Outcome 3. This sound like good news for you, since all treatment Bobs got better and no placebo Bobs did. Congratulations! The treatment might have caused these cures.
Or again the treatment might have done nothing and the placebo (somehow) blocked the natural cure. It is after all only a guess, with no evidence, the placebo has zero effect on the Bobs. Assuming placebos have no effect, then we might be tempted to say we know with certainty the presence of the treatment was needed for a cure in people exactly like Bob.
But it’s no so, because it could be that something which was not the treatment happened only to the treatment and not placebo Bobs and this something is what caused the cure. For instance, maybe the treatment Bobs coincidentally got zapped by a cancer-curing ray which missed each placebo Bob.
Yet we can rule this out because we assumed both sets of Bobs behaved and were situated identically after receiving their meds and up to the time of measurement. This is a tricky assumption because it means the treatment can only work if it does not cause any changes in behavior, that it only works by shifting stuff inside the treatment Bobs’ bodies, stuff that does not in any way contribute to behavior. Behavior includes thinking. So we exclude cases where the treatment is itself inactive except that it causes the treatment Bobs to think they received the treatment and perhaps enhancing the placebo effect (we are told this happens with real drugs).
Another possibility: the treatment itself did not work, meaning it was not an active agent, but was instead a catalyst which either unlocked some beneficial or blocked some harmful biological process which led to a cure. A catalyst could be a change in behavior (such as removing the desire to listen to NPR thus causing a reduction in stress).
We have ruled these out for a very important reason. If there is one or more behavior difference between the groups, any one of these differences, or some of them in combination, could have been the true cause of the cure. We wouldn’t know, not with the evidence we have, whether it was the treatment or the behavior which caused the cure.
And even if the treatment worked as advertised, we can only say it did so in Bobs. What about not-Bobs? Recall what we said about the closeness of not-Bobs to Bobs: that it is always an arbitrary measure. We just don’t know whether the treatment would have any effect on not-Bobs; not with the evidence we have.
Outcome 4. Bad news for your plans of going public. Your treatment either did no good or it caused harm, or possibly blocked a natural cure. The placebo might not have done anything either, except getting out of the way of a natural cure. Or it might have caused the cure, either by chemical means or by “releasing” the placebo effect. The latter is only possible if the treatment simultaneously held back the same placebo effect. Which of these combinations actually occurred? You get it by now. We can’t know, not with the evidence we have.
Everything said in Outcome 3 is the same here, but with the drugs reversed.
We’re almost in a position to figure what all this means to the proposition “My treatment cures cancer of the albondigas.” But first two terms.
A necessary truth says of a proposition that it must be true, that it could not possibly be false. In particular there is no observation which could refute a necessary truth. Truths which are necessary are true even if you don’t want them to be. Examples: it is a necessary truth that “Necessarily true propositions exist”1 and that “1 + 1 = 2”. The latter is so even if you collected two objects which, upon bringing them together, resulted in no objects (think of an electron plus positron). Mathematical propositions have nothing (directly) to say about real objects; they are entirely metaphysical.
It may not be obvious that the proposition “1 + 1 = 2” (or any proposition) is necessarily true. In these cases, the necessary truth might be the end result of a chain of reasoning, as mathematical proofs are, which ultimately rest on indubitable axioms, which are propositions which are true but are true for no other reason than we’re aware of their truth. All the truths in math, logic, and philosophy are like this.
This last sentence is not equivalent to the one which says “All the propositions in math, etc…” because there are many false propositions in these areas. Note too that a necessary truth, as used here, does not mean a formal logical truth, though there will be overlap, because we haven’t any interest whether any particular proposition can be shoe-horned into some schema which somebody might have shown has a counter example (think grue). Each proposition/argument is and must be taken on its own account.
Next time: the second term and complications.
1Roger Scruton said something to the effect that people who state the opposite of propositions like this are inviting us not to believe what they say, and that we should take them up on it. (I have to dig this quote up.)