On the (non-)surprising paradoxes
Topic: GENERAL LOGIC
Here are two related posts:
Paradox vs. Surprise
By Jon Kvanvig
Source: Certain Doubts
A paradox is different from a result that is merely surprising, but what is the difference? This question touches on matters beyond epistemology, but it is applicable to the major epistemic paradoxes, including preface, lottery, surprise quiz, and knowability. It is the latter that prompts my question.
In the knowability paradox, we purportedly demonstrate that if all truths are knowable, then all truths are known. There is no question that the result is surprising, but what makes it paradoxical? Compare it with Godel?s incompleteness theorems, for example, which are also quite surprising, but not paradoxical. Or compare it to the ontological argument, where it is purportedly shown that if a certain description is possibly exemplified, then it is necessarily exemplified. This, too, is quite surprising, but I doubt it is paradoxical. So, what is the difference?
Perhaps the difference is psychological. Logical results are suprising when they go beyond what we presently believe to be true, and when they concern issues of significance to us. We notice the result which, prior to the proof, we doubted; after seeing the proof, we are convinced and thereby surprised. When the result is paradoxical, however, something additional happens. The proof threatens our intellectual commitments in some way, it threatens our firmly held opinions on matters that are significant to us. Admitting the soundness of a proof to the contrary thus engenders a bit of mental apoplexy: we know something has to give, but it?s hard to see what.
Is there a different account of the distinction? I?m not sure; if you have a different account, please share it. But if we suppose that this account is on track, one has to dig a bit to find a paradox in the knowability result. The result is a conditional: if every truth is knowable, then every truth is known. That?s not a denial of any deeply entrenched viewpoint I hold on issues that are significant to me. So why the fuss? I think there is something paradoxical in the neighborhood here, and I think it has important lessons. But since it depends on the psychological account of the difference between surprising and paradoxical derivations, I?ll hold off to see if there might be a better account of the difference.
The Swedish Drill
By: Leon Felkins
Swedish civil defense authorities announced that a civil defense drill would be held one day the following week, but the actual day would be a surprise.
However, we can prove by induction that the drill cannot be held. Clearly, they cannot wait until Friday, since everyone will know it will be held that day. But if it cannot be held on Friday, then by induction it cannot be held on Thursday, Wednesday, or indeed on any day.
What is wrong with this proof?
This problem has generated a vast literature (see here). Several solutions of the paradox have been proposed, but as with most paradoxes
there is no consensus on which solution is the "right" one.
The earliest writers (O'Connor, Cohen, Alexander) see the announcement as simply a statement whose utterance refutes itself. If I tell you that I will have a surprise birthday party for you and then tell you all the details, including the exact time and place, then I destroy the surprise, refuting my statement that the birthday will be a surprise.
Soon, however, it was noticed that the drill could occur (say on Wednesday), and still be a surprise. Thus the announcement is vindicated instead of being refuted. So a puzzle remains.
One school of thought (Scriven, Shaw, Medlin, Fitch, Windt) interprets the announcement that the drill is unexpected as saying that the date of the drill cannot be deduced in advanced. This begs the question, deduced from which premises? Examination of the inductive argument shows that one of the premises used is the announcement itself, and in particular the fact that the drill is unexpected. Thus the word "unexpected" is defined circularly. Shaw and Medlin claim that this circularity is illegitimate and is the source of the paradox. Fitch uses Godelian techniques to produce a fully rigorous self-referential announcement, and shows that the resulting proposition is self-contradictory. However, none of these authors explain how it can be that this illegitimate or self-contradictory announcement nevertheless appears to be vindicated when the drill occurs. In other words, what they have shown is that under one interpretation of "surprise" the announcement is faulty, but their interpretation does not capture the intuition that the drill really is a surprise when it occurs and thus they are open to the charge that they have not captured the essence of the paradox.
Another school of thought (Quine, Kaplan and Montague, Binkley, Harrison, Wright and Sudbury, McClelland, Chihara, Sorenson) interprets surprise in terms of knowing instead of deducing. Quine claims that the victims of the drill cannot assert that on the eve of the last day they will know that the drill will occur on the next day. This blocks the inductive argument from the start, but Quine is not very explicit in showing what exactly is wrong with our strong intuition that everybody will "know" on the eve of the last day that the drill will occur on the following day. Later writers formalize the paradox using modal logic (a logic that attempts to represent propositions about knowing and believing) and suggest that various axioms about knowing are at fault, e.g., the axiom that if one knows something, then one knows that one knows it (the KK axiom). Sorenson, however, formulates three ingenious variations of the paradox that are independent of these doubtful axioms, and suggests instead that the problem is that the announcement involves a blindspot:
a statement that is true but which cannot be known by certain individuals even if they are presented with the statement.
This idea was foreshadowed by O'Beirne and Binkley. Unfortunately, a full discussion of how this blocks the paradox is beyond the scope of this summary.
Finally, there are two other approaches that deserve mention. Cargile interprets the paradox as a game between ideally rational agents and finds fault with the notion that ideally rational agents will arrive at the same conclusion independently of the situation they find themselves in. Olin interprets the paradox as an issue about justified belief: on the eve of the last day one cannot be justified in believing BOTH that the drill will occur on the next day AND that the drill will be a surprise even if both statements turn out to be true; hence the argument cannot proceed and the drill can be a surprise even on the last day.
For those who wish to read some of the literature, good papers to start with are Bennett-Cargile and both papers of Sorenson. All of these provide overviews of previous work and point out some errors, and so it's helpful to read them before reading the original papers. For further reading on the "deducibility" side, Shaw, Medlin and Fitch are good representatives. Other papers that are definitely worth reading are Quine, Binkley, and Olin.
Posted by Tony Marmo
at 01:01 BST
Updated: Sunday, 15 August 2004 08:40 BST