Craig DeLancey, PHL309: Past Assignments

PHL309 Logic, Language, and Thought Professor: Craig DeLancey
Office: Marano 212A
Email: craig.delancey@oswego.edu

Past Assignments
3 February
Two readings, and a quick assignment.

Reading 1: Read the Allegory of the Cave section of Plato's Republic. 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 dialogue The Meno by 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"!) "Men." is supposed to be short for "Meno," and of course "Soc." is supposed to be short for "Socrates."

Assignment: Art class! Make two drawings. (1) Can you draw the set-up of the cave allegory? (2) what is it that Socrates ends up drawing in the dirt with the boy, in the Meno? Can you reconstruct in drawings the steps of his proof? 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.

If you feel you need a resource for reviewing logic, at any time during the semester, you can use my book, which is free.
5 February
Please read Book 1 of Euclid'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. The links are clickable!

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?

8 February
Reading Read the selection from Galileo's Two 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.

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

Together in class we will read Book 1 part 2 of the Posterior Analytics. An online version can be found here.

If you can read it before class, and consider the following question: What is an axiom, according to Aristotle? We'll discuss this in class, and we'll also then give background as an introduction to Galileo.
10 February
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? On Blackboard, or Write up your answer and bring it to class.

Reading. Read from Galileo's Two New Sciences: First Day, from pages 28 [67] to 35 [73].

12 February
Read pages 35 [73] to 42 [80] of our Galileo selection. Pages 35 [73] - 38 [76] cover the soup dish, which we dicussed, so you really could skip to 38 [77] - 42 [80]. I think there are at least four additional 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 on Blackboard, or you can bring them on paper to class.
15 February
We begin our discussion of Kant!

A reminder that CircleIn is available for your use. This is an app that lets you help each other with the course, and to get rewards for doing so. The instructions for downloading CircleIn are here.
19 February
Practice/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:

a priori,
a posteriori,
synthetic,
analytic,
analytic a proiri,
synthetic a posteriori,
synthetic a priori.
If you disagree with Kant's notions, consider yourself as trying to find examples that he would accept.

Remember from our discussion:

"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.
Look at the first two pages of our Lecture Notes on Kant for examples.
22 February
We will continue our discussion of Kant.
23 February
I'll have office hours as needed, since I have a series of meetings all day. Just send me an email and I'll arrange a GoogleMeet for us, if you want to ask me anything.
24 February
No class.
1 March
In class, we will continue our discussion of Cantor.
3 March
Homework! 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}
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 proper subset 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 2ⁿ subsets.)
a. {a}
b. {a, b}
c. {a, b, 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 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....}
b. {1, 2, 3, 4, 5....} and {1, 3, 5, 7, 9, ....}
c. {1, 2, 3, 4, 5....} and {1}

(Note that for c, the second set contains only one element, namely the number one.)
In class we'll discuss whether there is just one kind of infinity.
5 March
We'll review the homework and introduce or review the Diagonal Argument.

This will also allow us to respond to all of Galileo's arguments against there being actual infinities.

8 March
Here is a copy of our set theory notes.

In class: we will discuss Cantor's theorem. (This may be helpful: here is both proofs side by side.)

10 March
Here are the new things we've learned which we want to be comfortable with:

Basics of natural set theory and functions
Definition of cardinality
The helpful claim
The Diagonal Argument
Cantor's theorem
All is well with the infinite, right? But.... Cantor's antinomy.

Some of us might need some extra practice with set theory. If you believe you do, do the following and I will grade it for you.

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.

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.
In class, we will continue to discuss Cantor's antinomy. What should our reaction to it be?

12 March
We've encountered non-Euclidean Geometry and Cantor's antinomy. How will we overcome the many questions and doubts that are starting to plague us? Let's continue with Frege's Logicism.
15 March
What comes next? Intuitionism, Logicism, Formalism.
In class today, we'll discuss Logicism.

Things you should know at this point:

The basic facts about functions and sets
Cantor's definition of cardinality
Cantor's Claim
The Diagonal Argument
Cantor's Theorem
Cantor's Antinomy
What Logicism is
Remember we have extensive class notes.

16 March
Office hours: today I'd like to revise my office hours time to 9-11 am. Here's the link: https://meet.google.com/djs-gdyf-xev?authuser=0.

Philosophy club is going to meet online at 6:00 pm, to watch and discuss an episode of The Good Place. I'll post a link here Tuesday. Why don't you join them? To join the video meeting, click this link: meet.google.com/wps-spos-syd.
18 March
In class, we'll review a bit, and then I'll tell you about formalism and the basic idea of intuitionism, in prep for attempting to read Brouwer.
19 March
Read: We are going to read a paper by Brouwer, the most famous of the intuitionists. It's a little bit challenging, but we'll take it slowly and we'll discuss each section together. Read the first half of it, pages 83-88. There is a lot of philosophy, squeezed in between the technical stuff. The paper is available on BlackBoard and I'll email it to you.

