Smullyan Fooled at Last

In a short article entitled “Was I Fooled?”, with which he opens his book What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles  (Prentice Hall, 1978), Raymond Smullyan reports on the problem which he claims introduced him to logic. We are led to believe that the problem is deep, that it is, as it turns out toward the end of the book, a version of the Liar Paradox. But is it really deep? And, if so, is it really the logical puzzle that Smullyan thinks it is? I suspect that the answer to the former question is “yes”, and I further suspect that, if that answer is right, the right answer to the second question is “no”.

But, first, here’s the relevant text from Smullyan’s report (which I should have the good sense not to paraphrase):

My introduction to logic was at the age of six. It happened this way: On 1 April 1925, I was sick in bed with grippe, or flu, or something. In the morning my brother Emile (ten years my senior) came into my bedroom and said: ‘Well, Raymond, today is April Fool’s Day, and I will fool you as you have never been fooled before!’ I waited all day long for him to fool me, but he didn’t. Late that night, my mother asked me, ‘Why don’t you go to sleep?’ I replied, ‘I’m waiting for Emile to fool me.’ My mother turned to Emile and said, ‘Emile, will you please fool the child!’ Emile then turned to me, and the following dialogue ensued:

Emile: So, you expected me to fool you, didn’t you?

Raymond: Yes.

Emile: But I didn’t, did I?

Raymond: No.

Emile: But you expected me to, didn’t you?

Raymond: Yes.

Emile: So I fooled you, didn’t I!

Well, I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled. On the one hand, if I wasn’t fooled, then I did not get what I expected, hence I was fooled. (This was Emile’s argument.) But with equal reason it can be said that if I was fooled, then I did get what I expected, so then, in what sense was I fooled [?] So, was I fooled or wasn’t I?

The problem that is explicitly identified in the excerpt should not, of course, be thought deep by 21st-century standards. Taken at face value, Emile’s puzzle for little Raymond is at best as deep as the Barber “paradox”. The Barber is useful as an introduction to the much deeper paradoxes in the Liar family (which, Barber aside, are not all equally deep, by the way). But, unlike what goes on with the deep ones, there is a familiar, fairly simple solution to the Barber: You use the reasoning leading to the contradiction as a reductio of the assumption that there is a barber who shaves all and only the non-self-shaving men in town. (There can be no such barber. You’ve failed to describe a possible barber. Yes, it looks like there could be such a person, but, alas, there cannot. Live with it!) In Smullyan’s April-1st. problem, the solution, likewise, seems simple enough: there can be no such fooling. Smullyan’s mother gets it right when she suspects that Emile is up to some kind of fancy nonsense. Somehow, as she understands, there is something improper in the kind of fooling that Emile is torturing the kid with. The fact that the kid plays along only makes for late-night nuisance, not late-night philosophy. Mother knows best.

Now, if I’m not mistaken, as educational as it may otherwise be, the setup leading to the contradiction is, to epistemological eyes, largely wasted by Smullyan, its epistemic depths remaining unplowed. But that setup can be rescued for a deeper lesson. (Footnote: I hope I’m not unfair to the venerable Smullyan, needless to say. But it is a bit disappointing that, writing in the late 70’s, he seems to miss the fairly obvious resemblance that his setup bears to the then-familiar Surprise Exam Paradox.) It takes minimal tweaking for us to do so. The original setup leads us to assume what our rescue mission should now turn into an explicit assumption: the assumption that the kid is required to have a rational expectation. To be sure, anyone can be fooled, if one’s being fooled simply requires one’s believing some falsehood on the basis of testimony. (Think of a gullible person falling prey to a false assertion the falsehood of which should be obvious to her.) But only rational acceptance of testimony (whatever the conditions for that may be) will allow for philosophically interesting surprise. So, the assumption we need is that the kid is rational in expecting to be fooled. Under this assumption, Smullyan has it that the kid is fooled by Emile iff the kid rationally expects to be fooled by Emile and is surprised by (i.e., does not rationally expect) something Emile does before midnight, something the kid could not, under those circumstances, have rationally believed that Emile would do.

