converting words into logic symbols

To run a marathon you must be over 5 feet and have a healthy heart. If you are not over 5 feet and do not have a healthy heart, you cannot run the marathon. Right this statement in symbols.

Let a(x) be the assertion "x can run the marathon"

For the first statement I got $\displaystyle (over 5 feet) \wedge (healthy heart) \rightarrow a(x)$

For the second statement I got $\displaystyle \neg (over 5 feet) \wedge \neg (healthy heart) \rightarrow \neg a(x)$

I have a feeling this is wrong because if the hypothesis is false then the statment is true, if I don't use implication what do I use?

Re: converting words into logic symbols

The first formula should be $\displaystyle \forall x\,(\mathop{\mbox{over 5 feet}}(x) \wedge \mathop{\mbox{healthy heart}}(x)\rightarrow a(x))$, and similarly for the second.

The use of implication is correct. These statements are true for all x, including those for whom the premises are false.

Re: converting words into logic symbols

Quote:

Originally Posted by

**emakarov** The first formula should be $\displaystyle \forall x\,(\mathop{\mbox{over 5 feet}}(x) \wedge \mathop{\mbox{healthy heart}}(x)\rightarrow a(x))$, and similarly for the second.

The use of implication is correct. These statements are true for all x, including those for whom the premises are false.

What does "premises" mean? Is that an synonym for hypothesis?

Re: converting words into logic symbols

Yes, a synonym. It's what is located left of the arrow.