Unless stated otherwise:
Determine whether these arguments are valid or invalid. Show all work.
For symbolic arguments, use at least two methods: by Definition, by Formula and/or by the
Valid Forms of Arguments and Invalid Forms of Arguments for each argument as applicable.
For syllogistic arguments, use Euler diagrams or Venn diagrams as applicable.
(1.)
p↔qp∨_q∴
p | q | p \leftrightarrow q | p \underline{\lor} q | \neg q |
---|---|---|---|---|
T | T | T | F | F |
T | F | F | T | T |
F | T | F | T | F |
F | F | T | F | T |
Premise 1 | Premise 2 | Conclusion |
p | q | p \leftrightarrow q | p \underline{\lor} q | (p \rightarrow q) \land (p \underline{\lor} q) | \neg q | [(p \rightarrow q) \land (p \underline{\lor} q)] \rightarrow \neg q |
---|---|---|---|---|---|---|
T | T | T | F | F | F | T |
T | F | F | T | F | T | T |
F | T | F | T | F | F | T |
F | F | T | F | F | T | T |
(2.)
p \lor q \\[2ex]
p \\
\rule{1.2in}{0.3pt} \\
\therefore \neg q
p | q | p \lor q | \neg q |
---|---|---|---|
T\checkmark | T | T\checkmark | F\times |
T\checkmark | F | T\checkmark | T\checkmark |
F | T | T | F |
F | F | F | T |
Premise 2 | Premise 1 | Conclusion |
p | q | p \lor q | (p \lor q) \land p | \neg q | [(p \lor q) \land p] \rightarrow \neg q |
---|---|---|---|---|---|
T | T | T | T | F | F |
T | F | T | T | T | T |
F | T | T | F | F | T |
F | F | F | F | T | T |
(3.)
p \rightarrow \neg q \\[2ex]
q \\
\rule{1.2in}{0.3pt} \\
\therefore \neg p
p | q | \neg q | p \rightarrow \neg q | \neg p |
---|---|---|---|---|
T | T | F | F | F |
T | F | T | T | F |
F | T\checkmark | F | T\checkmark | T\checkmark |
F | F | T | T | T |
Premise 2 | Premise 1 | Conclusion |
p | q | \neg q | p \rightarrow \neg q | (p \rightarrow \neg q) \land q | \neg p | [(p \rightarrow \neg q) \land q] \rightarrow \neg p |
---|---|---|---|---|---|---|
T | T | F | F | F | F | T |
T | F | T | T | F | F | T |
F | T | F | T | T | T | T |
F | F | T | T | F | T | T |
(4.)
p \lor q \\[2ex]
\neg p \\
\rule{1.2in}{0.3pt} \\
\therefore q
p | q | p \lor q | \neg p |
---|---|---|---|
T | T | T | F |
T | F | T | F |
F | T\checkmark | T\checkmark | T\checkmark |
F | F | F | T |
Conclusion | Premise 1 | Premise 2 |
p | q | p \lor q | \neg p | (p \lor q) \land \neg p | [(p \lor q) \land \neg p] \rightarrow q |
---|---|---|---|---|---|
T | T | T | F | F | T |
T | F | T | F | F | T |
F | T | T | T | T | T |
F | F | F | T | F | T |
(5.)
(p \lor q) \rightarrow r \\[2ex]
\rule{1.7in}{0.3pt} \\
\therefore (p \land q) \rightarrow r
p | q | r | p \lor q | (p \lor q) \rightarrow r | p \land q | (p \land q) \rightarrow r |
---|---|---|---|---|---|---|
T | T | T | T | T\checkmark | T | T\checkmark |
T | T | F | T | F | T | F |
T | F | T | T | T\checkmark | F | T\checkmark |
T | F | F | T | F | F | T |
F | T | T | T | T\checkmark | F | T\checkmark |
F | T | F | T | F | F | T |
F | F | T | F | T\checkmark | F | T\checkmark |
F | F | F | F | T\checkmark | F | T\checkmark |
Premise 1 | Conclusion |
p | q | r | p \lor q | (p \lor q) \rightarrow r | p \land q | (p \land q) \rightarrow r | [(p \lor q) \rightarrow r] \rightarrow [(p \land q) \rightarrow r] |
---|---|---|---|---|---|---|---|
T | T | T | T | T | T | T | T |
T | T | F | T | F | T | F | T |
T | F | T | T | T | F | T | T |
T | F | F | T | F | F | T | T |
F | T | T | T | T | F | T | T |
F | T | F | T | F | F | T | T |
F | F | T | F | T | F | T | T |
F | F | F | F | T | F | T | T |
(6.)
