To explain what I mean we first need to go back to Quine’s seminal paper Two Dogma’s of Empiricism. Even though I think Carnap clearly had the better of the argument about analyticity¹ I want to give Quine credit for pointing out to me the way in which the concepts we use depend on our background assumptions (how we model/conceptualize things) for their very coherence. Just as the concept of an equivalence class stops making sense once you start talking about non-transitive relations so too do many of our scientific and everyday notions cease to be well defined when we no longer accept the assumptions they were defined with respect to.

Stealing an example from Quine consider the Newtonian concept of kinetic energy. A good Newtonian physicist would have said that the kinetic energy is defined to be .5mv² where m is the object’s mass and v it’s velocity. However, kinetic energy is obviously also intended to be in some sense a measure of the work it would take to stop that object. Since these two notions coincide on the Newtonian picture there isn’t any problem. So long as we believe (or even use as an approximation) Newtonian physics there isn’t any question as to which is the right definition of kinetic energy. We are simultaneously committed to the concept capturing both notions. What Quine observed is that once we abandon the Newtonian conceptual framework there isn’t really any objective fact about which commitments we should honor and which we should discard. If scientists had responded to special relativity by using kinetic energy to describe the Newtonian formula and started theorizing about the conservation of Eisensteinian smenergy we couldn’t really accuse them of having made a mistake. Just as there is no right way to extend the notion of an equivalence class to non-transitive relations there often isn’t any right way to extend our scientific or everyday concepts outside of the frameworks they were conceived in.

The upshot of all this, in my view, is that our most useful concepts often presuppose certain assumptions. When these assumptions no longer hold the concepts themselves may cease to be coherent. So keeping this in mind let’s take a look at the assumptions that give rise to our concept of rationality.

Without going into too much detail I think it’s fairly safe to assume that a major (primary?) grip on (belief) rationality comes by way of postulating that people hold various beliefs where we take those beliefs to behave in some loose way like propositions. In other words we gloss over complexities like the effect of context and social situation on the views people express and simply pretend they either do or don’t believe some claim. Of course you can embelish this view a great deal and allow people to believe things in varying degrees or even take them to merely have some transitive implication relation. However, we can only stretch these concepts so far before they become unwieldy and useless, something we all implicitly recognize when we hesitate to attribute beliefs to ethnic groups, countries, or our computers.

So what? It’s hardly news that some aspects of people’s behavior won’t be well described by idealizing them as having something like beliefs. However, the point I want to press home is that rationality isn’t a property that big fleshy globs of atoms have. Rationality, is a concept grasped in terms of a certain kind of idealization about human behavior. It’s a useful concept and useful idealization but it’s still a type error to think of it as a property that applies to actual physical beings. We frequently forget this because in most contexts there is an obvious “right” way to idealize someone as an agent with certain beliefs so we talk about people having irrational beliefs and find it useful. However, it’s important to remember this shorthand only makes sense as long as this kind of idealization makes for a decent model of human behavior. Just like it’s simply confused to talk about the Newtonian kinetic energy of a particle traveling at .999c there are situations in which idealizing people as having something like belief is such a bad way to model their behavior that talking about rationality is similarly confused.

But the situation for a viewpoint independent concept of rationality applicable to real people only gets worse once you realize just how sensitive the ascription of rationality is to the way we choose to idealize the situation. Choosing to idealize a split brain patient as a single agent will yield very different judgments about his degree of rationality than idealizing his actions as the result of two seperate agents with distinct beliefs. It’s not that one of these idealizations is wrong and the other right (what could that even mean?) but just that in certain contexts one will be more useful than the other. And it’s not just split brain patients, Frued and many others have often taught that people were better modeled as the result of several competing agents or personalities. To really drive home the dependence of ascriptions of rationality on your choice of model just try to work out how you could make a principled application of the concept to a network of partial autonomous, partially integrated AIs.

My point is that it’s not just that we can’t ever be fully rational. It’s that the very notion as applied to living breathing people isn’t even coherent. Rationality is a concept that lives in an abstract idealize realm populate by agents possesing something like beliefs. It’s only applicable to real creatures insofar as this kind of idealization is useful and people largely agree on how it should be done. Step beyond that and it just doesn’t make sense anyone. I also think this realization can help alleviate some of the confusion over various paradoxes like the surprise quiz but that’s another post.

For concreteness sake let’s assume that every time you take MDMA² you have an outflowing of love and sympathy which makes the death penalty or any retributive (as opposed to deterrent or preventative) punishment seem to be a horrible moral abomination³. Or even just that you know that if you were to take the drug you would feel this intuition. Now intuitively one wants to say in these cases that such a drug induced intuition doesn’t count or at least counts much less but why? Well one reason we might wish to exclude such intuitions is the worry that there would be too many of them. Indeed if you believe experiences (or whatever intuitions are) supervene on local physical state (e.g. brain state) then your likely to think that some kind of brain intervention could create any moral intuition desired⁴. But this isn’t a (sound) argument that these intuitions aren’t equally valid it’s merely a wish that they aren’t. It would be nice to have access to moral facts but we can’t discredit the possibility that none of our intuitions, drugged or otherwise, give us any evidence just because we don’t like it.

