Tautology, contradiction or contingency: exercises with answers
Ten formulas to classify as tautology, contradiction or contingency: the law of excluded middle and its negation, Peirce's law, the contrapositive against the converse, De Morgan as an equivalence, and formulas that look plausible but are FALSE on every row. Decide by table or by argument, then check the classification line.
A tautology is TRUE on every row, a contradiction FALSE on every row, and a contingency TRUE on some rows and FALSE on others. The engine decides each one by an exact check over the whole table, and the classification line under every problem states the verdict. For a contingency the table shows at least one row of each value; for the other two, the solution names the reason the column cannot vary.
Two habits make the job fast. First, look for a row that falsifies the formula: for A -> B the row A = TRUE, B = FALSE already shows it is not a tautology, and the row A = TRUE, B = TRUE shows it is not a contradiction. Second, when the formula is an equivalence X <-> Y, it is a tautology exactly when X and Y have the same column, which is how the De Morgan and contrapositive problems are settled without filling the table.
1. Classify A OR NOT A. A OR NOT A
| # | A | f |
|---|---|---|
| 0 | FALSE | TRUE |
| 1 | TRUE | TRUE |
A OR NOT A- Complement ·
A OR NOT A = TRUEA OR NOT A→TRUEResult:TRUE
Classification: tautology
Tautology, the law of excluded middle: one of A, NOT A is TRUE on every row. The simplifier reduces it to TRUE by complement.
2. Classify A AND NOT A. A AND NOT A
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | FALSE |
A AND NOT A- Complement ·
A AND NOT A = FALSEA AND NOT A→FALSEResult:FALSE
Classification: contradiction
Contradiction, the law of non-contradiction: A and NOT A are never TRUE together. Complement reduces it to FALSE.
3. Classify A -> B. A -> B
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | TRUE |
| 1 | FALSE | TRUE | TRUE |
| 2 | TRUE | FALSE | FALSE |
| 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 Classification: contingent
Contingency: FALSE on A = TRUE, B = FALSE and TRUE on the other three rows.
4. Classify ((A -> B) -> A) -> A. ((A -> B) -> A) -> 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) -> A) -> A Classification: tautology
Tautology, Peirce's law. If A is TRUE the outer implication holds at once; if A is FALSE then A -> B is TRUE, so (A -> B) -> A is FALSE and the outer implication holds vacuously. The step list finds no law; the exact check settles it.
5. Classify A <-> NOT A. A <-> NOT A
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | FALSE |
A <-> NOT A- Complement ·
A <-> NOT A = FALSEA <-> NOT A→FALSEResult:FALSE
Classification: contradiction
Contradiction: a value never agrees with its own negation, so the biconditional is FALSE on both rows. The simplifier reduces it to FALSE.
6. Classify (A OR B) AND NOT A AND NOT B. (A OR B) AND NOT A AND NOT B
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | FALSE |
| 2 | TRUE | FALSE | FALSE |
| 3 | TRUE | TRUE | FALSE |
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 AND NOT B Classification: contradiction
Contradiction: NOT A AND NOT B is NOT (A OR B) by De Morgan, and a formula AND its negation is FALSE everywhere. The table has no TRUE row.
7. Classify (A -> B) <-> (NOT B -> NOT A). (A -> B) <-> (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 <-> NOT B -> NOT A Classification: tautology
Tautology: an implication and its contrapositive have the same column, both FALSE only on A = TRUE, B = FALSE, so the biconditional holds on every row.
8. Classify (A -> B) <-> (B -> A). (A -> B) <-> (B -> A)
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | TRUE |
| 1 | FALSE | TRUE | FALSE |
| 2 | TRUE | FALSE | FALSE |
| 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 <-> B -> A Classification: contingent
Contingency: an implication and its converse differ on the two rows where A and B differ, so the biconditional is FALSE there (A = TRUE, B = FALSE and A = FALSE, B = TRUE) and TRUE where they agree.
9. Classify (A -> B) AND A AND NOT B. (A -> B) AND A AND NOT B
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | FALSE |
| 2 | TRUE | FALSE | FALSE |
| 3 | TRUE | TRUE | FALSE |
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 AND NOT B Classification: contradiction
Contradiction: A and NOT B together make A -> B FALSE, so the three conjuncts never hold at once. This is the failing row of modus ponens asserted as a fact.
10. Classify NOT (A AND B) <-> (NOT A OR NOT B). NOT (A AND B) <-> (NOT A OR NOT B)
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | TRUE |
| 1 | FALSE | TRUE | TRUE |
| 2 | TRUE | FALSE | TRUE |
| 3 | TRUE | TRUE | TRUE |
NOT (A AND B) <-> NOT A OR NOT B- De Morgan's law ·
NOT A OR NOT B = NOT (A AND B)NOT A OR NOT B→NOT (A AND B)Result:NOT (A AND B) <-> NOT (A AND B) - XOR identities ·
A <-> A = TRUENOT (A AND B) <-> NOT (A AND B)→TRUEResult:TRUE
Classification: tautology
Tautology: De Morgan's law as an equivalence, both sides FALSE only on A = TRUE, B = TRUE. The simplifier reaches TRUE here through De Morgan and the XOR identity.
Questions
Can the simplifier settle the classification?
Sometimes: when its laws reduce the formula to the constant TRUE or FALSE, as for A OR NOT A and A <-> NOT A, the step list and the classification agree. Often they do not reach a constant, as for Peirce's law, and then the classification comes from the exact check alone; both are shown, and the check is the one that decides.
Related
Worked examples
Glossary
See also
- Tautology in logic: meaning and examples
- Contradiction in logic: meaning and examples
- Contingency in logic: meaning and examples
- Peirce's law: ((A → B) → A) → A is a tautology
- A contradiction: A AND NOT A
Practice
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