PHL309 Logic, Language, and Thought Professor: Craig DeLancey

Office: Marano 212A

Email: craig.delancey@oswego.edu

Past Assignments23 JanuaryIntroductions. Review of our themes. Reminders about the basics of logic.25 JanuaryTwo readings, and a quick assignment.

Reading 1:Read the Allegory of the Cave section of Plato'sRepublic. You can read this by reading the first half of book 7, which is available for free here. Or, I have a trimmed version here.

Reading 2:read the short dialogueThe Menoby Plato. If you are pressed for time, just read the part where Socrates talks to the slave boy. You can find this part by searching for "Boy." and you'll find where he first interacts with the child. (From: the line "Men. Certainly. Come hither, boy. " to the line: "Men. Yes, they were all his own."). (Noelle found a typo! "right feet" should be "eigth feet"!)

Assignment:what is it that Socrates ends up drawing in the dirt with the boy? Can you reconstruct in drawings the steps of his proof? Try to do so on a piece of paper, along with a brief explanation of the steps and what they show. Hand this in at the beginning of class. Imagine you are trying to explain to someone you know: what Socrates does in his discussion with the boy.

If you feel you need a resource for reviewing logic, at any time during the semester, you can use my book, which is free.27 JanuaryPlease read Book 1 ofEuclid's Elements. You are reading just to familiarize yourself with how Euclid thought and argued. You'll find you know everything he says; just note how he arrives at his conclusions. An online version can be found here.

Come to class prepared to answer these questions.

- Is Euclid a Platonist? That is, how do you think that he believes he knows the things he claims to know?
- Consider postulate 5. As we'll see this semester, this is a very important postulate. Do you agree with it? That is, in your view, is it true? If you believe it is true, how do you think you know that it is true? If you doubt it, why would you doubt it?
30 JanuaryReading. Read the selection from Galileo'sTwo New Sciences: First Day, from pages 11 [49] to 28 [67].

I'll email you the prefered translation. If you don't get that email let me know.

Also, a fair translation is freely available here.

Short reading question: A brief question for you to answer in a single page. Galileo considers a reason why some things may hold together. He uses the two slabs of marble example to illustrate the idea. What is this reason? Do we consider it today sufficient explanation of why things hold together?

1 FebruaryReading. Read from Galileo'sTwo New Sciences: First Day, from pages 28 [67] to 35 [73].

3 FebruaryReading. Read from Galileo'sTwo New Sciences: First Day, from pages 35 [73] to 44 [82].

This is the most important section, for our purposes.

Practice:answer the following question and bring it to class to hand to that guy up front: There are at least five arguments here in our Galileo selection against either infinitesimals or actual infinities. Describe one of these arguments carefully as a reductio ad absurdum argument. What is he trying to show? What is the assumption for reductio (the assumption of the opposite of what he's trying to prove)? What are the premises (the things he assumes). What is each step of Galileo's reasoning? What's the contradiction that he finds?

I apologize, but I cannot have my office hours today. I have an afternoon conflict.

Also, Philosophy Club will meet in MCC 211. Free pizza!

6, 8 FebruaryWe'll continue reviewing Galileo's arguments, talk about how lots of people worried about infinitesimals, and begin our discussion of Kant!

10 FebruaryWe will continue our discussion of Kant.

13 FebruaryPractice/homework!:Due at the beginning of class: A quick homework. This will require you to hand in 7 sentences. Give an example of a sentence for each of Kant's four kinds, and also an example of a sentence for each of the cross-category kinds. Each sentence example must be your own (no credit for an example we used in class). So, you must provide an example of your own of a sentence that is:If you disagree with Kant's notions, consider yourself as trying to find examples that he would accept.

- a priori,
- a posteriori,
- synthetic,
- analytic,
- analytic a proiri,
- synthetic a posteriori,
- synthetic a priori.

Remember from our discussion:Look at the first two pages of our Lecture Notes on Kant for examples.

- "Analytic" means that you can tell if the sentence is true or false based on the meaning of the parts of the sentence.
- "Synthetic" means the sentence combines (synthesizes) new things, so that you cannot tell if the sentence is true or false based on the meaning of the parts of the sentence.
- "A priori" means that you can know whether the sentence is true or false before having to check experience.
- "A posteriori" means that you can know whether the sentence is true or false only after having checked experience.

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Please don't hesitate to reach out with any questions. If you are able to pass this along please let me know. Thanks in advance for your help.

This email was sent by Climate Reality Project, located at 1 Rockridge Place, Oakland, California, 94618 (USA). Our telephone number is (908)-477-3964.20 FebruaryHomework! Working on your own, write up your answers to the following questions, and hand them in at the beginning of class.

