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<p> <!--Query ID="Q1" Text="Please check and confirm if the authors and their respective affiliations have been correctly identified. Amend if necessary" Resolved="yes"-->In the paper we discuss the problem of the probabilities of conditionals. We present a simple formal probabilistic model which allows one to give a natural interpretation to conditionals and to compute their probabilities. In order to achieve these aims, we define a hierarchy of languages and a corresponding hierarchy of probabilistic spaces suitable for interpreting the languages. The constructions are recursive in character. The mathematical tools are elements of the theory of Markov chains – which allow one to give a very intuitive graphical representation of the construction. The crucial step consists in defining (for a set of simple conditionals <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11229_2025_4984_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="238" /> </InlineMediaObject> <EquationSource Format="TEX">\({A_1} \to {B_1}\textit{;}\,{A_2} \to {B_2}\textit{;}\, \ldots {A_{\text{n}}} \to {B_{\text{n}}}\)</EquationSource> </InlineEquation>) a Markov graph (with a corresponding probability space) which allows one to compute the probabilities of all the Boolean combinations of these conditionals. This procedure can be iterated so as to provide interpretations to all conditionals. We restrict our attention to conditionals where the antecedent has a positive probability. The model allows one to show that the probability assignment concerning the initial, non-conditional beliefs has a unique, well-defined extension to the probabilities of all conditionals definable in the language. The simplicity of the model makes it a more convenient tool than, for example, Stalnaker Bernoulli spaces.</p>

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An inductive construction of a model for probabilities of complex conditionals

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  • 发表日期：

    2025年6月4日

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  • Krzysztof Wójtowicz,
  • Anna Wójtowicz

摘要

In the paper we discuss the problem of the probabilities of conditionals. We present a simple formal probabilistic model which allows one to give a natural interpretation to conditionals and to compute their probabilities. In order to achieve these aims, we define a hierarchy of languages and a corresponding hierarchy of probabilistic spaces suitable for interpreting the languages. The constructions are recursive in character. The mathematical tools are elements of the theory of Markov chains – which allow one to give a very intuitive graphical representation of the construction. The crucial step consists in defining (for a set of simple conditionals \({A_1} \to {B_1}\textit{;}\,{A_2} \to {B_2}\textit{;}\, \ldots {A_{\text{n}}} \to {B_{\text{n}}}\) ) a Markov graph (with a corresponding probability space) which allows one to compute the probabilities of all the Boolean combinations of these conditionals. This procedure can be iterated so as to provide interpretations to all conditionals. We restrict our attention to conditionals where the antecedent has a positive probability. The model allows one to show that the probability assignment concerning the initial, non-conditional beliefs has a unique, well-defined extension to the probabilities of all conditionals definable in the language. The simplicity of the model makes it a more convenient tool than, for example, Stalnaker Bernoulli spaces.