A more promising approach is to observe that we don’t credit the sensory experiences of inebriated people to the same degree we credit those of sober folks and argue that philosophical intuitions work similarly. While this sounds good the problem is that it’s just not true that we always trust sober perceptions more than chemically altered ones. For instance if a perceptual task requires great focus we very well might prefer the observation of someone taking a small dose of amphetamines than that of a sober person⁵. Certainly imagine drugs or other brain alterations that would improve our perceptual accuracy in some ways even while they might impair it in others. Thus it’s not that we have a blanket rule about trusting sober observations more, rather, we merely induct on prior observations about perceptual accuracy in different states. Without an independent check on moral facts we don’t have any reason to take our normal sober brain states as more reliable in this regard than others⁶.

More broadly one might observe that even without knowing anything about drugs or the effects of brain injuries one would probably believe that most modifications to the brain would degrade, rather than improve our perceptual abilities. However, we only believe this because we have reason to believe that evolution has tuned our brain for perceptual accuracy. Given a situation where we have reason to believe evolution would have tuned our perceptions to get an incorrect, rather than correct, result⁷ we should believe that random alterations to our brains would be likely to improve the result. After all if your brain is a reliable mispredictor (when X occurs we perceive ~X) then any alteration in that behavior would have to be an improvement. Thus whether or not we should assign a higher probability to our normal sober intuitions being correct or those induced by brain changes depends on whether or not we have reason to believe evolution favored accurate or inaccurate intuitions.

When our intuitions are not subject to an external check I really don’t think we have any reason to give more weight to our actual intuitions than those we would have if our brains were altered. In the particular case of moral intuitions I would argue that if anything we have reason to believe that our intuitions are, if anything, less reliable than those selected at random. We have plenty of meta-moral intuitions like ‘all people deserve equal moral consideration’ yet there seems to be no shortage of examples where evolution has favored more concrete intuitions in conflict with these principles, e.g., people tend to have different moral reactions when it’s a family member’s life on the line than a strangers. Thus any analysis that gives more weight to our actual intuitions than other possible ones must acknowledge the existence of evolutionary pressures to have inaccurate moral intuitions while their are both in principle (moral facts would seem to lack causal powers) and pragmatic (continued failure to show otherwise) reasons to think there isn’t any evolutionary pressure for our moral intuitions to match up with true moral facts.

I think this actually establishes an extremely strong negative result. In the absence of a plausible naturalized epistemology of morality (or philosophy of math, or knowledge of possible worlds) it’s irrational to use our intuitions as evidence. Without any justification of why our actual intuitions are more likely to be valid than any of those intuitions we could have had it’s an outright error to treat them as stronger evidence for their claims than the fact that we could have had some other intuition. However, even if you aren’t willing to take it this far it raises some very interesting questions. One that seems particularly challenging for the meta-ethicists is the following:

Suppose theoretical analysis (or survey of galactic civilizations) reveals that our moral intuition about the importance of life is actually an improbable fluke and evolution tends to equip any sentient being with the intuition that it’s the future of someone’s genetic line (or their happiness) that is morally salient not whether they live or die. Does that give us reason to believe that death isn’t morally salient? If not how can it be rational to believe something about moral facts on the basis of an accident without any connection to these facts?

While my intuition is that the idea behind this argument is correct I think the argument Professor Bostrom gives isn’t quite right. In particular the focus on what human civilizations are likely to do and ancestor simulation seems all wrong. There is no reason why totally alien beings could not simulate people nor to believe that our simulated universe resembles the real one in which the simulation is running much less that we are an earlier stage of the simulators history. Later I might think about how to fix this point but what made me want to write this post was the comment in the NYT that you could get out of the argument by either denying strong AI (a simulation wouldn’t be conscious) or by assigning a low probability to the chances that human beings will progress far enough to run such simulations.

This reminded me of the post I wrote a year ago about Sleeping Beauty in The Matrix arguing that a widely accepted solution to the sleeping beauty problem also implied we should believe the universe creates infinitely many individuals with the same memories and experiences as we have. Of course intuitively I think the conclusion of this argument is total crap but it’s tough to figure out why it’s wrong⁴. In short I think there is something very subtle going on in these sort of arguments that I don’t yet understand. If I ever figure it out I will post but until then I’m remaining skeptical.

The point I want to illustrate is that whether or not this is the actual world or a matrix style simulation is an instance of the sleeping beauty problem. In particular the same reasoning that supports the two-thirds solution to the sleeping beauty problems seems to guarantee we are in a repetitive simulation. I don’t know if this puzzle has appeared in the literature yet (it seems kinda obvious to me so I have a hard time believing it hasn’t) but it is new to me (no I haven’t read all the papers I link so it could be there). Since I find the reasoning supporting the two-thirds solution compelling I expect some other solution will ultimately be forthcoming but I expect that solution will be more interesting than the problem itself.

More concretely it certainly seems possible that we are actually in a virtual world like the matrix. It remains possible even if we suppose that our lives are in fact arbitrarily long (the ‘real’ world has conquered aging) and every time we die in the virtual world our memories are wiped and the simulation is restarted. While we might think ‘a priori’ such a world is unlikely since it seems possible we should assign it some non-zero probability. However, since this matrix world repeats the experience we are having infinitely many times it would seem that the same reasoning which allows us to support the two-thirds solution in the sleeping beauty problem requires that we find the matrix world arbitrarily more probable than our own world, i.e., the probability of the matrix world conditionalized on matrix world or real world is one (or arbitrarily close to one). Of course the virtual world aspect isn’t central to the problem. We could repeat the same argument in favor of Nietchzie’s world of eternal recurrence or any situation which involves making us experience our lives so far arbitrarily many times.

