# Conditional statements truth table [duplicate]

I have read in quite a few books that the proposition 'p->q' can be read as either 'if p then q' and 'p only if q'.

Let p = it rains, and q = take an umbrella.

Then with according to the first form the argument is 'If it rains, take an umbrella'. And according to the second form the argument is 'Take an umbrella only if it rains'.

Then the cases 'it rains - took an umbrella (T)', 'it did not rain - did not took an umbrella (T)', 'it rains- did not took an umbrella (F)' are trivial.

did not rain - took an umbrella.

Why is this true and not for example undefined or not known?

• Note that "take an umbrella only if it rains" is a gloss for q→p, not p→q. You want "it only rains if you take an umbrella", which would certainly be true if you took an umbrella every time it rained. Oct 24, 2012 at 13:06
• This reads to me like it covers very similar ground as other questions; let me know if think the problem here isn't being addressed in the other question and answer and we can maybe discuss a good reformulation of this which might address those concerns. Oct 30, 2012 at 18:30