The interesting problem here is whether anybody can rationally believe a person who promises to surprise the rational believer. How can the kid rationally believe that Emile will fulfill his promise? To feel the sting of the question, you have to think about what the rational doxastic attitude would be when the agent is confronted with the troubling promise. Suppose the kid thinks about the range of acts that Emile cannot rationally be expected to perform. For instance, Emile had never burned the house down, nor did he ever show any tendency to do so. The kid wouldn’t rationally take his doing so as a possibility. So, if Emile did end up burning down the house, his act would come as a total surprise. The arson scenario is one that the kid believes will not be actualized. He has numerous such beliefs (both occurrent and dispositional): Emile will not kill himself; Emile will not kill a neighbor; Emile will not produce a genuine bank statement showing a billion-dollar balance, etc. And then there are the acts about which the kid suspends judgment (either occurrently or dispositionally): “Emile might claim that I’m an adopted child (though I have no reason to believe that he will)”; “Emile might put salt in my coffee (though I have no reason to believe that he will)”; etc. If Emile’s action provided the kid with evidence for believing any of those propositions to which the kid is, prior to the prank, disposed to react with either disbelief or suspension of judgment, the kid would be surprised; and, so, Emile would have fulfilled his promise to fool the kid. But all of those acts are outside the range of the kid’s rational expectations. So, how would the kid rationally believe that he was going to be fooled? The promise that he will be fooled is an epistemic blindspot for the kid. (Footnote: For the concept of an epistemic blindspot, see Roy Sorensen’s book on the subject. Here, I borrow liberally from Sorensen and use the term “epistemic blindspot” as synonymous with “contingent proposition that a given person cannot rationally believe (or know) under any circumstance”. Footnote to the footnote: As I see it, self-refuting beliefs, for instance, a belief the content of which is the proposition expressed by “I have no beliefs now”, are not clear cases of rationally unbelievable propositions, though they are clearly not knowable. There is opposition to this claim in the literature.)

But, now, I hear two objections. First, there is the objection according to which there is no stable path to the conclusion that the fooling cannot rationally be anticipated. This is Emile’s own conclusion; it gives us the first half of Smullyan’s contradiction. (It’s also a popular reaction to the Surprise Exam.) Here, the problem seems to be that, if nothing in the range of rationally unpredictable acts takes place, fooling comes “from above”, as it were: higher-order fooling is the absence of first-order fooling. Some fooling will have taken place, if no obvious (first-order) fooling does. So, if one is not obviously fooled, then one is “higher-orderly” fooled. The objection would have it that this is deducible from the promise alone. So, necessarily, given a belief in the promise, fooling ensues. The truth of the promise is knowable a priori. And, if the promise is knowable, it’s not a blindspot. (Smullyan’s contradiction is then derived from the simple fact that, if he is necessarily fooled, the promise is fulfilled, as he expected, and, so, no fooling has ultimately taken place. If fooled, then not fooled. And this inference is, of course, okay.)

Isn’t this a little too easy? Do we really have the requisite concept of higher-order fooling? Let it be granted that we do. Even so, isn’t it open to little Raymond to object that there is, after all, a promise left unfulfilled by Emile, namely, the promise of first-order fooling? It takes only the weakest form of contextualism for us to see that little Raymond will legitimately object that it was the promise of first-order fooling that he was really excited about – and both will go to bed frustrated. The easy way out for little Raymond is that Emile was equivocating between senses of “fooling”, or that he simply was ambiguous when the promise was made. That’s the kind of frustration that bad philosophy and bad pranks are made of. (Incidentally, it should be noted that nothing like this gets us to the bottom of the Surprise Exam situation.)

So, the first objection is a huge letdown in its attempt to turn the promise into something knowable a priori. Common sense says “boo!”.

The second objection likewise denies that Emile’s promise is an epistemic blindspot for little Raymond, but does so with much more philosophical finesse. It will claim that the kid can rationally expect to be fooled, but the kid’s rational expectation arises from good induction. (That fooling will take place is not knowable a priori.) He’s been fooled by Emile many times before on April 1st. When obvious (first-order) fooling doesn’t come, some fooling – whatever you may want to call it – will have taken place. The kid will have been frustrated. His frustration naturally gives rise to some concept of fooling. The kid acquires the concept right there and then, and accepts that he was fooled. (And again, if he was fooled, then he did get what he expected, and, so, he wasn’t fooled after all.)

Against the objections, I would claim, decisively, that the promise is an epistemic blindspot. If it is a blindspot – if the kid cannot rationally believe it to begin with – then Smullyan doesn’t get his contradiction. He will have been fooled, the fooling will be paradoxical, and there will be no contradiction in sight. (The stage where one grapples with a contradiction is just too late.)

