Friday, March 27, 2009

AI Moral Issues

I recently had a discussion with Eliezer Yudkowski. I contacted him mainly because I thought from what I knew of his thinking that he would be interested in foundational logical issues, yet I saw no technical details of this sort in his many online writings (except for a cartoon guide to Lob's Theorem). I quickly learned the reason for this. Eliezer believes that there is great risk associated with AI, as I already knew. What I had not guessed (perhaps I should have) was that he therefore considers technical details too dangerous to release.

My natural disposition is to favor open research. So, I am very reluctant to accept this position. Of course, that is not in itself an argument! If I want to reject the conclusion, I need an actual reason-- but furthermore I should search just as hard for arguments in the other direction, lest I bias my judgement.

(That second part is hard... although, it is made easier by the large body of literature Eliezer himself has produced.)

The key question is the actual probability of Very Bad Things happening. If the chance of disaster thanks to AI research is above, say, .1%, we should be somewhat concerned. If the chances are above 1%, we should be really concerned. If the chances are above 10%, we should put a hold on technological development altogether until we can find a better solution. (AI panic puts the current probability at 26.3%!)

There are two issues here: the probability that AIs will become sufficiently powerful to pose a serious threat, and the probability that they would then proceed to do bad things. These topics are large enough individually to merit a long discussion, but I'll try to jot down my thoughts.

AI Power

From what I've read (not from my recent discussion with him), Eliezer's main argument for the first point is that AIs will, because they are software, have a far greater ability to self-improve than humans have. Thus, if they start out with approximately human-level intelligence, they could end up far smarter than us just by thinking for a while. "Thinking for a while" could mean a few days, or a few years-- the risk is still there, so long as the AIs would eventually reach superhuman intelligence.

Personally, I think the human mind has a fairly high ability to self-improve. Sure, we can't modify our neural substrate directly, but it can equally be said that an AI can't modify its silicon. Our software could in principle be equally maleable; and the possibility, combined with the assumption that self-improvement is highly effective, would suggest that evolution would have hit on such a solution.

However. Eliezer likes to say that evolution is not very smart. This is certainly true. So, I don't really know how to assign a probability to evolution having hit upon a comparatively flexible form of recursive self-improvement. So, I need to take into account more evidence.

  • I can't "examine my own source code". An AI could. (If humans could do this, AI would be an easy problem!)
  • I can learn new ways of thinking. I can even consciously examine them, and decide to accept/reject them.
  • I can't wish habits away.
  • An AI could run detailed simulations of parts of itself to test improved algorithms.
  • I can run mental simulations of myself to some extent (and do so).
Tentative conclusion: An AI could be a better self-improver than I, but it would be a difference in degree, not a difference in kind.

This conclusion is neither here nor there in terms of the question at hand.

Other arguments to the effect that AIs could gain enough power are ready at hand, however. The main one is that if a particular amount of computing power can create human-level intelligence, it seems pretty obvious that larger amounts of computing power will create larger amounts of intelligence. It also seems like people would willingly cross this threshold, even competing to create the largest AI brain. Even if self-improvement turns out to be a total flop, this would create superhuman intelligences. If one such intelligence decided to grab more hardware (for example using a virus to grab computing power from the internet) the amount of computing power avaliable to it would probably become rather large rather fast.

(All of this assumes human-level AI is possible in the first place: a notable assumption, but one I will not examine for the moment.)

The argument from superhuman intelligence to superhuman power is fairly straightforward, though perhaps not 100% certain. The AI could hack into things, accumulate large amounts of money through careful investment, buy a private army of robots... more probably it would come up with a much cleverer plan. How smart, exactly, does it need to get in order for this to be a danger? Estimates range from the level of a smart human (because a smart human can already be dangerous) to the intelligence of all humans combined (because that is what the AI would be up against). For the really bad scenarios to occur, it would seem, the AI needs to be capable of major scientific innovation; innovation on the order of at least a team of scientists. (I must admit that, here and now, a single scientist could potentially release a deadly disease from a lab-- this involves no innovation, since the diseases are already there. But, this is because a select few scientists have special access to these diseases. An AI might get such access, but that doesn't seem especially probable until a time when AIs are all around and being given all sorts of other responsibilities... at which point, if the AIs are actually unfriendly, the disease is only one of many worries.)

