Argument validity exercises with answers

Ten arguments, each written as one formula (premises) -> (conclusion). An argument is valid exactly when that formula is a tautology; decide which are valid, name the form, and for the invalid ones find a counterexample row, an assignment that makes every premise TRUE and the conclusion FALSE.

The classification line under each problem is the verdict: tautology means a valid argument, contingent means an invalid one. For an invalid argument the truth table shows the counterexample directly, since the FALSE rows of (P1 AND P2) -> C are exactly the rows on which the premises hold and the conclusion fails. The valid forms here are modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism, constructive dilemma and a reductio; the invalid ones are affirming the consequent, denying the antecedent and two slips, one with an inclusive OR and one with a shared consequent.

Write the argument carefully before judging it: the premises go inside one AND, the whole antecedent in parentheses, and the conclusion after the arrow. A common slip is to test the conclusion alone; B on its own is contingent for every argument, and that says nothing about validity. Validity is about the arrow from premises to conclusion, never about whether the conclusion happens to be true.

1. Premises: if it rains the street is wet (A -> B) and it rains (A). Conclusion: the street is wet (B). Valid? ((A -> B) AND A) -> B

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND A -> B

Classification: tautology

Modus ponens, valid: the formula is a tautology. The only row where A -> B and A both hold is A = TRUE, B = TRUE, and there B holds.

2. Premises: A -> B and NOT B. Conclusion: NOT A. Valid? ((A -> B) AND NOT B) -> NOT A

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND NOT B -> NOT A

Classification: tautology

Modus tollens, valid. With NOT B, the implication holds only on A = FALSE, B = FALSE, where NOT A is TRUE.

3. Premises: A -> B and B. Conclusion: A. Valid? ((A -> B) AND B) -> A

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	FALSE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND B -> A

Classification: contingent

Affirming the consequent, invalid: the formula is contingent. The counterexample is A = FALSE, B = TRUE: the implication holds vacuously and B holds, but the conclusion A is FALSE. The table shows that single FALSE row.

4. Premises: A -> B and NOT A. Conclusion: NOT B. Valid? ((A -> B) AND NOT A) -> NOT B

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	FALSE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND NOT A -> NOT B

Classification: contingent

Denying the antecedent, invalid. Counterexample A = FALSE, B = TRUE: both premises hold and NOT B fails; B can be true for other reasons.

5. Premises: A OR B and NOT A. Conclusion: B. Valid? ((A OR B) AND NOT A) -> B

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A OR B) AND NOT A -> B

Classification: tautology

Disjunctive syllogism, valid. The premises together hold only on A = FALSE, B = TRUE, where B is TRUE; the formula is a tautology.

6. Premises: A -> B and B -> C. Conclusion: A -> C. Valid? ((A -> B) AND (B -> C)) -> (A -> C)

#	A	B	C	f
0	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	TRUE	TRUE
2	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	TRUE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND (B -> C) -> A -> C

Classification: tautology

Hypothetical syllogism, valid: a tautology on all eight rows. On the four rows where both premises hold, A TRUE forces B and then C, so A -> C never fails.

7. Premises: A -> B, C -> D and A OR C. Conclusion: B OR D. Valid? ((A -> B) AND (C -> D) AND (A OR C)) -> (B OR D)

#	A	B	C	D	f
0	FALSE	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	FALSE	TRUE	TRUE
2	FALSE	FALSE	TRUE	FALSE	TRUE
3	FALSE	FALSE	TRUE	TRUE	TRUE
4	FALSE	TRUE	FALSE	FALSE	TRUE
5	FALSE	TRUE	FALSE	TRUE	TRUE
6	FALSE	TRUE	TRUE	FALSE	TRUE
7	FALSE	TRUE	TRUE	TRUE	TRUE
8	TRUE	FALSE	FALSE	FALSE	TRUE
9	TRUE	FALSE	FALSE	TRUE	TRUE
10	TRUE	FALSE	TRUE	FALSE	TRUE
11	TRUE	FALSE	TRUE	TRUE	TRUE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	TRUE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND (C -> D) AND (A OR C) -> B OR D

Classification: tautology

Constructive dilemma, valid: a tautology on all sixteen rows. Whichever of A, C the third premise provides, the matching implication yields B or D.

8. Premises: A OR B and A. Conclusion: NOT B. Valid? ((A OR B) AND A) -> NOT B

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	FALSE

(A OR B) AND A -> NOT B

Absorption · A AND (A OR B) = A
(A OR B) AND A → AResult: A -> NOT B

A -> NOT B · verified equivalent on every row · simplified, not minimal

Classification: contingent

Invalid: an inclusive OR does not exclude the other disjunct. Counterexample A = TRUE, B = TRUE, the one FALSE row of the table. The step list first absorbs A OR B into A, showing that the premises amount to A alone.

9. Premises: A -> B, C -> B and B. Conclusion: A. Valid? ((A -> B) AND (C -> B) AND B) -> A

#	A	B	C	f
0	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	TRUE	TRUE
2	FALSE	TRUE	FALSE	FALSE
3	FALSE	TRUE	TRUE	FALSE
4	TRUE	FALSE	FALSE	TRUE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND (C -> B) AND B -> A

Classification: contingent

Invalid, a three-premise version of affirming the consequent: B may come from C, or from neither. Counterexample A = FALSE, B = TRUE, C = TRUE; A = FALSE, B = TRUE, C = FALSE is a second one.

10. Premises: A -> B and A -> NOT B. Conclusion: NOT A. Valid? ((A -> B) AND (A -> NOT B)) -> NOT A

#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	TRUE

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

(A -> B) AND (A -> NOT B) -> NOT A

Classification: tautology

Valid, a reductio: if A led both to B and to NOT B, A cannot hold. The premises hold together only on the two rows with A = FALSE, where NOT A is TRUE, so the formula is a tautology.

Questions

Can a valid argument have a false conclusion?

Yes, when a premise is false: validity only promises that the conclusion is TRUE on every row where all premises are TRUE. Soundness, the truth of the premises, is a separate question that no truth table settles. And an argument with inconsistent premises is valid for every conclusion, vacuously, because no row makes all its premises TRUE.

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Argument validity exercises with answers

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