Truth operations for tautologies, contradictions and logic proofs are not different for fuzzy sets. But these can differ considerably in the result from those of classical logic.
The truth tables for approximate Modus Ponens and Modus Tollens are given below:
Approximate Modus Ponens
| A | B | A → B | A∩(A → B) | A∩(A→B)→B |
| 0.3 | 0.2 | 0.7 | 0.3 | 0.7 |
| 0.3 | 0.8 | 0.8 | 0.3 | 0.8 |
| 0.7 | 0.2 | 0.3 | 0.3 | 0.7 |
| 0.7 | 0.8 | 0.8 | 0.7 | 0.8 |
Approximate Modus Tollens
| A | B | A → B | A∩(A → B) | A∩(A→B)→B |
| 0.4 | 0.1 | 0.6 | 0.4 | 0.6 |
| 0.4 | 0.9 | 0.9 | 0.4 | 0.9 |
| 0.6 | 0.1 | 0.4 | 0.4 | 0.6 |
| 0.6 | 0.9 | 0.9 | 0.6 | 0.9 |
In the above table the last column which is (A∩(A→B)→B contains values other than unity, this represents a quasi tautology.
If propositions have truth tables all equal to 0.5 (half-truth) then the resulting truth value of the tautology is also half truth.