One question remains: is there enough computing power around currently to cause concern? This is something that did come up in the conversation. If current machines are not risky, then it could be possible today to hit upon the right AI design, yet not achaive human-level intelligence using it. Personally, I think this would be ideal. Such a scenario, with AIs gradually increasing in intelligence as the hardware increased in capability, would give humans time to experiment with AI technology, and also consider its consequences. (Indeed, some argue that this is already the situation: that the fundamental algorithms are already known, and the hardware just needs to catch up. I don't agree, although I can't deny that a large amount of progress has already been made.)

Eliezer argued that human-level intelligence on modern-day machines was plausible, because evolution is not a good engineer, so human-level intelligence may require far less hardware than the human brain provides. Estimates based on the brain's computing power vary quite widely, because it is not at all clear what in the brain constitutes useful computation and what does not. Low estimates, so far as I am aware, put the brain's computing power near to today's largest supercomputer. High estimates can basically go as far as one likes, claiming that chemistry or even quantum physics needs to be simulated in order to capture what is happening to create intelligence.

Of course, the internet is vastly more powerful than a single machine. But the risk of an AI escaping to the internet does not seem very high until that AI is at least near human level pre-escape. So, what is the probability that current machines could be human-level with the current algorithm?

My faith in evolution's engineering capabilities is somewhat higher than Eliezer's. Specifically, Eliezer is (from what I've read) quite fond of the study of cognitive bias that has become a popular subfield of psycholgy. While I enjoy Eliezer's writings on rationality, which explicates many of these biases, I am reluctant to call them design flaws. Upon reflection, there are better ways of doing things, and explicating these better ways is an important project. But my best guess is that each cognitive bias we have is there for a reason, essentially because it makes for a good pre-reflection guess. So, rather than design flaws, I see the cognitive biases as clever engineering tricks. (I do not know exactly how far from Eliezer's way of thinking this falls.) This is merely a default assumption; if I studied the cognitive bias literature longer, attempted to come up with explanations for each bias, and failed, I might change my mind. But, for now, I am not comfortable with assuming large amounts of mental inefficiency... although I admit I do have to postulate some. On the other hand, I also have to postulate a fairly high amount of inefficiency to human-made AI, because it is a hard problem.

So, again, this leads neither here nor there.

But, the real question is not whether today's computers are human level; the more critical question is quite a bit more complicated. Essentially:

Will there be a critical advance in software that occurs at a time when the existing hardware is enough to create an AI hazard, or will software advances come before hardware advances, such that humanity has sufficient time to get used to the implications and plan ahead?

Again, a difficult question to answer. Yet it is a really important question!

A historical view will of course show a fairly good match-up between amount of processing power and results. This old paper begins with such an account geared towards computer vision. Yet, there are real advances in algorithms happening, and they will continue to happen. A small but striking example of sudden improvement in algorithms is provided by Graphplan, an algorithm which changed the AI planning landscape in 1997. Of course, today, the algorithms are even better. So, hardware clearly isn't everything.

A proper estimate would involve a serous analysis of the pace of advance in computing-- how probable is it that Moore's law will keep its pace, speed up, slow down, et cetera-- and likewise an analysis of progress in AI algorithmics. But, equally important is the other side of the question; "such that humanity has sufficient time to get used to the implications and plan ahead". How much hope is there of this, assuming that the software is available before the hardware?

I've said that I think this is the ideal outcome-- that people have a while to first get used to near-human-level AI, then human-level, then superhuman level. Part of why I think this is that, in this scenario, there would probably be many superhuman AIs rather than just one. I think this would improve the situation greatly, but the reasons are more a topic for the section on whether an AI would in fact do bad things. In terms of AI power, the situation is not as persuasive. It seems perfectly possible that a society that became used to the presence of AIs would give them various sorts of power withought thinking too hard, or perhaps even thinking that AIs in power were safer than humans in power.

