“Negatives” aren’t special

The oft-repeated claim that you can’t prove a negative is, frankly, dumb. Really, really dumb. It implies that you can’t prove Jesus didn’t rise from the dead, but you can prove he stayed dead.

I’m not just being flippant. Negative statements always imply positive statements and vice-versa. If p is true, not-p must be false. If you prove p, you disprove not-p. I think this point has wide-ranging philosophical implications, which I’ve been thinking about lately because of two books I read recently.

The first is Nassim Nicholas Taleb’s The Black Swan. On the whole, I liked this book, and would recommend it to others. Taleb has a background in finance, and on that subject, he’s great.

The book’s key idea is that improbable events are often not quite as improbable as they seem. If your calculations tell you to be 99% certain, maybe you should allow for a 5% chance that the assumptions behind the calculation are significantly mistaken. So far, so good.

Taleb stumbles, though, when he tries to write about abstract philosophy. He declares himself a Popperian, after Karl Popper, who proposed that scientific theories can never be verified, only falsified.

This claim is not quite as stupid as the naive “you can’t prove a negative!” If you think science aims at universal generalizations, well, it’s true that no number of examples can logically prove a universal generalization, while a single counterexample can logically disprove one.

Only it’s not quite as simple as that. In the real world, how can you be certain that your counterexample is really a counterexample? Say you’re citing the proverbial black swan to disprove the claim that all swans are white. How do you know the black swan was there?

I mean, maybe you saw it. Or think you did. Do you imagine it’s a universally true generalization that every time you seem to see an animal of a given color, you’re really seeing one? But how can you prove that, if you’re a Popperian? That’s the problem.

In fact, sometimes, it’s not even clear Taleb is carefully limiting himself to Popperian claims about universal generalizations. He says things like, “Negative knowledge is more robust to error than positive knowledge” (actually from his later book Antifragile). That sounds a lot like the silly “you can’t prove a negative!” mistake.

The other book that’s had me thinking about the negative vs. positive distinction is Joshua Greene’s Moral Tribes. This may literally be one of the best books I’ve ever read, right up there with Dawkins’ and Pinker’s best work. It’ll probably inspire quite a few blog posts, time permitting, right now I want to focus on one point.

In Moral Tribes, Greene defends utilitarianism, a moral philosophy notorious for, among other things, making no distinctions between acts and omissions. That means that letting someone die when you could have saved them is just as bad as murder.

Most people think letting someone die when you could have saved them is bad, but if they think about it for awhile, they’ll also think the idea that it’s just as bad as murder is crazy. Greene responds to this objection by trying to debunk the act vs. omission distinction through work in moral psychology that seems to show people’s intuitive responses in this area make no coherent sense.

From personal experience, you don’t need a psychology lab to attack the act vs. omission distinction. A philosophy classroom might do. My first ethics course in college spent a lot of time on the act vs. omission distinction, and it seemed like the more we talked about it, the more confused we got.

Sure, it seemed straightforward at first. But then we started considering thought experiments that challenged whether the distinction was important, or even coherent. By the end, we were talking about things like trying to compare the sizes of infinitely large spaces of possible actions.

Greene proposes the distinction between acts and omissions seems natural because of how our brains are set up. It’s easier to evolve a mental module for thinking about something, than it is to evolve a module for thinking about everything that isn’t one particular thing. But that’s probably as far as the distinction goes.

  • Luke Breuer

    This would be much less confusing if you allow a trinary logic with:

         (1) true
         (2) false
         (3) unknown

    Much stupidity is had by ignoring (3). Indeed, Aristotle needed this for the problem of future contingents: will a sea battle happen tomorrow? Failing to properly recognize ignorance is, well, see the paper Ignoring Ignorance is Ignorant, which advances Dempster–Schafer theory as better than Bayesian inference.

    One has a burden of proof/reason to show (1) or (2), but not (3). Now, even this is not quite right, as (3), taken to its logical conclusion, would result in radical skepticism and be unescapable. It is as if you have to start with some sort of ‘seed’ in order to grow anything!

    • MNb

      Even this is a false trilemma. Our realitiy is probabilistic, allowing everything between 0 (false) and 1 (true). That even applies to something like “you can’t walk through a wall.”

      • Luke Breuer

        Ummm, once you measure something, [one account is that] it snaps to a distinct value. One way to think of pre-measurement superpositions of quantum state is probabilities, but it’s not the only one. We don’t measure probabilities, we infer them. We measure discrete things.