p \land q \\[2ex]
\rule{1.2in}{0.3pt} \\
\therefore p
p | q | p \land q |
---|---|---|
T\checkmark | T | T\checkmark |
T | F | F |
F | T | F |
F | F | F |
Conclusion | Premise 1 |
p | q | p \land q | (p \land q) \rightarrow p |
---|---|---|---|
T | T | T | T |
T | F | F | T |
F | T | F | T |
F | F | F | T |
p | q | \neg q | p \rightarrow \neg q | \neg p |
---|---|---|---|---|
T | T | F | F | F |
T | F | T | T | F |
F | T\checkmark | F | T\checkmark | T\checkmark |
F | F | T | T | T |
Premise 2 | Premise 1 | Conclusion |
p | q | \neg q | p \rightarrow \neg q | (p \rightarrow \neg q) \land q | \neg p | [(p \rightarrow \neg q) \land q] \rightarrow \neg p |
---|---|---|---|---|---|---|
T | T | F | F | F | F | T |
T | F | T | T | F | F | T |
F | T | F | T | T | T | T |
F | F | T | T | F | T | T |
p | q | p \rightarrow q |
---|---|---|
T\checkmark | T\checkmark | T\checkmark |
T | F | F |
F\times | T\checkmark | T\checkmark |
F | F | T |
Conclusion | Premise 2 | Premise 1 |
p | q | p \rightarrow q | (p \rightarrow q) \land q | [(p \rightarrow q) \land q] \rightarrow p |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | T |
p | q | r | p \leftrightarrow q | q \rightarrow r | \neg r | \neg p | \neg r \rightarrow \neg p |
---|---|---|---|---|---|---|---|
T | T | T | T\checkmark | T\checkmark | F | F | T\checkmark |
T | T | F | T | F | T | F | F |
T | F | T | F | T | F | F | T |
T | F | F | F | T | T | F | F |
F | T | T | F | T | F | T | T |
F | T | F | F | F | T | T | T |
F | F | T | T\checkmark | T\checkmark | F | T | T\checkmark |
F | F | F | T\checkmark | T\checkmark | T | T | T\checkmark |
Premise 1 | Premise 2 | Conclusion |
p | q | r | p \leftrightarrow q | q \rightarrow r | \neg r | \neg p | \neg r \rightarrow \neg p | (p \leftrightarrow q) \land (q \rightarrow r) | [(p \leftrightarrow q) \land (q \rightarrow r)] \rightarrow (\neg r \rightarrow \neg p) |
---|---|---|---|---|---|---|---|---|---|
T | T | T | T | T | F | F | T | T | T |
T | T | F | T | F | T | F | F | F | T |
T | F | T | F | T | F | F | T | F | T |
T | F | F | F | T | T | F | F | F | T |
F | T | T | F | T | F | T | T | F | T |
F | T | F | F | F | T | T | T | F | T |
F | F | T | T | T | F | T | T | T | T |
F | F | F | T | T | T | T | T | T | T |
(10.) You must eat well or you will not be healthy.
I eat well.
Therefore, I am healthy.
p | q | \neg q | p \lor \neg q |
---|---|---|---|
T\checkmark | T\checkmark | F | T\checkmark |
T\checkmark | F\times | T | T\checkmark |
F | T | F | F |
F | F | T | T |
Premise 2 | Conclusion | Premise 1 |
p | q | \neg q | p \lor \neg q | (p \lor \neg q) \land p | [(p \lor \neg q) \land p] \rightarrow q |
---|---|---|---|---|---|
T | T | F | T | T | T |
T | F | T | T | T | F |
F | T | F | F | F | T |
F | F | T | T | F | T |
(11.) Premise: If you live in Baltimore, then you live in Maryland.
Premise: Amanda does not live in Baltimore.
Conclusion: Amanda does not live in Maryland.
p | q | p \rightarrow q | \neg p | \neg q |
---|---|---|---|---|
T | T | T | F | F |
T | F | F | F | T |
F | T | T\checkmark | T\checkmark | F\times |
F | F | T\checkmark | T\checkmark | T\checkmark |
Premise 1 | Premise 2 | Conclusion |
p | q | p \rightarrow q | \neg p | \neg q | (p \rightarrow q) \land \neg p | [(p \rightarrow q) \land \neg p] \rightarrow \neg q |
---|---|---|---|---|---|---|
T | T | T | F | F | F | T |
T | F | F | F | T | F | T |
F | T | T | T | F | T | F |
F | F | T | T | T | T | T |
(12.) We must build a hydroelectric plant or a nuclear plant.