The Sleeping Beauty Problem

In the sleeping beauty paradox we drug the hapless heroine into a deep slumber on sunday. At this point we flip a coin. If the coin lands heads we wake her up on monday tell only giving her time to realize she is awake but not informing her of the day or any other information before we render her unconscious and wipe her memory of the event. In this case we do not wake her again till wednesday. If instead the coin lands tails we do the same thing on monday, wake her wipe her memory and put her back to sleep, but instead of letting her sleep will wednesday we do the same again on tuesday. The puzzle is when woken is it rational for sleeping beauty to believe the probability of tails is still 1/2 or should she now believe it is 2/3.

If you are interested in the reasons for one side or the other you can read the philosophical arguments about the sleeping beauty problem. I may write up my thoughts on the problem in more detail later but rather than get into the details here I will just assume Elga’s argument for the two-thirds answer is correct.

In effect Elga’s solution seems to mandate the following means of calculating the credence we should assign to the coin being tails upon waking up. Consider all the ways you could have the experience you just had (being woken up by the experimenter) where the coin lands tails and weight each of those by its probability of occurring. Divide this number by the sum of all the ways you could have this experience weighted by probability and divide. Since their are two ways you could have the experience of waking up when the coin is tails (it is monday or it is tuesday) and each of these has probability 1/2 (if the coin lands tails you are guaranteed to have both) the numerator is one. The only other way you could have the experience of waking up is if the coin landed heads and it is monday the denominator is 3/2. Hence giving the ‘right’ answer of 2/3.

This solution has great intrinsic appeal. If we want to calculate the probability of T given E we usually sum up all the ways T and E could happen weighted by their unconditional probability and divide by the sum of all the ways E could happen weighted by unconditional probability. Though in this case we may not technically be calculating the probability of tails given that you had the experience of waking up since this is the normal means of updating on new information it seems appealing to do the same in the sleeping beauty case.

Sleeping Beauty and Repeating Simulations

However, let us consider what happens if we apply this reasoning to the matrix example. In this case instead of the coin flip we have the choice of world (real or virtual) and instead of being woken up twice our simulation is repeated n times. So instead of H₁, T₁ and T₂ as Elga had we have the following outcomes.

• R₁: The world we experience is real and we are living our one and only life (after death we just cease to exist or go on to some other type of existence).
• V₁: The world we experience is just a simulation and we are on the 1st repetition of the simulation.
• V₂: The world we experience is just a simulation and we are on the 2nd repetition of the simulation.
• V_n: The world we experience is just a simulation and we are on the nth repetition of the simulation.

• V: The world we experience is just a simulation and we are on some repetition of the simulation, i.e., V=V₁v V₂…v V_n.

Since we only want to compare the probabilities of this world being real and being in this particular simulation we will conditionalize our probabilities on VvR, i.e., P(X) denotes the probability of X given ‘V or R’ and hence P(V)+P(R)=1. Importantly P(V) and P(R) do not give our current probabilities for V and R but the ‘a priori’ probabilities for these worlds before it is conditionalized on any experience. As we are simply postulating some possible world where we are thrust into simulators we can assume that all n repetitions of the simulation are guaranteed, i.e., there is no chance the simulation will break or we will die before all repetitions are complete.

Applying the same solution Elga does to this situation we calculate the current credence we should hold for being in situation V to be P(V)*n/(P(R)+n*P(V)). Note that if we hold P(V) and P(R) fixed and allow n to go to infinity our rational credence in V goes to 1. Of course one might have a lower probability for being in a situation where the simulation is repeated n+1 times than one where it is repeated n times but we can avoid this problem by considering a world where the simulation is repeated infinitely (omega) many times. Countable additivity guarantees the credence we should have in this case is just the limit of our prior answer as n goes to infinity. Since we seem happy to consider the possibility that our universe will keep expanding forever it seems perfectly possible that such a situation could exist. Heck one could just pick some system of particles and dynamical laws which repeat at regular intervals forever and postulate psycho-physical laws which makes the motion of these particles create the experiences of your life. So long as we assign non-zero ‘a priori’ probability to such a situation the above argument shows we should believe it is infinitely more likely we are in this repeating simulation than the real world, i.e., the credence we should give V given V or R is 1.

Potential Solutions

Obviously this result seems absurd. It seems totally unreasonable that we should be required to believe the world is infinitely more likely to be a simulation or even that we will repeat our current life after we die just because we are having some experience. So what moves can we make to avoid this result? Our first implication might be to discard the most abstract and non-intuitive principle we used, countable additivity. However, this really does no good as just using finite additivity we should be able to prove the credence we should assign to being in V is greater than any number below 1. Alright so what about giving up the assumption that P(V) isn’t 0? Remember P(V) refers to the conditional probability of V given V or R so this would require believing we have grounds to believe that the real non-recurring world was infinitely more likely than this world filled with simulations. Even if you are willing say this about a world running indefinite simulations of this one are you really prepared to (essentially) rule out a world of eternal recurrence?