So, why is the promise a blindspot? Simply because, again, from the fact that no act in the range of predictable acts can rationally be expected, we should infer that the promise is rationally unbelievable. The second objection rests on fallacious induction. It is fallacious to infer that, because something unpredictable has occurred many times, it will occur again. Ceteris paribus, the unpredictable does not become predictable just because it occurs over and over again. It remains unpredictable every time. This is reminiscent of the Gambler’s Fallacy. The fact that I’ve won a fair game of roulette 50 times in a row, doesn’t make me a favorite when the 51st spin comes around. A superficially similar fallacy is at work here. Every trick Emile performed on previous April Fool’s days was rationally unpredictable. The kid cannot rationally expect him to succeed again – not this year, not ever. (Not ever? Well, maybe not not ever. Once he acquires the concept of higher-order fooling, Smullyan’s reasoning may be applicable. I’m not pursuing the issue right now.) And the fact that – unlike the Gambler situation – Emile’s track-record should allow the kid rationally to infer that Emile will somehow do it again clashes with the conclusion that no first-order prank is predictable. We then resort to the claim that a non-existent prank will be as good as a prank. We are now positing a new kind of prank: the higher-order prank. Which reminds us of the Preface situation. No act in the range of Emile’s possible first-order surprises can rationally be expected. And yet we are tempted to say that some higher-order act by Emile should be thought surprising. But this is not the Preface in disguise either. In the Preface, you can have excellent prima facie justification for the prefatorial belief. And that evidence is independent of your evidence for each first-order belief. Here, we want to infer that a (higher-order) prank is coming on the basis of past unpredictable (first-order) pranks. There is, however, no evidence for expecting a higher-order prank. (And there is also the issue of whether the concept of a higher-order prank can be acquired prior to the absence of first-order pranks, which I’m putting aside.) All the evidence, if, per impossibile, there were any, would be for expecting a first-order prank. And yet, we can see that Emile is in an ideal position to strike, since, surely, we’re still in the grip of what seems to be good induction. (Nothing of the sort seems to be going on in the Preface scenario.)

If an epistemic blindspot is essentially involved in the situation, Emile’s trap is paradoxical – deeply so. But it should have introduced little Raymond to epistemology as well, not just logic.

Is there something I’m missing here?

Smullyan Fooled at Last — 9 Comments

1. Dear Claudio,
I’m not really satisfied with the account of fooling – it isn’t just a relation between epistemic attitudes. A proper April Fools’ spoof would require that somebody is lead to belief a false assumption about a state of affairs and to act on or to assert this judgment, at which instant he is to be demonstrated that he has been fooled. For example, if Emile only told him something unbelievable without real evidence, or something believable which would excite or affect Raymond, Raymond would just shrug of the first with an “I don’t believe you” as soon as Emile told him the lie and shrug of the other with “Whatever, I didn’t really care” when Emile would expose that he was lead to believe a falsehood. So the fooling has both to be convincing and relevant to Raymond.
But this is only half of the story: As someone accustomed to this tradition, little Raymond would be suspicious of anything his brother Emile would be doing and saying to him, iff it would lead to an assumption he really had to act or speak upon. April’s Fools is a game, in which one party has to convince or raise doubt to such an extend the other party has to act upon it, and the other party has to examine evidence closely and try to bare the doubt that there’s really something happening which would, under normal circumstances, require this party to act. Party A wants to fool Party B, and Party B wants to remain cool and to see through any deception Party A might instigate. In other words: Party A has to expose that he fooled B by making act foolish, and Party B must abstain from any foolish behaviour or assertion. So, even a trick question posed by A to B might count as well as any practical joke – as long as B is outwitted despite taking some precautions, as second-guessing everything A says and looking under sofa cushion before taking a seat A offered.
The promise of Emile simply raises the stakes – Raymond must have expected to fool his brother anyway on April 1st, and would have been somewhat distrustful. Emile now claimed a he end of the day that he has won the game by setting up the table and refusing to play at the same time. But has he really won? After all, he has excited Raymond enough to make him stay up late, waiting for his brother to make his move. On the other hand, this is in no way a reaction Raymond would show under normal circumstances, but only because he wants to participate in a round of the “April Fools” game with his brother. He isn’t acting on a false assumption which falsehood could be demonstrated to him, thereby showing, that he was fooled, in being expectant an distrustful, he es merely acting according to the rules of the game. But neither he nor his brother did make the rule of the game and get to decide when a round of the game would take place – this is done by tradition and the calendar, respectively – so Raymond could rationally expect his brother to try and fool him on this exact date regardless of what Emile promised. The promise is just a plain, and comparatively cruel lie, that Emile tries to cover up by claiming he has won the April Fools game in his “meta” way. But this claim itself is wrong, and there is no need to evoke something as abysmal as an epistemic blind spot, because this is about pragmatics, not semantics: Raymond simply hasn’t been acting foolish all day, until after Emile seemingly claimed his victory, that is, when he’s lying in his bed, unable to decide whether he was fooled or not. If Emile would now expose this behaviour as foolish, he would be right and would have won the April Fools game – he would have fooled his brother into an endless spiral of doubt by claiming to have already fooled him. But he still would have broken his promise, because neither the presumed falsehood nor Raymond’s actions are that much divergent: Big brothers often refuse to partake in games smaller children like, and a six-year-old might tend to obsess about happenings of the day they can’t quite make sense of. It would have been a rather meek April’s Day Fool – not at all what Emile promised: That he would make his brother believe something outrageous and make quite visibly an utter fool of himself.
Mother was indeed right.

