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According to SEP, Lewis's theory of counterfactual conditionals defines truth for counterfactuals as follows:

[...] the truth condition for the counterfactual “If A were (or had been) the case, C would be (or have been) the case” is stated as follows:

(1) “If A were the case, C would be the case” is true in the actual world if and only if either (i) there are no possible A-worlds; or (ii) some A-world where C holds is closer to the actual world than is any A-world where C does not hold.

We shall ignore the first case in which the counterfactual is vacuously true.

I am unable to understand, why the first case is "vacuously true". I could easily explain it by treating the counterfactual conditional as a material conditional, since material conditionals are always true if the antecedent is false. But I have read that counterfactual conditionals shall not be treated as material conditionals. I would appreciate anyone explaining to me why the above mentioned first case is true.

I would also appreciate, if you could use following example (and maybe correct it, if it is wrong) to exemplify your answer:

Real world case: I have eaten fish and my face got swollen.

I think the corresponding counterfactual conditional has to be: If I hadn't eaten fish, my face wouldn't have swollen.

(If I use my example, I would say that A = "I had not eaten fish" and C is = "my face wouldn't have swollen". If I now imagine that in all possible worlds A is false, this would mean that in all worlds [= real world + all possible worlds] I had eaten fish. This is the point, where I am stuck: Why does this mean that the counterfactual conditional is true?)

I thank you very much for any replies.

Rainer_Zoufal
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  • Isnt one of the worlds one in which not eating fish imples no swollen face. Another one being that there is no causal connection between not eating fish and no swoolen face. You could have been smacked in the face while eating fish or walking the street or get stung by a bee on the fish. – Deschele Schilder Jul 09 '21 at 15:36
  • Is your question why case (i) counts as an instance of vacuousness, or what the motivation is for permitting a case of vacuous truth? – Natalie Clarius Jul 09 '21 at 16:20
  • "But I have read that counterfactual conditionals shall not be treated as material conditionals." They aren't; even though counterfactuals can also become vacuously true, they still behave different from material implication in other respects. – Natalie Clarius Jul 09 '21 at 16:25
  • @lemontree: My question is mainly, why case (i) counts as an instance of vacuousness. You using the term "vacuousness" led me to look it up. Now things are starting to get more clear... It seems to me, the answer to my question is: The contrafactual conditional is considered true if the antecedent is false (for all possible worlds) - like for material conditionals. Is this correct? If yes, then I wonder, why it is written "if..., then..." (or "wenn..., dann..." in German). This is highly missleading, don't you think? – Rainer_Zoufal Jul 09 '21 at 16:45
  • And if I may add another question: Compared to the real world case, the antecedent of the counterfactual conditional has to written as the opposite of what was happening in the real world. Does the consequent also always have to be written as the opposite of what was happening in the real world? For example, would following CC also be correct? "If I hadn't eaten fish, my face would have swollen." – Rainer_Zoufal Jul 09 '21 at 16:59
  • "The contrafactual conditional is considered true if the antecedent is false (for all possible worlds) - like for material conditionals. Is this correct?" Yes. "why it is written "if..., then..." (or "wenn..., dann..." in German)" I'm not following - where is this written? Is your issue perphaps a similar one as [this one](https://math.stackexchange.com/questions/2210234/shouldnt-syntax-definitions-make-use-of-iff-rather-than-if) I had a while back? – Natalie Clarius Jul 09 '21 at 17:05
  • "Does the consequent also always have to be written as the opposite of what was happening in the real world?" Not sure what you mean by "have to". If A is impossible, then both A ☐→ B and A ☐→ ¬B are true; this is the same behavior as vacuous truth with material implication. So if it is a logical necessity that you ate fish there are no possible worlds where you didn't, then both "If I hadn't eaten fish, my face would have swollen" and "If I hadn't eaten fish, my face would not have swollen" are true. – Natalie Clarius Jul 09 '21 at 17:13
  • This constructed example probably seems strange mostly because one wouldn't normally consider "I didn't eat fish" to be impossible, so in reality there is no reason to believe that the counterfactuals in question would have to come out as (vacuously) true. – Natalie Clarius Jul 09 '21 at 17:19
  • "Does the consequent also always have to be written as the opposite of what was happening in the real world?" No, it doesn't. It could be that the consequent continues to hold if the antecedent is non-actual. Usually in such cases, in English you might insert the word 'even' or 'still' or both to indicate this. E.g., "if I had not eaten the oysters, I would still have got sick". For Lewis, this means that in the closest possible world in which your counterpart does not eat the oysters, they get sick, just like you did in the actual world. – Bumble Jul 09 '21 at 17:37
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    Another point to note about vacuousness is that in the years that have elapsed since Lewis wrote _Counterfactuals_, it has become much more common to allow that conditionals with impossible antecedents are not trivially true. Such conditionals are usually referred to as counterpossible conditionals, and the logic of such conditionals is an active area of research. There is more information about this in the SEP article about Impossible Worlds. https://plato.stanford.edu/entries/impossible-worlds/ – Bumble Jul 09 '21 at 17:38
  • @lemontree: "I'm not following - where is this written? Is your issue perphaps a similar one as this one I had a while back?" No, this is not my issue. My issue is, that in ordinary language sentences with "if..., then" imply causation - at least in German. When I say "If Paris is in Italy, Berlin is in Sweden" "sounds" wrong, but is true according to logics, because the antecedent is wrong. That is why I try to avoid saying "If A, then B", when I want to state a material implication and rather say "A is a sufficient condition for B instead. – Rainer_Zoufal Jul 10 '21 at 10:38
  • @lemontree: Saying "if A would've been, then B would've been" for "A ☐⇒ B" seems to create the same problem with ordinary language understanding. In the beginning, I did apply this understanding to the CC. The first quotation in your answer really cleared things up for me. – Rainer_Zoufal Jul 10 '21 at 10:45
  • Of course, no reputable logician claims that a natural language indicative conditional is equivalent to mathematical material implication, though I wouldn't be so sure that the former always implies causation. But there's no real need for a separate cumbersome terminology; everyone familiar with math speak knows exactly what is meant by an "if then" in a mathematical text but at the same time as no issue understanding an ordinary language indicative conditional, simply because it's obvious by context if you're dealing with a formal definition or a colloquial conversation. – Natalie Clarius Jul 10 '21 at 14:13
  • Thank you for pointing that out. For me this was not so obvious - you might have noticed I am a novice regarding this field. – Rainer_Zoufal Jul 10 '21 at 14:19
  • Sure; what I should have said is that it is no problem in everyday mathematical practice among professionals, but you are right that it may not at all be so obvious to beginners. – Natalie Clarius Jul 10 '21 at 14:21

