"Argghh! If I would have played my lucky number 666 yesterday, I would have won that jackpot today! Had I played the lottery, I would have bet on my lucky number. If I would have bet on my lucky number yesterday, then I would have won the jackpot since that number was selected in the lottery today. So I should have played my lucky number!"
—But is this sort of reasoning valid?
The argument above features claims about how the world could have been different than it actually is, since the course of history and the consequences of earlier events might have transpired quite differently. This is another way of saying that what is true might not have been true, had things gone differently for our lottery player. Wondering "What if I had played my lucky number?" generates claims about possible alternate ways events might have occured. Answers such as "If I had played my lucky number (which was selected in the lottery today), then I would have won a million dollars," or "Were I to have played my lucky number, I would have bought myself a Porsche," all express alternate possible states of affairs. We call such claims "contrary-to-fact conditionals" or "counterfactuals" because they describe hypothetical but plausible events that have not (but might have) happened. Sometimes, on the basis of such reasoning, we derive false conclusions (we should kick ourselves for not playing our lucky number), because we do not realize that whatever is possibly true might never have been actually true.
Counterfactual conditionals are those in which the antecedent is false, and which assert that one event would be, or would have been the case if another event were, or would have been the case. A counterfactual sentence has the form "If A were true, then C would have been true," where A, the counterfactual antecedent, describes an event that is contrary to what is actually true, and C, the counterfactual consequent describes a result expected to hold in an alternative world where the antecedent is true.
Counterfactuals have this form: If P were true, Q would be true.
E.g.: "If Kangaroos had no tails they would topple over."
This means that had the world been different than it actually is, that is, a world in which Kangaroos have no tails, then Kangaroos in that world would topple over.
Material conditional: If P is the case, then Q is the case.
A material conditional will be true whenever the antecedent is false, but this cannot be the case for counterfactual conditionals since their antecedents are always false, and so every counterfactual conditional would be true. But every counterfactual conditional is not true. Thus counterfactuals are not material conditionals.
"If Oswald did not kill JFK, then someone else killed JFK."
If it were the case that P, then it COULD have been the case that Q.
A counterfactual will not be true whenever the antecedent is false.
"If it were the case that Oswald did not kill JFK, then it COULD have been the case that someone else killed JFK."
I.e., If it were the case that Oswald did not kill JFK, then, POSSIBLY, someone else killed JFK.
If it were the case that P, then it WOULD be the case that Q.
"If it were the case that Oswald did not kill JFK, then it WOULD have been the case that someone else killed JFK."
I.e., If it were the case that Oswald did not kill JFK, then, NECESSARILY, someone else killed JFK.
| P could be true is logically equivalent to: | P must be true is logically equivalent to: |
| (1) possibly P | (1) necessarily P |
| (2) not necessarily not-P | (2) not possibly not-P |
| (3) P is not impossible | (3) not-P is not possible |
- Necessarily, if R then P.
- If it were the case that P, then it would have been the case that Q.
- Therefore, if it were the case that R, then it would have been the case that Q.
The Problem: Recall that we have an invalid pattern of reasoning if and only if all of the premises of an argument can be true and the conclusion false at the same time. This pattern of reasoning does not guarantee a true conclusion when the premises are true. We can imagine an exception to the pattern, which would illustrate that it does not guarantee a true conclusion given true premises. If we can produce such an example, we can never trust this pattern of reasoning as a guarantor of truth.
Scenario: The first premise is true and the second premise is true. Suppose I started out on a trip to Santa Barbara just before 6, tried out a new shortcut that cut an hour off the usual time for the trip, and arrived just before noon. Neverthless, the second premise may be true and the conclusion false. Now suppose that IF I had started at 5 (which is before 6) then I would have been too sleepy to remember to try the shortcut, then I would not have arrived before noon. Thus, while it is true that had I started before 6, I would have arrived before noon, it is not true that had I started at 5 this morning, I would have arrived before noon. This exception proves that this form of argument is invalid.
Analysis: In this scenario I assumed mistakenly that starting at 5 (which is well before 6) would get me to my destination as soon as or even earlier than starting just before 6 (but after 5). However, I neglected to consider the possibility that starting even earlier could also delay my arrival. I did not realize that getting up at 5 might get me there on time but it also might not, but I presumed that it would. By strengthening the antecedent condition (getting up earlier) the possibility that I arrive later than noon (an undesirable consequence) can become a reality and this possibility should not be overlooked.
- If it were the case that P then it would have been the case that A.
- If it were the case that A, then it would have been the case that M.
- Therefore, if it were the case that P, then it would have been the case that M.
The Problem: An argument is invalid if and only if all of the premises of the argument can be true and the conclusion false at the same time. The pattern of reasoning above does not guarantee a true conclusion when the premises are true. We can imagine an exception to the pattern, which would illustrate that it does not guarantee a true conclusion given true premises. If we can produce such an example, we can never trust this pattern of reasoning as a guarantor of truth.
Scenario: Anna loves Pablo and Pablo loves Anna, they do most things together, but their love is so secure that they do not do everything together. Sometimes Anna goes to a party without Pablo. Miguel loves Anna too and he chases after her whenever Pablo is not around. Pablo despises Miguel and threatens to hurt him because Miguel follows Anna around. Miguel fears Pablo, so Miguel never risks meeting Pablo. On the evening of the party Pablo was in jail on a DUI charge, so he is not able to go to the party with Anna as they had planned. Anna decided to go to the party anyway but did not (because her bicycle had a flat tire and it was too far to walk). Had Anna gone, Miguel would have gone since he knew Pablo was in jail.
Analysis: In this scenario, while it is true that had Pablo gone to the party, then Anna would have gone, and it is true that if Anna would have gone, then Miguel would have gone, it is not true that had Pablo gone to the party, then Miguel would have gone. If Anna had gone to the party, Pablo still would not have gone, but Miguel would have gone (because he heard about Pablo's arrest). The first premise is true and the second premise is true. This exceptional case proves that this form of argument is invalid because it overlooks the possibility that even if Pablo had gone to the party, Miguel would still not have gone to the party. Notice, however, we may avoid the fallacy if we could assume that if Anna would have gone, then Pablo would have gone. Sometimes by adding another premise we can rule out all cases where transitivity fails. But in this scenario, we need not make that assumption.
Conclusion: Transitivity does not always fail for counterfactual arguments, but since it does sometimes, hypothetical syllogisms are unreliable and thus invalid.
- If it were the case that P then it would be the case that A.
- Therefore, if it were not the case that A, then it would not be the case that P.
Analysis: Suppose Boris wanted to go to the party, but stayed away because he wanted to avoid Olga (who has a hopeless crush on him). If this is the case, then the conclusion is false even if the premise is true. Olga would have gone to the party all the more willingly if Boris had been there, so the premise is true. Thus, the original claim (1) is true but its contrapositive (2) is false (unlike the contrapositive of a material conditional which is its logical equivalent, i.e., it has the same truth-value).
Conclusion: Transposing (or replacing) a counterfactual-conditional with its contrapositive form does not preserve its truth-value.