A more substantive move might be to reject the idea that ‘a priori’ probabilities exist. After all most philosophers are skeptical of objective prior probabilities so don’t we get rid of this paradox for free when we give up objective priors? Even if true it would still be pretty cool to show that reasonable assumptions actually excluded objective priors. However, nothing in this argument required that these probabilities be objective only that we have some notion of likelihood about various ways the world might have turned out. I have been careful to put ‘a priori’ in scare quotes everywhere because this probability function can be the result of any model we adopt and need not be a consequence of logic alone. If the physicists come up with some theory which gives probabilities for various types of initial conditions (and at least some of these allow for a universe which lasts forever) it seems the problem reoccurs. Do we really want to be philosophically committed to the absence of such a physical theory?

Moreover, denying the existence of this sort of ‘a priori’ probability function leaves us at a loss about whether or not to believe we are in some sort of simulation. If I observed certain very bizarre or intuitively absurd events it would be perfectly rational of me to conclude I was in some type of simulation. For instance if a giant black unix terminal opened up in the sky and what appeared to be computer commands appeared inside with corresponding effects out in the world it seems reasonable I should accept that I’m in a simulation. Yet despite never having seen (or at least remembered) the real world I would judge certain ways the real world might be more likely than others, e.g., fundamental physical objects are particles not toasters or starfish. Moreover, if I was trying to predict what the real world was like I would first judge how probable that sort of world seemed and then modify that judgement by how likely that sort of world was to cause simulated experiences like mine. These judgments may not be a priori in the sense of being a priori truths but if does seem like our basic intuitions of probability give us the sort of probability which lead to all this trouble.

This seems to leave us only two possible options: bite the bullet or abandon the ‘two-thirds’ solution to the Sleeping Beauty problem. As both the limiting frequency and a number of compelling arguments favor the ‘2/3′ option I am loathe to give it up. Additionally biting the bullet may not be quite as unpalatable as it seems. It seems to be a valid scientific possibility that the universe will collapse and be recreated in a new big bang. If this happens indefinitely our world effectively is a recurring world and the world of repeated simulation seems to lose any apparent advantage. However, this draws the paradoxes of infinity deeply into our judgments about the world. In particular the sleeping beauty problem asks us to calculate the probability of being in a world where the coin lands heads versus one where it lands tails. Once we assume the world is recurring both possibilities present the same number of experiences of waking, countably many in both cases, nullifying the ‘two-thirds’ solution. Alternatively we might postulate that our world actually consists of countably many (or more) copies (as in many worlds interpretations of quantum mechanics). However, this still leaves the above problem and adds the additional puzzle of what it even means for the same experience to be happening countably many times.

Ultimately I’m pretty much at a loss about this problem. Maybe someone reading this will have some ideas.

Ultimately the question of which theory¹is better — that the grand canyon is the result of billions of years of geological change or that it is the work of god — is all about how we assign our prior probabilities. Clearly if we assign a very high prior probability to the existence of a god who would make the universe look like it had been created billions of years ago that would be the better theory. This seems exactly what the ID advocates are really doing, the attacks on the scientific explanation being just an after the fact justification.

Most philosophers I know seem to be convinced that ID advocates are not just wrong but somehow objectively wrong in a way which an impartial observer could determine. Yet this would seem to require some objectively correct assignment of prior probabilities like Carnap wished to find and most modern philosophers don’t believe such things exist. Now we could try and turn to some kind of demarcation between science and non-science like Popper offered but this would only tell us something about how we use the word science and not give as an objective reason to prefer one explanation to the other.

To be clear this doesn’t really affect the legal issues of teaching ID in schools. Regardless of it’s objective status we have a certain cultural agreement on what counts as religious belief. However, I thought the situation brought up some interesting philosophical issues and illustrated the reasons I have such a hard time dismissing Carnap’s program. In some sense I really want it to be true since it feels like I have objectively better reason to believe in geology than a divine trick but I don’t know if that is the sort of intuition that is evidentiary or not.

I’m skeptical of Weatherson’s analysis of the problem for slightly different reasons than those Kenny emphasizes. I’m skeptical primarily because I don’t think prior true belief is enough to block discovery (a fact Kenny seems to doubt as well). I’m also unsure if justified true belief is even necessary for mathematical discovery or philosophical discovery. I will discuss this question and try and iron out some of the further ambiguities in the ‘discovered blah’ usage. However, the real point of this post is to muse about the relationship between JTB and knowledge in the case of the a priori. Can these two concepts really come apart in situations involving purely deductive reasoning. In particular I will argue that in the Kripke situation under consideration knowledge and JTB simply don’t come apart so while Kenny’s comments may be correct they are mostly irrelevant to this case.

Further Ambiguities of ‘Discovery’

Before we can discuss the relationship between discovery and knowledge any further we need to clarify the ways that ‘discovered blah’ is used. In the comments that Weatherson makes on Kenny’s blog he discriminates between discovering X and discovering that ‘X is true.’ In particular Weatherson suggests Fermat may have discovered Fermat’s Last Theorem (FLT) but surely didn’t discover that ‘FLT is true.’ This is indeed an important distinction but I think we need to clarify several more uses of ‘discovered’ before we can fully analyze this issue.

