PHL309 Logic, Language, and Thought Professor: Craig DeLancey

Office: Marano 212A

Email: craig.delancey@oswego.edu

Past Assignments29 AugustTwo 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."). (Typo! "right feet" should be "eight 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.31 AugustPlease 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?
5 SeptemberReading. 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 his reason? Do we consider it today sufficient explanation of why things hold together? Write up your answer and bring it to class.

Fun to see: scan of the original "Sidereus Nuncius".

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

Philosophy club meets at 4:30pm in MCC 211. The meeting will be a showing of the 2017 movie Annihilation.12 SeptemberRead pages 35 [73] to 42 [80] of our Galileo selection. I think there are at least five arguments in these few pages meant to prove that we cannot reason about infinity or infinitesimals. This is the important part! Read it closely!

Homework:Identify and describe two (2) of Galileo's arguments that we cannot have or reason about infinity or infinitesimals. Each is a reductio ad absurdum argument. Describe it is such, identifying the contradiction that he identifies. Write your answers up to be handed in.

14 SeptemberConsider, and have an opinion about the following question: What is Galileo's solution to his paradoxes about infinity?

We'll consider that question as we continue reviewing Galileo's arguments, talk about how lots of people worried about infinitesimals, and then begin our discussion of Kant!

17 SeptemberWe begin our discussion of Kant!21 SeptemberPractice/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 or in the class notes!). 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.
24 SeptemberWe will continue our discussion of Kant.

Someone just pointed out to me this fun video series covering some of the stuff we're discussing:

- This one discusses postulate 5.
- This one discusses Bolyai on non-Euclidean geometry. It uses the word "true" in a problematic way (we'll discuss this) but it's funnish.
3 OctoberHomework! Working on your own, write up your answers to the following questions, and hand them in at the beginning of class.One of you was kind enough to send the following video. If you feel like jumping into a different perspective....

- 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}

e. {} ∈ {a, b, c}

- 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}

e. {} ⊆ {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}

e. {} ∪ {9, 7} 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}

e. {1, 2, 3, 4, 5, ....}- 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. {a}

b. {a, b}

b. {a, b, c}

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}

In class, we'll organize your table and see if that helps us with our next topic: Cantor's theorem.5 OctoberWe'll review the homework, the Diagonal Argument, and then Cantor's Theorem. What are the consequences of Cantor's theorem?

10 OctoberWe will continue to discuss Cantor's antinomy and Frege!

Things you should know:Remember we have extensive class notes.

- Basic facts about functions and sets
- Cantor's definition of cardinality
- Cantor's Claim
- The Diagonal Argument
- Cantor's Theorem
- Cantor's Antinomy

If you purchased the optionalLogicomix, it covers the period we are discussing now. Read it! (If anyone wants to borrow my copy, let me know.)

Also, here is our set notes handout.

12 OctoberWe agreed some more practice would help. So here is a brief homework for you to hand in at the beginning of class.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.

- Write out on a single page the following sets (where P(A) means the power set of A):

- P({1})
- P({2,3})
- P(P({}))
- P({{1, 2}}) (this is a trick question! Pay attention to what the set is, and what its member(s) are/is.)
- Intersection and union.

- What is {a, b} ∩ {b, c}?
- What is {a, b} ∩ {}?
- What is {a, b} ∪ {b, c}?
- What is {a, b} ∪ {}?
- General questions

- 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.

22 OctoberMidterm 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) Why does it arise? What does it mean for Frege's logicism?
- What are consistency, completeness, and decidability?

24 OctoberWe'll discuss Godel and how he made the Godel sentence. I will hand out a homework sheet for Friday's homework.

25 OctoberSteinkraus talk, "Climate Justice and Carbon Renewal" in MCC room 132 from 4-5 pm. Come along! It will be interesting to get a philosophical perspective on a problem that will define your future.