Practice: On BlackBoard, 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?

22 & 24 March
We'll discuss Hilbert's problems, and we may get a chance to talk about Russell's version of Logicism.
23 March
Wellness day, no classes. I'm not going to have open office hours, but if you want to talk just drop me an email and I'll arrange a meeting quickly.
26 March
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)? ["Infinite sets can have proper subsets of the same cardinality."] 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.
What is Cantor's Antinomy (or, if you prefer Cantor's Contradiction)? Derive it from Cantor's other claims.
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? (Also known as "Russell's Paradox," but I avoid that name because it's not a paradox, it's a contradiction!)
What is intuitionism? What part of Kant's intuition does Brouwer reject, and what does he keep?
Other questions I've been asked: Will we have to know Frege's axioms? No.
March 29
In class, we will discuss: how does Godel pull off his trick of making the sentence, "This sentence is not provable"?

March 31
Practice: Due at the beginning of class: homework on Hilbert's problems and Godel number. This is on a handout that I will gave you in class.

2 April
Godel's Second Theorem! And then, time allowing, introduction to Alan Turing.
5 April
Introduction to Alan Turing, continued.
7 April
Please take a look at Tursi. On the jar download, you should be able to download a copy of the program. Please do see if it will run on your preferred machine.

Then: the Halting Problem, proof 1.
9 April
In class: the Halting Problem, proof 2.

We will do another example in Tursi.

Re-do: If you want to rewrite any of the answers on your midterm, you may do so and turn them in during class this day. Answer the question correctly, but also explain what you did wrong in your previous answer, and why you got confused. Hand in your test and the new answers. I'll give you extra-credit. You can just hand it in at the beginning of class like a homework.

Here are some other Turing machine simulators:
http://math.hws.edu/eck/js/turing-machine/TM.html
http://morphett.info/turing/
12 April
We will discuss the Halting Result, and Conway's Life.

Here's a fun video of a kind of toy Turing machine.

You might enjoy playing with Conway's Game of Life at https://playgameoflife.com/.

This video will be relevant: a UTMs in the life world.
14 April
No class. Quest.

Three interesting Philosophy talks on the philosophy of science will be give today in Quest at 3:00 pm. Please join us!

Instead of my Tuesday office hours, I will have office hours in person in MCC212A from 10:00 am to 12:00 pm.
15 April
Today I'd like to have my office hours earlier. I hope that's OK. I'll be available from 9:30 - 11:30 at https://meet.google.com/hcj-ehdx-yvs
If that doesn't work for you, email me and we'll find another time.

FYI, I was asked about the midterm redo points. I will count them at 50%. That's a cut, but it still ensures that everyone who wants to in fact did great in the end.
16 April
If you cannot run Tursi, you could write the Turing machine "in your head." Tursi helps us to test program, and can prevent you from making some mistakes, and it lets you "watch" a turing machine running; but it should be possible to write the program as we did in class. In such a case, if it is more convenient for you, you could write the machine on paper and hand the paper in to me.

Homework: 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 DeLancey
# 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

The 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.

You are only handing in this text file; or, if you do the program by hand, then you can just hand in your code written out on a page. 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:

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

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 1
1 1 0 R 1
That's contradictory. Consider instead:
1 1 1 R 1
2 1 0 R 1

In class, we will introduce Descriptive Complexity.
19 April
Incompressibility result.

To 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.)
21 April
Extra-Credit: 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 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.
22 April
My office hours are 9-11 today at meet.google.com/ymu-urzm-nzd
23 April
Before class, read the selection from Kripke's "On Rules and Private Language."

Extra-credit: You 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 to approximate descriptive 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 means you 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:

"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".
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).
28 April
Reading: read parts 1-4 and 6 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.

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"?
We'll spend the beginning of the class discussing structuralism, before turning to Turing.
30 April
Read the first three pages of Searle's paper on BlackBoard, where he introduces the Chinese Room thought experiment. In class: structuralism described. Then: back to the question of AI.
2 May
For those who want to do blind peer review: Give me your first draft, without your name on the paper, electronically. If you do so, I will soon after send you an anonymous draft for you to review.
3 May
The Lucas-Penrose argument.
5 May
Alternatives to Turing AI.
7 May
Which theory is right?

& Review.
10 May
I'll be in my office in MCC212A from 10-12. I'll also keep online office hours at this time, at the following link: https://meet.google.com/svd-gajk-mgr?authuser=0