The thing is, they might be right-- depending on how well-designed the AIs were. Which brings us to:

AI Ethics

If an AI gained sufficient power, would it destroy humanity?

Of course, that depends on many variables. The real question is:

Given various scenarious for an AI of sufficient power being created, what would that AI do?

The major scenarios under consideration:

  • Many powerful AIs (an many not-so-powerful AIs) are developed as part of an ongoing, incremental process open to essentially all of humankind
  • A single powerful AI is developed suddenly, by a single organization, as a result of a similarly open process
  • A powerful AI is developed by a single organization, but as a result of a closed process designed to minimize the risk of AI methods falling into the wrong hands
Eliezer argues that the second scenario is not as safe as the third. Suppose the added effort to make a friendly powerful AI as opposed to just-any-powerful-AI is 1 year. Then, in an open process, an organization not very concerned with friendlyness will be able to create an AI 1 year before an organization concerned with friendliness.

This, of course, depends on the idea that it is easier to create an unfriendly AI than a friendly one. Eliezer has written at length on this. The key concept is that a mind can have any particular goal; that there is no system of ethics that will be universally accepted by any sufficiently intelligent mind, because we can quite literally program a mind that has the exact opposite set of values for any given set. (Just reverse the sign on the utility function.)

The argument I gave to eliezer for universal ethics is essentially the same as a view once argued by Roko. (Roko has since been convinced that Eliezer is correct.) He calls it the theory of "instrumental values". The idea is that most rational agents will value truth, science, technology, creativity, and several other key items. It is possible to contrive utility functions that will not value these things, but most will. Therefore, a future created by a bad AI will not be devoid of value as Eliezer argues; rather, unless it has a really weird utility function, it will look a lot like the future that a good AI would create.

This is a topic I want to go into a lot of detail on, and if the first part of the post hadn't ended up being so long, I probably would. Instead, I'll blog more about it later...

For now, it is important to observe (as Eliezer did in our conversation) that reguardless of far-off future similarities, a good AI and a bad AI have an immediate, critical difference: a bad AI will very probably consider humans a waste of resources, and do away with us.

Similarly, a badly designed but well-intentioned AI will very probably result in a future devoid of purpose for humans. Maximizing happiness will justify forced drugging. Maximizing a more complicated value based on what people actually consider good may easily result in a locking-in of currently enjoyed hobbies, art forms, et cetera. Maximizing instrumental values might easily lead to the destruction of humans.

The terrible, horrible dillemma (it seems to me) is that once a single AI gains power, be it good or bad, it seems that the single utility function that such an AI is programmed with becomes completely locked in. Any flaws in the utility function, no matter how small they may seem, will be locked in forever as well.

There are various ways of getting around this, to some extent. One approach I've been tossing around in my head for a while is that an AI should be uncertain about its own goals, similar to human uncertainty about what is ethical. This is entirelyadmissable within a Bayesian formalism. What, then, would the AI take as evidence concerning ethics? I visualize it something like this: the AI would have a fair amound of (hand-programmed) knowledge about what sorts of things are probably ethical, and it would search for simple rules that would meet these criteria. A better theory would be one that fit better to the patchwork of preconceptions about ethics. Preconceptions would include things like "what a human considers ethical, is more probably ethical..." "what a human, given large amounts of time to reflect, would consider ethical, is more probably ethical..." as well as simpler statements like "killing an unwilling victim is unethical with high probability", creating pain, and so on. A few different Three-Laws style systems could "fight it out", so to speak.

Eliezer suggests a different sort of solution: an AI should behave in a highly lawful manner, setting definite rules and consequences, rather than merely doing whatever it takes to do "good" as defined by humanity. He's suggesting this as a solution to a somewhat different problem, but it applies about as well here. An AI that booted up, calculated a good set of laws for utopia, set up physical mechanisms to enforce those laws, and then shuts off, will not lock the future into a single utility function. It will of course give it a huge push in a particular direction, but that is quite different. It is purposefully leaving the future open, because an open future is a plus according to (at least) the majority of humans.

