Syllogistic after Ivo Thomas

This blog post presents Ivo Thomas’s 1949 system of formal syllogistic as documented by Arthur Prior here, but with new nicer notation, and with a bunch of extra proofs of meta results thrown in for good measure since it doesn’t cost any extra to add those. The main goals of doing this are, first, to give me a license to use traditional syllogistic forms whenever I want, because it’s all formalized here, and second, to spread this nice notation for the propositions which I thought was good, and which is based on the notation used by Wolfgang Lenzen to write about Leibniz’s “Theory of the Syllogism and Universal Calculus” in the book The Rise of Modern Logic by Gabbay and Woods, page 56. Really it is Lenzen’s notation but with the obvious modification to allow negative terms.

The system (i) admits negative terms and (ii) forbids empty terms. It is layered over classical propositional logic (with connectives ¬, ∧, →, ↔), and uses only modus ponens and uniform substitution for term variables at the propositional level. We give the syntax and intended semantics up front, state the axioms and rules precisely, and then derive the standard theorems. Given the possible choices of systems, I figured not allowing empty terms is best because it is closest to Aristotle and to natural language (which is, after all, where all syllogisms come from); if someone says “All As are Bs” and then it turns out there are no As, the main issue with their argument isn’t logical validity, it’s that by usual interpretation standards, they lied to me when stating that premise. Syllogisms are for the Agora, not for mathematicians. It happens that nowadays you have to have it all nice and formal somewhere or else someone gets suspicious, so this is what this post is. The parenthesized notation also adds brevity for when mentioning syllogisms on social media, etc.

Unicode is used for the things that did not involve any superscripts or subscripts, since that makes it easier to copy and paste the results, especially the syllogistic forms, which are the most useful results. \( \LaTeX \) is used for some results which were mostly meta results which only serve for bookkeeping anyway.

How to forget your memories

If you have some memories that you’d rather forget, you can simply do so. This post will say how I did it.

My background: Years ago, (somehow, I still remember this, but obviously can’t be sure when precisely) I used to struggle with randomly recalling very embarrassing memories. Whenever I recalled one, this had a strong effect on me. It distracted me from what I was doing, it ruined my day, I kept mulling them over pointlessly. Then somehow I had the idea of making a focused effort to not think of them anymore.

How to do it: The way to do this is simply to notice whenever you’re about to think of those memories, and instead try to think of something else very fast. Obviously you shouldn’t try to not think of the memories, that’s like trying not to think of a pink elephant, which I bet you just did. No, actually change the subject, fast. Look for a specific thing as fast as you can, and think as deeply as you can about that thing.

Tips of things to think of: Memories can come to you anywhere. It’s great if you can always have some sort of very interesting content related to your personal interests that you can come back to instantly to get away from the memories, but personally I find it impossible to carry any such content everywhere. Besides, even when you’re looking at your favorite book, you might be currently at a not-so-great passage which isn’t good enough to distract you from the memories that you were about to think of. So what I recommend is to try to focus very hard on the first thing that you do see in the environment you’re currently in. (Or if that was what brought up the memories, some other thing. But if you have objects that reliably make you remember bad memories, consider getting rid of those objects.) For me, at least, the first thing I see around me is usually whatever furniture is in the room I’m currently in. If you go outside more than I do, then it might be other things, such as streets and buildings and trees. Try to think very hard about the thing you’re looking at from a new perspective, such as for instance: how it was physically built, the materials that it’s made of, its history, depictions of it in art, the types of activities that it’s used for, its geometrical shape and features, its deep metaphysical nature. Think very intensely about that, and if you stop being able to think of more stuff about that thing, then look around yourself harder for a second thing. If you apply yourself enough to this activity, you will hopefully forget which memory it was that you were about to remember, and be unable to come back to it.

Why this works: This works because it is the opposite principle to spaced repetition. Whenever you recall memories, this reinforces them, so what you need to do if you want to kill some memories is to prevent them being recalled.

Caveats: Memories are associated to other memories. Sometimes, you know recalling a good memory from your past will quickly remind you of a bad one. There’s no way to keep the good ones without the bad ones in that case, you have to kill the entire network. As a result of what I did, I basically cannot recall any of my past from before about 2017, when I turned 18 and got into college. I cannot remember anything about my school friends, or earlier online friends, even if I try.

