Sentential probability logic. Origins, development, current status, and technical applications.

*(English)*Zbl 0922.03026
Bethlehem: Lehigh University Press. London: Associated University Presses, 304 p. (1996).

The relation investigated here is this:
\[
P(\phi_1) \in \alpha_1, \ldots,P(\phi_m) \in \alpha_m \models P(\psi) \in \beta,
\]
where \(P\) is any probability function in a probability model of the sentences, and \(\alpha_i\), \(\beta\) are subsets of \([0,1]\). This is a relation that has been investigated before (for example by N. J. Nilsson [“Probabilistic logic”, Artif. Intell. 28, No. 1, 71-87 (1986; Zbl 0589.03007)]), but never with such generality, such thoroughness, or such an eye to applications. The issue is an important one, particularly when the \(\alpha_i\) and \(\beta\) are intervals. For example, it is of interest to know when \(P(h) \geq 1 - \varepsilon\) holds in every probability model in which \(P(h_1) \geq \varepsilon_1, \ldots , P(h_m) \geq \varepsilon_m\) holds. The author gives procedures for answering such queries in general.

The framework of the investigation is provided by a standard sentential language with an unbounded number of atomic constants. For an initial segment of such a language, involving only the constants \(A_1,\ldots,A_n\), a probability model is determined by the assignment of probabilities \(k_i\) to the \(2^n\) constituents that can be formed from the \(n\) constants. In this framework, the author proves (inter alia):

Theorem 4.61: For any given probability logical consequence \[ P(\phi_1) \in [a_1,b_1], \ldots P(\phi_m) \in [a_m,b_m] \models P(\psi)\in \beta \] the optimal set \(\beta\) is an interval whose end points are the minimum and maximum of a linear function subject to linear constraints.

Conditional probability logic is introduced by adjoining a special function on ordered pairs of sentences: the value assigned to \(\psi| \phi\) is the expected ratio when \(P(\phi) > 0\), and is the arbitrary constant \(c \in [0,1]\) otherwise. This leads to the development (in Chapter 5) of a conditional probability logic, in which the underlying structure is that of a three-valued logic, with a conditional \(\psi\dashv\phi\) whose truth value is \(u\) (the third value) whenever the truth value of \(\phi\) is 0. (Otherwise the truth tables are those of Kleene’s strong three valued logic.) This is called suppositional logic. A conditional probability-like function \(P^*_M\) is defined in the natural way, based on the standard logic for the atoms of the language. We replace “\(\dashv\)” by “\(\mid\)” and we have both a logic and a probability formalism that allow iterated conditionals of this form. Lewis’s trivialization results do not hold for this logic, since for compound sentences in the extended logic, additivity and the product rule do not always hold.

As an illustration of the type of result that is established, we cite Theorem 5.43, a modus ponens type of inference: \[ P(A_1| A_2) = p,\;P(A_2) = q \models P(A_1) \in [pq,1-\overline p q], \] where \(\overline p = 1-p\).

The book not only presents an elegant and concise treatment of both probability logic and conditional probability logic, as conceived by the author, but a thorough treatment (2/3 of the volume) of this historical background (as would be expected from the author of Boole’s logic and probability (1976; Zbl 0352.02002); 2nd revised and enlarged edition (1986; Zbl 0611.03001)). It is slightly marred by a number of typographical errors, but the overall layout is pleasing.

The framework of the investigation is provided by a standard sentential language with an unbounded number of atomic constants. For an initial segment of such a language, involving only the constants \(A_1,\ldots,A_n\), a probability model is determined by the assignment of probabilities \(k_i\) to the \(2^n\) constituents that can be formed from the \(n\) constants. In this framework, the author proves (inter alia):

Theorem 4.61: For any given probability logical consequence \[ P(\phi_1) \in [a_1,b_1], \ldots P(\phi_m) \in [a_m,b_m] \models P(\psi)\in \beta \] the optimal set \(\beta\) is an interval whose end points are the minimum and maximum of a linear function subject to linear constraints.

Conditional probability logic is introduced by adjoining a special function on ordered pairs of sentences: the value assigned to \(\psi| \phi\) is the expected ratio when \(P(\phi) > 0\), and is the arbitrary constant \(c \in [0,1]\) otherwise. This leads to the development (in Chapter 5) of a conditional probability logic, in which the underlying structure is that of a three-valued logic, with a conditional \(\psi\dashv\phi\) whose truth value is \(u\) (the third value) whenever the truth value of \(\phi\) is 0. (Otherwise the truth tables are those of Kleene’s strong three valued logic.) This is called suppositional logic. A conditional probability-like function \(P^*_M\) is defined in the natural way, based on the standard logic for the atoms of the language. We replace “\(\dashv\)” by “\(\mid\)” and we have both a logic and a probability formalism that allow iterated conditionals of this form. Lewis’s trivialization results do not hold for this logic, since for compound sentences in the extended logic, additivity and the product rule do not always hold.

As an illustration of the type of result that is established, we cite Theorem 5.43, a modus ponens type of inference: \[ P(A_1| A_2) = p,\;P(A_2) = q \models P(A_1) \in [pq,1-\overline p q], \] where \(\overline p = 1-p\).

The book not only presents an elegant and concise treatment of both probability logic and conditional probability logic, as conceived by the author, but a thorough treatment (2/3 of the volume) of this historical background (as would be expected from the author of Boole’s logic and probability (1976; Zbl 0352.02002); 2nd revised and enlarged edition (1986; Zbl 0611.03001)). It is slightly marred by a number of typographical errors, but the overall layout is pleasing.

Reviewer: H.E.Kyburg (Rochester)

##### MSC:

03B48 | Probability and inductive logic |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03B80 | Other applications of logic |

60A05 | Axioms; other general questions in probability |

03B30 | Foundations of classical theories (including reverse mathematics) |