How is inference with counterfactuals different than inference with material conditional statements?

Question

_{I abbreviate 'Counterfactual Conditional Statements' to CCS and 'Material Conditional Statements' to MCS.}

_{Source: p 338, A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley}

Subjunctive conditionals are often called counterfactual conditionals because their antecedents are typically false. As a result, the ONLY way of determining their truth value in ordinary language is through some kind of inference.

1. Why do only CCS necessitate some [exceptional] kind of inference? MCS do also, no?

Eg:  2. If A is true, then B is false.      3. If A were true, then B would be false.

In both 2 and 3, you must check whether A is true. If A is false, then both 1's and 2's antecedents are false. So how do 2 and 3 differ?

Answer 1

2. If A is true, then B is false.      3. If A were true, then B would be false.

If you reflect a bit on them you should find out that these two examples work very differently. If the premise is not true in (2), then according to classical logic the whole conditional is vacuously true, the inference scheme remains valid but its particular application in (2) is not sound. The interesting case occurs when A is true (or at least you think so), which then allows you to conclude that B is false if you accept the argument in the first place.

But this cannot be how (3) works, because the subjunctive conditional already presupposes (or conventionally or pragmatically implicates, in other theories) that A is actually false. If that would suffice to make the whole subjunctive conditional true, all subjunctive conditionals would be true by virtue of their own presupposition (viz., their conventional or pragmatic implicature). This can't be quite right. If they make sense at all, we'd like some subjunctive conditionals to be true, and others to be false given that their premises are actually false, and, of course, this should be so for a reason. The idea is that we sort of imagine what the world would be like if A were true, and then check on the basis of this 'knowledge' whether B or not B would hold in that world. And that is where the mess starts, for there is no universal agreement of how to spell this out in formal terms and there are many competing explanations for the meaning of conditionals like (3). These types of sentences and their mood are also expressed very differently in different languages, there is interplay with the tenses and a lot of cross-linguistic variation.

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To give you an idea of how different a semantics for (3) might be in comparison to an ordinary conditional, here is an example inspired by Lewis's conditional logic, though probably not identical to it. Order all possible worlds (=models of the logic) by some reflexive and transitive relation of closeness centred around the actual world. If w0 is the actual world, and w1, w2, w3 are other worlds, and e.g. w0 < w1 ~ w2 < w3, this means that w1 and w2 are equally close to the actual world and closer to it than w3. The counterfactual (3) then roughly means:

In all worlds in which A is true that are closest to the actual world (in which A is false), B is false.

If that is the case, the counterfactual is true, otherwise it is false.

People tried to spell out this closeness in terms of minimal change, e.g. by counting the number of changes to the assignments of truth and falsity to propositional variables you need to get from the actual world (viz. "the right model", current state of the universe) to the respective other world (viz. model, possible state of the universe). But this approach is highly disputed, because sometimes a very small change at a time t can have incredibly huge effects at a time later than t.

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A final caveat: In all of the above, I have silently presumed that the correct semantics for (2) is the ordinary standard conditional, which is only false when the antecedent is true and the subsequent is false. However, this is only a highly idealized and rough approximation to the meaning of English if-then clauses.

Answer 2

First, I would like to point out the two different meanings of 'infer', in case this ambiguity is the OP's real problem.

One can use it to mean "use true statements and a conditional with those statement as its premises and deduce the consequence." In that technical sense, all conditionals require inference. But that seems not to be what the author is talking about.

What he seems to mean here by 'infer' is to consider the context in which a statement is made and decide from that context that it is true.

IMHO, he should have been more careful not to use a lay sense of a word which also has a technical sense in the vocabulary of the subject he is teaching.

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A subjunctive construction indicates that you are supposed to imagine a specific scenario, usually a given change in the world.

If a speaker is asking you to imagine a given imaginary version of the world, it seems clear that in that imagined world, other things will also be different. You almost always have to guess or infer what the speaker assumes will not change. And you can never guess perfectly, so even if you know the basic things they imagine will not change, you have to avoid making deductions that rule out incidental or random changes that no one has any control over.

Anything they have not considered when they make the statement could also be different in the artificial world they are asking you to consider.

Statements in the indicative do not involve the same kind of guessing.

"If a polynomial is quadratic, it will have at most two roots." is virtually equivalent to "Our function were quadratic, it would have at most two roots" because you can safely assume the entirety of math is not allowed to change. We are just fairly convinced that our function is not quadratic.

"If Marcus is here then Genine is here" can only be said reliably if you know some real reason why they must be together, for instance you saw them get into the same car, and that car did not stop along the way.

"If Marcus were here, then Genine would be here." has a bunch of assumptions about what part of the other information we know about the current world would still have to be true in a world where Marcus were here. We do not know what background assumptions the speaker is making the implication under. (But you probably would, if you knew Marcus and Genine.)