2. Hi, Leif,

Thanks for commenting. I should think carefully about your criticism.

For now, some quick replies:

1. I can’t see that one’s rationally believing the claim that’s supposed to do the fooling is a necessary condition of being fooled. Look at what happens to us, philosophers. Some of us hold views on the basis of very intricate fallacies. Sometimes, the fallacies are exposed in such a way as to *prove* that you were irrational when you held those beliefs. That’s a way of being fooled. There are other ways. For instance, people routinely fall prey to mathematical fallacies. There, it’s clear that they can be *proven* irrational. (Moreover, given the proper setting, epistemic blindspots can satisfy your condition according to which “fooling has to be convincing”.)

2. Are you saying that my account of the situation conflicts with thinking (as you think we should) that Raymond and Emile are engaged in an April Fool’s Day *game*? I’m not sure it should be thought of as a game. Maybe it is a game. It surely is a *ritual*. But why is that a problem for my account?

3. What is the “spiral of doubt” that you see there? (Is a “spiral of doubt” a figurative description of plain doubt?) I can see how one irrationally gets to Smullyan’s doubt. You get there by following his reasoning all the way to the contradiction, and then failing to conclude, from that contradiction, that Emile’s fooling is a logical impossibility. (I definitely didn’t mean to suggest that Smullyan wouldn’t agree that the contradiction proves impossibility. So, the question “Was I fooled?” may just have been his way of leading the reader to a Barber-like conclusion.) But my main point is that you can’t go that far without having already been fooled. (So, maybe his implicit answer was “No, I wasn’t”, whereas mine is “You surely were”.)

3. Hi Claudio, thanks for your kind reaction. I’ll try quickly to answer ti your points:
1. I think there’s a huge difference between committing a fallacy an being fooled. If I’ll never be convinced that I’m wrong and have committed a fallacy and a neutral audience would never be convinced of that, I might be a fool in the eyes of someone that is convinced. But to be fooled as expected on an April fools day, my foolishness has to be exposed in a way that itself leaves no doubt whether I’ve been held a false belief or not. That includes a proper demonstration that I have held this belief, for otherwise I could always claim that I didn’t.
2. I called it a game to highlight that there are certain rules which shape the expectations the boys have regarding each others behaviour. Maybe its a ritual, but it is one in which the person trying to fool, an the person to be fooled are engaged in a competitive manner. Because of this, Emile’s initial announcement shouldn’t be counted as a move in the game, because he only restates the rules of the game (I will fool you) only adding (like you’ve never been before). He only ups the ante in a bet that is already active by the conventions of A.F.Day. Only when he seemingly calls of the game, the meaning of this first assertion is transformed. Now Raymond starts to doubt, and this doubt is foolish – even if we count Emiles broken promise as a legitimate act of fooling is still isn’t that extraordinary.
3. The spiral of doubt is indeed epistemologically speaking plain doubt, but – as per Smullyans description – doubt that can’t be mended by facts of the world. But the reasons why Raymond can’t decide are soundly non-epistemic: He won’t accept that his brother has broken his promise (to fool him in an extraordinary way), only fooling him into thinking a. that he would be fooled before bedtime b. that that could be an extraordinary way of fooling. But he won’t accept that because he holds his older brother in high esteem, and because he can’t distinguish this cognitive dissonance from the open embarrassment a proper Aprils’ fool would involve.

4. Leif,

The plot thickens. Our discussion got me thinking: Can you fool somebody without knowing it? I’m inclined to say “no”. And, if that’s the right answer, we’ve all been wrong: Little Raymond, Professor Smullyan, Emile, and me!