First though I would quibble with the idea that Fermat didn’t discover ‘FLT is true.’ Fermat didn’t have a proof of Fermat’s last theorem but quite likely he had done a fair number of calculations which gave inductive support to FLT. It would seem that if we are ever going to allow knowledge in the physical sciences when the most we can hope for is inductive support we should allow the same type of inductive inference to establish knowledge of mathematical truths. So I think it is quite plausible that Fermat did discover that ‘FLT is true.’ If we don’t accept this sort of non-proof based knowledge (or equivalently JTB in this case) this knowledge related theory of discovery would be totally inadequate to explain our usage of ‘discovered’ in cases like Ramanujan’s discoveries in number theory. Many of the discoveries made by Ramanujan were not supported by rigorous proofs but rather by intuition and calculation, i.e. by inductive means.

I think this confusion over whether Fermat discovered ‘FLT is true’ is really due to another ambiguity in our usage of ‘discover.’ In mathematics we often use ‘discovered X’ as a shorthand for ‘discovered X is provable.’ This explains our reluctance to call a purely computational observation a discovery in mathematics even though it clearly meets our standards for knowledge in the sciences and other empirical endeavors. It also comports with our willingness to call hand wavy or non-rigorous arguments in mathematics discoveries because they give us (inductive) reason to believe the statement is not only true but actually provable. Of course in some cases (some results of Ramanujan) we may be unsure whether someone really gave us reason to believe something was provable or just true explaining some of the disagreements about who discovered what occasionally seen in mathematics.

This use of ‘discovered blah’ in mathematics to mean ‘discovered blah is provable’ from some understood class of axioms is not the only specialized use of the word ‘discovered.’ In Weatherson’s original post he notes that, “One can discover lots of spiders without discovering that spiders exist.” While he uses this to suggest that there is a difference between discovering an instance of some principle and the principle itself and in particular how one might think Kripke discovered many necessary a posteriori truths without discovering that necessary a posteriori truths existed the grammar of this sentence should suggest something more complex is occurring. At the very least if we are going to take discovery to be some type of relation one can have with a piece of information we need to explain the apparently extensional usage in “discover lots of spiders.”

Just as their is a specialized sense of discovery present in mathematics I suggest that there is a special sense of discovery with respect to natural kinds totally distinct from this conception of discovery applying to pieces of information. Surely when Weatherson speaks of discovering lots of spiders he can’t mean discovering that they were spiders. Otherwise it would be impossible for someone to discover spiders before someone formulated the class arachnids or a precursor notion. It even seems wrong to suppose that someone must have discovered the fact that some creature existed with properties X, Y and Z where X, Y and Z would ultimately prove sufficient conditions to be classified as an arachnid. If the class mammal is first formulated as definitionally requiring members to have hair I still think it would be correct to speak of some alien discovering a mammal by observing a dolphin even if he did not observe the small hairs on its body. I expect some more realistic example can be given along these lines by someone who has more knowledge of taxonomy than myself.

My suspicion that this notion of discovery which grammatically appears to apply to objects not facts is related to some notion of natural kind comes from the following examples. At least to me it seems intuitively correct to speak of a color-blind man being the first person to have discovered green emeralds even if they did not know these objects were emeralds or green. It does not, however, seem right to speak of a color-blind man being the first to have discovered grue emeralds if he did not know the object he had was indeed green or an emerald. It could be that I’m just being mislead by the oddity of sentences about grue but I can’t see how we could make sense of this sense of discovery without some involvement of natural kinds. With the aid of natural kinds we can explain the discovery of spiders as discovery that there existed a creature with properties X,Y and Z and these properties would lead someone with knowledge of the natural kind in question to infer that the creature was a member of that natural kind. Thus hearing a skittering in the dark which happened to be a mouse would not constitute discovery of a mouse, though it would count as discovery of a small nocturnal creature. This seems to agree with my intuitions about usage but let me know if it strikes you differently.

I don’t think this undermines Weatherson’s point about Kripke possibly discovering that many sentences were a posteriori truths (I very much doubt he was the first person to utter those sentences). Though whether or not you find the spider analogy compelling depends on whether you think spiders and a posteriori truths are both natural kinds. However, it does call our attention to another complication of any analysis of discovery. A usage we may need to distinguish even in mathematics as I expect some kinds of Platonists will use discover in this natural kind sense.

Did Kripke Discover THAT Necessary A Posteriori Truths Exist?

While interesting this other use of the word discovered really has little impact on the question of whether Kripke discovered that there are necessary a posteriori truths. Actually I don’t think there are necessary a posteriori truths so Kripke certainly did not discover that they exist but for the moment I will assume his arguments are compelling. In this case my intuitions suggest that Kripke did discover that necessary a posteriori truths exist though someone earlier may have discovered necessary a posteriori truths in the natural kind sense.

I do not find the presence of someone merely having true belief in some fact enough to block later discovery. Imagine, for instance, that Kripke had mad the first serious philosophical mention of necessary a posteriori truths but that we later discovered the writings of some madman from the middle ages which showed he believed in necessary a posteriori truths with no justification whatsoever. Or maybe with some utterly absurd justification like, “the world would be more beautiful with necessary a posteriori truths therefore god must have made things that way.” I think it would be correct to say that Kripke discovered that necessary a posteriori truths exist.