26 OctoberDue at the beginning of class: the Godel Numbering homework.

Here is the homework handout.

In class, we'll continue to discuss the question: how did Godel pull this trick off?

31 OctoberRead "Intuitionism and Formalism" by L. E. J. Brouwer, pages 81-87. The article is available on BlackBoard. I will also email it to you.

(Page 88 includes a discussion of proving consistency that I think is on to an important point. You might give it a try.)

(Pages 88-90 describe the Burali-Forti paradox; we have not discussed ordinal numbers, so this is hard going. But note on page 90, first full paragraph, he describes the escape that the formalists came up with--a new set-formation principle.)

Either on BlackBoard, or by bring a page to class, answer the following questions.

- How does Brouwer characterize the different approaches to "mathematical exactness" (see page 83)?
- Where does he find the origin of intuitionism? (The "old form" of intuitionism.)
- What do you think Brouwer means by "consciousness of delight"? Why does he think the formalist does away with it?
- What does Brouwer think was the most serious blow to Kantian intuitionism?
7 NovemberPlease take a look at Tursi. On the download page, you should be able to download a copy of the program. Please do so and make sure that it will run on your preferred machine. It is also possible to make it run on any machine on campus university machine, I think....

Jim Spagnola has volunteered to be a help desk, when he can, for people struggling to get Tursi to run. Very often, he'll be at Shineman 425 or very nearby (on that hall). You can email him at jspagnol@oswego.edu. (You may recognize other folks from PHL309 in that same region of Shineman, who can also help you.)9 NovemberHomework:making two Turing machines. You may work in teams of 3 or fewer people. What you hand in will be two (or three) text files. In the comments section, include the team members and also how the tape must be prepared. (Some people are confused about this. A text file is a kind of file--Word files and rtf files will not open in Tursi. In Word, choose "save as" and then choose "text" to make a text file. If you are confused, email me.)

Thus, your text file can begin with something like:# My team includes: Craig DeLanceyThe first five lines are comment lines. They tell who is on the team and also how the tape must be prepared and how to find the answer. The sixth line specifies that the start state will be 0. The last says that the halting state will be H. You can of course choose your own symbols for start and halt states. Below all that, you list the rules.

# This addition machine adds two numbers represented in unary

# as a series of 1s, separated by a single space. The machine

# starts on the leftmost 1 of the left number. The machine ends

# under the the rightmost 1 of the answer number.

#! start 0

#! end H

You are only handing in this text file. The Tursi program is just to test your program. You don't write inside the Tursi program--except to put something on the tape that you can test.

(Here is the simple example we made together in class. Look at it for a model.)

The machines you make will be:EMAIL your files to me. That way, I can load them up myself. Please name your text file "LASTNAME.1.txt" and "LASTNAME.2.txt". If you are a team, just pick one last name. I just need to be able to keep them separate from other people's homework, so this is important.

- a machine to tell if a number is odd or even.
- a machine to tell if a number is evenly divisible by three.

I was asked for some advice on a few things.

- It's best if the answer is on the tape (as opposed to saying the answer is in the state of the machine itself). You can say how the answer is put on the tape.
- You can use whatever tape alphabet you want! You are not limited to 0 and 1!
- A common error is to give the machine conflicting messages. This will be rejected by Tursi. The whole point of the machine having internal states is so that you can do different things when handling the same input. Suppose sometimes you want the machine to write 1 when reading 0, and sometimes sometimes you want the machine to write 0 when reading 0. You cannot say:
1 1 1 R 1That's contradictory. Consider instead:

1 1 0 R 11 1 1 R 1

2 1 0 R 116 NovemberHomework:make two Turing machines. You may work in teams of 3 or fewer people. What you hand in will be two (or three) text files. In the comments section, include the team members and also how the tape must be prepared.