  • MNb

    “But how can you prove that, if you’re a Popperian? That’s the problem.”
    It isn’t a problem, simply because the Popperian argues that nothing can be proven. You write it yourself:

    “it’s true that no number of examples can logically prove a universal generalization.”
    You’re demanding from a heater that it keeps your drink cool.
    The real problem for the Popperian – and Popper himself admitted it – is that the falsification principle itself can’t be falsified by any means.

    It would be nice though if you – and Luke underneath – first told us what you (he) exactly mean(s) with “proof”. Do you mean 100% absolute eternal unchangeable certainty? Do you use the mathematical meaning? I can prove you Pythagoras’ Theorem both right and wrong (right: Euclidean axioms; wrong: non-Euclidean axioms). Or do you just mean a high probability, like with the higgs boson?

  • Luke Breuer

    I just randomly found the following in Michael Polanyi’s Personal Knowledge, under “Equivalence of Belief and Doubt”:

        The first point in my critique of doubt will be to show that the doubting of any explicit statement merely implies an attempt to deny the belief expressed but he statement, in favour of other beliefs which are not doubted for the time being.
        Suppose somebody says ‘I believe p‘, where p stands for ‘planets move along elliptic orbits’, or else for ‘all men are mortal’. And I reply “I about p‘. This may be taken to mean that I contradict p, which could be expressed by ‘I believe not-p‘. Alternatively, I may be merely objecting to the assertion of p as true, by denying that there are sufficiently good grounds to choose between p or not-p. This may be expressed by saying ‘I believe p is not proven’. We may call the first type of doubt ‘contradictory’ and the second ‘agnostic’.
        It is immediately apparent that an expression of contradictory doubt ‘I believe not-p‘ is of the same character as the affirmation ‘I believe p‘ which it calls in question. For between p and not-p there is no other difference than that they refer to different matters of fact. ‘I believe not-p‘ could stand for the allegation that planets move along orbits which are not elliptical. (272)

    Polanyi goes on to talk about how agnostic doubt also makes truth-claims (about constructability or lack thereof within some set of axioms), but I got tired of typing.

  • Alex SL

    This is a rather complex issue. On the one hand I would wholeheartedly agree that negatives aren’t special, and that at least very many positive statements can be rephrased into a negative statement, showing the absurdity of “you cannot prove a negative”.

    However, your argumentation around the black swan seems a bit odd to me. You are basically arguing from solipsism to show that in reality we can never know anything anyway, so proved positives are as unreliable as proved negatives. But that does perhaps prove too much. I’d prefer to point out that in science as well as everyday life we generally consider negatives proved “beyond reasonable doubt” in many cases where they have not been proved to the degree that a statement in formal logic could be. That means, yes, we can be reasonably sure that we have seen a black swan with our eyes, but also that yes, we can reasonably assume that there are no abominable snowmen if hundreds of years of searching around the globe have failed to turn one up.

    As for the ethical question of acts versus omissions, it would depend on how much of an omission it is. Yes, if somebody is dying in front of my eyes, I am the only one there to help, and I could very easily help at basically no cost to myself, then I am surely nearly as culpable for their death if I don’t help as if had killed them myself. However, if you are talking about somebody dying in Haiti because I didn’t give enough of my salary to charity, then I’d say my responsibility for aiding them is somewhat diminished compared to the first situation.

    We cannot be expected to care for all people on this planet equally. Yes, society as a whole should be organised to care for all its members equally (that why we have things like government!), but expecting an individual to care as much about a stranger they have never interacted with except across twenty intermediaries as about their own daughter is unrealistic, perhaps even inhumane.

    So the question is: if through pure reasoning you arrive at the conclusion that there is no difference between acts and omissions, but human moral intuitions scream that there is such a difference, should we (a) study more moral philosophy and override our intuitions or (b) conclude that maybe pure reasoning without taking our evolved intuitions into account is not a good guide to realistic ethical rules?

    It could also be pointed out that your experience of the lines between acts and omissions becoming blurred after long discussions does not demonstrate the absence of a clear difference between better defined cases. There is, after all, also a complete gradient between toddler and adult, but you still wouldn’t let the toddler work in a salt mine. The concept of adult is an important one despite the fact that it doesn’t have a clear categorical distinction from the non-adult stage.

    • http://patheos.com/blogs/hallq/ Chris Hallquist

      However, your argumentation around the black swan seems a bit odd to me. You are basically arguing from solipsism to show that in reality we can never know anything anyway, so proved positives are as unreliable as proved negatives.

      Sorry, that wasn’t the intent. I’m not actually advocating solipsism. It was more of a reductio: “Sure, you can argue for radical skepticism about induction, but you can also argue for radical skepticism about sense experience, so what?”