- For each claim, identify if it is true or false.
a. 2 ∈ {1, 2, 3}

b. {c} ∈ {{a}, {b}, {c}}

c. 1 ∈ {{1}, {2}, {3}}

d. {a} ∈ {a, b, c}

- For each of these pairs: are they identical? (You can answer "yes" or "no".)
a. {1} and {{1}}

b. {2, 3, 1} and {1, 2, 3}

c. {7} ∩ {9} and {7, 9}

d. {7} ∪ {9} and {9, 7}

- Give an example of a
propersubset of each of the following sets.a. {1, 2, 3}

b. {a, b}

c. {{1}, {2}, {3}}

d. {1}

- For each of the following sets, what is its Powerset? (Remember also our rule, if a set has n things in it, then it has 2
^{n}subsets.)a. {1, 2}

b. {c, d, e}

c. {}

d. {{}}

- Assume these sets continue in the simplest way as listed. For problems a and b, can you give an example of a function that relates the first set to the second set in a 1-to-1 correspondence (an on and onto function that would also be a function if it went backwards)? For problem c, can you find a function that is on the first set and onto the second set, even if it is not 1-to-1? (You can answer these using some basic arithmetic -- no need to get fancy; for example, the function relating the naturals to the even numbers would look like: f(x) = 2x.)
a. {2, 4, 6, 8, 10....} and {200, 400, 600, 800, 1000....}(Note that for c, the second set contains only one element, namely the number one.)

b. {1, 2, 3, 4, 5....} and {1, 3, 5, 7, 9, ....}

c. {1, 2, 3, 4, 5....} and {1}

Joseph was kind enough to send us the following video. I'm a little concerned that it called numbers cardinals (that's a little confusing), and that it often mixes ordinals and cardinals in explanations. I'm hiding ordinals from you, because we don't need them, and they are complex and challenging. But if you feel like jumping into a different perspective....

22 FebruaryWe'll review the homework, the Diagonal Argument, and then Cantor's Theorem.

24 FebruaryIn class: Let's review our road up to this point. Then, let's review the proof of Cantor's Theorem. What are it consequences?

27 FebruarySome more set theory review. We want to keep our brains sharp!Some folks have asked me for logic resources. My book is free and can be read here. The last chapter includes a brief overview of set theory.

- What is {a, b} ∩ {b, c}?
- What is {a, b} ∩ {}?
- What is {a, b} ∪ {b, c}?
- What is {a, b} ∪ {}?
- If x ∈ A, then is the following true? x ∈ A ∩ {} (Explain your answer.)
- If x ∈ A, then is the following true? x ∈ A ∪ {} (Explain your answer.)
- Infinite sets can have proper subsets of the same cardinality. Name a proper subset of the natural numbers that we have not discussed in class as an example, that has the same cardinality as the natural numbers. Describe the function that shows that it is the same cardinality as the Naturals.

In class, we're going to talk about Cantor's Antinomy.

1 MarchWe will continue our discussion of Logicism. ReadLogicomixif you have it.3 MarchOptional practice problems to help you for the midterm. If you want to bring me your answers I will review them.

- What are these powersets? That is, write out their contents.

- P({1})
- P({2, 3})
- P({4, 6, 8})
- P(P({}) (the power set of the power set of the empty set)
- P({{1, 2}})
- Why do I know that the set of odd numbers is a proper subset of the set of natural numbers?
- Is there a function
onthe odd numbers andinto(but notonto) the natural numbers that is one-to-one? If all you know is that there is a functiononthe odd numbers andintothe natural numbers, and this function is one-to-one, then what does this tell me about their respective cardinalities?- And, what are the respective cardinalities of the odd numbers and the natural numbers? Give the most precise answer (that is, maybe the best answer is not the answer to the previous question). How do you know? Explain.
6 MarchReview.

- Our guiding questions.
- Platonism, Kantianism, Logicism, Formalism, Intuitionism.
- Galileo's arguments against actual infinities and infinitesimals.
- Cantor: basic set theory, Cantor's Claim, Cantor's Theorem, The Diagonal Argument, the Cantorian Antinomy.
- Frege's project. Russell's paradox.
- Hilbert's questions. Consistency, completeness, decidability.
7 MarchI'll be around and available in my office from 9 a.m. -- 1:00 pm.

Here is a lecture on YouTube in which the first 32 minutes covers some of our Cantor material. Useful for a review using an alternative perspective perhaps. (I don't like his construction of the crazy number in the diagonal argument, though; we should be precise and say how we will construct such a number.)8 MarchRemember: resources to help you study include our class notes here.

Midterm exam in class. Here are some study questions.

- Reconstruct one of Galileo's arguments that we cannot have an actual infinity or that we cannot have actual infinitesimals. Make your reconstruction an explicit reductio ad absurdum argument, in which you make clear the contradiction, and the premise we reject because of the contradiction.
- Give an example of a sentence for each of Kant's four kinds: a priori, a posteriori, synthetic, analytic. Give an example of a sentence for each of Kant's complex kinds: analytic a priori, synthetic a priori, synthetic a posteriori.
- How do you and I have knowledge about geometry, according to Kant? Why is our knowledge about non-Euclidean geometry a problem for Kant's account?
- Answer some basic questions about set membership, subsets, powersets, the definition of cardinality. What is a powerset? Be able to apply these concepts!
- What is Cantor's Claim (about some proper subsets of infinite sets)? How we can use Cantor's Claim (assuming it works) to answer some of Galileo's arguments?
- Reconstruct Cantor's Diagonal Argument to prove that the cardinality of the reals is greater than the cardinality of the natural numbers.
- What is Cantor's Theorem? Prove it.
- How does Cantor's Theorem, and the claim that a set exists if we can determine its members, result in Cantor's Antinomy?
- What is Russell's contradiction? (AKA Russell's Paradox) In what system was it provable? Why was it a problem for that system that a contradiction was provable in that system?
- Bonus: what does it mean to say that a system is consistent? Complete? Decidable?