Nowadays, I feel like I have grown as a person to the extent that I am very emotionally resilient, and I could definitely simply deal with the memories if somehow I got them all back now. But they’re gone, I can’t get them back. In order to be able to tell people about my life more accurately, I have made a deliberate effort to reconstruct my past from whatever physical or digital evidence I had left from back then, which wasn’t that much, since I had wisely gotten rid of most of it. But there are a lot of things I simply don’t know.

I’m glad I did what I did. The memories were holding me back. They were actively preventing me from achieving the personal growth which I did achieve, over the years since I lost them.

It helps that the cut-off point, 2017, is also when I began losing interest in childish pastimes such as video games, and began getting more interested in intellectual matters of various sorts. I began reading a lot of serious nonfiction, which I had more or less never done before. The person I had been before had more or less nothing to contribute to developing the person I became.

Against Newcomb

This blog post defends my position that Newcomb’s problem never happens, and if you think you’re in it, you should think again.

“The paradoxical situation, in its intended interpretation, is not merely of a kind that we are most unlikely to encounter; it is of a kind that simply cannot occur.” —J.L. Mackie, Newcomb’s Paradox and the Direction of Causation

Contents

• Statement of Newcomb
• Ambiguity, and my position
• SilasBarta misconstrues Newcomb
• How Ahmed construes Newcomb as parallel to real life
• Bermúdez shows Ahmed’s construal prevents Newcomb from being a decision problem at all
• Horgan’s construal is best, but rules out real-life relevance
• Horgan’s view is enlightening
• It is irrational to believe oneself to be in Newcomb’s problem as Horgan construes it
• Horgan’s ultimate reason for one-boxing
• What to do if you think you find yourself in Newcomb’s problem

Statement of Newcomb

Newcomb’s problem is this hypothetical situation, now sometimes called “Standard Newcomb” to distinguish it from some different statements of it:

You must choose between taking (and keeping the contents of ) (i) an opaque box now facing you or (ii) that same opaque box and a transparent box next to it containing \$1000. Yesterday, a being with an excellent track record of predicting human behaviour in this situation made a prediction about your choice. If it predicted that you would take only the opaque box (‘one-boxing’), it placed \$1M in the opaque box. If it predicted that you would take both (‘two-boxing’), it put nothing in the opaque box. (Ahmed 2018, p. 1)

Sometimes a version is considered where this “being with an excellent track record of predicting human behaviour” is actually perfect at predicting human behavior; this version is called the “limit case” of Newcomb’s problem. However, in discussions about “the rational strategy to take”, it is generally agreed that it is indifferent how accurate the predictor has been, as long as his accuracy is slightly better-than-chance; according to Bermúdez, “> 0.5005, to be precise”. (Ahmed 2018, p. 37) Leaving the precise probabilities aside, then, you can draw a payoff matrix:

(Ahmed 2018, p. 19)	The Predictor has predicted two-boxing and so the opaque box contains $1,000,000	The Predictor has predicted one-boxing and so the opaque box is empty
Take just the opaque box	$1,000,000	$0
Take both boxes	$1,001,000	$1,000

Ambiguity, and my position

In discussions of Newcomb’s problem, everyone knows what Newcomb’s problem is (it is the hypothetical situation above), but people disagree about what the essence of the problem is, with respect to “the rational strategy to take”. This is shown when they try to explain their reasoning by making an analogy between Newcomb’s problem and a different situation: people who think different things are essential to the problem will make different analogies, keeping what they consider essential and changing what they do not consider essential.

I believe that there are no parallels to Newcomb’s problem in real life, so I believe that the essential parts of the problem are criteria that cannot be realized. I believe that if you believe that you find yourself in a Newcomb-like situation in the proper sense of that phrase, then you have fallen under some illusion, or committed a fallacious inference, or otherwise been irrational, and the rational thing to do is to revise your beliefs until you no longer believe yourself to be in such a problem.

In order to defend my position, I will argue against construals that defend the idea that Newcomb’s problem is relevant to real life.

First, I will claim that SilasBarta’s construal does not capture the essence of the problem. Then, I will explain (following Bermúdez) that Arif Ahmed’s construal, although it offers an attractive framing of the arguments about the different strategies, prevents Newcomb from being a decision problem at all, so it prevents there being any rational strategy to take in it. Finally, I will discuss Terry Horgan’s construal, which gives a better framing of Newcomb as a decision problem, but which, as I will argue, prevents it ever being realized in the real world.

