Probability logic investigates probabilistic inference and relates it to deductive and other inferential systems. It is challenging to relate human deductive reasoning to probability logic rather than to logic. Oaksford & Chater (O&C) were the first who systematically investigated human deductive reasoning within a probabilistic framework. Our approach to human reasoning is, in many respects, related to O&C's. However, our approach is closer to probability logic, especially with respect to the analysis of the experimental tasks. In commenting on Bayesian Rationality (Oaksford & Chater Reference Oaksford and Chater2007, henceforth BR), we discuss a selection of questions arising from these different perspectives.Footnote ¹

Common everyday reasoning requires us to process incomplete, uncertain, vague, and imprecise information. Artificial Intelligence (AI) has developed many different approaches for uncertain reasoning. Typical examples are belief functions, possibilistic models, fuzzy systems, probabilistic description languages, many-valued logic, imprecise probabilities, and conditional independence models. Of these approaches only conditional independence models are considered in the book. Why do O&C consider probability and not, say, belief functions as a normative reference system?

The probabilities are interpreted as subjective probabilities. The theory of subjective probability was conceived by Bruno de Finetti (Reference de Finetti1974/1975), and further developed by Coletti and Scozzafava (Reference Coletti and Scozzafava2002), Gilio (Reference Gilio2002), and many others. A central concept of subjective probability theory is coherence. A probability assessment is coherent if it cannot lead to sure losses. The theory does not require event structures that are closed under negation and conjunction. Conditional events are primitive. Conditional probabilities are not defined by absolute probabilities. When we talk about P(A|B), why should we assume that we also know P(A and B) and P(B)? These properties make the coherence based probability logic more appropriate as a framework for psychological research than other approaches, including the “pragmatic strategy of using the simplest probabilistic semantics” (BR, p. 75).

O&C argue that classical logic should be replaced by probability theory as a framework for human reasoning. This position is too radical. We should not “throw out the baby with the bath water.” Probability theory presupposes logic for operations on propositions. Simple logical inferences like And-Introduction or Modus Ponens (MP) are endorsed by practically all subjects. We do not see a dichotomy between logic and probability.

We fully support O&C's hypothesis, that human subjects understand the indicative “if A, then B” in the sense of a conditional probability, P(B|A), and not as the probability of a material conditional, P(A⊃B). Many empirical studies corroborate this hypothesis (Evans & Over Reference Evans and Over2004).

In subjective probability theory, conditional events are not truth-functional. If the antecedent is false, then the truth value of the conditional is undetermined. This corresponds to what is called a “defective truth table.” Considering P(B|A), we would not say the Ramsey Test adds P(A)=1 to one's stock of belief. Rather, A is assumed to be true. Probability 1 and the truth value TRUE are not the same. Ramsey's idea can be explained in the way Lindley (Reference Lindley2006) and several others introduce conditional probability. Subjective probabilities are assessed relative to a knowledge base K. The absolute probability P(A) is shorthand for P(A|K), and P(B|A) is shorthand for P(B|A:K). The colon separates the supposition from the knowledge base. The change from supposition to fact does not change the conditional probability, P(B|AC:K)=P(B|A:CK).

The core of O&C's analysis of the conditional inferences (Modus Ponens [MP], Modus Tollens [MT], Affirming the Consequent [AC], and Denying the Antecedent [DA]) is “that the probability of the conclusion … is equal to the conditional probability of the conclusion given the categorical premise” (BR, p. 119), P(conclusion|categorical premise). Thus, the consequence relation (denoted by a horizontal line and three dots in the book) is probabilistic. In our approach the consequence relation is deductive. Each premise obtains a probability. The probabilities of the premises are propagated deductively to the conclusion. Because the MP, MT, AC, and DA arguments consist of only two premises, only an interval probability can be inferred (Pfeifer & Kleiter Reference Pfeifer and Kleiter2005b; Reference Pfeifer and Kleiter2006). A conclusion with a point probability would require three premises. In this case, however, the argument does not “mimic” the logical forms of MP, MT, AC, or DA. O&C mention probability intervals, but they do not use them.

In Chapter 4, O&C oppose nonmonotonic reasoning and probabilistic reasoning, and advocate probabilities. We do not see why both approaches are incompatible. System P (Kraus et al. Reference Kraus, Lehmann and Magidor1990) is a basic and widely accepted nonmonotonic reasoning system. Several probabilistic semantics were developed for System P (Adams Reference Adams1975; Biazzo et al. Reference Biazzo, Gilio, Lukasiewicz and Sanfilippo2005; Gilio, Reference Gilio2002; Hawthorne & Makinson Reference Hawthorne and Makinson2007; Lukasiewicz Reference Lukasiewicz2005). We observed good agreement between the predictions of the coherence-based semantics and actual human inferences (Pfeifer & Kleiter Reference Pfeifer and Kleiter2005a; in press a; in press b).

O&C were the first who realized the importance of generalized quantifiers in psychology. Murphree (Reference Murphree1991) and Peterson (Reference Peterson2000) developed logical systems for syllogisms that involve generalized quantifiers (see also Peters & Westerståhl, Reference Peters and Westerståhl2006). On p. 219 of BR, O&C note that “without a notion such as p-validity [not in Adams' sense!], there is no way of defining the correct answers to these generalized syllogisms.” Peterson, however, investigates syllogisms that involve quantifiers like “Most,” “Few,” or fractionated quantifiers like “n/m.” The semantics of the quantifiers works by comparisons of the cardinalities of appropriate sets and by the use of relative frequencies. Thus, Peterson's semantics can easily be related to a probabilistic interpretation. Moreover, Atmosphere or Matching are easily generalized within this framework.

O&C use Bayesian networks to model syllogisms. Each vertex in the network corresponds to a term in the syllogism. The directed arcs represent conditional probability distributions. We are uncertain about the statement that in Bayesian networks the conditional independence is a “standard assumption” (BR, p. 222). Moreover, under the assumption of conditional independence (X independent Z, given Y), there exists only one probabilistic model. Three models (Figure 1, 3, and 4 in Fig. 7.3) are Markov equivalent; only the vee-structure (Figure 2 in Fig. 7.3) has a different factorization. It encodes a marginal independence (X and Z are independent).

Rational analysis puts the quest for cognitive processes and representations in the second line. This is fine if the normative models fit the empirical data. In this case the models are both normative and descriptive. A good theory requires: (1) a thorough task analysis, (2) a minimum of generality to avoid “adhoceries,” and (3) a connection with other theoretical concepts like language, memory, or attention.