Here's one that, like it or not, I think about all the time, owing to how I make my living:
The Surprise Quiz Paradox:
[Y]our teacher tells you (i) she's going to give the class a surprise exam next week, and (ii) you won't be able to work out beforehand on which day it will be. Using this information, you work out that it can't be on Friday (the last day), or else you'd be able to know this as soon as class ended the day before, contrary to the second condition. With Friday excluded from consideration, Thursday is now the last possible day, so we can exclude it by the same reasoning. Similarly for Wednesday, Tuesday, and finally Monday. So you conclude that there cannot be any such exam. This chain of reasoning guarantees that when the teacher finally gives the exam (say, on Wednesday), you're all surprised, just like she said you'd be. (The Surprise Examination Paradox)As a teacher, hearing about this paradox is a little like receiving a notification from your insurance company that your policy is cancelled now that your dead. You're pretty sure you're not dead. But then there's this notice. Cool.
Here’s a paradox (or a set of paradoxes) that I often refer to in my lectures:
Puzzles (paradoxes) attributed to Eubulides of Miletus:
The Heap: Would you describe a single grain of wheat as a heap? No. Would you describe two grains of wheat as a heap? No. ... You must admit the presence of a heap sooner or later, so where do you draw the line?*Students imagine that every word can be defined with a precise definition.
The Bald Man: Would you describe a man with one hair on his head as bald? Yes. Would you describe a man with two hairs on his head as bald? Yes. ... You must refrain from describing a man with ten thousand hairs on his head as bald, so where do you draw the line? (Sorites Paradox)
Nope. Imagine a series of slight modifications (removal of small amounts) of a chair. When does it cease to be a chair? Any answer will be unacceptable because it is arbitrary.
VOTING. More than one thing is referred to as the “paradox of voting.” The particular “paradox” I have in mind (roughly) is discussed in the encyclopedia entry below:
The paradox of voting … is that for a rational, self-interested voter the costs of voting will normally exceed the expected benefits. Because the chance of exercising the pivotal vote (i.e. in case of a tied election) is tiny compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the costs. The fact that people do vote is a problem for public choice theory, first observed by Anthony Downs. (Paradox of Voting)Now, don’t get me wrong. I believe in voting. I vote (most of the time). I think that I have good reasons to vote.
But, in my view, one reason that I do NOT have for voting is the one we most often hear, namely, that “one’s vote counts!”**
Unless one will countenance the fallacy of equivocation, one cannot really defend the notion that one’s vote “counts,” for, to say (in the relevant contexts) that one’s vote “counts” is to suggest that it will make, or it quite possibly will make, a difference to the election’s outcome (winning/losing). (I'm not particularly interested in the issue of self-interest; I'm interested in a point about efficacy.)
Here’s where the so-called paradox*** of voting comes in. Obviously, for typical government elections (I’m not referring to elections involving small numbers of people—department elections and the like), the chances that one’s vote will make a difference to the outcome are extremely small.
Consider the recent election for the “board of trustees” seat now held by TJ Prendergast, which was unusually close. The final tally was the following:
As it was, Prendergast received 4107 more votes than Muldoon did. Suppose that Smith voted for Prendergast. Had Smith not voted, the outcome would have been only very slightly different: Prendergast’s total would have been 115,303, not 115,304.
So, in fact, Smith’s vote, if it “counted,” it did not “count” in the sense that it made a difference (of any consequence) to the outcome. To be sure, his vote was “counted.” Nevertheless, it did not “count.” (Remember: equivocation is verboten.)
It is true, of course, that it could have counted, though, in fact, it did not count. But, clearly, the odds of one’s vote “counting” are infinitesimal. Very likely, all of you who read this (hundreds!) will go through your entire lives voting and, in the end, there will not have been even one election in which any of your votes “counted” or even came close to counting in any meaningful sense.
Some will respond to this by noting that, in recorded history, there have been elections in which a single person’s vote “counted” in the way I have in mind. (For an illustration, see Examples of Why Your Vote Counts.)
Of course this occurs. Given the great number of elections that occur, this goes without saying, I think.
But the occurrence of these events does not respond to the point at hand, namely, that, though it is possible that one’s vote will “count,” it is extremely unlikely that it will count. Given that one could live a great many lifetimes before encountering even one election in which one’s vote “counted,” in what sense is one being told anything true and motivating**** when one is told that one’s vote “counts”?
In my view, when we seek to persuade people to vote on the grounds that their “vote counts,” we are either confused (i.e., we think we have a valid point when we do not) or we are lying/manipulating (we know that we have no valid point, but we offer it anyway perhaps because [we think] our end is good).
My guess is that confusion more than lying is afoot.
On the other hand, there are so many instances in which our “teachings” are manifestly (or nearly manifestly) invalid, we should consider the possibility that, yes, we offer this false point not “knowing” that it is false—but, still, it must be said that we have good reasons to suspect that, often, what we “teach” is logically hinky at best, and so, quite possibly, this is logically hinky too.
Organic muffins, anyone?
*So what's paradoxical about this? Well, you start with a heap of sand. Plainly, after removing grains of sand for a sufficient period of time, you end with a non-heap (one grain). And yet there is no "line" that you cross to get from "heap" to "non-heap." You cross a line, but there is no line to cross.
**To act to influence large numbers of voters—something sometimes available to leaders—means the difference between a significant number of people voting for X or not. Here, whether or not “Smith’s vote counts,” the leader’s urgings might count a great deal. It will remain true, however, that not one of those votes counted.
“Yes, but what if everyone thought that way.” It is of course true that what (say) 50,000 voters in state X do during a particular election can make all the difference. And that is why those who care about the outcomes of elections rightly concern themselves with persons and events that influence large numbers of voters. But all of that can be acknowledged without falsely supposing that each voter’s vote “counts.” That the collective vote of 50,000 voters “counts” does not imply that each of those votes “counted.”
***What is “paradoxical” here? It is, I suppose, that, though it matters a great deal how everyone votes, in fact it matters not at all how any given voter votes. That Americans in general voted for candidate X matters to the outcome of the election. That any given voter voted for candidate X does not matter to the outcome of the election. To endorse both statements might seem to be the endorsement of a contradiction, but it is not.
****Obviously, one is not being told anything of significance if one is being told merely that one's vote could mean something like the difference between Prendergast's receiving 115,304 and 115,303 votes. Why would anyone care about that difference?