
Assignment 4
(I will post another file with 23 more problems for
Assignment 4, within hours of posting this one. At least
with this one you can get started.)
This is due in class between 6:50 and 7 p.m. on 19 March.
Same rules as for Assignment 2 (look at the Asst. 2 file linked
on the course page to remind yourself of those rules.)
There are other instructions below. People lost lots of marks
on earlier assignments and on Test 1 by ignoring instructions.
It's easy enough to lose marks without doing it this way.
In bullet proofs on this assignment, you may use any theorems on
the axiom/theorem sheet we have been using in class for a long time
now (theorems up through 2.4.26).
Problem 1:
Give a Hilbert proof that
 P > (Q > R) > (P > Q) > (P > R).
To get any credit for this problem, you must use the
Deduction Theorem exactly three times. The Hilbert proof
I have in mind has exactly six lines. Then you'll need,
I believe, three English sentences at the end of your proof.
(Write up the proof neatly and carefully enough to convince
the grader that you understand the D.T. perfectly.)
Problem 2:
Bulletprove that  (P /\ Q) > (P \/ Q).
You may NOT use the Deduction Theorem in your proof.
This is a ridiculously short proof.
Problem 3:
Now prove again the same theorem as in Problem 2, but this time
you must use the Deduction Theorem. Begin with a sentence of the
form "Assume _____________" (where you fill in the blank with the
appropriate wff). Then give a bullet proof of P \/ Q (exactly
one of your steps should use the symbol <===> (talked about in
Γ
a file posted on the course page a long time ago, and mentioned in
class as well), where Γ is the set consisting of the wff
you filled in the blank with above  be sure to say explicitly
what the set Γ is, in your proof).
In addition to all the theorems on our usual class list, you may
use in this problem, as annotation for a step in your bullet proof,
any metatheorems proven in class up through the end of class on
5 March. That includes what we called "merging" and "splitting".
(I give such detailed instructions for a few reasons  to
force people to do well on the assignment; to produce, ideally,
only one solution, so that the grader does not have to work with
five different marking schemes for as many different proofs; to
speed up the assignment for you since time is short till it is
due.)
Problem 4:
Bulletprove that
 (P == Q) == ( (P /\ Q) \/ (¬P /\ ¬Q) ).
(This is also very short and easy.)
Problem 5:
Do Exercise 20 on page 87 of our text.
Note that the wff there is really
(A > C) > ((B>C) > ((A \/ B) > C)).
For credit on this problem
you must give a Hilbert proof. I think it should have
exactly five lines in it, and I suggest using theorem
scheme 2.4.24 somewhere in it. The proof I have in mind
also has an English sentence or two at its end.
