The problem I am working on is:
Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.
a) $∀x(C(x)→F(x))$
b)$∀x(C(x)∧F(x))$
c) $∃x(C(x)→F(x))$
d)$∃x(C(x)∧F(x))$
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Here are my answers:
For a): For every person, if they are a comedian, then they are funny.
For b): For every person, they are both a comedian and funny.
For c): There exists a person who, if he is funny, is a comedian
For d): There exists a person who is funny and is a comedian.
Here are the books answers:
a)Every comedian is funny.
b)Every person is a funny comedian.
c)There exists a person such that if she or he is a comedian, then she or he is funny.
d)Some comedians are funny.
Does the meaning of my answers seem to be in harmony with the meaning of the answers given in the solution manual? The reason I ask is because part a), for instance, is a implication, and "Every comedian is funny," does not appear to be an implication.
Best Answer
You literally wrote down symbol by symbol what the statements were. But languages used for everyday communications only rarely uses quantifiers, and does not have absolute truth/falsehood. Pretend you are talking to a friend. If you gave your answer to say a), the friend would probably look at you as if you were crazy. We do not talk that way. If you used the manual's answer, the friend might say 'No way' or 'if they are not they become unemployed rather soon" or 'well most'. You are communicating, but reality is not even close to the precision of mathematical statements. Mathematics models the real world but it is not the real world. People simply do not talk that way. They talk the way the manual answers, and that is much more fuzzy than math.