The third option that I can think of is one I've already mentioned: have several powerful AIs rather than one. This still carries a large risk. 20 AIs that decide humans are useless are just as bad as 1 AI that decides humans are useless. However, 20 AIs with well-intentioned-but-wrong utility functions are probably much better than 1, so long as they all have different well-intentioned utility functions.

The AIs would probably have incentive to enforce a balance of power. If one AI becomes obviously more powerful than the others, the others have incentive to gang up on it, because that one persuing its utility function to the utmost is probably far worse for the others than whatever the group consensus is. That consensus should be something favoring humans, since the individual goals are all random variations of that theme... if we look at all the goals, and ask what they have in common, favoring humanity should be the answer.

Of course, that result isn't entirely certain. First, the average of many mistaken goals is not necessarily a good goal. Second, the average is not necessarily the sort of compromize that would result. Third, once a compromize has been agreed upon, the AIs might (rather than maintaining their standoff) all rewrite their utility functions to reflect the consensus, and effectively merge. (This would be to avoid any future defectors; the utility of stopping other possible defectors might be higher than the utility of keeping your ability to defect by keeping your utility function, thus making it rational to agree to a group rewrite.) In this case, the lock-in that I'm afraid of would happen anyway (although we'd be locked in to a probably-less-terrible utility function). Fourth, the situation might not result in a standoff in the first place. Even with several AIs to begin with, one could gain an upper hand.

Friendly AI (as Eliezer calls it) is a hard question. But, it is not the question at hand. The question at hand is:

Which is better, an open research process or a closed one?

I've touched on a few of the factors, but I haven't come close to a definite answer. A proper answer requires an examination of the friendliness issue; an analysis of the curve of technology's growth (especially as it relates to computing power); an examination of what sort of theoretical advances could create a powerful AI, and in particular how suddenly they could occur; an idea of AIs future place in society (both sub-human, human-level, and superhuman AI), which requires a socio-economic theory of what we will do with AI; and a theory of AI psychology, mapping the space of possible minds (focusing on which minds are friendly to humans).

I'll try to address each of these issues more closely in the next few posts.

Sunday, March 22, 2009

After having thought about it, I am becoming more convinced that the correct approach to the issues mentioned in these two posts is to accept the line of reasoning that deduces an enumerator's truth from the truth of all its consequences, thus (as mentioned) inviting paradox.

Consider the original stated purpose of my investigation: to examine which statements are probabilistically justifiable. To probabilistically accept one of the enumerators simply because (so far) all of its implications appear to be true would be a violation of the semantics, unless the truth value of an enumerator really was just the conjunction of the enumerated statements. So to satisfy my original purpose, such a thing is needed.

As I said, paradox can be avoided by allowing enumerators to be neither true nor false. In particular, it is natural to suppose a construction similar to Kripke's least-fixed-point theory of truth: an enumerator is "ungrounded" if it implies some chain of enumerators which never bottoms out to actual statements; ungrounded enumerators are neither true nor false.

The problem is that we will now want to refer to the ungrounded-ness of those sentences, since it appears to be a rather important property. For this we need to augment the language. This can be done in multiple ways, but it will ultimately lead to a new reference failure. And filling in that hole will lead to yet another hole to fill. In general I would deal with this by using my theory that almost works. This entails building an infinite hierarchy of truth values which starts:

True, False, Meaningless1, Meaningless2, Meaningless3, MeaninglessInfinity, MeaninglessInfinity+1...

I am generally interested in investigating whether this hierarchy is equal in power to the usual Tarski hierarchy, namely,

True1, True2, True3, ... TrueInfinity, TrueInfinity+1, ...