SilasBarta misconstrues Newcomb

LessWrong is a webforum dedicated to fostering wrong and irrational beliefs, chiefly revolving around unrealistic mathematical models (“unrealistic” not in the sense merely of “idealized” but in the sense of “actively reality-denying”) being somehow applicable to practical situations. In a 2011 post to LessWrong, the user SilasBarta pretends that anything in the world is anything like Newcomb’s problem by explaining it as follows:

The common thread across all the Newcomblike problems I will list is this: “You would not be in a position to enjoy a larger benefit unless you would cause [1] a harm to yourself within particular outcome branches (including bad ones).” Keep in mind that a “benefit” can include probabilistic ones (so that you don’t always get the benefit by having this propensity). Also, many of the relationships listed exist because your decisions are correlated with others’.

Emphases original. The little box “[1]” refers to a footnote where SilasBarta clarifies the sense of “cause” involved, which he thinks of in a certain precise way, but this is not relevant here. What is relevant is that absolutely nothing else in this definition of the essence of the problem is thought of in any precise way at all, which is why he includes, for instance, this example as an instance of Newcomb’s problem:

Cheating on tests: You would not be in the position to reap the (larger) gains of being able to communicate your ability unless you would forgo the benefits of an artificially-high score. (Kant/Categorical Imperative name-check)

As many commenters point out, obviously many people successfully cheat on tests and continue to be in the position to take further tests, possibly cheating on them again and again. Many people’s experience of school is that of a wild place where any oversight is minimal and nominal, and the rules are just pieces of paper. The “would” here is extremely idealized, but it is claimed to be essentially the same as the “would” in Newcomb’s problem, where, as SilasBarta says:

Newcomb’s problem: You would not be in the position of Box #2 being filled unless you would forgo the contents of Box #1.

Despite how the vagueness and broadness here beggars belief, SilasBarta goes on to apply such paraphrases to other petty infractions such as shoplifting, as well as other examples that will be brought up later. I do not consider this approach productive and I will not address it further, although I have noted it because I consider it a helpful way to emphasize how many different construals of Newcomb there can be.

How Ahmed construes Newcomb as parallel to real life

In Arif Ahmed’s introduction to the 2018 collection of essays on Newcomb’s problem, the essence of the problem is claimed to be that there is a conflict between these two principles:

Causal Principle: A rational agent does what she thinks will cause her to realize her aims.

Evidential Principle: A rational agent does what constitutes her best evidence that she will realize her aims.

This does give a compelling framing of the arguments for the two different strategies: someone following the evidential principle will believe that taking only the opaque box (one-boxing) is the action that gives you the best evidence that you will get the \$1M. Since the predictor’s track record is excellent, this is believed to be evidence that, if you one box, this makes it likely that the predictor predicted one-boxing, and so the \$1M is almost always in the opaque box for one-boxers. On the other hand, someone who follows the causal principle reasons differently: once the boxes have been filled, the contents of the opaque box are fixed and independent of your choice now. Whether or not you choose both boxes cannot causally affect the predictor’s earlier action. So, since the transparent box always contains \$1000, a person who follows the causal principle argues you should always take both boxes (two-boxing), because that guarantees you an extra \$1000, whatever is in the opaque box (the strategy is dominant).

Ahmed assumes that this is the essence of the problem, and basing himself on this, claims that many other situations are analogous, such as Fisher’s smoking case:

Suppose that what explains the correlation between smoking and lung disease is not (as everyone now thinks) that smoking causes lung disease, but rather that both have a common cause: an innate predisposition towards lung diseases that also, and separately, predisposes its bearers to smoke. Suppose you are wondering whether to smoke, but you don’t know whether you have the predisposition. You know that you would like smoking, but you like good health very much more. (Ahmed 2018, p. 4)

With the idea being that smoking would be evidence that you have the predisposition to smoking. Following this same idea about the causal principle and evidential principle, Ahmed cites some other analogies in the literature, such as: voting in an election if you think your vote won’t by itself causally affect the outcome, but counts as evidence of how other voters will behave; being vicious or virtuous in the context of Calvinist predestination; macroeconomic policy choices when the public has rational expectations about how the central bank will behave; bets about experiments involving non-causal quantum correlations; and Prisoners’ Dilemma (from game theory) in a context where each prisoner is confident enough that both reason alike (this idea is due to David Lewis). (Ahmed 2018, p. 5–6)