Still I agree with Weatherson’s sentiment that one can’t go discovering philosophical truths by going through published works and fixing faulty justifications. My inclination is to dispute the idea that discovery is a nice pretty notion analyzable just in terms of temporal priority, knowledge, justification and true belief. I suspect intervening attitudes make a difference when trying to decide if some unjustified true belief counts as a discovery, or at least precludes later discovery. My solution would be to say that you can’t discover a truth that is already widely believed, though you could discover a valid justification of that truth. You can, however, discover a truth that is currently not accepted even if someone previously had an unjustified belief in it. In other words a sufficient condition for discovering something is being the first person to know it (or have JTB plus some other condition) and first knowing it at a time when it currently isn’t accepted as true. I’m unsure whether one should be regarded as discovering something when you don’t know that fact but on the basis of your discovery the truth of that fact is continuously accepted until someone does come to know that fact.

Gettier Cases?

Whatever you think the correct thing to say about these examples may be I don’t think one can maintain they constitute anything like a Gettier case. Perhaps the FLT case might be a Gettier situation if one supposes Fermat based his belief in the theorem on some false, but inductively supported, generalization about numbers. However, if Aquinas made a mistake in his supposed justification for a posteriori truths it was a deductive mistake. As Gettier cases seem to rely on an essentially inductive, or at least probabilistic, type of inference it seems likely that here we either have knowledge or don’t have JTB.

The essential aspect of all the Gettier cases I have seen is a probabilistically supported but ultimately wrong step in the inference. So long as every step in a justification is purely deductive the truth of the premises should absolutely guarantee the truth of the conclusion. It would seem, therefore, that so long as the dispute is over the deductive steps in an argument JTB and knowledge don’t come apart. Though of course it is always possible there is some other way knowledge and JTB come apart that I’m not thinking of.

At least in the case of the Aquinas situation it seems Kenny’s suggestion of identifying discovery with JTB that gives others a good reason to believe the statement will not save us from the conclusion that Aquinas did not discover that necessary a posteriori truths exist. If Aquinas did not know that necessary a posteriori truths exist it was not because some probabilistically valid inference in his justification turned out to be false. Rather it would be because he made some deductively invalid step in his reasoning preventing him from having either knowledge or JTB.

Gettier Cases and Mathematics

Just because it seems impossible to have a Gettier type case arise in deductive reasoning should not lead us to conclude they cannot happen in mathematics. I’m unsure if Kenny wants actual historical examples (who cares if it actually happened?) or just plausible candidates for Gettier cases in mathematics but the later are quite easy to create.

There are, for instance, several simple algorithms which appear to generate only primes on casual inspection. There are even some simple formulas which give the n-th prime evaluated at n for all values of n reasonably computable by hand. Someone might easily try such a formula for many values of n gain a high degree of inductive confidence in its validity and use that to prove some face about primes like the Riemmann hypothesis. Supposing they observed enough values of n to establish a high level of inductive confidence in the formula, despite the fact that it is not in fact valid, this would seem to constitute a Gettier case in mathematics.

If we want a historically more plausible situation where what is discovered is not just that something is true, but that it is actually provable we can look to the four color theorem. Suppose the computer the four color theorem was first verified on had a tiny bug which altered a calculation occurring in part of the computer aided proof. If the theorem later turned out to be true but the computer aided theorem false it would seem we would have a valid Gettier case in mathematics. However, I think we are going to need some kind of inductive or probabilistic inference any time we want to create a Gettier case like this one.

Given the type of examples Weatherson deploys I suggest the answer lies in some application of charity to the believer. Even if the justification an agent is disposed to offer for his belief cites some false fact if there is a true fact in the general region which would suffice in the justification we should credit that as knowledge. For instance (stealing from Weatherson’s example) if an agent justifies his belief that X by citing p&q but q is false and all the agent needs is p we should still count that as knowledge. So while I think this point is worth noting I guess I don’t feel it is of great significance because I tend to think with all of these disposition type terms (belief, knowledge etc..) there is some ambiguity which must be addressed by an application of charity.

However, my goal in this post wasn’t to attack or even respond to Weatherson’s point but to offer my very simplistic analysis of knowledge. This is something that has been floating around in my mind for awhile and it wouldn’t surprise me if it is also somewhere out in the literature but I haven’t seen it there. In short, modulo this issue of charity, why not just define knowledge as justified true belief where every necessary step along the justification is also true? So first I should perhaps explain how I view a justification. Basically I assume a justification is a lot like an inductive version of a proof. Thus it will proceed from certain assumptions moving through intermediate conclusions along the way before reaching the statement in question. Of course since the steps are only inductively and not necessarily deductively valid all the premises can be true and each step be valid yet reach false conclusions (intermediate or final). The Gettier cases rest on the possibility that the final conclusion is true but that some of the intermediate conclusions are in fact false.

It seems to me the obvious answer to this worry is just to require each intermediate step along the justification to be true as well as validly inferred from the preceding deductions. For instance suppose Fred goes to visit Susan. Fred observes a silver Prius in Susan’s driveway and concludes that Susan owns a silver Prius. If, however, the silver Prius in the driveway actually belongs to Susan’s friend Josephine but Susan really does own a car of a similar type and color we have ourselves a Gettier case.

Presumably Fred reasons thusly. There is a silver Prius in Susan’s driveway. It is likely that the car parked in Susan’s driveway is Susan’s. Thus the silver Prius in Susan’s driveway is Susan’s. Hence Susan owns a Silver Prius. Notice how the requirement that every intermediate step in the deduction be true nicely prevents this case of justified true belief from being knowledge. Since ‘the silver Prius in Susan’s driveway is Susan’s’ is false we don’t have a case of knowledge. If on the other hand Fred had known that Susan only let people who owned the same kind of car as her park in her driveway and used this fact to conclude Susan owned a silver Prius it would count as knowledge.

