Pascal's Mugging Solved

0 common_law 27 May 2014 03:28AM

Since Pascal’s Mugging is well known on LW, I won’t describe it at length. Suffice to say that a mugger tries to blackmail you by threatening enormous harm by a completely mysterious mechanism. If the harm is great enough, a sufficiently large threat eventually dominates doubts about the mechanism.

I have a reasonably simple solution to Pascal’s Mugging. In four steps, here it is:

  1. The greater the harm, the more likely the mugger is trying to pick a greater threat than any competitor picks (we’ll call that maximizing).
  2. As the amount of harm threatened gets larger, the probability that the mugger is maximizing approaches unity.
  3. As the probability that the mugger is engaged in maximizing approaches unity, the likelihood that the mugger’s claim is true approaches zero.
  4. The probability that a contrary claim is true—that contributing to the mugger will cause the feared calamity—exceeds the probability that the mugger’s claim is true when the probability that the mugger is maximizing increases sufficiently.

Pascal’s Mugging induces us to look at the likelihood of the claim in abstraction from the fact that the claim is made. The paradox can be solved by breaking the probability that the mugger’s claim is true into two parts: the probability of the claim itself (its simplicity) and the probability that the mugger is truthful. Even if the probability of magical harm doesn’t decrease when the amount of harm increases, the probability that the mugger is truthful decreases continuously as the amount of harm predicted increases.

Solving the paradox in Pascal’s Mugging depends on recognizing that, if the logic were sound, it would engage muggers in a game where they try to pick the highest practicable number to represent the amount of harm. But this means that the higher the number, the more likely they are to be playing this game (undermining the logic believed sound).

But solving Pascal’s Mugging also depends on recognizing that the evidence that the mugger is maximizing can lower the probability below that of the same harm when no mugger has claimed it. It involves recognizing that, when it is almost certain that the claim is motivated by something unrelated to the claim’s truth, the claim can become less believable than if it hadn’t been expressed. The mugger’s maximizing motivation is evidence against his claim.

If someone presents you with a number representing the amount of threatened harm 3^3^3..., continued as long as a computer can print out when the printer is allowed for run for, say, a decade, you should think this result less probable than if someone had never presented you with the tome. While people are more likely to be telling the truth than to be lying, if you are sufficiently sure they are lying, their testimony counts against their claim.

The proof is the same as the proof of the (also counter-intuitive) proposition that failure to find (some definite amount of) evidence for a theory constitutes negative evidence. The mugger has elicited your search for evidence, but because of the mugger’s clear interest in falsehood, you find that evidence wanting.

The concept of belief and the nature of abstraction

4 common_law 31 March 2014 08:14PM

[Cross-posted.]

Belief, puzzling to philosophy, is part of psychology’s conceptual framework. The present essay provides a straightforward yet novel theory of the explanatory and predictive value of describing agents as having beliefs. The theory attributes full-fledged beliefs exclusively to agents with linguistic capacities, but it does so as an empirical matter rather than a priori. By treating abstraction as an inherently social practice, the dependence of full-fledged belief on language resolves a philosophical problem regarding its possibility in a world where only concrete particulars exist.

 

The propositional character of belief


It can appear mysterious that the content of epistemic attitudes (belief and opinion) is conveyed by clauses introduced by that: “I believe that the dog is in his house.” If beliefs were causes of behavior, our success in denoting them gives rise to an apparently insurmountable problem: how do propositions—if they exist at all—exist independently of human conduct, so as to be fit for causally explaining it?

While belief ascriptions figure prominently in many behavioral explanations, their propositional form indicates that they pertain to states of information. My belief that my dog is in his house consists of the reliable use of the information that he’s there. Not only will I reply accordingly if asked about his location; in directing other my conduct, I may use that information. If I want the dog to come, I will yell in the direction of his house rather than toward his sofa. Yet, I won’t always use this information: I might absent-mindedly call to my dog on the sofa despite knowing (hence believing) that he is in his house. Believed information can be mistakenly disregarded.

Belief “that p” is a propensity to take p into rational account when p is relevant to the agent’s goals. But taking certain information into account involves also various skills, and it must be facilitated by the appropriate habits. The purposeful availability of believed information is also affected by, besides skills, inhibitions, habits, and desires.

What becomes striking on recognizing beliefs as propensities to use particular information is that behavior can be so successfully explained, when we know something of an agent’s purposes, by reference to the information on which we can predict the agent’s reliance.