Bermúdez shows Ahmed’s construal prevents Newcomb from being a decision problem at all

The essay by José Luis Bermúdez, which is the first essay in the same volume after the introduction, agrees with Ahmed that part of the essence of Newcomb’s problem is a conflict between causal and evidential reasoning. He frames it as a payoff matrix with these parameters:

(Bermúdez 2018, p. 22)	S1	S2
A	a1	a2
B	b1	b2

Where “CDT” stands for “Causal Decision Theory” and “EDT” stands for “Evidential Decision Theory”, Bermúdez believes that the five essential features of Newcomb’s problem, after it is framed as such a payoff matrix, are these:

1. b₁ > a₁ and b₂ > a₂
2. a₁ > b₂
3. Each of S1 and S2 must be causally independent of both A and B
4. S1 and S2 must be probabilistically dependent upon A and B, respectively
5. EDT and CDT must yield conflicting prescriptions, with EDT recommending A and CDT recommending B.

Bermúdez argues in his essay that a strategy called the “Tickle Defense” always works to cut off any parallel between a real-life situation and Newcomb’s problem. As Ahmed had framed the tickle defense, it was like this:

If you are predisposed to smoke, then presumably you already like the idea of smoking (you have a “tickle” or urge to smoke), and whether you do is something that you already know. But the predisposition only makes you smoke by making you like the idea, and since you already know about that, your actual choice reveals no more about the presence or absence of the predisposition. From the perspective of the agent herself, smoking is therefore not any sort of evidence of a state that it doesn’t cause. The Fisher smoking case is therefore not a Newcomb Problem. (Ahmed 2018, p. 9)

This framing of the tickle defense allows Ahmed to (among other objections to it) say, citing Lewis [pdf], that “we might question the quasi-Cartesian assumption that you know your own motivational state”, since “subconscious desires and beliefs can play the same role in motivation as familiar conscious ones” and this is something that “we cannot simply assume away”. But Bermúdez clarifies that “knowing that you like the idea of smoking” is no more knowledge than is contained in the payoff matrix for the problem. And as Bermúdez says in a footnote, you must know this payoff matrix to be in a Newcomb problem at all:

To be in a (putative) Newcomb Problem is not just to be in a situation where a third-person observer might observe that CDT and EDT give conflicting recommendations. Newcomb’s Problem is supposed to be a first-person dilemma – a situation where the conflict between CDT and EDT is manifest to the decision-maker. For that to be the case, however, the decision-maker must herself be able to reason her way to each of the conflicting recommendations, which in turn requires that she know her probability and utility assignments and know that she is a maximizer of expected utility. So, the assumptions in the text are really necessary idealizations. (Bermúdez 2018, p. 29)

So, Bermúdez spends the first part of his essay saying that the tickle defense applies not only to medical examples like the smoking case, but also to the economic example regarding public expectations about money supply. In the final part, he says that, although it would certainly support the idea about Newcomb’s problem (NP) existing in real life if the Prisoner’s Dilemma (PD) were parallel to it (“given how many different types of social interaction can profitably be modeled as PDs”), it isn’t. As he argues, it is essential to the prisoner’s dilemma that the other prisoner’s choice is independent of yours. When Lewis adds the assumption that “each prisoner is confident enough that both reason alike”, this makes the problem no longer be even a game as defined by game theory—it is transformed “from a problem of strategic choice into a problem of parametric choice” (Bermúdez 2018, p. 39), due to the probabilistic dependence between your choice and the other person’s choice. So NP and the PD are fundamentally different.

Bermúdez concludes that there is no real-life application of Newcomb’s problem in real life, since all the claimed real-life parallels were false. He has more to say about the supposed parallel to voting in a footnote:

