cbostrum (08/01/82)
Here are the interesting results on the food poll. The question was: I am very interested in knowing what people think about the relative plausibilities of the following two statements about restaurant food: A: Good food is not cheap. B: Cheap food is not good. Is it your opinion that: I) A is more plausible than B II) B is more plausible than A III) A and B are roughly equally plausible There were not many answers (about 15) but I solicited many more in person around here. The mailed in answers were often confused, and the types of confusion were interesting, but I wont go into that here, but get to the main point. As some may have guessed, the survey was just a test of an entirely different matter from food, which is whether or not people will recognise logically equivalent statements. Almost everyone would agree that A and B are respectively equivalent to: A': (x:Food) Good(x) -> ~Cheap(x) B': (x:Food) Cheap(x) -> ~Good(x), which we regonise immediately as being logically equivalent, using traditional Tarskian semantics for first order logic. Thus A and B say the same thing, and are therefore equally plausible. (One philosopher I know defended herself by interpreting the statements in an intuitionist logic, where contrapositon is not generally valid, thus denying that "A' iff B'" although accepting them as translations.) Most people, as expected, claimed that the statements were both "not plausible" or "roughly equally" plausible, but showed no sign that they realised that the statements were in fact logically equivalent. I take this as meaning that most people did not, in fact, realise the logical equivalence. Of course, it is not quite right to do this, but obviously I couldnt just ask them straight out. More evidence for the lack of realisation came from the sort of justification people gave for their votes: I received some interesting analyses of how restaurants (both "good" and "cheap") worked, and why the statements thus have any plausibility that they do, as well as vigourous denials of one or both statements, or simple counterexamples. The real point is that the relative plausibilities of the statements can be decided without recourse to the world at all, and only 2 people pointed this out. (One said the statements were equivalent, and I am giving him the benefit of the doubt in assuming he meant *logically* equivalent. The other was the net.misc communication that exposed the logical equivalence (despite my attempts to not have such a thing occur)). I expect I might receive some argument about the putative translations from A to A' and B to B'. The only plausible argument, in my opinion, is that the first ocurrences of the descriptions of foods in each statement were meant to be interpreted referentially (A la Keith Donellan) rather than attributively. Thus, we get: A'': The food served in ritzy restaurants is not cheap. B'': The food served in greasy spoons is not good. or something similar. These two statements (espcially B'') seem to be the ones that people were more often debating the plausibility of. Of course, they are not logically equivalent. I would say the first is very plausible, and the second is much a matter of taste (cf: a response about truck drivers). Anyway, thank you to those who participated for giving us some insight into the often non-logical way our minds work.
ark (08/01/82)
No the statements are not equivalent, in the sense that the formulae do not capture all the information usually inferred from the English. When I say "good food is not cheap," what I am really saying is: "good food is usually not cheap." Similarly, when I say "cheap food is not good," I am really saying "cheap food is usually not good." These two statements are definitely NOT equivalent. Consider a situation in which all food is good, and a tiny bit of it is cheap too. Then the first statement is true but the second is not. To convince yourself that the "usually" belongs there instead of "always," suppose someone told you that good food is not cheap and the next day you found a restaurant that did indeed serve cheap good food. Would you think that that single counterexample demolished the claim? Probably not: you would think: "remarkable -- here's a restaurant that serves cheap good food. Let's file that with the rare exceptions."