We are nearly at the point where we can say something about the candidate proposition of interest: “Treatment cures cancer of the albondigas.” It is obviously contingent, therefore it cannot be necessarily true. It may be contingently true or false, or have a non-extreme probability. This depends on which information we condition the proposition on.
First recognize the central limitation of the proposition. If it does cure, whose cancer is it curing? Well, the cancer of the Bobs. What about non-Bobs? We don’t know. So what was the purpose of the experiment? Just that: to learn whether the treatment cured Bobs. About not-Bobs we know nothing, unless we condition on evidence we did not see or on assumptions which are not proved.
This make strike you as odd since usually, or at least as it appears, experiments are meant to provide information about extensions to the experiment. We don’t do real drug trials to say things only about the patients in the trial, but about new possible patients. We don’t test material coming off a production line to say things only about the tested substances (which are commonly destroyed), but about the remainder of the output. And so on.
Repeat: speaking of extensions is not what we’re doing at the moment, though we will come to it. This is emphasized because one must keep these activities separate. For now, consider only the evidence of the experiment itself.
Let’s recall what this could be: (1) Every Bob is cured, (2) no Bob is cured, (3) every treatment Bob is cured and no placebo Bobs are, (4) no treatment Bob is cured and every placebo Bob is. After the experiment, we will have seen one of these outcomes, and when we do we invoke the conscious decision to add the outcome to our list of conditioning evidence. Remember that it is we who decide which evidence/premises to consider in any contingent proposition. Change the evidence/premises, change the probability of the proposition. This principle is key.
Outcome 1 What is the probability of “Treatment cures cancer of the albondigas” given the observation “Every Bob is cured” and given what we know about the experimental set up—all Bobs are identical, the measurement time, etc.? Part of this evidence is tacit: knowledge of English words and grammar, which can sometimes be ambiguous, but which is crucial in understanding how evidence relates to our proposition of interest.
Now “cures cancer” can mean that the treatment was the active, efficient cause, that somehow (it’s not important here how) the drug interacted with the body in some biochemical manner which eradicated the cancer. Or “cures” could mean the mere presence of the treatment was needed, but that the treatment itself did nothing active. To rework an example from the comments, suppose Bob went into a room and switched on light. Bob is the cause of the switch being thrown, which in turn caused the light to illuminate. This is different then if Bob threw the switch only because he saw Alice enter. Alice is not an active cause, but her presence was needed (we wouldn’t punish Alice for the switch throwing if, say, such a thing were illegal).
Was the presence of the treatment needed to cure the Bobs? Because all the Bobs are identical and all—both placebo and treatment Bobs—were cured, then if “cure” means “presence needed”, and since we assumed the presence of the placebo, we deduce our proposition is contingently false, or has probability 0. We don’t need the treatment: the placebo is enough. If we gave the placebo to the treatment Bobs we know, via our assumptions, they would have been cured, too. How we don’t know; but that they would be cured, we do.
What happens if we remove the evidence of presence of the placebo; that is, if the experiment only consisted of the treatment “arm”? Then all we can say is that the proposition is contingent, that it does not have an extreme probability, that its probability lies between 0 and 1. The observational evidence boils down to “The treatment either cured or it did not”; or put another way, “The Bobs got better on their own or the treatment ‘worked’.” Both sentences are tautologies, thus both are necessary truths, and hence provide no information to the proposition “Treatment cures cancer of the albondigas.” (Adding necessary truths to premises never changes the probability of a proposition.)
History assures there is more than one way to skin a cat. Perhaps there is more than one way to cure cancer of the albondigas. The placebo might have cured one way and the treatment another. That is, it could be the presence of the placebo was needed to cure the cancer, or the placebo was active in its cure, but in a different way than the treatment. (We only assume the placebo is not itself active, of course.) Perhaps the cancer would have been cured by eliminating biochemical A or B and placebos knocked out A, and the treatment B (or whatever).
Now if we assume the treatment interacted with the Bobs’ bodies differently than the placebos, for which there is no proof, then we open the possibility the treatment actively cured. But we must also assume that not only did the treatment interact differently, but that the interaction was such that it allowed the treatment to “do its thing.” And if we assume that, then we are arguing in a circle (with regards to our proposition).
We could go on with this imaginings indefinitely, but really we are left with nothing to say about the proposition, where cure means “active”, except that it is contingent, that is does not have a non-extreme probability, which is not saying much.