Arif Ahmed (2014a: 117–19) has suggested that voting in large elections can count both as a multiagent PD and as an NP, provided that the following three conditions hold: (a) nonvoting dominates voting, because of the inconvenience of voting; (b) any voter in a large election should be to all intents and purposes certain that their vote will not make a difference; (c) voters often take their vote to be diagnostic. As Ahmed observes, many people vote, despite (a) and (b). He proposes (c) as a possible explanation. If so, there is potentially an argument to be made that conditions (1) through (5) are satisfied. Giving this proposal the attention it deserves would take us too far afield. I hope to address it in later work. However, here are two comments. The voting scenario does not have a predictor, and the other voters certainly have preferences over the possible outcomes. So, the first and third reasons for not taking NP to be a strategic choice problem do not apply. But the second does. To take your vote to be diagnostic is incompatible with taking other voters to be independent. And for that reason, the voting case cannot be a PD, in my view. But still, you might think that leaves open the possibility that it counts as a real-life NP. I wonder, though, about the payoff table. Attitudes to voting are very complicated, bringing into play all sorts of loyalties, obligations and perhaps what Nozick has called symbolic utility. So, I wonder about the assumption that nonvoting dominates voting. I am also not convinced that generally speaking people do take their votes to be diagnostic. Ahmed cites evidence that students planning to vote Yes in a 1992 Canadian referendum estimated that a higher proportion of the electorate would vote Yes than students planning to vote No. But that does not show that they take their vote to be diagnostic. They could, after all, be planning to vote Yes because they hope to ‘surf the wave,’ as it were. This is another case where we need more detail about the backstory. (Bermúdez 2018, p. 40)

I think this is fair. If you’re claiming that voting is a real-life example, you had better not add a bunch of questionable assumptions about voting. At any rate, given my preferred construal of NP, which will come next, it will turn out that no case of voting is actually an NP, ever.

Horgan’s construal is best, but rules out real-life relevance

In Terence Horgan’s Essays on Paradoxes, the first three essays are about Newcomb’s problem. Horgan’s goal is to defend one-boxing, which he ultimately finds himself unable to vindicate as the only rational approach. He is not particularly concerned with the question whether the Newcomb problem is relevant to any real-life situation, and he does not try to construct an analogy between Newcomb and a different situation. His proposal, however, captures what I believe is the essence of the problem.

In the first two essays, Horgan draws on David Lewis’s discussion of the semantics of counterfactuals. In Counterfactual Dependence and Time’s Arrow, Lewis had argued that counterfactuals are vague, but that there is a standard resolution of this vagueness which applies in most contexts (as in, “if the match hadn’t been struck, it wouldn’t have lit”), but that some special contexts call for a backtracking resolution of the vagueness (as in, “if the match hadn’t lit, it wouldn’t have been struck”). In the course of defending that the Newcomb context calls for such a backtracking resolution, Horgan develops his notion of “act-independent knowledge”. In the third essay, Horgan simplifies his argument to rely only on the idea of act-independent knowledge, without relying on counterfactuals at all.

As defined in Horgan’s third essay, then, “act-independent knowledge (for short, AIC knowledge), for a given decision problem P, [is] knowledge that is possessed by the chooser in P in a way that does not depend on any evidence that the chooser in P might possess concerning which act the chooser will perform.” (Horgan, p. 46) In Horgan’s formulations of the problem in that essay, we have it as a premise of the limit case that “I have act-independent knowledge that I will act in the manner predicted”, and we have it as a premise of the standard case that “I have act-independent knowledge that it is extremely probable that I will act in the manner predicted.” (Horgan, p. 47) This is the issue, and it allows Horgan to infer, roughly speaking, that scenarios where he does not act as predicted can be rationally disregarded for the purposes of decision-making (although he finds himself unable to claim that they rationally must be so disregarded).

Horgan’s view is enlightening

Horgan’s parallel arguments from the notion of power seem to put the explanatory power of his construal on display. Regarding the limit case, Horgan gives two arguments based on the notion that you should only think about outcomes that are “within your power”, one for each strategy:

• Two-boxing argument from power: Either (1) I have the power to choose both boxes and receive \$1,001,000, and also the power to choose the second box and receive \$1 million, whereas (2) I lack either the power to choose both boxes and receive \$1,000 or the power to choose the second box and receive \$0; or (3) I have the power to choose both boxes and receive \$1,000, and also the power to choose the second box and receive \$0, whereas (4) I lack either the power to choose both boxes and receive \$1,001,000 or the power to choose the second box and receive \$1 million. Hence the outcome I have the power to achieve by choosing both boxes is preferable to the outcome I have the power to achieve by choosing the second box—whatever those outcomes are. And if this is so then I ought to choose both boxes. Hence I ought to choose both boxes. (Horgan, p. 42)
• One-boxing argument from power: Either I will choose both boxes and then obtain \$1,000, or I will choose only the second box and then obtain \$1 million; and this proposition follows from propositions which I know are true and which say nothing about which act I shall perform (or about the probability of either act). Hence I lack the power to falsify the being’s prediction. But I have the power to take both boxes, and also the power to take only the second box. Hence I have the power to choose both boxes and then obtain \$1,000, and also the power to choose the second box and then obtain \$1 million; while I lack either the power to choose both boxes and then obtain \$1,001,000 or the power to choose the second box and then obtain \$0. So the outcome I have the power to achieve by choosing only the second box is preferable to the outcome I have the power to achieve by choosing both boxes. And if this is so then I ought to choose only the second box. Hence 1 ought to choose only the second box. (Horgan, p. 43)

