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Probability Theory and Probability Semantics

Probability Theory and Probability Semantics

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    Probability Theory and Probability Semantics
    Book Description:

    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.

    eISBN: 978-1-4426-7878-1
    Subjects: Philosophy

Table of Contents

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  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Acknowledgments
    (pp. ix-xii)
    H. Leblanc and P. Roeper
  4. Part One: Probability Theory

    • Introduction to Part I
      (pp. 3-4)

      Probability values can be attributed to a variety of items such as event types, properties, sets of various kinds, and statements. From a mathematical perspective sets are the preferred arguments of probability functions. From a logical point of view it is primarily statements which can be said to be more or less probable and, since we are concerned here with probability logic, we opt for statements. This choice is even more appropriate in so far as a matter particular interest for us is the relationship between probability and logical notions like logical consequence and consistency. Part II is devoted to...

    • Chapter 1 Probability Functions for Propositional Logic
      (pp. 5-25)

      In a language of finitary prepositional logic only finite conjunctions and disjunctions are recognised; indeed the connectives ‘∧’ and ‘∨’ are 2-place connectives. Best known of course are languages for classical prepositional logic, and probability functions defined on the statements of such a languageLPoccupy a central place in probability theory. But we also introduce here probability functions defined on sets of statements rather than statements and probability functions for intuitionistic rather than classical prepositional logic. The reason for including such functions in our study is in both cases our interest in tracing the connection between semantics and probability...

    • Chapter 2 The Probabilities of Infinitary Statements and of Quantifications
      (pp. 26-44)

      Probability functions for a quantificational language require constraints which link the probability of quantified statements to the probabilities of their instances. Attention is thereby focussed on substitutional accounts of quantification. Such accounts view universal quantifications as conjunctions of their instances, which means, in case there are infinitely many instances, that they function as if they were statements of an infinitary language, i.e. a language permitting conjunctions of infinitely many conjuncts and disjunctions of infinitely many disjuncts.

      The substitutional account of quantification is not the only reason for investigating probability functions for an infinitary language. It is also of interest that...

    • Chapter 3 Relative Probability Functions and Their ${\top}$–Restrictions
      (pp. 45-58)

      A one-to-one correspondence between absolute and relative probability functions exists in the case of Kolmogorov functions. We noted in Section 2 of Chapter 1 that ifPis a relative probability function forLPwhich meets constraint

      RKIf${P(B, \top)=0}$,then P(A,B) = 1for every statement A of LP,

      and if${{P \intercal}}$(the${\top}$-restriction ofP) is the absolute probability function such that\[{{P \intercal}}(A)=P(A,{\text T}){{,}^{1}}\]then\[P(A,B)=\left\{ \begin{array}{*{35}{l}} {{P \intercal} (A\,\wedge \,B)/{P \intercal}}(B) & if \ {{P \intercal}}(B)\ne 0 \\ 1\quad otherwise. & {} \\ \end{array} \right.\]

      Kolmogorov functions constitute only a subclass of the relative probability functions forLP, but for any relative probability functionP, the Τ-restriction${{P \intercal}}$stands out as a counterpart ofP. The present...

    • Chapter 4 Representing Relative Probability Functions by Means of Classes of Measure Functions
      (pp. 59-77)

      Any Kolmogorov relative probability functionPcan be represented by its${\top}$-restriction${{P \intercal}}$, which is an absolute probability function, in the sense thatPand${{P \intercal}}$determine one another:\[{{P \intercal}}(A)=P(A,{\top}),\]and\[P(A,B)=\left\{ \begin{array}{*{35}{l}} {{P \intercal} (A\,\wedge \,B)/{P \intercal}}(B) & if \ {{P \intercal}}(B)\ne 0 \\ 1\quad otherwise. & {} \\ \end{array} \right.\]

      We also found in Chapter 3, Section 2, that any absolute probability function represents in this sense a Kolmogorov relative function.

      Here we deal with the question whether relative probability functions generally can be represented by absolute ones. It is clear that only in the case of Kolmogorov functions will a single absolute probability function be sufficient. In the other cases a multiplicity of unary functions...

