Counterfactual Conditionals: Truth Conditions and the Semantics of Possible Worlds. On R. Stalnaker's and D. Lewis's Therories

Abstract

In modern logic there are many papers on the relevant role that counterfactual conditionals play in regard to the main problems in philosophy of science and whit respect to their use in non-scientific contexts. These papers show, from different points of view, the impossibility of a truth-functional analysis of counterfactual conditionals and the difficulty to precise their significance by means of truth conditions.
In this paper are discussed two theories about counterfactual conditionals and upon “possible words” and “comparative similarity” concepts. They are R. Stalnaker’s theory in “A Theory of Conditionals” (in Causation and Conditionals, edited by E. Sosa, Oxford Univ. Press, 1975) and David Lewis’s theory in Counterfactuals (Oxford, 1973). Especially, we study the truth conditions for counterfactual propositions and their logical consequences with respect to natural language.
Stalnaker’s theory intends the same truth conditions for both conditional propositions and counterfactual propositions, because they deal with non-actual possible situations and therefore both conditionals are propositions about counterfactual worlds. He states the truth conditions for the conditional connective “>” as follows:
A > B is true in i if B is true in f (A,i)
A > B is false in i if B is false in f (A,i)
f is the selection function which takes a proposition and a possible world as arguments and a possible world as its value. This function selects for each antecedent A and world j, a particular possible world i in which A is true. Thus, the conditional sentence is true in the actual world when its consequent is true in the selected world. For selecting a world by the selection function, Stalnaker formulates four further conditions: (1) For any antecedent A and base world i, A must be true in f (A,i). This condition requires for the antecedent to be true in the selected world. (2) For any antecedent A and base world i, f (A,i) = λ if and only if there is not a possible world accessible to i in which A is true. This condition requires that the absurd world “λ” would be selected only when there is an antecedent that is impossible. Conditions (1) and (2) require that the world selected differ minimally from the actual world and this implies that there are no differences between the actual word and the selected world excepting those that are required implicitly or explicitly by the antecedent. (3) For all base world i and all antecedents A, if A is true in i, then f (A, i) = i. This condition requires that the base world would be selected if it happens to be among the worlds in which the antecedent is true. (4) For all base world i and all antecedents B and B´, if B is true in f (B, i) and B´ is true in f (F, i), then f (B, i) = f (B´, i). This last condition ensures that the ordering among the worlds is established in a way that if any selection function establishes B as prior to B´ in the ordering, then no other selection function may establish B´ as prior to B. Conditions (3) and (4) together establish a total ordering of all selected world with the actual world preceding all of them.
Summarizing the most important objections to this theory:
1) The unified treatment of both types of conditionals (the indicative and the counterfactual) makes the truth value the same for any given pair of conditionals, when in fact one of them could be false as the example of Ernest Adams shows:

a) If Oswald did not kill Kennedy, then somebody else did it.

b) If Oswald had not killed Kennedy, then somebody else would have done it.

Obviously, the first conditional is true and the second may be false.

2) In Stalnaker’s theory it is impossible (except in the vacuous case) to distinguish between “necessary” and “possible” counterfactuals:

a) If Oswald had not killed Kennedy, then somebody else would have done it.

b) If Oswald had not killed Kennedy, then somebody else might have done it.

Again, the first conditional may be false and the second may be true.

