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Fuzzy Tautologies - Contradictions and Logic Proofs

Fuzzy Tautologies – Contradictions and Logic Proofs

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

ABA → BA∩(A → B)A∩(A→B)→B
0.30.20.70.30.7
0.30.80.80.30.8
0.70.20.30.30.7
0.70.80.80.70.8

Approximate Modus Tollens

ABA → BA∩(A → B)A∩(A→B)→B
0.40.10.60.40.6
0.40.90.90.40.9
0.60.10.40.40.6
0.60.90.90.60.9

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.