The difference basically springs out of the use of a loopy truth predicate (a truth predicate that can apply truly to sentences which contain the same truth predicate). Asking for a single truth predicate appears to force the existence of infinitely many meaninglessness predicates. Is a loopy truth predicate automatically more powerful? Not obviously so: the loopy truth predicate will have a maximal amount of mathematical structure that it implies (the least fixed point), which does not even include uncountable entities. The Tarski hierarchy will continue into the uncountables, and further. (I should note that that statement is not universally accepted... but if someone suggests that the Tarski hierarchy should stop at some well-defined level, can't I just say, OK, let's make a truth predicate for that level? Shy not keep going? And we don't actually need an uncountable number of truth predicates: we need an ordinal notation that can refer to uncountable ordinals, and we just make the truth predicate take an ordinal as an agrument.)

So the question is: is there an isomorphism between the mathematical structures captured by the Tarski heirarchy of truth, and my hierarchy of nonsense.

Thursday, March 19, 2009

Maximally Structured Theory

The proof-theoretic concept of a theory's "strength" has little to do with the sorts of considerations in the previous two posts. Proof theory normally worries about how much a system proves, while I've been worried about how much a system can say.

The sort of proof-theoretic strength I've read about (I admit there are at least two, and I don't know much about the other one, known as ordinal strength) is defined in terms of interpretations. One system can be interpreted in another if, roughly, we can "translate" the provable statements in one into provable statements in the other, and the translation preserves the basic logical operators (so and -> and, or -> or...). There is no requirement that the translation of an unprovable statement be likewise unprovable, which means that the logic we're translating into may add more provable statements. This is, of course, the point: of two systems A and B, B is stronger than A if B interprets A but A does not interpret B. This indicates that B proves more statements than A.

Now suppose that system C is also stronger than A. It may be the case that system C is not comparable to system B: neither can interpret the other, so there is no way to say that one is more powerful than another. We can think of this as a disagreement between the two systems; both prove more than A proves, but B might prove some statement where C instead proves the negation of that statement.

Is one system right and the other system wrong? Well, if both systems are supposed to be talking about the same mathematical entity, then yes. (Or they could both be wrong!) But if not, we can instead think of the different systems as describing separate mathematical entities. It is then natural (in my opinion) to assert that the proper foundation of mathematics should interpret both. This would mean that it contained both structures within it.

I am somewhat surprised that this is not the current approach. Instead, people seem to be concerned with whether or not the axiom of choice is actually true or not. I suppose this is my first deviation from the classical 'platonist' view :).

So, suppose we restrict ourselves to finite first-order theories. How much structure do we get?

First, just because we restrict ourselves to finite theories doesn't mean we can't interpret an infinite theory. Peano arithmetic, which I've been talking about for the last two posts, is a perfect example; it has an infinite number of axioms, but they are computably enumerable, so we can create an axiom system that enumerates them by allowing the use of extra function symbols.

Second, we can interpret far more than first-order theories; higher-order theories can be interpreted as first-order theories with restricted quantifiers.

What we can't interpret is a fundamentally infinite theory, such as... well, the truths of arithmetic. No finite theory will interpret the infinite theory containing all those truths. (Notice, now I'm being Platonist, and claiming that there actually is a fact of the matter.)

What will the maximal theory look like, the theory that can interpret any others? Will it be finite?

Let's first construct an infinite theory that interprets all finite theories. We can't interpret inconsistent theories without being inconsistent, so let's throw those out... then, we enumerate all the consistent theories (in order of size, let's say, using some other rule to break ties), but using a new set of function symbols for each theory, so that they don't interfere with eachother. (Each finite theory can only use a finite set of function symbols, so there is no trouble here.) Using different function symbols allows us to simply take the union, forming an infinite theory.

This infinite theory cannot be converted into a finite form, unfortunately, because the enumeration is not computable: we can't simply distinguish between theories that are consistent and theories that are not.

Now... if we take a logic wich lacks the rule of explosion, ie, a paraconsistent logic, we wouldn't need to worry so much about avoiding inconsistent theories. The inconsistent theories would be isolated, and wouldn't "infect" the others. In this case we could computably enumerate all theories (still making sure to take different function names for each), and therefore we could make a finite theory which performed the enumeration. (The finite theory would use only a finite number of function symbols, but it would provide an interpretation for an infinite number of them, via some encoding.)

