Assignment for 2/29

Please read Max Black's article, "Achilles and the Tortoise". It is available from the course website. In it he claims (and subsequently argues for) the following:

"(Though a) mathematical solution (to Zeno's paradox of Achilles and the Tortoise) has behind it the authority of Descartes and Pierce and Whitehead--to mention no lesser names--. . . I cannot see that it goes to the heart of the matter. It tells us, correctly, when and where Achilles and the tortoise will meet, if they meet; but it fails to show that Zeno was wrong in claiming that they could not meet."

(Black 1951: 122; editing by Chris Tillman, emphasis in the original.)

Is Black's suggestion that a purely mathematical solution is inadequate correct? Why or why not? For next class, please briefly state and explain/defend your answer.

Assignment for 2/29

Please read Max Black's article, "Achilles and the Tortoise". It is available from the course website. In it he claims (and subsequently argues for) the following:

"(Though a) mathematical solution (to Zeno's paradox of Achilles and the Tortoise) has behind it the authority of Descartes and Pierce and Whitehead--to mention no lesser names--. . . I cannot see that it goes to the heart of the matter. It tells us, correctly, when and where Achilles and the tortoise will meet, if they meet; but it fails to show that Zeno was wrong in claiming that they could not meet."

(Black 1951: 122; editing by Chris Tillman, emphasis in the original.)

Is Black's suggestion that a purely mathematical solution is inadequate correct? Why or why not? For next class, please briefly state and explain/defend your answer.

Assignment for 2/27

Please extract an argument from Zeno and present it in numbered premise-conclusion form. State which premise is least plausible and why. Extra credit will be given to extractions of more difficult arguments.

Assignment for 2/22

For Wednesday, please read the Zeno fragments. (Pages 7-9 of Reading 2.) Also, please extract an argument from Aristotle against Parmenides. (Pages 8-11 of Reading 3. Do not use the same argument that you used for your paper.) Present the argument against Parmenides as a valid argument in numbered premise-conclusion form. State which premise you find least plausible and why you find that premise least plausible. Feel free to post your argument or any questions if you get stuck.

Parmenidean Arguments

Existence and Thought
1. There are no non-existent objects.
2. If (1), then there are no non-existent objects that have any properties.
3. Therefore, there are no non-existent objects that have any properties. (1,2 MP)
4. If (3), then whatever has any properties exists.
5. Therefore, whatever has any properties exists. (3,4 MP)
6. If (5), then if A has the property of being spoken or thought of, then A exists.
7. Therefore, if A has the property of being spoken or thought of, then A exists. (5,6 MP)
8. If (7), then if A can be spoken or thought of, then A exists.
9. Therefore, if A can be spoken or thought of, then A exists. (7,8 MP)
10. If it’s true that if A can be spoken or thought of, then A exists, then it’s true that if A does not exist, then A cannot be spoken or thought of. (Law of logic)
11. Therefore, if A does not exist, then A cannot be spoken or thought of. (9,10 MP)

No Generation Argument
12. If (11), then it’s not the case that what is came from what is not.

Explanation: Call what is ‘Gary’ and what is not ‘Bob’. Then the idea that Gary was generated/born/had a first moment is the idea that Gary came from Bob. But now suppose (11) is true. Then Bob is nothing; i.e., Bob is non-existent. So Bob cannot be spoken or thought about. So there is no proposition that Gary came from Bob. So it’s not the case that Gary came from Bob. So (12) is true.

13. Therefore, it’s not the case that what is came from what is not. (11, 12 MP)


Parallels to the No Generation Argument may be given for the conclusion that what is is indestructible.

No Change/Motion Argument
19. If what is changes, then what is goes from not being F to being F.
20. If what is goes from not being F to being F, then there is a state of affairs that what is is not F.
21. Therefore, if what is changes, then there is a state of affairs that what is is not F. (19, 20 HS)
22. If (11), then there is no state of affairs that what is is not F.
(Explanation: Call the fact or state of affairs of what is being F ‘Roy’. Then the thought that what is is not F is the thought that Roy does not exist. But by (11), it cannot be thought that Roy does not exist.)
23. Therefore, there is no state of affairs that what is is not F. (11, 22 MP)
24. Therefore, what is does not change. (21, 23 MT)
25. If what is does not change, then what is does not move.
26. Therefore, what is does not move. (24, 25 MP)

