I'm thinking of a number.
 if it is not a multiple of 4, then it is between 60 and 69
 if it is a multiple of 3 it is between 50 and 59
 if it is not a multiple of 6 it is between 70 and 79.
What is the number?
(In reply to
Solution + Explanation by nikki)
"Therefore, by the converse of the second statement, we know that the number is not a multiple of 3."
Actually it is the contrapositive of the second statement that leads to this conclusion. When the original premise is "If A then B", the contrapositive, "If not B then not A" is a valid conclusion, and was what was used here. The converse, "If B then A", is not a valid conclusion. So the reasoning was valid, but the terminology wrong.
However, the claim "Next, let’s look at the 6069 range. In order for the number to be in this range, it must NOT be a multiple of 4, but it MUST be a multiple of 6." does invalidly use the converse in reasoning. It assumes that statement 1 can be construed as "If the number is between 60 and 69 then it is not a multiple of 4." This is the converse of what is said, and is therefore not a valid conclusion. It is the valid remainder of this paragraph that excludes this range: that by the contrapositives of statements 2 and 3 that it must be a multiple of 6 without being a multiple of 3, an impossibility.
Similarly the first statement "In order for the number to be in the 5059 range, the number must both be a multiple of 3 and a multiple of 4. ": the 3 actually devolves from the 6 in the third statement, and is better stated as "a multiple of 6 and of 4", which still leads to being divisible by 12, but its origin is in the contrapostive of statement 3 rather than the converse of statement 2.

Posted by Charlie
on 20040924 10:02:37 