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?
20 MarchLogicism revisited: Russell's Principia. Then: Godel.

As for midterm grades: I try to make this course accessible to people who don't have as strong a logic background by making the practices count for much of the grade. The downside of this is that those who skipped some practices may have received a midterm grade that seems very low relative to your test performance. We'll have more practices, and and another test, so a missed assignment or two will not have such a strong effect in the end.22-24 MarchI'll be out of town giving a talk. Here is an assignment that you can tackle on this day. Bring your write-up to me on Monday the 27th.

Assignment:complete the problems on the handout.

27 MarchWe will discuss and review our background, before we introduce Godel's Theorem I and Godel's Theorem II.

29 MarchReview and discussion of possible implications of Godel.

Turing!

31 MarchMore turing!

Note: Philosophy Club meets Friday, March 31, 2017, from 4:00 PM - 6:00 PM, in Marano Campus Center Room 258 to watch and then discuss the filmEx Machina. There will be pizza!3 AprilPlease 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....

As you know, I usually have my office hours 1:40-3:00. But today there is a meeting 1:30-2:30 I need to attend. So, I'll have office hours from 2:30 to 4:30 to make up for that. I hope this is convenient for you.7 AprilHomework: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 email "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.

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

Note:if you are very comfortable with Tursi, and wouldn't mind helping others, let me know. Some people are not familiar with programming but don't have a team or know anyone in class. It might be nice to tell them other people they can talk to besides just me.

In class, we'll discuss The Halting Problem: Impossible Machine Proof.12 April, or thereaboutsHomework: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.

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.14 AprilNo class.

To get a sense of Gregory Chaitin's work, you might 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.)19 April, or thereafterPractice:You can work in teams of 4 or less people for this one and hand in a single homework for the whole team. We're going toapproximateKolmogorov Complexity. We will do two cheats to approximate. First is, for some strings we'll let you print forever. Second is, we don't have a UTM, so we'll instead count characters on the tape before the machine starts (including blanks if you use some specific number of them) and rules in the rule table. Email a text file.

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 that you can have whatever you want on the tape at the start, but we count those as cost for your program). Try to do so with as short a program and/or as few things on the tape as you can manage; extra credit to the team that has the lowest total count for a problem. The three strings are:HINTs: I've been asked if 2 and 3 are impossible. For 2, it can be done relatively easily. Look at the number behind you, copy it, add one. Repeat. For 3: consider, what if it were Chaitin Random? Then you could not compress it. Remember you can both write rules and put something on the start tape. And all you need to ensure is that the tape has that string on it when the machine stops. So what will your start tape look like?

- "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".

Remember where you can get Tursi. On the download page, you should be able to download a copy of the program.21 AprilTursi workshop!

During class, in our classroom (Friday F 12:40 pm - 1:35 pm), several of your peers who have experience with programming are going to be available in class to help/advise/ tutor, or just praise the Tursi program. Come to class to help brainstorm Tursi solutions. If Tursi is easy for you, come to class to help others!24 AprilRead the selections from Wittgenstein. We will discuss radical conventionalism.25 AprilReview hour. I'll be in MCC 232, from 3:40-4:45pm, to review our grand tour of the limits of reason and the sources of reason. I can answer any questions you have, or do our potted history and results in a fast-talking hour! Or we can just discuss where these questions go next, if that is of greater interest to you. (My apologies if the time is not convenient--I picked the latest available time hoping that was best for the greatest number.)