No Plurality Argument
27. If there is more than one thing, then something is true of one thing and not another.
(Explanation: Suppose that there are at least two things, A and B. If A is distinct from B, then A must have some property that B does not have. One plausible candidate for the difference is that A must be located where B is not. An even more plausible candidate is that A has the property of not being B. A must have this property, given our assumption. So (27) is true.)
28. If something is true of one thing and not another, then there is a state of affairs that something is not F.
29. Therefore, if there is more than one thing, then there is a state of affairs that something is not F. (27, 28 HS)
30. If (11), then there is no state of affairs that something is not F.
31. Therefore, there is no state of affairs that something is not F. (11, 30 MP)
32. Therefore, there is not more than one thing. (29, 31 MT)

Writing Resources

I have created a page with writing resources that you should use in constructing your final prose draft of essay 1. The page is also available from the course website.

Aristotle and Pseudo-Aristotle readings are up on the course website. They are not pretty but they are readable. Additional copies are available at your friendly neighborhood library.

First Parmenidean Argument

1. There are no non-existent objects.
2. If (1) is true, then there are no non-existent objects that have properties.
3. Therefore, there are no non-existent objects that have properties.
4. If (3) is true, then whatever has any property exists.
5. Therefore, whatever has any property exists.
6. If (5) is true, then if A has the property of being spoken or thought of, then A exists.
7. Therefore, if A has the property of being spoken or thought of, then A exists.
8. If (7) is true, then if A can be spoken or thought of, then A exists.
9. Therefore, if A can be spoken or thought of, then A exists.

Here is the proper symbolization of (1-9):
1. A
2. A --> B
3. Therefore, B
4. B --> C
5. Therefore, C
6. C --> (D --> E)
7. Therefore, D --> E
8. (D --> E) --> (F --> E)
9. Therefore, F --> E

Here are explanations for each of the premises:

Explanation of (1): First a word about existence and identity. If A exists, then something is identical with A. In addition, if x = y, then x is F if, and only if, y is F. Finally, 'there are Fs' means that there exist Fs. Now for the reasoning in favor of (1). Suppose (1) is false. Then there exist objects that are non-existent. But that is impossible, since it is contradictory for there to exist objects that are non-existent. So (1) is true.

Explanation of (2): A property is any feature, attribute, or way that something is. Now note that (2) is a conditional ("if-then") premise. So, as with explaining all conditional premises, we must assume the "if"-part and give a reason for accepting the "then"-part. So assume that (1) is true. Then there exist no objects that are non-existent. But if that's so, then there exist no objects that are non-existent that have properties. (Compare: if there are no dogs, then it follows that there are no dogs that have fleas. If I have no dinner, then I have no steak dinner. Etc.) So (2) is true.

Explanation of (3): (3) follows from (1) and (2) by Modus Ponens.

Explanation of (4): (4) is another conditional premise. So assume that (3) is true. Suppose now that A has properties. Then either A exists or A does not exist. If A does not exist, then there is a non-existent object that has properties. But this contradicts our assumption, (3). So A exists. But there is nothing special about A. The same reasoning applies to any other thing. So (4) is true.

Explanation of (5): (5) follows from (4) and (3) by Modus Ponens.

Explanation of (6): Assume (5) is true. Assume A has the property of being spoken or thought of. Then A must exist; otherwise, we contradict our assumptions. So (6) is true.

Explanation of (7): (7) follows from (5) and (6) by Modus Ponens.

Explanation of (8): Assume (7) is true. Assume A can be spoken or thought of. Then A has the following property: being something that can be spoken or thought of. But then, by (7), it follows that A exists. So (8) is true.

Explanation of (9): (9) follows from (7) and (8) by Modus Ponens.

For next class (Wednesday 2/8) please workshop your peer's Parmenides argument. This involves doing the following:

1. Symbolize your peer's argument. (Refer to the argument handbook if you're not sure how to do this.)

2. Determine whether your peer's argument is formally valid.

3. Determine whether you think Parmenides would accept the premises that your peer attributes to him.

4. On the basis of (1-3), give suggestions to your peer for improving his/her argument.

Proof vs. Justification

I would like to try to clarify a crucial point that came up in class today. Consider the following argument for an anti-Thalean conclusion:

1. Water is H2O.
2. Some things are not H2O.
3. Therefore, (1) and (2). (from 1,2 by Conjunction)
4. If (3), then not everything is water.
5. Therefore, not everything is water. (from 3,4 by Modus Ponens)

(1-5) is a valid argument. Its premises are plausible--we do not have good reason to doubt them. So we are plausibly justified in believing that they are true. But since the argument is valid, we are plausibly justified in believing that the conclusion is true as well. So barring a good argument that one of the premises in (1-5) is false, we are plausibly justified in accepting the conclusion. So we should accept the conclusion.