Is this successful reliance a unique feature of human cognition? We can use belief ascriptions to describe nonhuman behavior, but we can do the same for machines. The concept of belief, however, isn’t essential to describing nonintelligent machine behavior. When my printer’s light indicates that it is out of paper, I might say it believes it is, particularly if, in fact, the tray is full. But compare it to what is true of me when I run out of paper, where my belief that I have exhausted my supply can explain an indefinitely large set of potential behaviors, from purchasing supplies to postponing work to expressing frustrated rage—in any of an indefinitely large variety of manners. The printer’s “belief” that it is out of paper is expressed in two ways: it refuses to print and a light turns on, and I can refer to these directly, without invoking the concept of belief.

Applying the concept of belief to nonhuman animals is intermediate between applying it to machines or to humans; it can be applied to animals more robustly than to machines. It isn’t preposterous to say that a dog believes his bone is buried at a certain location, particularly if it’s been removed and he still tries to retrieve it from the old location. What can give us pause about saying the dog believes arises from the severely limited conduct that’s influenced by the dog’s information about the bone’s location, as is apparent when the dog fails, except when hungry, to behave territorially toward the bone’s burial place.

Humans differ from canines in our capacity to carry the information constituting a belief’s propositional content to indefinitely many contexts. This makes belief indispensable in forecasting human behavior: without it, we could not exploit the predictive power of knowing what information a human agent is likely to rely on in new contexts.

This cross-contextual consistency in the use of information seems to rest on our having language, which permits (but does not compel!) the insertion of old information into new contexts.

 

The social representation of abstractions


Explaining our cross-contextual capacities is the problem (in the theory of knowledge) of how we manage to mentally represent abstractions. In Kripke’s version of Wittgenstein’s private-language argument, the problem is expressed in the dependence of concepts on extensions that are not rule governed. The social consensus engendered by how others apply words provides a standard against which to measure one’s own word usage.

Abstraction relies, ultimately, on the “wisdom of crowds” in achieving the most instrumentally effective segmentations. The source of abstraction—a form of social coordination—lies in our capacity to intuit (but only approximately) how others apply words.

The capacity to grasp the meanings of others’ words underlies the fruitfulness of using believed propositions to forecast human behavior. With language we can represent the information that another human agent is also able to represent and can transfer to all manner of contexts. But this linguistic requirement for full-fledged belief does not mean that people’s beliefs are always the beliefs they claim (or believe) they have. Language allows us our propositional knowledge about abstract informational states, but that doesn’t imply that we have infallible access to those states—obviously not pertaining to others but not even about ourselves. Nor does it follow that nonlinguistic animals can have full-fledged beliefs limited only by concreteness. Nonlinguistic animals lack full-fledged beliefs about even concrete matters because linguistic representation is the only available means for representing information in a way allowing its introduction to indefinitely varied contexts.

This account relies on a weakened private-language argument to explain abstraction as social consensus. But I reject Wittgenstein’s argument that private language is impossible: we do have propositional states accessible only privately. Wittgenstein’s argument proves too much, as it would impugn also the possibility of linguistic meaning, for which there is no fact of the matter as to how society must extend the meaning to new information. The answer to the strong private-language argument is the propositional structure of perception itself. (See T. Burge, Origins of Objectivity (2010).) What language provides is a consensual standard against which one’s (ultimately idiosyncratic) personal standard can be compared and modified. (Notice that this invokes a dialectic between what I’ve termed “opinion” and “belief.”)

This account of the role if language in abstraction justifies the early 20th-century Russian psychologist Vygotsky’s view that abstract thought is fundamentally linguistic.

Buridan's ass and the psychological origins of objective probability

1 common_law 30 March 2013 09:43AM

[Crossposted]

The medieval philosopher Buridan reportedly constructed a thought experiment to support his view that human behavior was determined rather than “free”—hence rational agents couldn’t choose between two equally good alternatives. In the Buridan’s Ass Paradox, an ass finds itself between two equal equidistant bales of hay, noticed simultaneously; the bales’ distance and size are the only variables influencing the ass’s behavior. Under these idealized conditions, the ass must starve, its predicament indistinguishable from a physical object suspended between opposite forces, such as a planet that neither falls into the sun nor escapes into outer space. (Since the ass served Buridan as metaphor for the human agent, in what follows, I speak of “ass” and “agent” interchangeably.)

Computer scientist Leslie Lamport formalized the paradox as “Buridan’s Principle,” which states that the ass will starve if it is situated in a range of possibilities that include midpoints where two opposing forces are equal and it must choose in a sufficiently short time span. We assume, based on a principle of physical continuity, that the larger the bale of hay compared to the other, the faster will the ass be able to decide. Since this is true on the left and on the right, at the midpoint, where the bales are equal, symmetry requires an infinite decision time  Conclusion: within some range of bale comparisons, the ass will require decision time greater than a given bounded time interval. (For rigorous treatment, see Buridan’s Principle (1984).)