We won't build a nuclear plant, so we must build a hydroelectric plant.
p | q | p \lor q | \neg q |
---|---|---|---|
T | T | T | F |
T\checkmark | F | T\checkmark | T\checkmark |
F | T | T | F |
F | F | F | T |
Conclusion | Premise 1 | Premise 2 |
p | q | p \lor q | \neg q | (p \lor q) \land \neg q | [(p \lor q) \land \neg q] \rightarrow p |
---|---|---|---|---|---|
T | T | T | F | F | T |
T | F | T | T | T | T |
F | T | T | F | F | T |
F | F | F | T | F | T |
(13.)
p \rightarrow q \\[2ex]
q \land r \\
\rule{1.2in}{0.3pt} \\
\therefore p \lor r
p | q | r | p \rightarrow q | q \land r | p \lor r |
---|---|---|---|---|---|
T | T | T | T\checkmark | T\checkmark | T\checkmark |
T | T | F | T | F | T |
T | F | T | F | F | T |
T | F | F | F | F | T |
F | T | T | T\checkmark | T\checkmark | T\checkmark |
F | T | F | T | F | F |
F | F | T | T | F | T |
F | F | F | T | F | F |
Premise 1 | Premise 2 | Conclusion |
p | q | r | p \rightarrow q | q \land r | p \lor r | (p \rightarrow q) \land (q \land r) | [(p \rightarrow q) \land (q \land r)] \rightarrow (p \lor r) |
---|---|---|---|---|---|---|---|
T | T | T | T | T | T | T | T |
T | T | F | T | F | T | F | T |
T | F | T | F | F | T | F | T |
T | F | F | F | F | T | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | F | F | T |
F | F | T | T | F | T | F | T |
F | F | F | T | F | F | F | T |
(14.)
p \rightarrow q \\[2ex]
\neg p \\
\rule{1.2in}{0.3pt} \\
\therefore q
p | q | p \rightarrow q | \neg p |
---|---|---|---|
T | T | T | F |
T | F | F | F |
F | T\checkmark | T\checkmark | T\checkmark |
F | F\times | T\checkmark | T\checkmark |
Conclusion | Premise 1 | Premise 2 |
p | q | p \rightarrow q | \neg p | (p \rightarrow q) \land \neg p | [(p \rightarrow q) \land \neg p] \rightarrow q |
---|---|---|---|---|---|
T | T | T | F | F | T |
T | F | F | F | F | T |
F | T | T | T | T | T |
F | F | T | T | T | F |
p | q | r | p \rightarrow q | q \rightarrow r | p \rightarrow r |
---|---|---|---|---|---|
T | T | T | T\checkmark | T\checkmark | T\checkmark |
T | T | F | T | F | F |
T | F | T | F | T | T |
T | F | F | F | T | F |
F | T | T | T\checkmark | T\checkmark | T\checkmark |
F | T | F | T | F | T |
F | F | T | T\checkmark | T\checkmark | T\checkmark |
F | F | F | T\checkmark | T\checkmark | T\checkmark |
Premise 1 | Premise 2 | Conclusion |
p | q | r | p \rightarrow q | q \rightarrow r | p \rightarrow r | (p \rightarrow q) \land (q \rightarrow r) | [(p \rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r) |
---|---|---|---|---|---|---|---|
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | T | F | T |
T | F | F | F | T | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | T | F | T |
F | F | T | T | T | T | T | T |
F | F | F | T | T | T | T | T |
p | q | r | p \lor q | q \lor r | p \lor r |
---|---|---|---|---|---|
T | T | T | T\checkmark | T\checkmark | T\checkmark |
T | T | F | T\checkmark | T\checkmark | T\checkmark |
T | F | T | T\checkmark | T\checkmark | T\checkmark |
T | F | F | T | F | T |
F | T | T | T\checkmark | T\checkmark | T\checkmark |
F | T | F | T\checkmark | T\checkmark | F\times |
F | F | T | F | T | T |
F | F | F | F | F | F |
Premise 1 | Premise 2 | Conclusion |
p | q | r | p \lor q | q \lor r | (p \lor q) \land (q \lor r) | p \lor r | [(p \lor q) \land (q \lor r)] \rightarrow (p \lor r) |
---|---|---|---|---|---|---|---|
T | T | T | T | T | T | T | T |
T | T | F | T | T | T | T | T |
T | F | T | T | T | T | T | T |
T | F | F | T | F | F | T | T |
F | T | T | T | T | T | T | T |
F | T | F | T | T | T | F | F |
F | F | T | F | T | F | T | T |
F | F | F | F | F | F | F | T |
p | q | p \rightarrow q |
---|---|---|
T\checkmark | T\checkmark | T\checkmark |
T | F | F |
F | T | T |
F | F | T |
Premise 2 | Conclusion | Premise 1 |