This provides an interesting argument in favor of paraconsistent logic. But, not necessarily a definitive one. It seems to me that one could claim that the paraconsistent logic is not really interpreting the classical theories, since it is putting them into a logic with a nonclassical semantics. Perhaps one could define what the paraconsistent logic is really doing, and construct a classical theory that does that much. I don't know.


Don't think that it ends here, with one infinite (unspecifiable) theory to rule over all finite theories. We can talk about other such infinite theories, such as the theory containing all the truths of first-order arithmetc, the theory containing all the truths of second-order arithmetic, et cetera, the truths of set theory... (Actually, I don't think just "the truths of set theory" is well specified. "The truths of set theory under the iterative conception of set" is one way of being more specific.) So, we'll want to talk about whether these infinite theories can interpret eachother, and hence we'll have an increasing chain of "maximally structured" systems with respect to particular reference sets... the maximally structured theory with respect to finite theories is countably infinite; so is the set of truths of first-order arithmetic; the max. struct. theory w.r.t countably infinite theories will be uncountably infinite; and so it goes.

Alternatively, the paraconsistant version of this hierarchy appears to stop right away, with a finite theory that interprets all finite theories. This might be taken as a strength or a weakness...

Wednesday, March 18, 2009

More on Last Time's Subject

The system I suggested last time, at the end, can actually be stated in a much simpler (more standard) way.

Rather than adding in a totally new notation for representing enumerations of sentences in a Turing-complete manner, all we really need is to allow the use of arbitrary function symbols when making assertions. This sounds trivial! However (as the considerations from last time show) it makes a big difference in the actual behavior of the system.

Added function symbols are a standard part of the machinery of first-order logic, so in some ways it seems only natural to allow their use in first-order arithmetic. However, they are not normally considered a part of that language; functions in first-order arithmetic must be constructed rather than defined.

Considering functions part of the language obviously proves no new theorems about arithmetic, because we don't add any axioms that include the function symbols. The only thing that changes is what we can say. So what can we say?

Well, it allows us to say a whole lot. With function symbols, we can go as far as defining a truth predicate for arithmetic! This means that we can build the metalanguage for arithmetic. We can also go further, building the metametalanguage (the language containing the truth predicate for the metalanguage), et cetera. However, don't be fooled; there is an important difference between being able to build it and having it as part of the system. Normally when we talk about a language containing the truth predicate we mean that we've already got the rules to make it work in the axioms, not that we can add them if we like.

Adding function symbols to arithmetic follows the advice of the edit to the previous post, namely, we don't include any claim that the added statements are true just because all the sentences of arithmetic that they imply are true. This is essentially because the logical system is treating the function-symbols as actual things, not just as notational conveniences. The logic thinks there is a truth of the matter about how "F" behaves, even before we add axioms to define its behavior. It has no idea what that truth is, but it assumes that the truth exists. So even if a particular function-based statement implies only true statements about arithmetic, and so looks totally un-risky, the logic still will think that it implies a bunch of risky statements about the function symbols involved.

It would be interesting to develop a semantics for the function-symbols that did not result in this conclusion... however, as mentioned in the previous post, in that direction there is a strong risk of paradox. Some sort of non-classical system of truth values will almost certainly be needed.

Sunday, March 15, 2009



More logic stuff.

I've spoken in the past about using probabilistic justifications (or, when simplicity is desired, nonmonotonic justufucations) for mathematical belief. This provides at least some account of how a system can meaningfully refer to uncomputable concepts such as the natural numbers. (Calling the natural numbers "uncomputable" is sure to raise a few eyebrows. What I mean is that there are undecidable propositions concerning the natural numbers.) The natural question is: how much can be probabilistically justified?

To classify one statement as computably decidable, or probabilistically decidable, or neither, is quite accurate; any one statement or its negation could be simply added as an axiom, and thus made decidable (or more realistically, something else might be added as an axiom, rendering the statement decidable). Similarly, a statement might be made probabilistically decidable with the addition of an axiom that caused it to be equivalent to some probabilistically decidable statement. (Most trivially, the axiom "A <-> B" where "A" is a probabilistically jsutifiable statement and "B" was previosly not. Such an axiom might seem contrived, but it is true if A and B are both true!)