Buridan’s Principle is counterintuitive, as Lamport discovered when he first tried to publish. Among the objections to Buridan’s Principle summarized by Lamport, the main objection provides an insight about the source of the mind-projection fallacy, which treats probability as a feature of the world. The most common objection is that when the agent can’t decide it may use a default metarule. Lamport points out this substitutes another decision subject to the same limits: the agent must decide that it can’t decide. My point differs from that of Lamport, who proves that binary decisions in the face of continuous inputs are unavoidable and that with minimal assumptions they preclude deciding in bounded time; whereas I draw a stronger conclusion: no decision is substitutable when you adhere strictly to the problem’s conditions specifying that the agent be equally balanced between the options. Any inclination to substitute a different decision is a bias toward making the decision that the substitute decision entails. In the simplest variant, the ass may use the rule: turn left when you can’t decide, potentially entrapping it in the limbo between deciding whether it can’t decide. If the ass has a metarule resolving conflicting to favor the left, it has an extraneous bias.

Lamport’s analysis discerns a kind of physical law; mine elucidates the origins of the mind-projection fallacy. What’s psychologically telling is that the most common metarule is to decide at random. But if by random we mean only apparently random, the strategy still doesn’t free the ass from its straightjacket. If it flips a coin, an agent is, in fact, biased toward whatever the coin will dictate, bias, here, means an inclination to use means causally connected with a certain outcome, but the coin flip’s apparent randomness is due to our ignorance of microconditions; truly random responding would allow the agent to circumvent the paradox’s conditions. The theory that the agent might use a random strategy expresses the intuition that the agent could turn either way. It seems a route to where the opposites of functioning according to physical law and acting “freely” in perceived self-interest are reconciled.

This false reconciliation comes through confusing two kinds of symmetry: the epistemic symmetry of “chance” events and the dynamic symmetry in the Buridan’s ass paradox. If you flip a coin, the symmetry of the coin (along with your lack of control over the flip) is what makes your reasons for preferring heads and tails equivalent, justifying assigning each the same probability. We encounter another symmetry with Buridan’s ass, where we also have the same reason to think the ass will turn in either direction. Since the intuition of “free will” precludes impossible decisions, we construe our epistemic uncertainty as describing a decision that’s possible but inherently uncertain.

When we conceive of the ass as a purely physical process  subject to two opposite forces (which, of course, it is), and then it’s obvious that the ass can be “stuck.” What miscues intuition is that the ass need not be confined to one decision rule. But if by hypothesis it is confined to one rule, the rule may preclude decision. This hypothetical is made relevant by the necessity of there being some ultimate decision rule.

The intuitive physics of an agent that can’t get stuck entails: a) two equal forces act on an object producing an equilibrium; b) without breaking the equilibrium, an additional natural law is added specifying that the ass will turn. Rather than conclude this is impossible, intuition “resolves” the contradiction through conceiving that the ass will go in each direction half the time: the probability of either course is deemed .5. Confusion of kinds of symmetry, fueled by the intuition of free will, makes Buridan’s Principle counter-intuitive and objective probabilities intuitive.

How do we know that reality can’t be like this intuitive physics? We know because realizing a and b would mean that the physical forces involved don’t vary continuously. It would make an exception, a kind of singularity, of the midpoint.  

 

Infinitesimals: Another argument against actual infinite sets

-21 common_law 26 January 2013 03:04AM

[Crossposted]

Argument

My argument from the incoherence of actually existing infinitesimals has the following structure:

1. Infinitesimal quantities can’t exist;

2. If actual infinities can exist, actual infinitesimals must exist;

3. Therefore, actual infinities can’t exist.

Although Cantor, who invented the mathematics of transfinite numbers, rejected infinitesimals, mathematicians have continued to develop analyses based on them, as mathematically legitimate as are transfinite numbers, but few philosophers try to justify actual infinitesimals, which have some of the characteristics of zero and some characteristics of real numbers. When you add an infinitesimal to a real number, it’s like adding zero. But when you multiply an infinitesimal by infinity, you sometimes get a finite quantity: the points on a line are of infinitesimal dimension, in that they occupy no space (as if they were zero duration), yet compose lines finite in extent.

Few advocate actual infinitesimals because an actually existing infinitesimal is indistinguishable from zero. For however small a quantity you choose, it’s obvious that you can make it yet smaller. The role of zero as a boundary accounts for why it’s obvious you can always reduce a quantity. If I deny you can, you reply that since you can reduce it to zero and the function is continuous, you necessarily can reduce any given quantity—precluding actual infinities. When I raise the same argument about an infinite set, you can’t reply that you can always make the set bigger; if I say add an element, you reply that the sets are still the same size (cardinality). The boundary imposed by zero is counterpoint for infinitesimals to the openness of infinity, but the ability to demonstrate actual-infinitesimals’ incoherence suggests that infinity is similarly infirm.

