The test is open-everything with the sole limitation that you neither solicit nor give help while the exam is in progress.

This is an exam, so you have a severe time limit. Please make sure your answers are brief. Do all work on this exam sheet.

Choose any 10 of the 11 problems. Write the numbers of the 10 problems you want graded in the obvious boxes below.

1. What is the characteristic difference between the way a human would go about proving that UIIM were or were not a theorem of the MIU-system and the way a naïve computer program would attempt such a proof?
2. What does it mean for a formal system to be undecidable?
3. Consider the following formal system:
   • if x ∈ {a,b,c}, then x and xx are theorems
   • if x ∈ {a,b,c} and y is a theorem, then xyx is a theorem
   1. Characterize the set of theorems of this system.
   2. Give a derivation of abccba.
4. Where does meaning come from, according to GEB?
5. If the figure is BCDFGHJKLMNPQRSTVWXYZ what is the ground? How did you know that? Why did you know that?
6. How many possible sentences are there in the English language? Why?
7. Achilles and the Tortoise look at the following recursive picture. Achilles says, "this figure is made up of three copies of itself." The Tortoise says, "this figure is made up of nine copies of itself." Circle the copies seen by the Tortoise.

   

8. Suppose you saw the following writing on a piece of paper:

   circle centered at (0,0) radius 5
   line from (0,5) to (0,-5)
   line from (0,0) to (-2.5,-4.33)
   line from (0,0) to (2.5,-4.33)
   

   What is the frame message here? What is the outer message? What is the inner message? (To answer the latter question, you should have passed high school algebra or geometry....)

9. Translate the following into logical notation.
   1. If the moon is shining on the lake, then this mind is Buddha.
   2. All Italians except Romeo like gelato.
   3. Juliet will visit the capital of Greece.
   4. Some guitarists can play all the chords.
   5. Juliet, Romeo, and Guido are possibly all Italian.
10. What logical rule of inference is the Tortoise refusing to employ in Chromatic Fantasy and Feud?
11. Give a derivation of <<p∧~p>⊃~q> (Hint: something very, very similar to this was proven on the course notes, and this exam is open notes.)