Let’s see how each of our characters goes wrong.

Little Raymond believes the promise. The promise is an epistemic blindspot for him (as explained above, in the original post). He goes wrong when he believes the promise and stays wrong up until the moment when he finds fault with Emile’s reasoning. Then he goes wrong again by suspending judgment on whether he was fooled. Only higher-order fooling could conceivably have taken place. But his late-night inference indicates that higher-order fooling is impossible (the Barber-like conclusion). So, he’s wrong again: he has no business suspending judgment on the matter.

Professor Smullyan concludes – through Barber-type reasoning – that Little Raymond was not fooled. He’s wrong (as explained in the original post): Little Raymond was fooled by the promise (or so it still seems). He was fooled into believing irrationally that he was going to be fooled. As a consequence of that philosophical mistake, Smullyan is wrong again in thinking that Emile’s reasoning for the first half of the contradiction is sound. It is not, as we will now see.

Emile was wrong twice – and this is the new twist. He was wrong the first time when he falsely believed that Little Raymond would rationally believe that first-order fooling was coming. Little Raymond did believe the promise, but did so irrationally. So, Emile was wrong there. And he was wrong again, as Smullyan shows us, in thinking that higher-order fooling did take place. It did not, since it was competently blocked by Little Raymond’s late-night inference. (Little Raymond’s belief in the promise was a mistake, but he competently uses that original mistake when he turns it against Emile’s conclusion that higher-order fooling did take place. Surely, Emile was in no position to complain about Little Raymond’s late-night inference.)

And I was wrong myself – amazingly! 🙂 I was wrong in thinking that Little Raymond had, indeed, been fooled right at the outset, when he believed the promise. But it now looks like it’s at best doubtful whether his mistake (in believing the promise) suffices for Emile’s success. Since Emile falsely believes that the promise can rationally be believed, and, on that basis, believes that fooling will take place, he cannot know that it will (on that basis). And we now suspect that, if he was not in a position to know that fooling would take place, fooling did not take place. And, so, I was wrong for missing that twist in the original post.

The philosophy in the original post remains unaffected, however. That has to do with the crucial role of epistemic blindness in the story (and with Sorensenian epistemological encroachment on logical-paradox terrain).

I’ll think some more about your points (old and new).

5. Claudio,

I agree with your last analysis, but I don’t really know about the epistemic blind spot. As I have tried to argue before, Raymond had to expect Emile to fool him even without the promise – because it was April 1st. He really was fooled when Emile told him that his promise was fulfilled – but that was only a mediocre fooling, thus Emile’s promise hasn’t been fulfilled at all.

Regards, Leif

6. Thanks, Leif!

Under pressure from your latest reply, I think I see why my case for epistemic blindness may not be fully intelligible, or, in any case, not fully persuasive. We’ve been moving in opposite directions. You seem to think that there’s too much philosophy in my discussion of the Little Raymond case – or, at any rate, too much philosophy of the wrong kind. Myself, I think we don’t yet have enough of the very same kind. So, I’ll give you my no-holds-barred case for epistemic blindness as the key to the problem.

Consider Emile’s promise: “Raymond […], I will fool you as you have never been fooled before!” *Everything* that matters here revolves around the issue of whether Little Raymond can rationally believe the promise. I’m afraid the original post gives undue attention to the issue of whether there can be good induction, on Raymond’s part, to the conclusion that Emile will fulfill his promise. You go even further in the wrong direction, I’m afraid, in pursuing a rational belief for Raymond based only on induction from previous pranks (and eschewing the promise altogether). But the issue of whether there can be good induction by Raymond is largely a distraction. The most fundamental issue here is whether it is possible for the promise to be a case of knowledge for its addressee, Raymond. The weird thing about epistemic blindness is that a blindspot belief cannot be turned into knowledge under any circumstance, despite having a contingent proposition as content. No amount of good induction in support of the blindspot belief can make it *knowable* by the epistemically blind agent. (In fact, as I have claimed in a recent Synthese paper, deduction from known premises fares no better than good induction when it comes to the epistemization of a blindspot belief, though I don’t quite put it in those terms in the paper.) There can be no quick offer of examples here. We need to be shown how, exactly, the relevant blindspot belief can be known a priori not to be a case of knowledge. That’s what I’ll try to do now.

If I’m not much mistaken, we can safely paraphrase Emile’s promise as follows:
(1) I will do something to you, and you will not be able to hold a rational belief to the effect that the specific act that I will be performing is coming; so, when the act is performed, it will be surprising, that is, you won’t have known beforehand that that was the surprising act; you will, at most, have truly *guessed* that the act would be what it turned out to be.