Can more be said to establish that the conclusion about actual infinitesimal quantities also applies to actual infinite quantities? Consider again the points on a 3-inch line segment. If there are infinitely many, then each must be infinitesimal. Since there are no actual infinitesimals, there are no actual infinities of points.

But this conclusion depends on the actual infinity being embedded in a finite quantity—although, as will be seen, rejecting bounded infinities alone travels metaphysical mileage. For boundless infinities, consider the number of quarks in a supposed universe of infinitely many. Form the ratio between the number of quarks in our galaxy and the infinite number of quarks in the universe. The ratio isn’t zero because infinitely many galaxies would still form a null proportion to the universal total; it’s not any real number because many of them would then add up to more than the total universe. This ratio must be infinitesimal. Since infinitesimals don’t exist, neither do unbounded infinities (hence, infinite quantities in general, their being either bounded or unbounded).

 

Infinitesimals and Zeno’s paradox

Rejecting actually existing infinities is what really resolves Zeno’s paradox, and it resolves it by way of finding that infinitesimals don’t exist. Zeno’s paradox, perhaps the most intriguing logical puzzle in philosophy, purports to show that motion is impossible. In the version I’ll use, the paradox analyzes my walk from the middle of the room to the wall as decomposable into an infinite series of walks, each reducing the remaining distance by one-half. The paradox posits that completing an infinite series is self-contradictory: infinite means uncompletable. I can never reach the wall, but the same logic applies to any distance; hence, motion is proven impossible.

The standard view holds that the invention of the integral calculus completely resolved the paradox by refuting the premise that an infinite series can’t be completed. Mathematically, the infinite series of times actually does sum to a finite value, which equals the time required to walk the distance; Zeno’s deficiency is pronounced to be that the mathematics of infinite series was yet to be invented. But the answer only shows that (apparent) motion is mathematically tractable; it doesn’t show how it can occur. Mathematical tractability is at the expense of logical rigor because it is achieved by ignoring the distinction between exclusive and inclusive limits. When I stroll to the wall, the wall represents an inclusive limit—I actually reach the wall. When I integrate the series created by adding half the remaining distance, I only approach the limit equated with the wall. Calculus can be developed in terms of infinitesimals, and in those terms, the series comes infinitesimally close to the limit, and in this context, we treat the infinitesimal as if it were zero. As we’ve seen, actual infinity and infinitesimals are inseparable, certainly where, as here, the actual infinity is bounded. The calculus solves the paradox only if actual infinitesimals exist—but they don’t.

Zeno’s misdirection can now be reconceived as—while correctly denying the existence of actual infinities—falsely affirming the existence of its counterpart, the infinitesimal. The paradox assumes that while I’m uninterruptedly walking to the wall, I occupy a series of infinitesimally small points in space and time, such that I am at a point at a specific time the same way as if I were had stopped.

Although the objection to analyzing motion in Zeno’s manner was apparently raised as early as Aristotle, the calculus seems to have obscured the metaphysical project more than illuminating it. Logician Graham Priest (Beyond the Limits of Thought (2003)) argues that Zeno’s paradox shows that actual infinities can exist by the following thought experiment. Priest asks that you imagine that rather than walking continuously to the wall, I stop for two seconds at each halfway point. Priest claims the series would then complete, but his argument shows that he doesn’t understand that the paradox depends on the stopping points being infinitesimal. Despite the early recognition that (what we now call) infinitesimals are at the root of the paradox, philosophers today don’t always grasp the correct metaphysical analysis.

Distinguishing actual and potential infinities

Recognizing that infinitesimals are mathematical fictions solidifies the distinction between actual and potential infinity. The reason that mathematical infinities are not just consistent but are useful is that potential infinities can exist. Zeno’s paradox conceives motion as an actual infinity of sub-trips, but, in reality, all that can be shown is that the sub-trips are potentially infinite. There’s no limit to how many times you can subdivide the path, but traversing it doesn’t automatically subdivide it infinitely, which result would require that there be infinitesimal quantities. This understanding reinforces the point about dubious physical theories that posit an infinity of worlds. It’s been argued that the many-worlds interpretation of quantum mechanics, which invokes an uncountable infinity of worlds, doesn’t require actual infinity any more than does the existence of a line segment, which can be decomposed into uncountably many segments, but this plurality of worlds does not avoid actual infinity. We exist in one of those worlds. Many worlds, unlike infinitesimals and the conceptual line segments employing them, must be conceived as actually existing