, in world i, a considers world j as possible; and V is a valuation-function assigning to each proposition p relative to each world i a truth-value V(i,p)∈{t,f}. Thus C(a,p) is true (in world i∈I) iff p itself is true in every possible world j which is considered by a as possible (relative to i). The probabilistic "definition" POSS-C together with some elementary theorems of the theory of subjective probability immediately entails the validity of the subsequent laws of conjunction and non-contradiction. If a is convinced both of p and of q, then a must also be convinced that p and q: (C1) C(a,p) ∧ C(a,q) → C(a,p∧q). For if both Prob(a,p) and Prob(a,q) are equal to 1, then it follows that Prob(a,p∧q)=1, too. Furthermore, if a is convinced that p (is true), a cannot be convinced that ¬p, i.e. that p is false: (C2) C(a,p) → ¬C(a,¬p). For if Prob(a,p)=1, then Prob(a,¬p)=0, and hence Prob(a,¬p)≠1. Just like the alethic modal operators of possibility, ◊, and necessity, , are linked by the relation ◊p ↔ ¬ ¬p, so also the doxastic modalities of thinking p to be possible-formally: P(a,p)-and of being convinced that p, C(a,p), satisfy the relation (Def. P) P(a,p) ↔ ¬C(a,¬p). Thus, from the probabilistic point of view, P(a,p) holds iff a assigns to the proposition p (or to the event expressed by that proposition) a likelihood greater than 0: (PROB-P) V(P(a,p))=t ↔ Prob(a,p)>0. Within the framework of possible-worlds semantics, one obtains the following condition: (POSS-P) V(i,P(a,p))=t ↔ ∃j(iRj ∧ V(j,p)=t), according to which P(a,p) is true in world i iff there is at least one possible world j-i.e. a world j accessible from i-in which p is true. 1 Cf., e.g., Hintikka [1970]. 2 Clearly, since C(a,p) ∨ ¬C(a,p) holds tautologically, C10 and C11 entail that C(a,C(a,p)) ∨ C(a,¬C(a,p)) is epistemic-logically true. So either way there exists a q such that C(a,q).

Forthcoming in the Oxford Handbook of Probability and Philosophy, edited by A. Hájek and Chris Hitchcock.

This chapter is about probabilistic logics: systems of logic in which logical consequence is defined in probabilistic terms. We will classify such systems and state some key references, and we will present one class of probabilistic logics in more detail: those that derive from Ernest Adams' work.

Journal of Applied Logic, 2009

Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations. A novel notion of probabilistic entailment enjoying desirable properties of logical consequence is proposed and shown to collapse into the classical entailment when the language is left unchanged. Motivated by this result, a decidable conservative enrichment of propositional logic is proposed by giving the appropriate semantics to a new language construct that allows the constraining of the probability of a formula. A sound and weakly complete axiomatization is provided using the de-cidability of the theory of real closed ordered fields.

Erkenntnis, 2001

The logical interpretation of probability, or “objective Bayesianism” – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a “principle of insufficient reason” inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability.

In this paper the strategy for the eliminative reduction of the alethic modalities suggested by John Venn is outlined and it is shown to anticipate certain related contemporary empiricistic and nominalistic projects. Venn attempted to reduce the alethic modalities to probabilities, and thus suggested a promising solution to the nagging issue of the inclusion of modal statements in empiricistic philosophical systems. However, despite the promise that this suggestion held for laying the 'ghost of modality' to rest, this general approach, tempered modal eliminativism, is shown to be inadequate for that task. 2 See Beirmann and Faak 1957.

1992

Abstract Possibilistic logic has been proposed as a numerical formalism for reasoning with uncertainty. There has been interest in developing qualitative accounts of possibility, as well as an explanation of the relationship between possibility and modal logics. We present two modal logics that can be used to represent and reason with qualitative statements of possibility and necessity. Within this modal framework, we are able to identify interesting relationships between possibilistic logic, beliefs and conditionals.

Synthese

Some propositions are structurally unknowable for certain agents. Let me call them ‘Moorean propositions’. The structural unknowability of Moorean propositions is normally taken to pave the way towards proving a familiar paradox from epistemic logic—the so-called ‘Knowability Paradox’, or ‘Fitch’s Paradox’—which purports to show that if all truths are knowable, then all truths are in fact known. The present paper explores how to translate Moorean statements into a probabilistic language. A successful translation should enable us to derive a version of Fitch’s Paradox in a probabilistic setting. I offer a suitable schematic form for probabilistic Moorean propositions, as well as a concomitant proof of a probabilistic Knowability Paradox. Moreover, I argue that traditional candidates to play the role of probabilistic Moorean propositions will not do. In particular, we can show that violations of the so-called ‘Reflection Principle’ in probability (as discussed for instance by Bas van Fraassen) need not yield structurally unknowable propositions. Among other things, this should lead us to question whether violating the Reflection Principle actually amounts to a clear case of epistemic irrationality, as it is often assumed. This result challenges the importance of the principle as a tool to assess both synchronic and diachronic rationality—a topic which is largely independent of Fitch’s Paradox—from a somewhat unexpected source.

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&apos;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.

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters. Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.