3) In Stalnaker’s theory, the possible world selected by the selection function is a single world because it is supposed that there is only one world most similar to the actual world of all the worlds in which the antecedent holds. But, can a possible world differ from the actual world only with respect to what is implied by the antecedent and for the rest remain as it actually is? There are many ways in which, perhaps, the possible worlds can differ from the actual world, and nevertheless they can have the same degree of similarity with the actual world. If there are at least two possible worlds in which the antecedent holds with the same degree of similarity, the unicity of the selected world results in a problem.
David Lewis’ theory is a very much elaborated and fruitful conception and therefore it overcomes the difficulties above mentioned, but it states others. In Lewis’ theory there is not a single conditional that can appear as an indicative or as a counterfactual conditional from the speaker’s viewpoint about the truth value of the antecedent.
There are two different types of conditionals and his theory only deals with counterfactual conditionals. Lewis introduces two counterfactual operators that pretend to embrace both counterfactuals “necessity” and “possibility”, which were lost in Stalnaker’s theory. They are the “would” operator “□→”, that must be read: “If it were the case that __________, then it would be the case that __________”; and the “might” operator “◊→”, that must be read “If it were the case that ___________, then it might be the case that __________”. These operator are inter-definable and Lewis takes the “would” as primitive.
A □→ B = df ~ (A ◊→ ~ B)
A ◊→ B = df ~ (A □→ ~ B)
This theory is based on the semantic of possible worlds. Thus, there is a set of possible worlds one of which is the actual world. Each world i has a single sphere of accessibility Si, whose elements are the worlds that are accessible to i and differ from it just within certain limits. The accessible worlds more singular to i belong to inner spheres and those of less similarity belong to outer spheres. The set of all spheres of accessibility of a world i, forms a system of spheres $i around i, conforming a structure similar to Ptolemaic astronomy. The truth conditions of the “would” operator are as follow:
A □→ B is true at a world i (according to a system of spheres $i) if and only if either
(1) no A-world belong to any spheres S in $i, or
(2) some sphere S in $i does contain at least one A-world and A B holds at every world in S.
Condition (1) expresses the vacuous case. A counterfactual sentence is vacuously true if there is no antecedent-permitting sphere. Condition (2) gives the principal case: a counterfactual sentence is non-vacuously true if there is some antecedent-permitting sphere in which the consequent holds at very antecedent world, and it is false otherwise.
There are some important points to consider:
1) According to the truth conditions in Lewis’ theory, all counterfactual conditionals with impossible antecedents are true (vacuously true). It seems correct that some counterfactuals with impossible antecedents are true (namely counterfactuals with logically impossible antecedents). But other counterfactuals with impossible antecedents are false, as is the case of some counterfactuals whose antecedents deny what is affirmed by a physical law (physically impossible). For example: “If Uranus and Neptune had not been submitted to gravitation, then Leverrier would have discovered Neptune from the irregularities of Uranus orbit”;
2) In Stalnaker’s theory the Principle of Excluded Middle Conditional is valid:
(A □→ B) ˅ (A □→ ~ B)
In Lewis’ theory this principle is not valid because both disjuncts might be false. Nevertheless, Lewis affirms that the acceptation of this principle is plausible because in common language we cannot distinguish between the following expressions:
(1) ~ (A □→ B) (2) (A □→ ~ B)
Contrarily, if this principle is accepted, the difference between the “would” and the “might” counterfactuals is lost.
Nevertheless I think that the invalidity of the Principle of Excluded Middle Conditional is not as lamentable a consequence as Lewis has thought, because natural language sometimes distinguishes between expressions (1) and (2). If these expressions are interpreted in terms of Adam´s example, it is observed that the first expression is true and the second is false. Moreover, with the same example both disjuncts of the Principle of Excluded Middle are false.
3) In Lewis’ theory (likewise in Stalnaker’s theory), the interference is not valid by strengthening the antecedent:
A □→ B
.. (A.C) □→ B
It is possible to form consistent sequences of counterfactuals and their negated opposites with stronger and stronger antecedents and alternative consequents between a sentence and its negations:
A □→ B (A.C) □→ ~ B (A.C.D) □→ B (A.C.D.F) □→ ~ B etc.
This is so because the counterfactuals are in Lewis’ theory a variable strict conditional and not a strict conditional. Thus, the premise might be true and the conclusion false. There are many examples in natural language that are evidence of the fallacy of strengthening the antecedent, as it is done in the subjunctive and the indicative moods.
4) The inference of transitivity is considered by Lewis as a generalization of the fallacy of strengthening the antecedent and is therefore also invalid.
In opposition with the above case, natural language seems to observe the transitivity among counterfactual propositions. Michelson’s experiment in order to prove the existence of ether provides us with a good example: “If ether had existed, then it would have presented and obstacle to light; and if ether had presented an obstacle, then the velocity of light would have experimented a drop. Therefore, if the ether had existed, then the velocity of light would have experimented a drop”. Other more popular example is: “If Holland had scored a goal in the 43th minute of the second half, then it would have gained the match; and if Holland had gained the match then Argentine would not have been champion; therefore if Holland had scored a goal in the 43th minute of the second half, then Argentine would not have been champion.”
These and other examples show that both theories are far away from the use of counterfactual conditionals in natural languages. The invalidity of transitivity is caused by the fact than counterfactual operators are not strict conditionals; i.e., it is permitted to report different accessible worlds in the same counterfactual inference to validate the premises. So it is possible for the counterfactual inference to validate the premises. So it is possible for the counterfactual A □→ B and the counterfactual B □→ C to be true and for the counterfactual A □→C to be false, because the B-world “selected” to validate B □→ C would not be an A-world.
Contrarily, I argue that the truth conditions of counterfactuals are not independent from each other in the same counterfactual context. I assume that the world selected for validating the second premise B □→ C also must be an A-world. So this will be an A-world and also an A-B-world. Since in this A-B-world, C is true, then A □→ C is also true and the inference by transitivity is not yet a fallacy.
Now a new problem is raised: how is it possible that the strengthening of the antecedent and the transitivity can be invalid? It is right that, by transitivity, (A.C) □→ B follows from (A.C) □→ A and A □→ B. But since (A.C) □→ A is a logical law it can be omitted and then (A.C) □→ B follows only from A □→ B. Nevertheless this argument is wrong accordingly with our restriction. It is true that it is impossible for (A.C) □→ A, A □→ B and (A.C) □→ B to have the truth values true-true-false. It is also true that (A.C) □→ A cannot be false (it is always true). But it is possible for A □→ B to be true and for (A.C) □→ B to be false, because A □→ B may change its truth-value when it changes from being isolated to a context in which (A.C) □→ B figures.

[G.P.]