If that’s the promise, then, if Raymond believes it, there isn’t a single hypothesis about what the surprising act will be that he can rationally believe. His hypotheses about the upcoming prank will have (somehow) been inferred from his belief in the promise (plus unproblematic assumptions about the situation). We are led to believe that the promise plays a crucial role in his inferences. So, every one of his hypotheses will be of the form
(2) Emile will do x, but I don’t rationally believe that he will.

It may look like the general form of those hypotheses is not (2), but
(3) Emile will do x, but I won’t rationally believe that he will.
But (3) merely predicts ignorance. If (3) is true, there is a future time when ignorance finally takes place. So, if Raymond believes the promise, there will be a time when he will be thinking about values of “x”, but if, at that time, he hasn’t ceased believing the promise, he’ll be thinking about instances of (2), since he has been led to believe that the upcoming prank is unpredictable. (And we are led to believe that he can’t rationally stop believing the promise before it’s fulfilled, or before midnight, whichever comes first.)

(2) is an instance of
(4) P, but I don’t rationally believe that P.
So, the issue is: why is (4) a case of epistemic blindness for the believer?

Here’s a very simple proof that no belief of the form (4) can be knowledge, even though it will be contingent and non-self-refuting. You only need the following epistemic principles:

Knowledge-distribution (KD): If S know that P&Q, S knows that P and S knows that Q.

Rationality condition for inferential knowledge (RC): If S inferentially knows that P, then S rationally believes that P.

Truth condition for knowledge (TC): If S know that P, then it is a fact that P.

Now, assume, for reductio, that I know that (4), some instance of (4), of course, by inference from Emile’s promise – which is the only way (4) can get into the picture. (So, in what follows, “I know that (4)” is short for “I inferentially know that (4)”. By KD, if I know that (4), I know that I don’t rationally believe that P. And by TC, if I know that I don’t rationally believe that P, then I don’t rationally believe that P. So, I don’t rationally believe that P. By KD, however, if I know that (4), I know that P. But, by RC, if I know that P, then I rationally believe that P. So, I rationally believe that P. So, I both do and do not rationally believe that P. So, I don’t know that (4).

It’s as simple as that, really. And it should have looked as simple as that in the original post. Unless Emile’s promise can rationally be believed by Raymond, or unless the promise is irrelevant to Raymond’s expectations and to the whole issue of whether he was fooled, epistemic blindness is essential to the story.

Is this better?

7. Aaarrrgh!… Have I done enough to counter the most obvious objection to the epistemic blindness view of the problem? Maybe not. So, please, bear with me some more while I try to fine-tune the explanation.

As one looks at Emile’s promise, it may be tempting to object that there is a simple fallacy in the way I react to it: “From the fact that no specific prank can rationally be regarded as being the upcoming prank, it surely doesn’t follow that one can’t rationally believe the promise that *some* prank is coming.”

This objection would saddle Raymond with the following belief (the belief whose content is expressed by the following sentence):

(E)[mile] Some prank is coming, though I can’t rationally believe of any possible prank that it is coming.

And, granted, such a belief is not obviously irrational. It is analogous to believing something like: “Some catastrophe will make the headlines next year, though I can’t rationally believe of any possible catastrophe that it will make the headlines.”

(E) is of the form “There is an x which is such that x is coming, but, for any given x, it is not the case that I can rationally believe that x is coming”. Now, assuming that “I can’t rationally believe that P” implies “I don’t rationally believe that P”, (E) would seem to imply

(I)[mplication] Some x is coming, but I don’t rationally believe, of any given x, that it is coming.

But, crucially, I don’t see why (I) might be thought not to imply the epistemically blinding
(EB) Some x is coming, but I don’t rationally believe that some x is coming.

And (EB) takes us back to

(4) P, but I don’t rationally believe that P.

Is there a fallacy here?

8. I my view, Smullyan could have fooled his brother Emile by surprise for not expecting any surprise from him on April fool’s day.

9. I think Smullyan intends this story to be an example of self-referential paradox with facets of “Is the answer to this question no?” I think the exchange between Claudio and Leif is engendered by the flip-flop of perspectives induced by self-referential questions. Is a self-referential paradox identical to an epistemological blind spot?
Or is it in some way analogous; I think Claudio raised a question which was interesting because of the inherent paradox which satisfies Smullyan’s purpose.