1. Affirm A: If A then B; A; therefore B — valid.
2. Deny B: If A then B; not B; therefore not A — valid.
3. Affirm B: If A then B; B; therefore A — generally invalid.
4. Deny A: If A then B; not A; therefore not B — generally invalid.
The last two conclusions might happen to be true in a particular situation, but they do not follow from the conditional alone. Truth by coincidence must be distinguished from logical entailment.
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Exceptive Syllogism: Valid Inferences from Connected and Disjunctive Conditionals
1. Affirm A: If A then B; A; therefore B — valid.
2. Deny B: If A then B; not B; therefore not A — valid.
3. Affirm B: If A then B; B; therefore A — generally invalid.
4. Deny A: If A then B; not A; therefore not B — generally invalid.
The last two conclusions might happen to be true in a particular situation, but they do not follow from the conditional alone. Truth by coincidence must be distinguished from logical entailment.