Some ideas around “all other things equal” ([1]) apply in part to other situations, including assumption and falsification.
I originally began this text using “premise” throughout. The result was many clumsy and unfortunate formulations, and I revised this to, in most cases, use a greater variety of words and to prefer “assumption” (“assume”, etc.) instead, even in some cases where “premise” can be argued to be the better word.
Also note that other words (and associated concepts) might be relevant even when not present in this text, e.g. “postulate”. (The difference in meaning between such words is often subtle and not necessarily important in a given context.) In the specific context of science, it might be most common to “falsify a hypothesis”, the hypothesis having been stated in a definite form ahead of experiment (unlike many of the silent assumptions involved).
As in [1], let us start with an old joke:
A man believes that he is dead and is sent to a shrink. The shrink has him repeat, again and again, that “Dead men do not bleed!”. After some time, he pricks the finger of the patient. They both observe a drop of blood leaving the finger.
“What does this prove?” asks the shrink.
“That dead men do bleed!” answers the patient.
Here there are several assumptions in play that can be falsified, most notably that (a) the man is dead, (b) that dead men do not bleed. (Some words on other assumptions follow below.) By pricking the finger and causing bleeding, the psychiatrist has (to some approximation) falsified the conjuncture of these assumptions. However, he failed to consider that the problem might equally be with (b) instead of (a). Correspondingly, he assumed that the falsifying experiment would disprove (a) to the patient, while the patient stuck to his guns and saw (b) as falsified.
In this case, if transplanted to the real world, the shrink might have been justified, because there are no credible observations of e.g. dead men walking around, talking, and/or visiting shrinks (and a real-life colleague might well have begun with such observations). However, in many other cases, there really are two or more assumptions that all can be the cause of the falsification of their sum. Consider, for a trivial example, “Tom is taller than Dick; Dick is taller than Harry; ergo, Tom is taller than Harry”: If it now turns out that Harry is as tall as or taller than Tom, then it might be that Tom was, in fact, not taller than Dick—but equally that Dick was not taller than Harry.
This can be particularly problematic when someone sees a particular outcome as ideologically dictated, has a strong interest in one premise being seen as falsified and another as still valid, or similar. Note e.g. how various pseudo-sciences, charlatans, and whatnots can be very selective about interpretation or, even, in a reversal of the above, introduce implausible premises to “take the fall”.
Consider e.g. the repeated case of a “gender scientist” being confronted with evidence of differences in behavior between boys and girls in infants—and refusing to even consider the possibility that there might be an aspect of “nature” involved. Instead, behavioral differences must be explained by “nurture” and “ergo, socialization into gender roles begins much earlier than we thought—we must fight the Patriarchy even harder!”. (Note that the fundamental error here does not arise from being open to a “nurture” explanation but from ruling out that an ideologically dictated “nurture” could be false. This the more so, as the low age would make the relative likelihoods of “nature” and “nurture” as explanations much larger resp. smaller than for similar observations in, say, teenagers.)
Or, for “take the fall”, consider a psychic who introduces the idea that the presence of a skeptic would cause his powers to malfunction. While this, in the spirit of this text, cannot be categorically ruled out, it is both far-fetched and so self-serving that it should be under suspicion on that count alone. This the more so, should the idea be introduced after the fact, in a manner that seems geared at preventing scientific observation, or in an inconsistent manner (e.g. in that one psychic resorts to it and that another does not).
A particular complication is that there often are silent assumptions, of which not even the holder is necessarily aware. For instance, immediately above, it could also be that “taller than” is not transitive (in other words, that the “ergo” would be logically incorrect). This does not match real-life experiences with height, but could very much be an issue in a mathematical setting where more abstract comparisons are made. Likewise, there are similar real-life constellations where such transitivity does not hold, notably in the classic case of “Tom beats Dick; Dick beats Harry; Harry beats Tom” (in e.g. boxing). Indeed, the last points to a further complication of instability, in that what holds in one given (literal or metaphorical) measurement need not hold in another. It might e.g. be that Tom beat Dick when they boxed today but that the outcome is different tomorrow. We must, then, be careful about what we assume about measurements—and, indeed, it might well be that a seeming paradox around height legitimately arises if Tom was found to be (slightly) taller than Dick because they were compared in height when Tom was fresh out of bed and Dick came home from a night shift. (Humans tend to slightly and temporarily lose height throughout the day.) Even highly controlled scientific experiments can run afoul of such issues, because some confounding factor is not considered, because an instrument malfunctions, because the user of the instrument makes a faulty reading or enters the read data in a faulty manner, whatnot. (We then have silent assumptions like “we considered everything, the instrument worked correctly, the reading was correct, the entry was correct, ...” in addition to whatever assumptions the experiment was intended to test.)