My point in class is that, though I believe the foregoing is correct, we should not believe either (a) that (1-5) is a *proof* of its conclusion, in the mathematical or strictly logical sense, or (b) that the argument is no good unless Thales (or a Thalean) would be *forced* to accept all of its premises. Anyone can be so stubborn so as to resist a premise. But they are not playing the same game we are. The game is not to force others to believe so and so; rather, it is to figure out what we are plausibly justified in believing, given our evidence.

Some further comments are in order.

First, even if we accept (1-5), we are not entitled to *ignore* those who do not. Recall that it's plausible to accept the premises in (1-5) provided we lack good reasons to doubt them. Now if Thales, or a Thalean, comes along and gives a valid argument for the conclusion that some premise in (1-5) is false, and backs up the premises with reasons, then we no longer lack good reasons to doubt the premise. Thus, we should either no longer accept the premise in question or we should find what is wrong with the objection to it. A nice thing about validity is that we know that if the conclusion of the objection is false, one of the premises must be false as well. Takeaway point: THe goal of rational inquiry is to determine what is most plausible. The goal is not to make those who disagree cry "Uncle".

Second, different people are justified in believing different things. You have a justified belief about the color of your underwear, and I do not have a similarly justified belief, since you have evidence about the color of your underwear that I lack. What is reasonable to believe is what is best supported by your evidence. That evidence is different for different people. So two people can reasonably believe different things. But when the question is important for rational inquiry, our goal is to find out what is correct. The best way to proceed in figuring out what is correct is to consider arguments. Those for whom the evidence supports answer A should give their reasons (arguments) for thinking that A is the case. Those for whom the evidence supports answer B should give their reasons (arguments) for thinking B is the case. If it cannot be the case that both A and B are correct, the arguments will help us sort out whether it is more plausible that A is the case or that B is the case. Takeaway point: Rational inquiry is about sharing evidence to better justify our beliefs in answers to the questions that we are interested in (be they scientific, philosophical, or what have you).

Third (and finally!), given the above, it is hopefully clear that the following objection is no good:

Argument A does not prove its conclusion. Therefore, argument A is no good.

In the mathematical/logical sense of "proof", there is precious little that can be proven. (This is so despite the fact that infinitely many propositions can be proven in the strict mathematical/logical sense! What I mean is that few of the conclusions that we encounter on a day to day basis or in science or philosophy admit of anything like mathematical/logical proof.) In the strict sense of "proof", no one can prove that you have a face or that the sun will come out tomorrow. But that no one can prove such things is pretty clearly a poor reason for you to believe that you have no face and that the sun will not come out tomorrow.

I apologize about the length of this post; I think these issues are of vital importance, however. While the picture of rational inquiry I have outlined might be wrong in detail, and is certainly debatable (recall that we can rationally inquire about rational inquiry itself!), I believe it is substantially correct. Please feel free to comment about any aspect that you agree with/disagree with/find unclear/etc.

Today (2/1) we considered an objection to Thales on behalf of Anaximander. We thought it might work pretty well against the view that everything is made of water but not so well against the view that everything comes from water. Anaximander seems to have thought that the argument against Thales was sound and that his view--that everything came from something with no definite properties--avoids the objection he raised against Thales as well as relevantly similar objections. Is Anaximander right about this? Why or why not?

You are encouraged, but not required, to present and explain a valid argument for your conclusion. If you opt not to present and explain a valid argument, you must write a paragraph or two stating and defending your answer. You need not write more than that.

For next class, please read the Melissus fragments.

Also, please feel free to post or comment on any of the issues raised in class today. That is in large part what the blog is for. Contrary to popular belief, philosophy is not best done in an attic by one's self. It is best done when there is an open exchange of ideas. (If you need a less idealistic motivation, remember as well that you are required to make a certain number of substantial posts/comments and that those who make more than the required amount will be rewarded for their efforts.)