Conditional sentences pose well-known problems for semantic analyses that treat `if A then C' as a truth-functional sentential connective, that is, as equivalent to the disjunction `not-A or C.' I will illustrate the inadequacy of such an account with examples that appear to instantiate logically valid inference patterns, but are intuitively invalid: Strengthening of the antecedent, Contraposition, Hypothetical syllogism (also known as Transitivity,) and Vacuous truth.

The data suggest that while classical logic is only concerned with the preservation of truth (from premises to a conclusion,) more is required for an inference pattern to be plausible in natural-language reasoning. I will briefly outline the virtues of an alternative to the logical treatment. This alternative, entertained in various forms by a number of authors and most fully developed by Ernest Adams, makes two basic assumptions:

(1) What is preserved in natural-language inference is not truth, but (high) probability.
(2) The probability of a conditional `if A then C' is the conditional probability of C, given A.

Probability is harder to preserve than truth, and the conditional probability of C, given A is not equivalent to that of `not-A or C.' As a welcome consequence of (1) and (2), precisely those inference patterns for which there are counterexamples in the data are predicted to be invalid.

The main part of the talk is concerned with the question of how to define the semantic values of conditionals in the familiar possible-worlds semantics in such a way as to exploit the merits of the probabilistic approach. Sentences are typically modeled as denoting functions from possible worlds to truth values. This treatment is easily extended to yield a probability distribution over sentences, at least those that do not contain conditionals. The question then becomes what values conditionals should receive at individual worlds.

I will approach this task by questioning the often-made assumption that conditionals, like other sentences, have truth values at all worlds. Truth values are defined by reference to facts. In the case of a conditional `if A then C' at a world where A is false, there does not seem to be a decisive "fact of the matter" that truth conditions could refer to. This argument can be made, without artifice, using naturally occurring examples like (3).

(3) If the flood crest reaches the level projected, much of the city will be under water.

To say that a conditional does not have _truth_values_ at some worlds is not to say that it does not have _values_ a those worlds. The fact that the flood crest did not reach the level projected renders (3) neither true, nor false, nor undefined. Instead, I argue that (3) should receive an intermediate value (between 0 and 1) corresponding to the probability that it "would have been" true. As this paraphrase suggests, I maintain that (3) is interpreted in much the same way as (4).

(4) If the flood crest had reached the level projected, much of the city would have been under water.

The intermediate value is obtained by examining those worlds in the model where the antecedent is true. This resembles the evaluation procedure of the counterfactual logics of both Stalnaker (1968) and Lewis (1973), here recast in a probabilistic setting. Care must be exercised, however, in choosing the "right" subset from those worlds: Facts of w that are not causally affected by the antecedent must be held constant. Those causal relations are encoded as a partial order on propositions (more accurately, a partial order on partitions of the set of worlds) and used in such a way that facts that are "prior" (in a precise sense) to the false antecedent serve to delimit the set of accessible antecedent worlds. In the probabilistic context, these restrictions are similar in effect, but less stipulative, than certain ingredients of logics of counterfactuals (similarity between worlds, Lewis' hierarchy of "miracles").

Time permitting, I will close with some remarks on work in progress on related linguistic questions raised by the probabilistic approach, such as how to relate uncertainty about sentence denotations compositionally to uncertainty about word denotations.

The observation that there is not always a "fact of the matter" is not new (Ramsey, 1929) but rarely adequately addressed. Pretending that there is, that is, stipulating that conditionals must have truth values at all worlds, leads to insurmountable technical problems (summarily known as "triviality results" - cf. Lewis, 1976, 1986; Hall, 1994; Hajek and Hall, 1994) to which the present approach is immune. Those problems have led some to postulate a mixture of classical truth conditions and probabilistic constraints on use (Jackson, 1987; Lewis, 1991). An alternative reaction is to abandon the notion that conditionals ever have truth values, claiming instead that they are used solely as statements about beliefs (Adams, 1975 and elsewhere.) Both of those approaches are plagued by a number of shortcomings.

My proposal has precedents in the writings of van Fraassen (1976), McGee (1989) and Jeffrey (1991) (cf. also Stalnaker and Jeffrey, 1994.) Those authors converge on a similar conclusion with different motivations, but have been rightfully criticized for making counterintuitive predictions in certain cases (Lance, 1991; Edgington, 1995.) Those predictions are corrected here by the restricting reference to causal relations. The way those relations are encoded - as a partial order on partitions - is the essence of Pearl's (2000) theory, which is, however, couched in a quite different formal framework and addresses only simple counterfactuals.