1 Answers1

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Edit: If I understand you correctly now, the motivation for vacuousness is not your main issue with it, but by the time you wrote your clarification comment I had already written up this answer, perhaps it is of interest anyway:

The motivation for why counterfactuals are permitted to be vacuously true is given in the original Lewis (1973) "Counterfactuals", ch. 1.6 "Impossible Antecedents":

There is at lesat some intuitive justification for the decision to make a 'would' counterfactual with an impossible antecedent come out vacuously true. Confronted by an antecedent that is not really an entertainable supposition, one may react by saying, with a shrug: If that were so, anything you like would be true!

Further, it seems that a counterfactual in which the antecedent logically implies the consequent ought always to be true; and one sort of impossible antecedent, a self-contraadictory one, logically implies any consequent.

Moreover, one sometimes asserts counterfactuals by way of reductio in philosophy, mathematics, and even logic. [...] these counterfactuals are asserted in an argument, and must therefore be thought true; but their antecedents deny what are thought to be philosophical, mathematical, or even logical truths, and must therefore be thought not only false but impossible.

Here he concedes that these statements may not actually be vacuously true counterfactuals, but something else, either subjunctive conditionals of some other kind or non-vacuously true counterfactuals under the assumption that there is also such a thing as impossible worlds to be taken into consideration.

He also argues that one might want to discriminate between counterfactuals with impossible antecedents that are sensible to assert, such as

If there were a largest prime p, p!+1 would be prime
If there were a largest prime p, p!+1 would be composite

and such ones that one would have no good reason to assert anyway, like

If there were a largest prime p, pigs would have wings

and believes that

we have to explain why things we do want to assert are true [...], but we do not have to explain why things we do not want to assert are false.

Neverthless he proposes an alternative, stronger counterfactual that can not be vacuously true (definition simplified):

A ☐⇒ B is true if and only if there is at least one A-world and all worlds at least equally close to the actual world as that world have A ⊃ B true.

He finds his original pair of 'would' and 'might' counterfactual to be more intuitive but says that one could opt for the modified one if one has concerns against vacuous truth.

Natalie Clarius
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  • I understand my mistake now: I was trying to make CC with a non-nonsensical antecedent to fit case (i). The antecedent in "If I hadn't eaten fish, my face wouldn't have swollen" does not fit case (i), because there are a lot of worlds, where I do eat fish. Case (i) is only fulfilled by nonsensical antecedents like "2x2=5". Lewis argues that in that case, the CC shall be true, because "If that were so, anything you like would be true!" But one thing I do not understand: Why do we consider the CC to be true in such cases? Would it hurt to consider the CC to be wrong in such cases? – Rainer_Zoufal Jul 10 '21 at 10:51
  • I am not familiar with etiquette here, but I'd like to thank you very much for your help. – Rainer_Zoufal Jul 10 '21 at 10:53
  • You're saying that you're still not convinced why one would want a counterfactual to be true in such cases, while in another comment you're saying the first quote, which directly addresses this question, cleared things up for you. I reported what Lewis himself had to say about the issue; of course one may well hold different beliefs. – Natalie Clarius Jul 10 '21 at 14:25
  • As a last clarification to make sure we don't misunderstand each other: Classical material implication is not meant to capture the meaning of ordinary indicative conditionals ("If it rain*s*, the streets *are* wet"). But Lewis' counterfactual theory *is* intended to account for ordinary subjunctive conditionals ("If it rain*ed*, the streets *would* be wet"). One may or may not be convinced of his truth conditions. – Natalie Clarius Jul 10 '21 at 14:30
  • Correct, things got cleared up for me now in a sense that I _accept_ (or can follow) that per definition, if case (i) occurs, Lewis considers the CC to be true - very similar to how a material conditional behaves. I just do not understand the advantage of defining the CC to be true in such cases. – Rainer_Zoufal Jul 10 '21 at 14:52
  • Okay, glad I could clear things up for you. If this answers your question you could mark it as accepted. – Natalie Clarius Jul 10 '21 at 15:48