    • Chapter 5 The Recursive Definability of Probability Functions
      (pp. 78-98)

      A probability function is at times fully determined by its restriction to a certain subset of its arguments. This chapter investigates when this is the case. Of particular interest are those cases in which the restriction in question can be independently characterised, i.e. as other than just the restriction to a certain class of arguments. We shall investigate primarily probability functions defined on sets of statements. The corresponding results for probability functions defined on statements are then easily obtained by virtue of the one-to-one correspondences noted in Chapter 1, Section 3.

      In Section 1 we establish that absolute probability functions...

    • Chapter 6 Families of Probability Functions Characterised by Equivalence Relations
      (pp. 99-108)

      In this chapter we identify a number of families of relative probability functions forL,LbeingLP,LQ, orLω, and investigate their classification. The unifying feature of this brief study is that the constraints by means of which a family of functions is characterised are throughout formulated in terms of equivalence relations on statements, i.e. relations that are reflexive, symmetrical, and transitive. Each of the constraints in fact stipulates that two such equivalence relations coincide. We begin with a list of equivalence relations which will play a role in this chapter. All but the first relation are induced...

  5. Part Two: Probability Logic

    • Introduction to Part II
      (pp. 111-113)

      Part Two investigates the connection between probability functions and semantic notions. The idea of probabilistic semantics as an alternative to classical semantics was first introduced by Field in (1977) and was pursued by several writers, notably Leblanc in (1983a). It is the idea of explaining semantic notions, such as logical truth and logical consequence, in terms of probability functions. The idea is prompted by Popper's succeeding to formulate constraints for probability functions that make no use themselves of any semantic notions, in contrast to the statement counterparts of Kolmogorov's constraints which crucially involve the mutual substitutability of logically equivalent statements....

    • Chapter 7 Absolute Probability Functions Construed as Representing Degrees of Logical Truth
      (pp. 114-141)

      In this chapter we investigate the connection between probability and semantic notions in the case of absolute probability functions. We pursue the thought that absolute probability functions generalise logical necessity so that for a central category of functions the value of a functionPfor a statementAcan be interpreted as the degree of necessity assigned toA. And in the general caseP(A) can be understood as the degree of necessity ofArelative to certain assumptions specific toP

      Our approach is somewhat indirect. We take as our starting point the relation of logical consequence in the...

    • Chapter 8 Relative Probability Functions Construed as Representing Degrees of Logical Consequence
      (pp. 142-166)

      Having interpreted absolute probability functions as generalising the property of logical necessity, it is natural to construe relative probability functions as generalising the relation of logical implication. Since logical consequence is a relation between a set of statements and a single statement, it is convenient to concentrate on relative probability functions defined on sets of statements, rather than individual statements. It turns out that a relative probability functionPfor a languageLencapsulates a consequence relation ⊢p, defined

      XPA if and only if P({A},XZ) = 1for every set Z of statements,

      and a...

    • Chapter 9 Absolute Probability Functions for Intuitionistic Logic
      (pp. 167-181)

      Although we introduced intuitionistic probability functions in Chapter 1, Section 4, we did not offer any justification there for the groups of constraints said to characterise those functions. Indeed the task of developing intuitionistic probability theory and of formulating constraints for intuitionistic probability functions has not been well defined. Most writers on the matter, working within the paradigm of “probabilistic semantics”, seem to have assumed that the aim is, in the case of absolute probability functions, say, to describe a classIof unary functions for which it holds that

      P(A) = 1for every function P in I if...

    • Chapter 10 Relative Probability Functions for Intuitionistic Logic
      (pp. 182-190)

      Whereas logical consequence, classically understood, obtains when the truth of the premisses guarantees the truth of the conclusion, intuitionistically logical consequence is a matter of the premisses providing conclusive evidence for the conclusion; more precisely, logical consequence obtains when any conclusive evidence for the premisses constitutes conclusive evidence for the conclusion. Hence it is intuitionistically even more plausible than classically to regard relative probability functions, which are naturally taken to register evidential support for a statement relative to stated (and unstated) assumptions, as generalisations of the consequence relation.

      Moreover, since the significance of a connective is typically explicated in terms...

  6. Appendix I
    (pp. 191-222)
  7. Appendix II
    (pp. 223-224)
  8. Notes
    (pp. 225-230)
  9. Bibliography
    (pp. 231-234)
  10. Index
    (pp. 235-238)
  11. Index of Constraints
    (pp. 239-240)