I've just started a logic and proof class, and I'm confused about what we learned.

Given a proof of the form $$(P \lor Q) \rightarrow R$$ why is it true that you only have to show $$P \rightarrow R$$ or $$Q \rightarrow R$$ to satisfy the original proof? Why even have a disjunction as a premise if you only need one of the statements? Please help ease my confusion.

  • $\begingroup$ Not so: you have to show both implications in order to conclude the statement with the disjunction as "antecedent". See Clive Newstead's answer. $\endgroup$ – BrianO Feb 5 '16 at 5:13

Actually, you need both to hold; in classical logic, $P \vee Q \to R$ is equivalent to $(P \to R) \wedge (Q \to R)$.

The intuition is as follows: assuming $P \vee Q$, you know at least one of $P$ or $Q$ is true, but you don't know which. Thus, to deduce $R$ from the assumption $P \vee Q$, you need to be able to deduce $R$ from whichever one is true—since you don't know which it is, you have to be able to deduce $R$ from both possible cases.


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