PHL309 Logic, Language, and Thought Professor: Craig DeLancey

Office: Marano 212A

Email: craig.delancey@oswego.edu

Past Assignments27 JanuaryRead the Allegory of the Cave section of Plato'sRepublic. You can read this by reading the first half of book 8, which is available for free here.

Optional: if you are a virtuous human being and want to impress me, read also the short dialogueThe Menoby Plato.29 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.

Answer the following questions; you can either hand me hardcopy, or answer them on BlackBoard (I'm not sure I set up the BlackBoard correctly! Let me know!).

- 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?
1 FebruaryRead the selection from Aristotle. Come to class prepared to explain what Aristotle's view is on the infinite.3 FebruaryReading. Read the selection from Galileo'sTwo New Sciences: First Day, from pages 11 [49] to 28 [67].

I emailed to you the prefered translation. If you didn't get that email let me know.

Also, a fair translation is freely available here.

A brief question for you to answer: 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?

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

8 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, either on BlackBoard or you can print your answer and bring it to class if you prefer: There are at least five arguments here 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?

15 FebruaryDue at the beginning of class: A quick homework. Give an example of a sentence for each of Kant's four kinds: a priori, a posteriori, synthetic, analytic. Each sentence example must be your own (no credit for an example we used in class). Then, can you give an example of an analytic a proiri sentence; an example of a synthetic a posteriori sentence; and a synthetic a priori sentence. (If you disagree with Kant's notions, consider yourself as trying to find examples that he would accept.) So, that's seven sentences in total.

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.

26 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. 3 ∈ {1, 2, 3}

b. 1 ∈ {{1}, {2}, {3}}

c. {a} ∈ {a, b, c}

d. {a} ∈ {{a}, {b}, {c}}

- For each of these pairs: are they identical? (You can answer "yes" or "no".)
a. {2, 3, 1} and {1, 2, 3}

b. {1} and {{1}}

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}}

- For each of the following sets, list all of its subsets. (Remember: if we do not require that the subsets be proper, you can include as a subset the set itself. Remember also our rule, if a set has n things in it, then it has 2
^{n}subsets.)a. {}

b. {a, b}

c. {{}}

d. {c, d, e}- 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 to the second 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 and onto, even if it were 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}

7 MarchSome practice to help us prepare for the midterm.

- 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 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.
9 MarchReadLogicomix, if you got it.

In class, we'll review and then go back to Frege.

Review:Continuing our history:

- The homework reviewed
N, Q, Z, R- From, On, Into, Onto
- The diagonal argument
Frege. Russell's postcard.11 MarchLet's review the proof of Cantor's theorem before discussing Russell's version of logicism.

If you have Netflix, I've discovered that much of the BBC's documentary "The Story of Maths" season 1 episode 4 is about Cantor, Hilbert, and other folks we've discussed. It streams on Netflix. Much better than The Walking Dead.14 MarchLet's review our various positions and discoveries, before the midterm on Wednesday. Soon we'll start our next topic: a strange answer to our question, how do we know truths of reason?16 MarchMidterm 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?
18 MarchThe Kripkenstein Paradox. Read the handout that you were given in our last class.21-25 MarchBreak!28 MarchThe Kripkenstein Paradox, reviewed and continued. Radical conventionalism as an answer.30 MarchTuring! Start reading Casti'sGodelif you have it. We will introduce the idea of The Turing Machine.1 AprilHere is a fun video of a machine that is meant to be like Turing's imagined abstract machine. Fun to watch for a minute.

We're going to be using a program in these next weeks (although it might be possible to get by without the program). For those of you that have your own computer, can I ask you to try to download and start up the program, to ensure it works for you? It is called "Tursi," and you can download it at:http://ais.informatik.uni-freiburg.de/tursi/downloads.htmlLet me know if it runs or doesn't run for you. Not sure if it will run on pads. I'll try it in our labs also.... If you're using a mac, you'll download it, but if you then click on it you'll get a message denying you access. Press control while you click on it; you'll get a pull down menu; select open.4 AprilThe Halting Problem. Diagonal Proof.6 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. (I tried to set up a drop in BlackBoard, but I don't know if it worked or if it allows you to drop 3 files. Please let me know.)

Thus, your text file can begin with:# 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.

# 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 number.

#! start 0

#! end H

In class, we'll discuss The Halting Problem: Impossible Machine Proof.

- 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.)
8 AprilIn class: Implications of The Halting Problem. Conway's Game of Life and UTMs in the lifeworld. Predictable ≠ Deterministic. Review of our results and practice of Turing Machines.

We'll review the homework.

Then we turn to Godel.11 AprilSecond homework on Turing machines.

Homework:making two Turing machines. You must work alone on this homework. What you hand in will be two text files. In Tursi, make the following:In class, we will discuss Godel's First Theorem.

- a machine that will tell whether a number is even or odd.
- a machine that will double a number.

If you got the book by Casti,Godel, read chapters 1-4. Additional resources include the Stanford Encyclopedia entry on Godel; this is often technical, but the first few sections provide some accessible background.

Review. Then: Godel's Second Theorem.13 AprilQuest!15 AprilHomework on Godel due: You can download it here.

21 AprilK-complexity and randomness

25-29 AprilI will be in Arizona at the big Consciousness Conference. However! We will still have class online--we'll log into blackboard, I'll record a lecture, and we can discuss in a usergroup. So don't plan to use our class hour for anything else!

Our topic will be thought.25 AprilReading: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 some questions that are on blackboard or could be emailed to me:Later, I'll post some notes, including regarding Descriptive Complexity.

- 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"?

27 AprilReading:read part 6 of Turing's "Computing Machinery and Intelligence." A version is available here.

Practice: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 or use BlackBoard to send 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? If you need more time, I will accept machines until Friday.

- "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".

29 AprilRead The Lucas Argument.

2 and 4 MayFirst: are we computers? Reviewing Turing's speculations about artificial intelligence. The Lucas-Penrose Argument. We might review the Chinese Room argument. Quantum Computation?

Then, or perhaps at the same time: a bit of review and summary. Our results on limits, and what they seem to mean for human reasoning. Review of the impossible machine proof of the Halting Result, and review of the Incompressibility Result. Review of your Turing machines!6 MayWe will review the Lucas Argument, and discuss the Penrose argument. Then: What have we learned about logic, language, and thought?

Please complete today the online evaluation of my professoring. You should have received a link in email. I don't see your evaluations until after final grades are turned in.9 MayI'll have office hours 1:30 - 4:00 pm.