Symbolizing and proofing

A. True or False (1 pt each)
1. The following formula is a standard contradiction – (P&S) v (P&S)
2. A valid argument can have false premises and a false conclusion.
3. The formulas that make up a disjunction are called conjuncts.
4. The formal proof system allows us to determine the validity and
invalidity of arguments.
5. A?B is logically equivalent to (the same as) (A ? B) & (B ? A).
6. An invalid argument can have true premises and a true conclusion.
B. Short Answer (2 pts. Each)
7. What is the main connective of this formula? {[A v (B & D)] ? P}
8. Why does any statement entail itself?
9. What type of sentence is the following (e.g. conjunction, conditional, disjunction, negation, or
biconditional) and what is its main connective?
– [ R ? – (F v S)]
C. Truth Tables (5 pts each) Construct a full truth table for each of the following sequents (fill in
the tables provided). Indicate the main connective of each formula with an asterisk. Indicate
whether the argument is valid or invalid.
10. A ? B, – (A & – B) + – (A v B)
A B A ? B – (A & – B) – (A v B)
T T
F T
T F
F F
11. R v – R, R ? (F & S), – R ? – (F v S) + – (F & – S)
R F S R v – R R ? (F & S) – R ? – (F v S) – (F & – S)
T T T
F T T
T F T
F F T
T T F
F T F
T F F
F F F
12. Construct a truth table for the following formula and use it to indicate which kind of
statement (logical truth, contradiction, or contingent statement) the following formula is.
[(AvB)?A]?A
13. Construct a truth table to indicate whether the following two formulas are logically
equivalent.
– (A & B)
– A v – B
D. Symbolize the following sentences using propositional logic (1 pt each)
The statements indicated by the all-caps sentences are the following.
J = Johnson is guilty.
W = Wilson will get the death penalty.
L = Lewis will confess.
T = Johnson pulled the trigger.
S= Johnson is of questionable mental stability.
14. LEWIS will confess or WILSON will get the death penalty, but not both.
15. LEWIS will confess, unless Johnson pulled the TRIGGER and JOHNSON is guilty.
16. Assuming that JOHNSON is not guilty, WILSON will get the death penalty if and only if
LEWIS does not confess.
17. If Johnson pulled the TRIGGER, JOHNSON is guilty or he is of questionable mental
STABILITY.
E. Construct propositional logic proofs for the following sequents, using only primitive
inference rules (10 pts each)
18. P v Q, – Q + P
19. A ? B, A, C + B & C
20. –A ? B, (B & Q) & C + -A
21. A ? (B&C) + (A ? B) & (A ? C)
22. P & Q + P v Q
F. Use predicate logic to symbolize the following sentences (1 pt each)
23. There are (p)oliticians who take (b)ribes.
24. “(D)runkards tell the (t)ruth.”
25. Not everyone in this (c)lass loves (l)ogic..
26. [B]arack Obama is the current (p)resident.
27. Some (p)lanets are not (h)abitable.
28. Only U.S. (c)itizens can run for (p)resident.
29. Everyone in this (c)lass loves (l)ogic.
G. Construct a predicate logic proof for the following sequent (7 pts)
30. ?x (Lx ? Ax), ?x (Ax ? – Rx) + ?x (Lx ? – Rx)

 

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