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I'm currently reading Methods of Logic (Fourth Edition) by W. V. Quine. On p. 23 of Chapter 3 on The Conditional, he says

Whatever the proper analysis of the contrafactual conditional may be, we may be sure in advance that it cannot be truth-functional; for, obviously ordinary usage demands that some contrafactual conditionals with false antecedents and false consequents be true and that other contrafactual conditionals with false antecedents and false consequents be false. Any adequate analysis of the contrafactual conditional must go beyond mere truth values and consider causal connections, or kindred relationships, between matters spoken of in the antecedent of the conditional and matters spoken of in the consequent. It may be wondered, indeed, whether, any really coherent theory of the contrafactual conditional of ordinary usage is possible at all, particularly when we imagine trying to adjudicate between such examples as these:

  1. If Bizet and Verdi had been compatriots, Bizet would have been Italian;
  2. If Bizet and Verdi had been compatriots, Verdi would have been French.

The problem of contrafactual conditionals is in any case a perplexing one, and it belongs not to pure logic but to the theory of meaning or possibly the philosophy of science. We shall not recur to it here.

My understanding of this is that because the sentence "Bizet and Verdi had been compatriots," is false, sentences 1 and 2 must both be true, but then both Bizet and Verdi would have simultaneously been both Italian and French, which cannot happen. Is this correct?

It also seems to me that this can be the case with other sentences with false antecedents. For instance, how are sentences 1 and 2 any different from

  1. If Biden and Macron were compatriots, then Biden would be French;
  2. If Biden and Macron were compatriots, then Macron would be American.

and from

  1. If Biden and Macron are compatriots, then Biden is French;
  2. If Biden and Macron are compatriots, then Macron is American.

Moreover, it seems a bit strange to me to consider the sentences "Bizet and Verdi had been compatriots," and, for instance, "Bizet would have been Italian," by themselves. It seems that one wouldn't use the words "had been" or "would have been" without some extra context, as in "Bizet and Verdi had been compatriots before such and such thing happened," "Bizet would have been Italian had he not been French," or "Bizet would have been Italian if he had not been French," but I digress.

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The two sentences given by Quine are counterfactual conditionals. Unlike the material conditional, which is always true if its antecedent is false, these conditionals are not trivially true just because their antecedent is false. Counterfactuals express a connection between the antecedent and consequent and so they are not truth functions. Indicative conditionals also usually express a connection between the antecedent and consequent, but counterfactuals are distinguished by the fact that we allow their antecedents to range over possible situations that are not actually true.

The point of Quine's example is not that the sentences 1 and 2 are both true. Rather, he is claiming that there is no fact of the matter as to which is true, or maybe that it is impossible to judge correctly which is true. In elementary classical logic we can state the truth conditions of conjunction, disjunction, negation, etc., but counterfactuals do not have truth conditions in such a straightforward way, and maybe do not have truth conditions at all.

Quine's objection is rather simplistic, and you should bear in mind that Quine has an extremely minimalistic approach to logic. He considers anything outside classical first-order logic not to qualify as logic at all. Also, at the time he wrote the first version of Methods of Logic little was known about the logic of counterfactuals, or indeed about the logic of conditionals generally. I would say that the study of conditionals didn't gain serious momentum until the late 1960s with the work of Robert Stalnaker, David Lewis, Ernest Adams and Peter Gärdenfors.

As to the sentences 1 and 2, David Lewis considers that they are both false. Stalnaker considers them to be indeterminate. Unlike Lewis, Stalnaker holds to the law of conditional excluded middle, i.e. that (A □→ B) ∨ (A □→ ¬B), where □→ is the counterfactual conditional. He justifies this on the basis of supervaluationism. So for Stalnaker, "if Bizet and Verdi had been compatriots they would have been Italian or if Bizet and Verdi had been compatriots they would not have been Italian" is true.

The same goes for 3 and 4. Conditionals of this kind are often expressed in English using the auxiliary verb 'would'. They are sometimes called subjunctive though in reality the subjunctive mood is rare in English. For the purposes of logic, these are usually lumped together with counterfactuals.

Your 5 and 6 are expressed indicatively. Usually with indicative conditionals, the antecedent is epistemically possible. These conditionals are used where the antecedent might be true for all we know, or where we are at least willing to grant that it might be true for the sake of argument. Such conditionals are typically used to reason about what might actually be the case rather than about what holds in some hypothetical situation. So, uttering 5 suggests that the speaker doesn't know Biden's nationality but accepts that on the supposition that Biden is actually a compatriot of Macron then Biden is in fact French.

As to your final comment, yes, context matters when evaluating conditionals. Stalnaker and Angelika Kratzer in particular are keen on the idea that the truth conditions of a conditional are context-sensitive.

There is a huge literature on conditionals. There are some introductory articles in the Stanford Encyclopedia, particularly on Indicative Conditionals, Counterfactuals Conditionals, and The Logic of Conditionals.

Some other references are:

  • Jonathan Bennett, Conditionals: A Philosophical Guide (2003).
  • David Sanford, If P then Q (2003).
  • Ernest Adams, The Logic of Conditionals (1975).
  • David Lewis, Counterfactuals (1973).
  • Robert Stalnaker, several papers collected in Knowledge and Conditionals (2019).
  • Dorothy Edgington, “On Conditionals”, Mind, Vol. 104, pp. 235–329, (1995).
  • Angelika Kratzer, Modals and Conditionals (2012).
  • Nicholas Rescher, Conditionals (2007).
  • Michael Woods, Conditionals (1997).
  • Igor Douven, The Epistemology of Indicative Conditionals (2015).
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Seems to me that any counterfactual of this type calls upon the powers of imagination rather than logic. And there is no limit to imagination. So for instance, if Bizet and Verdi had been compatriotes, they could have both been German, or Chinese. Or they could have been from Mars, and sent to Earth to teach us music.

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