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.
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.
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The conclusion is that (2) only applies when ‘X’ actually refers to something.
Worry: But, then,
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.
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[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
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…
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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 is false. Whoa…]
Reply: Buridan points out that if then it follows that . 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.
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Reply: No. I only owe you ALL of the horses in this sense: Every single horse in existence is such that, if I were to deliver that horse to you, I would fulfill my obligation to you.
[What do you think? Are you satisfied with this solution? Perhaps ‘I owe you a horse’ should not be understood as entailing that ‘There is some horse that I owe you’?]
5. I Say Something False: These days, the paradox is usually put like this:
This statement is false.
The statement above must be either true or false. So which is it?
(1) It is true: If so, then it is false.
(because, if true, it follows that what it asserts is correct; and what it asserts is that it is false.)
(2) It is false: If so, then it is true.
(because, if false, then the assertion that it makes is correct; in which case it follows that it is true.)
Uh-oh. This is the famous ‘Liar Paradox’ (some state it as, ‘This statement is a lie’).
[If you think that the paradox is ONLY generated because the statement refers to itself, then consider this variant: The green statement is true. The blue statement is false.]
Reply: Buridan concludes that the statement is false. Why? His answer rests on the following move: Buridan claims that every statement, , ENTAILS (or, as he puts it, “virtually implies”) a second statement of the form <’A’ is true>.
On Buridan’s view, is false if it either (a) signifies something false, or (b) virtually implies something false! i.e., is true if and only if it meets these TWO conditions:
(a) Things must really be just as signifies, and (b) Things must really be just as <’A’ is true> signifies.
Normally, adding this second criterion makes no difference. For instance, consider a mundane proposition like
(a) Things really are just as
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Since both conditions are met, the statement is true. But, for self-referential statements like the Liar Paradox, something weird happens: Conditions (a) and (b) come apart.
Let us return to the statement,
(a)
Now,
Suppose that: <’This statement is false’ is true> In this case, criterion (b) is clearly met—i.e., <’This statement is false’ is true> signifies correctly (for, we’ve stipulated that this is what we’re supposing).
However, criterion (a) is NOT met. For, things are NOT just as
Suppose that: <’This statement is false’ is false> In this case, criterion (a) is met—i.e.,
However, (b) is NOT met, for things are NOT really just as <’This statement is false’ is true> signifies. For, by supposition, we’ve stipulated that the very opposite is the case.
Either way, the statement is false (for, either way, it fails to fulfill Buridan’s two criteria for truth). In short, Buridan would respond to the initial accusations as follows:
(1) If we suppose that <‘A’ is true>, this entails that is false. YES! (2) If we suppose that <‘A’ is false>, this entails that is true. NO! is still false
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6. You Will Throw Me In The Water: Consider this case:
Plato is guarding a bridge which crosses a river, where Socrates arrives and wants to cross. Plato makes a vow: “If the very next proposition that you utter is true, I will let you cross,” he says. “But, if it is false, I will throw you into the water.” Socrates thinks for a moment and then says, “You will throw me into the water.”
What should Plato do in order to keep his vow?
Let Socrates pass: If Plato does this, then Socrates had said something false, in which case Plato should have thrown Socrates into the river.
Throw Socrates into the water: If Plato does this, then Socrates had said something true, in which case Plato should have let Socrates cross.
Reply: Buridan asks three questions:
(1) Was Socrates’ assertion true or false? Note that, since it is about a future event, there is no way to know whether it is true or false at the time Socrates utters it.
(2) Was Plato’s vow true or false? Plato made a conditional claim (i.e., an ‘if-then’ statement), and conditionals are only false when their antecedent is true (the part right after the ‘if’) and their consequent is false (the part right after the ‘then’). For instance, if I say to you, “If you pass the exam, then I will give you an A”, I have said something false only if you do pass the exam (i.e., the antecedent is true) but I do NOT give you an A (i.e., the consequent is false). In short, I have lied.
Note that Plato has really made a DOUBLE-conditional claim. Buridan notes that, either way, Plato has lied. For, either way, Socrates has fulfilled the condition (i.e., the antecedent) of one of the two conditionals, but whichever one he fulfills will be the one that has a false consequent. In other words, one of the following claims is a lie:
“If you say something true, then I will let you pass.” (If the antecedent is fulfilled, then the consequent is false, because Plato does NOT let Socrates pass).
“If you say something false, then I will throw you into the water.” (If the antecedent is fulfilled, then the consequent is false, because Plato does NOT throw Socrates into the water.)
(3) What should Plato do to keep his vow? He can’t do anything to keep it. You shouldn’t make vows like this.(Though, some have suggested that Plato should first let Socrates cross and then drag him back and THEN throw him into the river!)
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7. Buridan’s Ass (Or, Al-Ghazali’s Dates): Buridan is most famously known for the ‘Buridan’s Ass’ puzzle. Traditionally, the puzzle is as follows:
Buridan’s Ass: There is a donkey standing in between two equally appealing piles of hay. Not having any reason to prefer one over the other, the donkey cannot decide which one to go to, and so instead stands forever paralyzed between them, and starves to death.
Though Buridan penned dozens of puzzles involving donkeys, this particular one is not actually among them—though, it does seem plausible that he would say this, as it fits with his views about choice and action. As it turns out, the best example in medieval philosophy of this sort comes instead from Al-Ghazali (though it seems to originate in Aristotle). Al-Ghazali asks us to imagine the following case:
Ghazali’s Dates: There is a hungry man, and before him are two equally ripe dates. Neither of them is prettier than the other; or nearer; or better in color; and so on. In short, there is nothing to distinguish them. What does he do?
Ghazali says that we must respond in one of the following three ways:
(1) Claim that the man takes one of the two dates arbitrarily.
Problem: Every choice is made for some REASON. But, there is NO reason to prefer one over the other (the balance of reasons between them is EXACTLY 50-50), so there is no reason to explain why the man chooses one date and not the other.
(2) Claim that the man is unable to prefer one over the other, and so remains paralyzed by indecision—motionless until he starves to death.
But, Al-Ghazali dismisses this possibility as absurd.
(3) Object to the example on the grounds that it is impossible, claiming that there could never be a situation where two options provide one with an EXACT balance of TOTALLY equal reasons in favor of each.
Al-Ghazali dismisses this too, claiming that he has just STIPULATED that it is so.
Ghazali was interested in this question because philosophers had asked whether God could have created the universe one year sooner? Or two? God seems to have no reason to have created the universe at exactly the moment He did. Yet He DID. Therefore, it MUST be the case that arbitrary decisions between identical options is possible. In short, Ghazali endorses option (1), above. [What do you think? Is that the correct answer?]
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