Now a plausible objection is that an invalid chain of reasoning can be massaged into a valid chain of reasoning. For instance if we replace the step in Fred’s reasoning where he infers that the car in Susan’s driveway is the car she drives by the inference that people frequently drive the same type of car that is parked in their driveway it would seem we have a case of justified true belief. However, far from a deficit I think this is actually a strength of this analysis. It is a feature of a proper analysis of knowledge that it should allow the same evidence to constitute knowledge seen in one way and not seen in another one. If Fred is disposed to cite the first inference to justify his belief when asked he doesn’t have knowledge. On the other hand if he really grounds his knowledge in the connection between the car in someone’s driveway and what type of car the individual owns, and isn’t just inferring it from the connection between owning a car and parking it in your driveway, then he does have knowledge.

Still one might protest that an inference like this shouldn’t count as knowledge. After except for the relationship between ownership and parking in the driveway this probabilistic relationship might not exist. However, if one takes this line it is difficult to see how any non-certain knowledge might exist. Given any sort of probabilistic inference about some object it is quite likely that there is some additional information one could specify about the object to make the inference invalid. For instance if you purport to know that someone is dead because they haven’t breathed for 5 minutes likely there is some additional fact, say the number of hairs in their beard or their age in seconds, which they share only with survivors of icy water immersions.

One question I have been puzzling over for awhile is whether allowing random phenomena in scientific theories allows us to create a theory explaining any collection of observations. That is if the world is actually too complicated to be described by a theory comprehensible to us can we make up a theory which appears to describe it by postulating randomness in the basic behavior of the universe? The answer, which was a lot simpler than I had been making it, is yes. No matter how complicated the universe turns out to be we can always come up with some theory which makes it look understandable but with some random elements.

This has some serious implications for realism and philosophy of science. In challenges the idea that the success of probabilistic theories like QM is evidence for their truth. It also illustrates the need for philosophy of science to address the question of what conditions can justify the retreat to a probabilistic theory. Unless we are to abandon the realist agenda or stipulate faith in the computable nature of the laws of nature we can’t accept a probabilistic explanation just because no deterministic explanation suffices. Thus putting the realist credentials of Quantum Mechanics in even more trouble.

I think this argument is interesting enough that it should be out in the literature so I’m particularly eager for comments. If anyone has any suggestions about how it could be fleshed out or how/if I should try to make this into a paper let me know. Any references to related literature would also be quite helpful (Kelly too since I haven’t read all his stuff).

Really Short Summary for Logic Student

The short version of the argument is as follows. If this goes over your head read the rest of the post.

We can idealize scientific observations as a real (infinite binary sequence). You can either think of this real as giving the list of all future observations or as coding a function between initial conditions and experimental results. The problem of finding a scientific theory thus becomes finding some algorithm which computes (predicts) the real since it is reasonable to assume testable scientific theories must make predictions computably. However, since there is a Martin-Lof random computing every degree if the universe we wish to describe is not computable we can always find some apparently random sequence relative to which it looks like we have a successfully sscientific theory. Since all we ultimately observe are the results of experiments this situation appears to be one in which some theoretical entity exhibits random behavior (given by the random real) which ultimately have observational consequences (with a mechanism given by the Turing reduction).

The Long Version

In order to apply the machinery of computability theory to the problems of scientific inference we first need to make some idealizations. Admittedly some of these idealizations are a bit crude. This is one of the areas I think needs work. However, I think they are accurate enough to allow us to draw real conclusions from them. If you are unfamiliar with computability theory this entry in the Stanford Encyclopedia of Philosophy gives a fairly complete introduction.

We idealize observations about the world as an infinite (length w) binary string (a real) or equivalently as an infinite string of integers. Call this real O. I think the best way to think of this idealization is as a mapping between the initial conditions of every experiment that will ever be conducted (actual physical experiment) and the result of that experiment. Alternatively we could use a real coding the sequence of all human sense data (i.e. experiences not some Carnapian sense data language) or we could restrict either representation to a particular class of observations, say those which count as physics experiments. The important point is that we can idealize all the observations that can every be made about the universe as a countable set on integers. Since it is reasonable to believe we could capture more and more subtle details of scientific practice by choosing better ways to code things into this real it is reasonable to think conclusions from this idealizations apply to actual science.

We take the test of a scientific theory is the ability to predict results of experiments from initial conditions. Modulo finite amounts of information which can just be embedded into the theory this ultimately amounts to computing the real O we mentioned above. Thus we can take a correct scientific theory to be a function giving us the values of O. Clearly though our scientific theories can’t just be arbitrary functions. It must actually be possible for us to use this function to predict results. Unless you believe Penrose and think humans can intuit things in principle beyond the capability of Turing machines this means the scientific theory must be computable.

If the universe is nice and has computable dynamics (i.e. the future states of the universe are computable from initial conditions) everything is good and we have no problem. O is actually computable and some computable scientific theory calculates it, i.e., makes the right predictions about the universe. What happens if the universe follows non-computable pattern? Certainly in this case we couldn’t come up with a scientific theory which can completely predict every observation, i.e. every bit of O. However, in modern quantum mechanics we don’t predict the outcome of every experiment with certainty. Instead we predict them probabilistically theorizing some inherently random events to explain the results of our observations.