The machines you make will be:

- an addition machine; the machine must be able to handle 0+0, 0+n, n+0, and any other two positive numbers.
- a subtraction machine for two number n and m where n ≥m (this restriction makes it easier!).
- extra credit! A multiplication that multiplies two numbers n*m. (This one is much easier if you allow yourself a bigger alphabet than just 1 and 0; but I'll be very impressed if you can do it with 1 and 0.)
Name your text file that you hand in: LASTNAME.3.txt and LASTNAME.4.txt and even LASTNAME.extra.txt. Remember, you can use whatever alphabet you want. Sometimes a problem is much easier when you add some things to your alphabet.

We had discussed Conway's life. Here is a video of one of the UTMs in the life world.19 NovemberRead the handout from Wittgenstein that I will give you. This will be the last answer that we will consider to our question, How do we know truths of reason?19 NovemberTo get a sense of Gregory Chaitin's work, read this popular piece he wrote for Scientific American. (FYI: Chaitin's claim that Godel's Theorem is related to his complexity work is controversial.)26 NovemberReading:read parts 1-4 of Turing's "Computing Machinery and Intelligence." A version is available here. Part 5 is optional. What does Turing mean by "the imitation game"? Be able to describe it. Here are the quiz questions you'll be asked in the class quiz at the beginning of class:

- Why does Turing want to avoid trying to define intelligence? What challenges do you think that there might be to defining "intelligence"?
- Describe the imitation game (now called the Turing Test).
- What does Turing mean by "machine"?
28 NovemberYou can work in teams of 3 or less people for this one and hand in a single homework for the whole team. We're going toapproximatedescriptive complexity. We don't have a UTM, so we'll instead count characters on the tape at the beginning, and states and rules in the machine. Using for each as small an alphabet as you can manage, make three turing machines and start tapes that can "print" (that is, will leave the tape such that on it there is) the following three strings. Of course, that meansyou hand in (1) a rule table and (2) a start tape condition for each string (including the contents of the tape at the start of the program--that is, you can put whatever you want on the tape at the beginning). Try to do so with as short a program and as few things on the tape as you can manage; extra credit to the team that has the lowest total count for a problem. If you like, assume that the tape comes completely full of 0s before you add anything to it (that is, each square has on it already a 0, so they don't count unless they are between other characters -- as if you bought the tape pre-zeroed). The three strings are:Name the files LASTNAME.6.txt, LASTNAME.7.txt, and LASTNAME.8.txt. The comments MUST explain how to prepare and interpret the tape, and what if anything should be on the tape. Extra credit to the simplest machines (defined by rules+square on tape used in preparing to run the machine).

- "10101010..." forever. (Extra credit, print also just "1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010", that is, make a machine that will instead just print "10" fifty times only.)
- "10110111011110111110..." forever.
- "10010101010010111110010000001110110110010010010110100101001000011001001001011101001001010110100100010".

Extra credit!58+67 extra points if you write a machine to compute the quus function. Call it LASTNAME.QUUS.txt.Also on 28 NovemberReading:read part 6 of Turing's "Computing Machinery and Intelligence." A version is available here.30 NovemberWe will continue with the Turing test, and then discuss the Lucas-Penrose argument.3 DecemberWe will discuss the Lucas-Penrose argument, and time allowing, we can discuss quantum computation.5 DecemberWe will discuss quantum computation, from 10,000 feet, as part of our discussion of alternatives to the Turing claim that computation can be thought.

You may find it interesting to watch the video here is a popular video with Penrose, in which he gives a version of his idea that computers cannot think. (Also interesting is the interview here.)

Time allowing, we can discuss practical limits to reason (as opposed to in-principle limits).6 DecemberI will be in my office from 9-12 and 1-4. If I can help in any way, stop in and see me.7 DecemberReview! What have we learned?

FYI: James S was kind enough to share the following fun videos on material relevant to our class:

This one on Non-Euclidean Geometry.

This one on Quantum computation.10 DecemberI'll have office hours 10:00 - 11:00 am.