Logic 3003, Spring 2005: Test 1
I.
Definitions and concepts: Answer 3 in each section.
A.
1.
2.
3.
4.
5.
Define completeness.
Define formal system.
State the rules that define the wffs of our language.
Explain how distinguishing the binding powers of different connectives allow us to
reduce the parentheses we actually bother to write.
What is an L-proof of a wff A from an ensemble of wffs ?
6.
7.
8.
9.
10.
What is an L-theorem?
What is a substitution instance?
State the rule of Theorem Introduction.
Define validity.
Define contingent wff.
B.
II.
Proofs:
1.
2.
3.
III.
Prove PQ, P  Q using the base rules
Prove P  (Q  R), P  R P  Q
Prove (P  Q) S  R  P  (S  (Q R)
Valuations:
Appealing to the soundness metatheorem, prove that the following sequents have no proofs:
1.
2.
3.
IV.
P (Q  R), R  P  Q
(A  B)  A, A  B  C
 A  C, C  (A  B)  B
Metatheory: Answer 3.
1.
2.
3.
4.
5.
Prove Metatheorem 3.2: The set of provable sequents is closed under uniform
substitution.
Explain the awkward technical maneuver that first assures us that, if there is a proof
of a wff A from some ensemble , we can (in principle) find it.
Explain how the combination of the strong soundness theorem with the strong
completeness theorem provides us with a decision procedure for whether a particular
sequent,   A, can be proven.
Explain the difference between strong completeness and weak completeness.
Explain how having a proof for every canonical sequent tightly links the
interpretation of our language provided by in truth tables to our ability (in this formal
system) to build proofs.