Still, it is convenient to talk in this way, and can be made meaningful. One way is by fixing the axioms we're dealing with; so, for example, we might be working just with robinson arithmetic or peano arithmetic. (I would prefer robinson arithmetic, since I would like to leave the epistemic status of mathematical induction open, but peano arithmetic assumes it to be true.) But this is not the only possibility. For now, I prefer something like the following:

  • Basic functions of arithmetic are computable. These vary from axiomatixation to axiomatization. At one extreme, we could provide axioms giving the behavior of all the common functions on natural numbers; at the other extreme, we could have no functions in our language, instead having relations sufficient to specify the structure. Most common is to take only the sucessor function as basic, and define addition, multiplication, et cetera from these. These differences may cause trouble for me (ie, result in non-equivalent definitions), but I'll ignore that for now...
  • Basic predicates and relations are computably decidable. Again, there is some variation about which relations are taken as basic, and which are taken as defined. Usually equality is taken as basic, but I've heard of greater-than being taken instead.
  • Statements whose truth is a function of a finite number of computably decidable statements are computably decidable. This lets us form larger decidable statements with the normal boolean functions, and or and not. It also allows bound quantifiers to be used, such as "For all numbers less than 100..."
  • Statements whose truth is a function of an infinite number of computably decidable statements are probabilistically decidable if any reasonable probabilistic decision process is guaranteed to converge.
A "reasonable probabilistic decision process" is intended to be something like the procedure described in this post. Obviously I need to make that a bit more well-defined, but it should be good enough for now.

It might also be convenient to define "computably justifiable", meaning that if a statement is true, it is computably decidable. (For example, "there exists a place in the decimal expansion of pi at which the number 9 is repeated 24 times" is verifiable if it is true (we can simply calculate the digits of pi until we find the spot), but if it is false, it is not.) "Computably falsifiable" similarly means that a statement can be falsified finitely if it is indeed false. "Probabilistically justifiable" would mean a statement was probabilistically decidable if it was true, and "probabilistically falsifiable" would mean that it was probabilistically decidable if false.

The basics are all set up, so let's get to the main show.

The question is: what statements are probabilistically justifiable?

I think I have mentioned before that the delta-2 statements on the arithmetic hierarchy are definitely probabilistically decidable, since they are limit-computable. For any statements not in delta-2, there is no guarantee that they will converge to the correct value (they might not converge, or worse, they might converge to the wrong value). Still, it is an interesting question whether *some* of these converge, and whether applying probabilistic methods is still better than guessing randomly (if there are more that converge correctly than converge incorrectly). But that is not the question for today...

Previously, I have restricted this question to statements in arithmetic. This seems sensible. However, lately I've been looking at the concept of proof-theoretic strength. Roughly, a logic is considered stronger if it proves the same truths plus more. This brought my attention to the idea of probabilistically justifying statements in higher logics based on their relevance to arithmetic. Could set theory, for example, be justified by its usefulness in describing true statements of arithmetic?

This led to a question: are there any statements that can be made, which purely assert things about arithmetic, but cannot be formulated in the language of first-order arithmetic? Obviously arithmetic can assert any single thing about arithmetic, so the question is whether all combinations of arithmetic statements that can be asserted by some logic can be asserted in arithmetic. ("Combinations of statements" refers to the same thing that "truth functions of statements" referred to in the earlier definition. Now, obviously, arithmetic can specify any finite truth-functional statement; so that challenge comes when we examine infinite truth-functional statements, such as those that are probabilistically justifiable.)

More formally:

Definition: Logic B is "effectively complete" with respect to logic A if and only if for every computably enumerable set S of statements in logic A, there exists a statement in logic B which imples exactly S closed under the deductive consequence relation of B. (We need to close the set, because we can't penalize logic B for not being able to imply x&y without automatically implying y, for example.)

Conjecture: Arithmetic is effectively complete with respect to itself.

