Qualitative Counterfactual Reasoning Research Paper

If this glass had been struck, then it would have shattered.

The implication is that, in fact, the glass was not struck and did not shatter; hence the term counterfactual, or contrary-to-fact, conditional. However the same kind of reasoning applies in situations where the truth– value of the antecedent is an open question, as for example in the open subjunctive conditional:

If this glass were struck, then it would shatter.

This suggests that the terms subjunctive conditional and suppositional reasoning might more appropriately delineate the topic. But since the term ‘counterfactual’ is well entrenched, it will suffice to stipulate that one may reason ‘counterfactually’ about hypotheses that turn out to be true.

The philosopher Frank Ramsey once remarked: ‘If two people are arguing ‘‘If p will q’’ and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q’ (Ramsey 1990). Ramsey seems to have had open conditionals in mind, but the comment has been taken by some to suggest a general pragmatic account of conditionals, that has become known as the Ramsey test: a subjunctive conditional is accepted in a belief state if and only if the consequent of the conditional is accepted when the belief state is adjusted so as to accommodate the hypothesis that the antecedent be true.

‘Accommodating’ the hypothesis must be more than just a matter of adding it to one’s stock of beliefs, since to extend Ramsey’s idea beyond the open case is to admit that the hypothesis may be inconsistent with one’s prior beliefs. The ‘adjustment’ should, at the very least, leave the belief state consistent. But the adjustment might also reasonably allow for what Isaac Levi (1996) calls inductive expansion, the addition of further beliefs that are not entailed by the hypothesis. This might suggest that the problem of counter-factual reasoning is one and the same as the problem of rational revision of belief. But there is a catch. Rational revision of belief is generally held to be subject to the following condition:

Preservation: If the evidence to be accommodated is consistent with the prior belief state, then any proposition accepted in the prior belief state will still be accepted in the revised belief state.

Example: if you come to believe that Phar Lap won the 1930 Melbourne Cup, and this piece of information is consistent with everything you previously believed, then you should not respond by rejecting any of your old beliefs. Information is valuable and should not be discarded unnecessarily.

The catch is this: the Preservation condition and the Ramsey test are inconsistent with one another on pain of triviality. This result is known as Gardenfors’ theorem (Gardenfors 1988, Sect. 7.4). It is closely related to the Lewis trivialization theorem described in the companion article.

The moral of Gardenfors’ theorem is that counterfactual reasoning has a logic of its own, distinct from the theory of the rational revision of belief. Supposition is not the same thing as belief revision. Entertaining the hypothesis A counterfactually is a matter of determining what one believes would be true, were A true. This must be distinguished from the different process of determining what one would believe to be true if one came to believe that A is true. Consider, for example, the difference between reasoning about what would have happened if Oswald had not killed Kennedy, and reasoning about what did actually happen if (surprise!) Oswald did not kill Kennedy.

2. Imaging

Revision is a matter of moving from a prior belief state to a revised belief state so as to accommodate input in the form of propositional evidence. Widely accepted criteria for the rational revision of belief have been developed by Alchourron, Gardenfors, and Makinson (Gardenfors 1988). These criteria are usually referred to in the literature as the AGM axioms. Among them is the Preservation rule, along with the following natural conditions:

(a) The proposition to be accommodated must be accepted in the adjusted belief state.

(b) The adjusted belief state should be consistent if both the prior belief state and the proposition to be accommodated are consistent.

Gardenfors’ theorem tells us that supposition can-not satisfy all of the AGM axioms, since it is a method of belief adjustment that fails to satisfy Preservation. So what can be said about the logic of supposition?

Agents typically agree on some matters of belief and disagree on others. In general, given two agents, we may talk about the beliefs that they share. A shared belief is simply a proposition accepted by both agents. Those shared beliefs may themselves be thought of as forming a potential belief state; the strongest belief state compatible with what both agents believe. Now if you and I both adjust our beliefs in order to accommodate the same proposition there will be an adjustment to the beliefs that we share. How might the adjustment to our shared beliefs be determined? One simple answer to that question would be provided by the following principle:

Composition: the adjustment to what is shared between belief states is what is shared by the adjustments to those belief states.

Call a method of belief adjustment that satisfies (a), (b), and the Composition principle an imaging method. The central result of the logic of supposition may now be stated: in the presence of conditions (a) and (b), the principle of Composition is equivalent to the Ramsey test condition. So, by the Gardenfors theorem, AGM revision cannot be compositional. But supposition will be. Supposition—that is to say, counterfactual reasoning—is always a matter of imaging.

