Some Logical Paradoxes from Jean Buridan 1. A Chimera is a Chimera: A chimera is a mythological creature with the head of a lion, the body of a goat, and the tail of a snake. Obviously, chimeras do not exist. So, it should be obvious that any statement which asserts something about a chimera is false. Consider: A chimera is in my pocket. A chimera is lying down. A chimera is breathing fire. But, it gets a little less clear with statements like the following: A chimera is a mythological creature. A chimera has the head of a lion. A chimera is pictured above. And still LESS clear when ‘chimera’ is the subject of a tautology: A chimera is a chimera. For, logically, any statement that has the form ‘X is X’ is necessarily true. For instance: A table is a table. A duck is a duck. Aquinas is Aquinas In short, Buridan has identified a tension between these two claims: (1) Any assertion made about something that does not exist is false. (2) Any assertion of the form ‘X is X’ is true. <A chimera is a chimera> fits both (1) and (2). So which is it? Is it true or false? Reply: Buridan concludes that the statement is false. This is because its subject term “supposits for nothing” (i.e., REFERS to nothing). Basically, we should interpret the statement as making the claim that “Something exists that is a chimera, and that thing is a chimera.” But it is FALSE that something exists that is a chimera. 1 The conclusion is that (2) only applies when ‘X’ actually refers to something. Worry: But, then, <Aristotle’s horse was brown> is also false? After all, his horse is long dead—i.e., it no longer exists—so ‘Aristotle’s horse’ doesn’t refer to anything. Reply: Aristotle’s horse DOES exist at some time or other (in this case, it exists in the past). So, it DOES refer (to something in the past). But, chimeras don’t exist at ANY time. [But, then, did I say something false when I said that a chimera is a mythological creature that has the head of a lion, body of a goat, and tail of a snake? Do fictions exist in SOME sense at least (e.g., as concepts, or abstract things, or ideas?)] 2. Today You Ate Raw Meat: Imagine that you bought some raw meat yesterday. Today, you cooked it and ate it. Did you actually eat RAW meat? Consider: 1. Whatever you bought yesterday is what you ate today. 2. But, yesterday you bought raw meat. 3. Therefore, today you ate raw meat. Reply #1: The thing you ate today is NOT the thing you bought yesterday! For, the thing you bought yesterday had the property of rawness (a property which the thing that you ate today lacked). (He also mentions that some of the matter has probably evaporated.) It is commonly thought that if two things are numerically identical, then they must share all and only exactly the same properties. (For instance, compare the properties of Samuel Clemens with the properties of Mark Twain.) So, since one has a property that the other lacks (namely, rawness), they cannot be the same object. [Problem: Is the cell phone in your pocket the one that you bought? After all, surely the one in your pocket has many properties that the phone you bought did NOT have. Even worse: Do YOU continue to exist over time? You probably think that you do. But, at noon you are sitting and at 1pm you are standing—different properties. How, then, can these both be one and the same individual? This is ‘the problem of temporary intrinsics’] Reply #2: When we predicate properties of things, we really are saying that thing has that property AT A TIME. So, imagine that today is Tuesday and that you bought the meat on Monday. What premise 2 really says is that you bought meat that had the property of Monday-Rawness. But, what the conclusion says is that you ate something that had the property of Tuesday-Rawness. Not only is this false, but the argument is no longer valid—the conclusion does not follow from the premises. 2 [These days we say that we have ‘time-indexed properties’. Perhaps the thing that you buy and the thing that you eat ARE one and the same object. Only, the property of rawness is “indexed” to Monday. For instance: The argument would be valid if we interpret the conclusion as saying merely that, ‘Today (Tuesday) you eat something that has the property of Monday-Rawness’. Note that this solves the problem of temporary intrinsics. You do not have conflicting properties at noon and 1pm because they are time-indexed. In other words, what you REALLY have are the properties of sitting-at-noon and standing-at-1pm.] 3. You Believe That You Are A Donkey: Imagine that you see a figure in the distance walking on all fours. It appears to be a donkey. Unbeknownst to you, it is your father, wearing the hide of a donkey and crawling around on all fours. You’re a donkey: 1. Whoever believes that her father is a donkey believes herself to be the offspring a donkey (and therefore herself a donkey). 2. You believe that your father is a donkey (in the story above). 3. Therefore, you believe that YOU are a donkey. Reply: Buridan rejects premise 2. In the story, you do not believe the proposition <My father is a donkey>. Rather, you believe <That thing in the distance is a donkey>. (However, you do not also believe that the thing in the distance is your father.) To illustrate the distinction: You can imagine that your dad comes closer and you suddenly realize it’s him and say, “Whoa! I thought you were a donkey!” But, Buridan is pointing out here that, strictly speaking, this is not the right way to put it. You did NOT think that your DAD was a donkey. What you SHOULD have said is, “Whoa! I thought I was looking at a donkey, but I was actually looking at you!” 4. I Owe You a Horse: We made a deal and I promised to give you one good horse. I do not own any horses, but I promised to make good on the deal by Easter. (Buridan also considers an alternative scenario where you sold me some mustard for $1, but I haven’t paid you yet.) The following seems true: I owe you a horse. (or $1) But is it? Let us consider all of the horses that exist. There is the king’s horse (Blackie). But, I do not owe you Blackie. There is the pope’s horse (Tawny). But I do now owe you Tawny. There is Peter’s horse (Brownie). But I do not owe you Brownie. And so on… 3 1. I do not owe you Blackie; or Tawny; or Brownie; etc. 2. If it is true of every horse that I do not owe you that horse, then I do not owe you a horse. 3. Therefore, I do not owe you a horse. [If you have taken Symbolic Logic, the problem can be stated as follows: Let Hx = x is a horse ; Oxyz = x owes y to z ; c = Chad ; a = Agatha (your name is now Agatha) ‘I owe you a horse’ translates into predicate logic as: (ꓱx)(Hx Ocxa) In English: There exists an ‘x’ such that x is a horse, and Chad owes x to Agatha. But, existentially quantified statements are really just giant disjunctions in disguise. For instance, if there were only THREE horses in existence (Blackie, Tawny, and Sapphire), the formula above could be re-written as follows: [(Hb Ocba) (Ht Octa)] (Hs Ocsa) In English: Either Blackie is a horse that Chad owes to Agatha OR Tawny is a horse that Chad owes to Agatha OR Sapphire is a horse that Chad owes to Agatha. But a disjunction is only true if at least one of its disjuncts is true. But, NONE of the individual disjuncts are true. Therefore, the entire disjunction is false. Or, in other words, the statement <I owe you a horse> is false. Whoa…] Reply: Buridan points out that if <I owe you a horse is true> then it follows that <I owe you Blackie, or I owe you Tawny, or I owe you Brownie, etc.>. But the only way that this disjunction is true is if at least one of its disjuncts is true. However, it would be completely arbitrary to claim that some SINGLE disjunct is true while the others are false. (For instance, we might say that I owe you Tawny, but not Blackie, Brownie, or any of the others—but this seems false, for there is no more reason to think that I owe you Tawny than any of the other horses.) So, if it is true that I owe you a horse, the only reasonable conclusion is that ALL of the disjuncts are true; that is, I owe you ALL of the horses. This ends up being Buridan’s solution. The alternative is that I owe you NONE of them— in which case, I do not owe you a horse at all. But this is false. So, I owe you ALL of them. Worry: I cannot fulfill my obligation unless I give you EVERYTHING that I owe. In that case, I won’t have fulfilled it unless I give you each and every horse.