In the overlap between some of the above, we have the worse case of silent assumptions that differ between the involved parties. In interpersonal settings this is often compounded by silent assumptions about the assumptions (silent or otherwise) of others. For instance, parts of what I write in [2] and [3] can be explained by such different assumptions and assumptions about assumptions, e.g. in that I might assume that a request for a time of day has no meaning beyond wanting to know the time of day, while the counterpart might see it as a conversation starter—and assume that I, too, would see it a conversation starter.
Of course, even overt assumptions might differ and be subject to assumptions by others. For instance, someone innocently suspected of a crime might assume with great certainty that he is innocent, while others might assume that he is (a) guilty and (b) lying about his innocence. (But note that it is almost always “great certainty”—not “perfect certainty”. Cf. the below discussion of the claim by Sherlock Holmes and note e.g. the possibilities, no matter how low, of memory loss and sleep walking.)
To return to the original joke, the shrink seemed to assume that merely repeating a particular claim would make the patient a believer in this claim (an assumption that was promptly falsified). Such crude methods are often used, especially, in a Leftist and/or indoctrinating context (note e.g. the case of Luke Salmonse, who observed pro-Islam chanting by tutors during a police training session) but are ultimately of dubious effectiveness outside morons and are, of course, of even more dubious ethics. A better way to proceed would have been to first come to an agreement of what outcome would have had what implication, thereby removing silent assumptions and unifying differing assumptions (or, as case may have it, show that the assumptions were so differing that the experiment was futile). Likewise, it was far from a given that a single drop of blood would invalidate the idea of “dead men do not bleed”, and someone else might have objected on this ground, e.g. because such a single drop might be compatible with a dead body (in general) or because special circumstances (notably, physical movement) might have allowed bleeding even where a regular dead body would not have bled. (And chances are that other assumptions yet applied that I have not identified, e.g. relating to the possibility of someone undead.)
Proof-reading, I spot a silent assumption of my own that might be wrong, namely, that the shrink had overlooked the possibility of “Dead men do bleed!” above. It might be that he was actually well aware of that interpretation of the falsification and that the repetitions were chosen exactly to reduce the risk that the patient would chose that interpretation.
To finally end with some words on the claim by Sherlock Holmes that once one has eliminated the impossible, whatever remains, however improbable, must be the truth:
That something is impossible, is very rare, and it is more often a matter of varying degrees of improbability (at least, if we look at events somewhat relevant to e.g. a criminal investigation) and it is important to have an insight into what the probabilities are, how they compare to each other, and the difference between “improbable” and “impossible”. (Even many truly far-fetched explanations, like a suspect being innocent because a previously unknown twin had committed a crime, are merely very, very improbable—not outright impossible.) This, of course, includes the risk of faulty assumptions, e.g. that “Tom has an alibi for the murder of Dick; ergo, he cannot be guilty”, which ignores that the alibi could be false, that he could have hired someone else to do the actual deed, and that the time of murder might be something other than believed—and, likely, some further assumptions. (In effect, we have a set of assumptions and an ensuing reasoning of “Tom did the deed, only the one who did the deed is guilty, the one who did the deed was present in person at the time of the murder, we know the time of the murder, Tom has an alibi for the time of murder, we can trust this alibi; ergo, specifically the assumption that Tom did the deed must be faulty; ergo, Tom is not guilty.) A particular problem with real-life criminal investigations (in my outsider’s/layman’s impression) is that the possibility of a framing by a third party is virtually never seriously entertained—there is a silent assumption of “not a frame job”. Now, a framing might be far rarer in real life than on TV, but it is certainly naive not to entertain the possibility, as it would rule out as “impossible” a very real possibility. On the trial level, it might be even worse, as frame jobs by, say, a DA seeking reelection and needing a conviction, or falsifying of evidence by police officers who “know” that someone is guilty but lack real evidence, is not unheard of. (Even if, again, far rarer than in fiction.)
With matters of crime, the standard level of proof is “beyond reasonable doubt” and I do not claim that a mere slight possibility would give cause for reasonable doubt. A thorough discussion of what should be considered, what actions and non-actions might be warranted, and similar, in light of an “unusual” defense and at various stages at the process from first investigation to verdict, is far beyond the scope of this text. However, I do note that what is reasonable and unreasonable doubt is ultimately for a judge or jury (depending on system used) to decide and that care must be taken not to rule something out prematurely. Likewise, any claims raised by a defendant must be judged on their merits, not on e.g. a prejudice that “no one is ever framed” or “the police would never falsify evidence”.
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