      • Alex SL

        I never thought you were seriously advocating solipsism, but I merely felt that it came across as a bit more as “all knowledge claims are equally weak” and not so much as “we can be reasonably sure even about negative claims”. I mostly agree with you anyway.

        Indeed, in a way I see an asymmetry that is kind of the other way, if anything: There are many cases where it is easy to see that somebody is definitely wrong about an issue even without being sure what the right answer is…

  • eric

    Greene responds to this objection by trying to debunk the act vs. omission distinction through work in moral psychology that seems to show people’s intuitive responses in this area make no coherent sense.

    It makes sense when you factor in economic or other costs. An act has an associated resource cost to me. An omission does not. The notion ‘acting to kill someone is evil while letting them die is not’ is just two cases of the fully self-consisent concept “one ought not expend resources so that another person dies.”
    This is also – ironically – consistent with the utilitarianism that Greene is trying to defend, which must consider resources like time, effort, etc.. in terms of happiness or personal value. An act costs more than an omission, so when the two have the same outcome, the act should be considered the less moral option under a utilitarianism framework.

    • Luke Breuer

      An act has an associated resource cost to me. An omission does not.

      Failing to sufficiently protect yourself from robbery is an act of omission with a potential cost.

      • staircaseghost

        You’ve misapprehended the target of the distinction.

        For all we know, in a given domain the net result of omissions always cost more than commissions.

        But in all domains, actions by definition entail expenditure of resources, even if it is de minimus time expenditure, whereas omissions don’t, as a rule.

        • Luke Breuer

          I prefer to think both forward and back in time when I consider expenditures, in which case acts of omissions cost. For example, in software, if you choose the quick-and-dirty route, you save in the short-term, but you rack up technical debt. This focus on next quarter’s profits destroys innovation and will wreck the US, if it continues long enough.

  • L.Long

    “Cannot prove an negative” is just one of those philo arguments and in reality this is not a real problem. “prayer does not work, so prayer is BS” now I have done many experiments that shows the negative statement is true. If some dim-bulb says ‘You can’t prove a negative’ then fine do a REAL experiment (duplicated & verified a couple of times) and prove the negative wrong, i.e. Pray around an amputee and when his leg grows back (NO FAIR if you use ANY kind of real medicine) we can talk, till then my proof of the negative stays!!!

    • ZenDruid

      The null hypothesis is not negative. If you prayed for an amputee and eir stump commenced to deteriorate further, that would be negative.

      • L.Long

        Disagree the statement is a negative and the stump deteriorating would be evidence of harmful effects of the praying.

        • ZenDruid

          …the stump deteriorating would be positive evidence of harmful effects of the praying…?

          • L.Long

            Correct! IF it could be duplicated and verified. Because correlation is not necessarily cause as it could be some other effect.

  • Beth Clarkson

    You can’t prove some negatives. Some concepts of god are easily rejected as disproven. Others – not so much. You can’t prove the universe wasn’t created last Thursday with everything – including all human memories – as they are.

    We can reject these not-unproven ideas about how the universe was created for any number of reasons, just like we can reject solipsism. But we cannot disprove them.

    The comparison to omissions versus acts is interesting. I’ll let that ruminate in my head for a while.

  • Anonymous Coward

    I share your criticism of the idea that you can’t prove a negative. I think it’s even worse than you say. It’s not only false that you can’t prove a negative, but in fact, the kinds of problems that people point out for “proving negatives” aren’t really about _negative_ness at all, but rather, _universality_. What they should have been saying is “You can’t prove a universal.” But then, of course even at that they’d be wrong.

  • jg29a

    Perhaps you should also address Goodman’s “new riddle of induction” here. Every white swan that I’ve experienced has also been a shmwhite swan, where “shmwhite” means “black iff the current U.S. president is female; else white”. Fast forward a couple of years. Hillary gets elected, and soon after I see a swan that is either another boring example of “all swans are white” or an amazing disconfirmation of “all swans are shmwhite”

    The important point about Goodman is that “white” is not *logically* privileged over “shmwhite”.”White” is not, for example, formally simpler, since we could just as easily define “white” as “shmwhite iff the current U.S. president is female; else shmblack”, or what have you. The new riddle of induction is that inductive evidence always provides us with an infinite number of predicates that fit the data, nearly all of which are not actually projectable, and yet logic alone cannot distinguish the subset whose member terms *are* projectable.

    My takeaway is that we have to rely upon both our evolved predispositions and the words and constructions in our evolved natural languages, and that rational empiricism is actually about iteratively tweaking that starting framework when problems appear, rather than thinking you can start from outside of it.

