Hypothetical Propositions:
Nonstandard Forms
"p only if q"

What about a statement with the form, "p only if q?"


I'll stay home tomorrow only if I'm sick.

Am I saying that if I am sick I will stay home tomorrow? No--I am saying that if I am not sick, then I will not stay home. Being sick is the only thing that would keep me home. Think of this as a promise I've made to you. Now suppose I am sick, but I show up anyway. Have I broken my promise? No--but I would break the promise if I stayed home and was not sick.

Thus, the statement, "p only if q" can be translated in either of two equivalent ways:

1. If not-q, then not-p.

2. If p, then q.

The first translation sounds more natural in some contexts, and the second in others. But because they are contrapositives, they are logically equivalent.

"p if q" | "p only if q" | "p if and only if q"
"p unless q" | "Whenever p, q" | "Without p, then q"

Hypothetical Propositions

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