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