Intro Logic Homework #1 SOLUTIONS - Ted Sider

Intro Logic Homework #1 SOLUTIONS
A. Determine whether these formulas are tautologies, contradictions, or contingent formulas
by using truth tables. Explain your answers (e.g. “this is a tautology because in the final
column of the truth table there are only Ts”.)
NOTE: I write the final column in boldface, and write the secondto-last
columns in italics.
1. (P!Q)"P This is a contingent formula because the final
column has a mixture of Ts and Fs.
( P ! Q ) " P
T T T T T
T T F T T
F T T F F
F F F T F
2. #P$P This is a contradiction because the final column
has all Fs.
~ P $ P
F T F T
T F F F
3. ~(P!Q)$(~P&~Q) This is a tautology because final column has all Ts.
~ ( P ! Q ) $ ( ~ P & ~ Q )
F T T T T F T F F T
F T T F T F T F T F
F F T T T T F F F T
T F F F T T F T T F

4. ~(P"Q)"(~Q"~P) This is a contingent formula because the final column
has a mixture of Ts and Fs.
~ ( P " Q ) " ( ~ Q " ~ P )
F T T T T F T T F T
T T F F F T F F F T
F F T T T F T T T F
F F T F T T F T T F
5. [(P$Q)&(P"R)] ! [~P"(Q&R)] This is a contingent formula because the final
column contains a mixture of Ts and Fs.
[ ( P $ Q ) & ( P " R ) ] ! [ ~ P " ( Q & R ) ]
T T T T T T T T F T T T T T T
T T T F T F F T F T T T T F F
T F F F T T T T F T T T F F T
T F F F T F F T F T T T F F F
F F T F F T T T T F T T T T T
F F T F F T F F T F F F T F F
F T F T F T T T T F F F F F T
F T F T F T F T T F F F F F F

6. ~[~P"(Q!R)] " [~P&Q] This is a contingent formula because the final column
has a mix of Ts and Fs.
~ [ ~ P " ( Q ! R ) ] " [ ~ P & Q ]
F F T T T T T T F T F T
F F T T T T F T F T F T
F F T T F T T T F T F F
F F T T F F F T F T F F
F T F T T T T T T F T T
F T F T T T F T T F T T
F T F T F T T T T F F F
T T F F F F F F T F F F
B. In each case, for the two given formulas, % and &, use truth tables to answer the following
questions (explain your answers): i) does % logically imply &? ii) does & logically imply
%? iii) are % and & logically equivalent? You may use the short cuts discussed in class.
7. %: P"R &: ~R"~P
i) % does logically imply & because the formula “%"&” is a tautology:
( P " R ) " ( ~ R " ~ P )
T T T T F T T F T
T F F T T F F F T
F T T T F T T T F
F T F T T F T T F
ii) by reversing the two italicized columns we can see that “&"%” is a tautology;
therefore & does logically imply %:

(~R"~P) " (P"R)
T T T
F T F
T T T
T T T
iii) since % logically implies & and & logically implies %, % and & are logically
equivalent.
8. %: [(P&Q)!R] &: [P&(Q!R)]
i) Since the formula “%"&” is not a tautology, % does not logically imply &:
[ ( P & Q ) ! R ] " [ P & ( Q ! R ) ]
T T T T T T T T T T T
T T T T F T T T T T F
T F F T T T T T F T T
T F F F F T T F F F F
F F T T T F F F T T T
F F T F F T F F T T F
F F F T T F F F F T T
F F F F F T F F F F F

ii) by reversing the two italicized columns we can see that “&"%” is a tautology;
therefore & does logically imply %:
[P&(Q!R)] " [(P&Q)!R]
T T T
T T T
T T T
F T F
F T T
F T F
F T T
F T F
iii) % and & are not logically equivalent, because % does not logically imply &;
and for formulas to be logically equivalent, each must logically imply the
other.
9. %: Q"R &: R"Q
i) % does not logically imply & because the formula %"& is not a tautology:
( Q " R ) " ( R "
T T T T T T T
T F F T F T T
F T T F T F F
F T F T F T F
Q )

ii) By reversing the italicized columns, we see that & does not logically imply % because
the formula &"% is not a tautology:
(R"Q) " (Q"R)
T T T
T F F
F T T
T T T
iii) In order for % and & to be logically equivalent, each would have to logically imply
the other. Since % didn’t logically imply & (nor for that matter did & logically imply
%), these formulas are not logically equivalent.
10. %: P$Q &: (P"Q) & (~P"~Q)
i) Since %"& is a tautology, % logically implies &:
[ P $ Q ] " [ ( P " Q ) & ( ~ P " ~ Q ) ]
T T T T T T T T F T T F T
T F F T T F F F F T T T F
F F T T F T T F T F F F T
F T F T F T F T T F T T F
ii) Since &"% is a tautology, & logically implies %:
[(P"Q) & (~P"~Q)] " [P$Q]
T T T
F T F
F T F
T T T
iii) Since each logically implies the other, % and & are logically equivalent.

C. Use truth tables to decide whether the following arguments are valid or invalid; explain
your answers. I'll write the arguments in the following form: Premise 1; Premise 2; ...
Premise n / Conclusion
NOTE: I’ll put the final columns of the premises and conclusion in
boldface.
11. P"~Q; ~Q"P / ~P$Q. The argument is valid because there is no case where the
premises are all true while the conclusion is false.
P " ~ Q ; ~ Q " P / ~ P $ Q
case 1 T F F T F T T T F T F T
case 2 T T T F T F T T F T T F
case 3 F T F T F T T F T F T T
case 4 F T T F T F F F T F F F
12. P"Q; Q"R ; ~R / P This argument is invalid because in case 8, all the
premises are true while the conclusion is false.
P " Q ; Q " R ; ~ R / P
case 1 T T T T T T F T T
case 2 T T T T F F T F T
case 3 T F F F T T F T T
case 4 T F F F T F T F T
case 5 F T T T T T F T F
case 6 F T T T F F T F F
case 7 F T F F T T F T F
case 8 F T F F T F T F F

13. P!Q; P"R; Q"R / ~~R This argument is valid because there is no case in which
all the premises are true while the conclusion is false.
P ! Q ; P " R ; Q " R / ~ ~ R
case 1 T T T T T T T T T T F T
case 2 T T T T F F T F F F T F
case 3 T T F T T T F T T T F T
case 4 T T F T F F F T F F T F
case 5 F T T F T T T T T T F T
case 6 F T T F T F T F F F T F
case 7 F F F F T T F T T T F T
case 8 F F F F T F F T F F T F