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 Invalid: A⇒B. If the hen lays an egg, then we cook omelettes.
 * C⇒B. If we eat omelettes, then we cook omelettes.
 * ∴ A⇒C. Therefore, if the hen lays an egg, we eat omelettes (invalid; eating omelettes is not necessarily related to the hen’s laying).

Invalid: B⇒A. If we cook omelettes, then the hen lays an egg.
 * B⇒C. If we cook omelettes, then we eat omelettes.
 * ∴ A⇒C. Therefore, if the hen lays an egg, we eat omelettes (invalid, not because the first premise is absurd but because the hen’s laying and our omelette eating are not necessarily related).

Valid: A⇒B. If the hen lays an egg, then we cook omelettes.
 * A. The hen laid an egg.
 * ∴ B. Therefore, we cook omelettes.

Valid: A⇒B. If the hen lays an egg, then we cook omelettes.
 * -B. We are not cooking omelettes.
 * ∴ -A. Therefore, the hen did not lay an egg.

Invalid: A⇒B. If the hen lays an egg, then we cook omelettes.
 * -A. The hen did not lay an egg.
 * ∴ B. Therefore, we are not cooking omelettes. (invalid; maybe we can get eggs elsewhere)

Invalid: A⇒B. If the hen lays an egg, then we cook omelettes.
 * B. We are cooking omelettes.
 * ∴ A. Therefore, the hen laid an egg. (invalid; maybe we can get eggs elsewhere)

The last two fallacies above are so obviously wrong that we might dismiss them as irrelevant to scientists. When couched in technical terms, however, these invalid syllogisms do appear occasionally in print. Both fallacies imply confusion between necessary and sufficient conditions. Both are deductively invalid, but they may have some inductive validity: Valid: If the hen lays an egg, then we cook omelettes.
 * The hen did not lay an egg.
 * Therefore, we may not cook omelettes.
 * (the hen’s failure is a setback to our omelette plans, but maybe we can get eggs elsewhere)

Valid: If the hen lays an egg, then we cook omelettes.
 * We are cooking omelettes.
 * Therefore, the hen may have laid an egg. (true, but maybe we got eggs elsewhere)

This second hypothetical syllogism is a cornerstone of scientific induction: “If hypothesis (H) entails Evidence (E), and E is true, then H is probably true.” It is fallacious to conclude that H is definitely true, but the evidence is relevant to evaluation of the hypothesis.

Pitfalls: Fallacious Arguments
After a bit of practice, one can readily recognize syllogistic arguments that are expressed in ordinary language, and one can evaluate them by examining their structures. Many arguments can appear to be structurally valid and yet be fallacious; such arguments yield a false conclusion even if the premises are true. These fallacies exhibit an error in execution, such as subtle problems in their premises, use of apparently relevant but logically irrelevant evidence, an incorrect connection of premises to conclusion, and grammatical errors or ambiguities. Many of these fallacies are genuine