Example: it is a sunny day and Alice and Bob are driving to Versailles. They have been there twice before. They share the belief that it was raining on exactly one of those occasions. Alice believes it was raining on their first visit, but Bob believes it was raining the second time. Alice says: ‘If it had rained the second time we were there, then since it rained the first time too, it would have been raining on both occasions.’ Bob does not have to adjust his beliefs to entertain the hypothesis that Alice considers counter-factual; he thinks it is true. Cher is sitting in the back seat. She does not know whether Alice or Bob is correct, but she is tired of their argument. Cher thinks to herself, ‘If Bob is right then if it had rained the first time they were there, it would have rained both times, but if Alice is right, then it would have still have been fine the second time.’ Cher concludes, ‘So, if it had rained the first time they were there it might have rained the second time and it might have been fine the second time.’ But Cher also thinks, ‘If it did rain the first time they were there, then Alice is right and it was fine on their second visit.’ Cher’s belief state consists of the beliefs shared by Alice and Bob. When Cher supposes that it rained the first time, the suppositional state is obtained by counterfactual adjustment of the shared beliefs, which by Composition is the state that consists of those beliefs common to the states obtained after Alice and Bob make that supposition separately. This suppositional state is distinct from the revised belief state that Cher would be in if she were to learn that it is Alice who is remembering events correctly.

This section concludes with a demonstration of the close connection between counterfactual supposition as imaging and the Stalnaker–Lewis semantics for the subjunctive conditional described in the companion article.

Consider the subjunctive conditional: ‘If this glass were struck, then it would shatter.’ By the Ramsey test, I accept this conditional when my supposing that the glass be struck leads to my concluding that the glass would shatter. Since supposition goes by imaging, this means that the image of my belief state on the hypothesis that the glass be struck entails that the glass shatter. But imaging is compositional, so taking any more detailed account of the way things might be that is consistent with what I believe, and forming the image of that on the hypothesis that the glass be struck, must also result in a state that entails that the glass shatters. This remains true for the most detailed accounts of all, which might as well be called ‘possible worlds.’ To accept the conditional is to believe that the actual world is such that its image on the hypothesis that the glass be struck entails that the glass shatters. Call those worlds compatible with the image of the actual world on the antecedent the closest worlds at which the antecedent is true. Then to accept the conditional is to believe that all the closest worlds at which its antecedent is true are worlds at which its consequent is true. The conditional whose acceptance conditions are given by the imaging Ramsey test is revealed as none other than the Stalnaker–Lewis conditional.

3. Subjunctivity

The role of subjunctivity in nonconditional contexts is illuminated from a rather different angle by attempts to correct the expressive limitations of conventional formal languages. Crossley and Humberstone (1977) observe that one sense of the three-way ambiguous sentence:

It is possible for every red thing to be shiny

cannot be rendered in the standard language of modal logic. The possibility operator may be positioned inside or outside the scope of the universal quantifier, but neither formulation captures that third sense whose expression seems to require recourse to the logic of actually:

It is possible for everything that is actually red to be shiny.

If the modal operator ‘possibly’ functions like an existential quantifier over possible worlds, allowing us to roam freely over the domain in search of a world at which the embedded sentence is true, then ‘actually’ functions to switch off that operator temporarily. In the example just given, it forces us to return to our own world when it comes to determining what the relevant red things are. Humberstone (1982) remarks that ‘actually’ may be viewed as an inhibitor: a sentential operator that shields whatever is within its scope from the influence of other external operators.

Thus ‘actually’ has a function that is the exact opposite of the kind of semantic role Castaneda (1967) ascribes to subjunctivity. As Humberstone puts it, expressive inadequacies like the above are overcome in languages in which the subjunctive mood functions to switch on or to activate an otherwise dormant external operator. So, to adapt an example of Quine (1956), the ambiguity of the English sentence:

I am looking for a dog that talks

is resolved in Spanish via the indicative subjunctive distinction. ‘Busco un perro que habla’ (‘habla’ in-dicative) means that there is a particular dog I am looking for that just happens to talk. ‘Busco un perro que hable’ (‘hable’ subjunctive) suggests that I do not have any particular dog in mind, any talking dog will do. The subjunctive element activates the ‘looking for’ operator at that point in the sentence, signaling that the dog’s talking is to be understood as part of the content of my seeking. In Spanish (at least) desire contexts are among those naturally activated by the subjunctive.

4. Desire

DAB: To desire something is to believe it desirable.

An argument similar to the proof of the Gardenfors theorem establishes that DAB also conflicts with Preservation (Collins 1988). But DAB is the simplest form that a cognitivist theory of motivation might take. So if cognitivism is not to be ruled out on purely logical grounds, ‘desirability’ must be interpreted suppositionally. Thus: a possibility is believed desirable just in case the supposition that the possibility be true leads to the conclusion that ones desires would be satisfied. Similarly, the expected desirability of an option is a belief-weighted average of the desirabilities of more specific ways the option might be realized. A suppositional decision theory obtains the expected desirability of an option by imaging on the supposition that the option be chosen. But this is none other than the causal decision theory (described in ‘Newcomb’s problem’). There is an argument for causal decision theory that bypasses the usual discussions of Newcomb’s problem.

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