So at the moment we have this picture where given any scientific experiment with particular initial conditions we can look up the experiment in our real O and retrieve the results of that experiment, i.e., O represents how every experiment scientists perform turns out. Thus a successful scientific theory is a computable function calculating O. If we want to allow a theory to postulate some kind of randomness as QM does we need to allow a scientific theory to predict results only relative to the outcome of these random events. The picture now is that given any experiment our scientific theory does not attempt to predict a definite result but instead identifies some collection of random events and tells us how the experiment will turn out for every way the random events could turn out. Doing the same thing we did with our experimental observations we can code the results of all the random events into a new real R. Even though an actual theory is likely to have random events with many different probabilities and possible outcomes these can all be coded into sequences of fair binary random variables¹. Thus we can pretend all random events are coin flips and code them into the infinite binary sequence R.

Now for a probabilistic scientific theory to appear to describe reality we only need a computable function calculating O from R, the outcomes of the supposed random events. Given this understanding when will it appear as though we have a valid scientific ttheory? Whenever there is some computable function calculating O from R where R looks random. So long as R looks random and we compute O correctly the situation appears exactly as if we had a valid scientific theory describing a universe with some inherently random events.

There is a great deal of literature about what it means for a real to look random but certainly a Martin-Lof random should count. A paper introducing the various definitions can be found here but the key point is that a Martin-Lof random real is completely unpredictable by any computable means. Importantly a Martin-Lof random is not even statistically predictable in a computable fashion, i.e., to any computable process a Martin-Lof real is indistinguishable from a series of truly random coin flips. Or more precisely no computable function betting on the bits of a Martin-Lof random can make arbitrarily large amounts of money. Thus anytime we can find a Martin-Lof random real R and computable function calculating O from R we have an apparently perfect probabilistic scientific theory.

Now the key point is this. Every real turns out to be computable in some Martin-Lof random. In other words no matter what the true nature of natural laws they always appear to be either computable or computable consequences of random events. Induction in some sense is always guaranteed to succeed so longs as it gets to consider probabilistic theories. If we live in a non-computable world we can always stipulate theoretical random events and create a theory which explains our observations in terms of these random events. Of course in reality what happens is the opposite, we look at the observational results and infer the values of the random events from these and then check that the supposedly random events occur with the right frequencies and without apparent pattern. The above argument establishes that we can always find some computable theory whose random events have the right frequencies and show no detectable patterns.

Repercussions for Realism

The result seems very damaging for realism, especially when our best scientific theory is probabilistic. Of course one could always try and take the hardliner and claim that if we have to resort to random events in our ideal theory their really are actual random events in the actual world. Yet these seems untenable as their are plenty of non-computable ways for the universe to work which don’t intuitively involve any randomness. For instance the actual natural laws could have some short and simple definition in terms of 0”. Since 0” can be expressed in a very simple and totally definite manner we could even live in a world which worked on very simple but non-computable natural laws. However, our scientific search for natural laws wouldn’t discover these laws but instead tell us that there were in fact random events and predict the results of experiments only probabilistically. Intuitively it seems there are a great many ways the world could be which would cause induction to ‘get it wrong’ but appear as if it got it right.

Troublingly there seem to be deep problems if you go either way. So first suppose that we just require the theory to be as simple as possible. Well then I can give you a good canidate right now.

Each particle has both a particle type given by an integer and a code for a turing machine e. The dynamics just says that each particle (or wavefunction if you prefer) just moves according to lim_r T_e(p1,p2…p_n) s.t. p1…p_n give all the positions and properties of any particles less than r cm away. Unless you want to nitpick and worry about convergence issues you can ignore that recursion theory bit and imagine that each particle has it’s own little computer in it that is running the ‘true’ laws of physics and predicting where it should go.

This theory seems clearly unsatisfactory but it is undeniably more simple than modern Quantum mechanics. Quantum mechanices requires book length tretises to describe what is going on here I just described it in one paragraph using no specially designed technical terms. However, this theory is clearly going to give us the right predictions with some initinal conditions, i.e., those initial conditions where each particle is of the correct type, i.e., the turing index assigned to the particle is of the type that makes it compute its true dynamics. Since the initial conditions for modern physics requires initial position and type of particle information this seems a perfectly acceptable initial condition (and it needs to be very precisce and detailed to predict all the features of the world we observe).

Conversely though if we take the position that both the data and initial conditions should be represented as simply as possible we run into other problems. First of all this just doesn’t seem to be what is done in science. What happens instead is that scientists rate how likely it would be for a ’similar’ universe to arise with a variation of the initial conditions. Also in this case it is unclear if we will ever have any reason to expect a stable theory. That is we very well may be forced to keep ditching theories. That is the most efficent mechanism to compress the initial condition information will never settle down. We would probably keep generating new theories that somehow compress the initial conditions more efficently given the new data. Yet this would seem to be a serious blow to realism. If at any point in time it was very likely you would change your theory in the future makes it difficult to argue that it gives you insight into what actuallly exists. I believe this is a very wide class of potential initial conditions which give rise to this problem realism would have to assert we had some way to know we weren’t in such a world.

So what gives?