This difference is explained very well by the essence of the problem being that one-boxers regard themselves as rationally taking into account their act-independent knowledge that the predictor’s prediction will be (very likely) right, while two-boxers either regard themselves as not having such knowledge or find such knowledge irrational to take into account against the dominance argument.

It is irrational to believe oneself to be in Newcomb’s problem as Horgan construes it

There is no such thing as act-independent knowledge that “I will act in the manner predicted”. If I know “I will act in the manner predicted”, this is necessarily an inference from these premises:

1. It was predicted that I will act in the manner M.
2. I will act in the manner M.
∴ 3. I will act in the manner predicted.

By the definition of act-independence, the second premise is not act-independent, and hence, there is no act-independent knowledge of the conclusion. Any belief in the conclusion on other grounds is irrational, and in particular, it is irrational to believe the conclusion on act-independent grounds. However, since Newcomb’s problem is constituted by the act-independent knowledge that I will act in the manner predicted, it is then irrational to believe oneself to be in Newcomb’s problem.

In order to be rational, the belief that “the prediction will match the act” must either be grounded on knowledge of both the prediction and the act or, alternatively, grounded on some bridge-law or mechanism linking the determinants of the prediction to the determinants of the act (together with enough information about those determinants to fix how the bridge-law applies in the present case). But in order for this to even be a decision problem, my choice must be open to rational deliberation in light of the payoff matrix. If the bridge-law fixes how the determinants of the prediction are linked to the determinants of the act in a way that screens off deliberation, then the act is not up for rational choice in the sense required by decision theory, and there is, as in the “tickle defense” cases, no decision problem at all. If, on the other hand, the bridge-law leaves deliberation efficacious—i.e., it leaves open which act will be selected after one has considered the very payoff table that motivates one- vs two-boxing—then, once I condition on this deliberative situation, I can no longer have act-independent knowledge that the prediction will match my act. Either way, I cannot have act-independent knowledge of prediction accuracy in a decision problem.

Horgan’s ultimate reason for one-boxing

Horgan concludes that Newcomb’s problem is a “deep antinomy of practical reason”, one in which “distinct normative principles that really are each partly constitutive of pragmatic rationality come into direct conflict with one another”, namely, expected-utility and dominance. (He discusses different versions of dominance, such as “qualitative dominance” and “quantitative dominance”, and he discusses different versions of expected utility, such as Gibbard and Harper’s counterfactual version. This is not relevant here, except insofar as, following a description of Newcomb that goes back to Nozick’s original paper, Horgan believes dominance favors two-boxing but expected utility favors one-boxing. Note that Michael Huemer has argued, against Nozick, that the correct interpretation of expected utility favors two-boxing.)

Horgan remains personally a one-boxer on the emotional grounds that he would feel more regret if he two-boxed. As he says near the end of the third essay:

Speaking for myself, consistent one-boxing wins the psychological tug of war. Here is why. Regret is virtually inevitable in this decision situation: either I will take only the second box and then end up regretting having passed up \$1,000 that I knew all along was there for the taking in addition to the contents (if any) of the second box, or I will take both boxes and then (very probably) end up regretting that I am the kind of person about whom the being has predicted that I will take both boxes. Since I strongly prefer the first kind of regret to the second, I will take only box 2, collect my \$1 million, and then regret that I did not take both. (Horgan, p. 59)

Of course, he cannot, and does not defend, that he is being rational here. The two-boxer may fairly think that he is passing up \$1,000 for no good reason. Near the end of the second essay, Horgan pushes further:

Again I see no way to avoid stalemate. But let me conclude by trying to make the one-boxer’s notion of power more vivid. Imagine being in a Newcomb situation with the following features. (1) You are a hungry prisoner, condemned to die tomorrow. (2) You are completely certain that the being has correctly predicted what you will do. (The limit case.) (3) Box 1 contains a delicious meal, which you may eat immediately if you choose both boxes. (4) If the being predicted that you will choose only box 2, then he put a note into box 2 which will cause the authorities to cancel your execution and set you free. (5) If the being predicted that you will choose both boxes, then he put nothing into box 2. (6) You know all these facts.

