# Equivalent assertion

I have the following statement:

"Only those who play in orchestra are blonde."

Does it tell me something about the whole group of people who play in orchestra? For example, why can't I say that all who play in orchestra are blonde?

According to the book, from which it was taken, it is said that the statement

"All people who are not blonde don't play in orchestra."

is false. So my question is - why? If it is false then there are some other groups of people with different hair color who can play orchestra. But it is a contradiction. For example, if the statement was: "Only those who have the ticket can buy TV", it is right to say that "All who can't buy TV don't have the ticket" (regardless the real life and based only on the statement/condition that is given). We are told in the first statement that only members of some group (those who play orchestra) have some characteristic (are blonde). So logically, those who don't have this specific characteristic don't belong to that group. Don't they?

Another example that I thought about: say that the base statement was:

"Only the smartest students get 100 points out of 100 in the exam."

So according to what the books says regarding the statement in the previous example, the following statement will be false:

"All those who don't get 100 points are not the smartest students."

So it seems that the main problem here is that I can't determine whether all the smartest students get 100 or not. These "only" and "all" are really confusing.

Sorry if it sounds stupid, but I'm stuck with that.

• It's really annoying that one can not use Latex in here to answer logic questions. I know, that's meta, just saying. Dec 16, 2012 at 19:35
• Clearly what you're confused about is 1. the quantifiers 2. the conditions, that make an implication "true". There are several questions you could check for clarification on that: 1 2 3 4 Dec 16, 2012 at 20:56
• And yet more: 5 6 7 Dec 16, 2012 at 20:57

"Only those who play in orchestra are blonde." Does it tell me something about the whole group of people who play in orchestra? For example, why can't I say that all who play in orchestra are blonde?

Compare the following sentences: "Only women give birth." Does it tell you that all women give birth? "Only those with PhDs teach at Harvard." Does it tell you that all people with PhDs teach at Harvard?

• Thank you very much sir. I appreciate the other answer as well. It seems like I've got a brain stalled for a while on this one. Thanks for clearing the matter. Dec 17, 2012 at 12:30

Peter Smith's answer is completely correct, but to expand:

"Only those who play in orchestra are blonde."

What this tells you is that all blonds play in orchestra. It does not say, however, whether or not there are non-blondes in the orchestra.

Consider a universe of a thousand people, with an orchestra composed of a hundred. Now say that there are only 20 blondes in the universe, and all of them are in the orchestra. Then it would in fact be the case that "Only those who play in orchestra are blonde." There are, however, 80 more non-blonde members of orchestra, thus rendering the statement "All people who are not blonde don't play in orchestra" false, for there are non-blondes who play in orchestra.

Your other example is similarly afflicted.

Only those who have the ticket can by TV

This says that having the ticket is necessary for buying the TV, but it says nothing about whether or not it is sufficient. It is entirely possible that one must have the ticket AND have a thousand dollars, thus those who have the ticket but DO NOT have a thousand dollars would be excluded. Thus, it is in fact INCORRECT to say that "All those who can't buy TV don't have the ticket," because there could easily be people with the ticket but without other requirements.

• I received a downvote on this answer, and I'm very interested in knowing why so that I may give better answers in the future. Dec 17, 2012 at 16:01