  • tmalatesta

    Maybe it’s just me, but I’m still waiting for someone/anyone to address how you prove a negative. Just throw some references at me and I’ll go look myself. I’ve got no problem with that. I do have “Moral Tribes” on my shelf, though I’ve never picked it up yet. I suppose I’ll start there until someone can point me something more specific to my question. Thanks.

    • MNb

      “There is no luminerous ether” was proven by the Michelson/Morley experiment.

      • tmalatesta

        Prove a negative?

        Failed to prove a positive.

        From what I can gather, the Michelson Morley experiment did not disprove the existence of luminiferous aether. That is, it did not prove the negative proposition that “An Aether does not exist”. Rather it failed to prove a positive assertion, that by observing the speed of light relative to the Earth’s movement through the aether, the movement of the Earth through the aether could be detected and measured. In fact, Michelson himself, believed for the rest of his life that his experiment was a failure.

        This experiment’s failure to substantiate its hypothesis led to the Lorentz Ether Theory, the first recognized theory of Relativity, which simply worked to fine-tune concept of an aether. But the failure of the Michelson Morley experiment, calling into question the existence of an aether, as it did, also led to other experimentation which sought to observe and measure the existence of an aether, and all of which met with similar results.

        Once the theory of Special Relativity was introduced, explanations of how light moved through empty space (the basis of the luminiferous aether theory) was accomplished without reference to an aether at all, though various attempts by Lorenze, Einstein, and others to reformulate concepts of the aether continued — Newton’s absolute space as “Aether of Mechanics”, Maxwell’s and Lorentz’s “Aether of Electrodynamics”, as an absolute state of motion — even through these concepts had very little in common with the earlier mechanistic theories of aether.

        In the end, science as a whole had taken on the attitude that the theory of an aether had no useful theoretical definition and no predictive value, and that phenomena historically attributed to aether theory were found to be either contradictory to the mechanistic aether theory or could otherwise be explained without reference to an aether at all. As a result the aether theory came to be seen as unusable and superfluous, and simply fell into disuse, but has never been considered “disproved”:

        Did these experiments “disprove the ether”? I wouldn’t put it that way. None of these, or any other experiment could disprove the general idea of an all-pervading ether. The point is that there is simply no experimental evidence for the ether, and no need for it in any of our physical laws or theories. You can’t disprove something that isn’t there, or something that doesn’t affect anything material in any way. Yet even today, there are non-scientists who desperatly seek to revive this 19th century ether concept. Why? The idea seems so “right” to them. They cannot imagine light moving through “nothing”. They are a classic case of people who have an emotional commitment to an appealing naive concept, and will bend physics and logic in order to justify that idea.”

        - Lock Haven University


        Whereas there may be a general “lack of belief” in the concept of an aether, there is, by no means, a general theory as to the non-existence of aether. It is simply that concepts of the existence of an aether no longer matter to the scientific world, or the world at large.

    • Caravelle

      “There is no largest prime”.
      Outside of mathematics you can’t “prove” a negative but you can’t “prove” a positive either. It’s all about evidence, and evidence can lean both ways.

      • tmalatesta

        That, I do agree with. And I will read up on that some, too, though I don’t expect understand much of it. :)

        • Caravelle

          I think one reason it makes intuitive sense that you can’t prove a negative, especially when picturing “do black swans exist, depending on whether I see one or not” scenarios, is that in those cases the version of the evidence that’s in favor of the negative statement is extremely weak. It’s a bit like in the raven paradox that says seeing a green apple is evidence for the statement “all ravens are black”. It’s true but sounds wrong, and there are many reasons it does but one of them is probably that it’s extraordinarily weak evidence for that statement.

          At the end of the day whether it’s a positive or a negative statement, the question is what kind of evidence you’d expect if the statement were true or false vs what evidence you see. While it’s hard to prove there is no black swan in the world just from looking at swans (though there can be other ways – for example if a physical, chemical, biological or evolutionary property of swans made it impossible for them to be black), it’s really easy to prove by the same method that there are no black swans in your bedroom. This is because you only see a tiny fraction of things in the world at large and there are many ways for things to go unnoticed, so “I see no black swans” is almost equally likely whether black swans exist or not. On the other hand you could hardly miss a black swan in your bedroom, so the odds of “I see no black swans” are very different under the hypothesis that there is a black swan in your bedroom and the hypothesis that there isn’t; this makes it very strong evidence for distinguishing the two hypotheses. Note that it is evidence in both cases (it’s still slightly more likely that you won’t see black swans in the world if they don’t exist than if they do), the difference is how strong a piece of evidence it is.