If you choose both boxes, you will do so in the full knowledge that you will be executed tomorrow. Likewise, if you choose only the second box, you will do so in the full knowledge that you will be set free. Now surely, in such a situation you would have a strong tendency to view yourself as having the power to choose your own fate— notwithstanding the fact that your choice will not causally influence the contents of box 2. Two-boxers seem to predominate among those who are currently working on the foundations of decision theory. But I think it is not unreasonable to speculate that most of them, if faced with the situation just described, would swallow hard and choose one box. No doubt they would grumble afterwards about having irrationally passed up a chance for a good meal when their happy fate was sealed in advance. But would you really choose two boxes in the certain knowledge that you will subsequently die, just to prove you mean business? (Horgan, p. 45)

I will emphasize how irrational Horgan is being here. If I do not mean business in this sense, then my life is not worth living. Abstract problems should assume a rational chooser, since it is very easy to make the correct choice arbitrarily scary in irrational ways (ways unrelated to your choices’s actual nature and outcomes), and although the scariness of the alternative may reduce your moral culpability for acting irrationally, it will not make your action right.

But Horgan’s solution does follow given the “certain knowledge” he claims exists in the problem, via the principle of explosion. In this context, I would also like to push against a strawman which is often given of the two-boxing choice in Newcomb’s problem. The SEP says that “causal decision theorists respond that Newcomb’s problem is an unusual case that rewards irrationality.” I have never seen any theorist actually argue this (the SEP does not cite any who do), and if any did, they are not presenting the best case for two-boxing. Newcomb’s problem is not an unusual case that rewards irrationality, because it is not a possible case of a decision problem at all, and it is not possible to describe a case where a valid and sound argument concludes that you will get more money by one-boxing. It is wrong to think there are such unusual cases.

If you cannot remove the absurdity from your belief system, however (on which see the next section), then you must believe trivialism is true, so reasoning doesn’t work and you can only do whatever feels emotionally compelling. But this is not decision theory anymore, whatever it is. In any possible case of a decision problem, the analogue of two-boxing is correct, possibly with some preliminary feinting to fool the predictor—which, in real life, you can always do, without exception.

What to do if you think you find yourself in Newcomb’s problem

As I said before, my position is that, if you think you find yourself in Newcomb’s problem, you should understand that you find yourself believing the absurd (“I find myself in Newcomb’s problem” ↔ ⊥), and you should remove that belief from your belief set. Using AGM belief revision, this is done as follows. Let $K$ be your deductively closed belief set, and let $N$ be your belief that “I am in a genuine Newcomb problem”, understood here via Horgan’s AIC. Then:

1. Find $S=\{s_1,\dots,s_m\}\subseteq K$ with $S \vdash N$. These would be anything that leads to your belief in the key premise that “I have act-independent knowledge that I will act in the manner predicted.”
2. Fix the epistemic entrenchment order $\preceq_E$. Highly entrenched would be ordinary causal structure, efficacy of present deliberation, the payoff table, base rates, known incentives, institutional facts. Minimally entrenched would be $N$ and any of its supports that don’t support anything else.
3. Contract by $N$: compute $K' = K \div N$. This drops $N$ and, if needed, the least-entrenched items in $S$.
4. Revise by $\neg N$, so you don’t slide back into the paradox by a different route: set $K'' = K' * \neg N = (K' \div \neg\neg N) + \neg N$.
5. If, after doing this, you still think you’re in a decision problem, then it is a genuine decision problem and you can solve it with $\neg N$ in place, which probably means doing some analogue of trying to fool the predictor (if there’s still time for that) and then two-boxing.

There are other constructions of AGM and other models of belief revision, but this AGM-compatible construction conveys the broad idea I’m recommending: remove your Newcomb-entailing beliefs, whatever they cost, and keep only the others. This allows you to remain rational. Belief in Newcomb’s problem is rationality-destroying, so no sense can be made of the idea of a rational decision within it; the rational decision given belief in it is to destroy the belief. In this case, unlike in “Schelling’s Answer to Armed Robbery” cases, so-called practical (pragmatic) rationality requires theoretical (epistemic) rationality.