Whether this is true or false, there are some interesting consequences. I'll note some of them in both directions, but keep in mind that (since only one direction is true) half of the consequences I mention will be false.

If the conjecture is false, then even if we are suspicious of mathematical entities other than numbers, it seems that we'll have good reason to want a more mathematically expressive language then first-order arithmetic. The idea of "effective completeness" as I defined it is fairly weak; it does not seem at all implausible to say that if we care about a domain, we should want a logic which is effectively complete with respect to it. So, if arithmetic is not, it seems like another is needed. Furthermore, these new statements may have the hope of probabilistic justification.

This does not necessarily justify set theory. It most certainly doesn't justify a Platonist attitude towards sets (ie, believing in sets as actual entities rather than mere symbol-manipulation). Still, it starts the climb towards abstraction, and that's a slippery slope. (ha ha, mixed metaphor.)

If the statement is true, on the other hand, that means first-order arithmetic can state anything that can be effectively stated. (That qualification, "effective", means: arithmetic can state any combination of statements that some computer program could generate in a (possibly nonterminating) sequence.) This would mean that any higher logic, in its capacity to refer to combinations of arithmetic statements, is unnecessary; in some sense it is already hiding within arithmetic, so it doesn't need to be invented seperately. Any higher-logic statement that could be justified probabilistically via its relationship to arithmetic could actually be interpreted as some statement in arithmetic: once we found the sentence's interpretation, we could not distinguish between the two in terms of their meaning.

So which is true?

I spent quite a while pondering it, and went down a few wrong paths trying to find the answer, but I won't waste time describing those. Eventually, I realized that I had an example right in front of me of a sentence that is not ever stated in the language of first-order arithmetic, yet assers an effectively enumerable set of such statements: mathematical induction!

Mathematical induction is always axiomized in peano arithmetic via the aid of an axiom schema. An axiom schema is an axiom with a hole in it, into which is supposed to be placed every possible statement that can be made in the system. Here is the induction schema:

"If X[0] and X[n]->X[n+1] for all n, then X[n] for all n."

In other words, if we can prove that X holds for zero, and that whenever X holds for a number it holds for the next number, then we have proved that X holds for all numbers.

"X" is the hole into which any particular statement that can be made about a number is to be dropped. This technique, making a statement about all statements, isn't actually avaliable to us in first-order arithmetic. Officially, the axiom schema is a stand-in for an infinite number of axioms.

So, you see, if everyone uses an axiom schema, I realized that it's probably because we can't summarize that infinite statement in the language of arithmetic alone. Furthermore, I realized that someone had probably proven that fact by now. So I looked it up, and it's true. My conjecture was false.

This means that higher logics are necessary, in a fairly strong sense. Which logics? Well, adding the truth predicate for arithmetic (ie jumping to the metalanguage for arithmetic) gets us effective completeness for first-order arithmetic statements. If we really were only concerned with arithmetic, we would stop there. But this new language will (I suspect) still not be effectively complete with respect to itself. So we could add a third layer to the cake, using a truth predicate for the second logic to form a meta-meta-language that would be effectively complete for it... probably, we can continue in this manner for the entire Tarski hierarchy of truth predicates. But instead, we could tackle head-on the problem of creating a language that is effectively complete with respect to itself.

Such a think might be impossible, I admit... it might lead to paradox in the same way that a language containing its own truth predicate can.

My idea is that, in addition to some standard stock of standard machinery (conjunction, negation, a domain such as arithmetic to talk about...), we add a turing-complete representation for constructing statement-enumerators. A statement-enumerator would be treated as the combined assertion of all statements it generated. These generators would be capable of generating other generator-sentences (otherwise it would just be another metalanguage).

A generator would be true if all its generated sentences were.

Sure enough, we can construct a generator that generates only its own negation. Assuming classical truth values, paradox ensues. So, we need to allow some sentences to be neither true nor false...

Same old game.


Actually, this scheme might fail only because I'm overstepping what is needed for effective completeness. Specifically, the sentence "A generator would be true if all its generated sentences were" seems convenient, but is not necessary.