p | q | p \rightarrow q | (p \rightarrow q) \land p | [(p \rightarrow q) \land p] \rightarrow q |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | F | T |
F | F | T | F | T |
p | q | p \rightarrow q |
---|---|---|
T\checkmark | T\checkmark | T\checkmark |
T | F | F |
F\times | T\checkmark | T\checkmark |
F | F | T |
Conclusion | Premise 2 | Premise 1 |
p | q | p \rightarrow q | (p \rightarrow q) \land q | [(p \rightarrow q) \land q] \rightarrow p |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | T |
p | q | r | p \rightarrow q | \neg p | \neg r | \neg p \rightarrow \neg r |
---|---|---|---|---|---|---|
T | T✓ | T✓ | T✓ | F | F | T✓ |
T | T | F | T | F | T | T |
T | F | T | F | F | F | T |
T | F | F | F | F | T | T |
F | T | T | T | T | F | F |
F | T | F | T | T | T | T |
F | F | T | T | T | F | F |
F | F | F | T | T | T | T |
Conclusion | Premise 3 | Premise 1 | Premise 2 |
p | q | r | p \rightarrow q | \neg p | \neg r | \neg p \rightarrow \neg r | (p \rightarrow q) \land (\neg p \rightarrow \neg r) | (p \rightarrow q) \land (\neg p \rightarrow \neg r) \land r | [(p \rightarrow q) \land (\neg p \rightarrow \neg r) \land r] \rightarrow q |
---|---|---|---|---|---|---|---|---|---|
T | T | T | T | F | F | T | T | T | T |
T | T | F | T | F | T | T | T | F | T |
T | F | T | F | F | F | T | F | F | T |
T | F | F | F | F | T | T | F | F | T |
F | T | T | T | T | F | F | F | F | T |
F | T | F | T | T | T | T | T | F | T |
F | F | T | T | T | F | F | F | F | T |
F | F | F | T | T | T | T | T | F | T |
Tautology |
p | q | r | p \rightarrow q | q \rightarrow r | p \rightarrow r |
---|---|---|---|---|---|
T | T | T | T\checkmark | T\checkmark | T\checkmark |
T | T | F | T | F | F |
T | F | T | F | T | T |
T | F | F | F | T | F |
F | T | T | T\checkmark | T\checkmark | T\checkmark |
F | T | F | T | F | T |
F | F | T | T\checkmark | T\checkmark | T\checkmark |
F | F | F | T\checkmark | T\checkmark | T\checkmark |
Premise 1 | Premise 2 | Conclusion |
p | q | r | p \rightarrow q | q \rightarrow r | p \rightarrow r | (p \rightarrow q) \land (q \rightarrow r) | [(p \rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r) |
---|---|---|---|---|---|---|---|
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | T | F | T |
T | F | F | F | T | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | T | F | T |
F | F | T | T | T | T | T | T |
F | F | F | T | T | T | T | T |
p | q | r | p \rightarrow q | q \rightarrow r | p \rightarrow r |
---|---|---|---|---|---|
T | T | T | T\checkmark | T\checkmark | T\checkmark |
T | T | F | T | F | F |
T | F | T | F | T | T |
T | F | F | F | T | F |
F | T | T | T\checkmark | T\checkmark | T\checkmark |
F | T | F | T | F | T |
F | F | T | T\checkmark | T\checkmark | T\checkmark |
F | F | F | T\checkmark | T\checkmark | T\checkmark |
Premise 1 | Premise 2 | Conclusion |
p | q | r | p \rightarrow q | q \rightarrow r | p \rightarrow r | (p \rightarrow q) \land (q \rightarrow r) | [(p \rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r) |
---|---|---|---|---|---|---|---|
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | T | F | T |
T | F | F | F | T | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | T | F | T |
F | F | T | T | T | T | T | T |
F | F | F | T | T | T | T | T |
p | q | p \rightarrow q | \neg q | \neg p |
---|---|---|---|---|
T | T | T | F | F |
T | F | F | T | F |
F | T | T | F | T |
F | F | T\checkmark | T\checkmark | T\checkmark |
Premise 1 | Premise 2 | Conclusion |
p | q | p \rightarrow q | \neg p | \neg q | (p \rightarrow q) \land \neg q | [(p \rightarrow q) \land \neg q] \rightarrow \neg p |
---|---|---|---|---|---|---|
T | T | T | F | F | F | T |
T | F | F | T | F | F | T |
F | T | T | F | T | F | T |
F | F | T | T | T | T | T |