Quantified Statements To English

Question

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.

Answer 1

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.

Answer 2

This is because the mathematical language is more accurate than the usual language. But your answers are right and they are the same as the book.

Answer 3

Your answers to parts a), b), and d) are OK; in part c) you've reversed the roles of "comedian" and "funny". The book's answers are correct and, in a couple of cases, closer to how people would ordinarily express these statements. One can make other equivalent statements that sound (to me) even more like ordinary usage, for example in d) I'd say "there is a funny comedian" and for b) I'd say "everybody is a funny comedian".

About the last sentence in your question, "Every comedian is funny" is, despite how it looks to you, an implication --- it says that being a comedian implies being funny. More generally, consider any restricted universal quantification, i.e., a situation where you assert some property $P(x)$ not for absolutely all $x$'s but only for those that satisfy some restriction $R(x)$. Such a restricted universal quantification is expressed by a universally quantified implication, $(\forall x)\,(R(x)\to P(x))$. I suspect that part of the purpose of part a) was to lead you to observe this connection between restricted universal quantification and implication.

Similarly, I suspect the purpose of part d) was to lead you to observe the connection between restricted existential quantification and conjunction. And the purpose of parts b) and c) is to show that if you mix these things up, using implication with existential quantification or using conjunction with universal quantification, you get some rather unnatural-sounding statements, and in any case you do not get the effect of just restricting the quantifier.

Answer 4

The homework question said "Translate these statements into English". And that means normal English, such as a native English speaker might use. Now, is

For every person, if they are a comedian, then they are funny.

normal English in that sense? Would you ever say that sort of thing outside the logic classroom?? Of course not!

What you have written down is a Logic-English mix (Loglish if you like), with something of the syntactic structure of the language of first-order logic, and the vocabulary of English. Now, that's a very useful thing to do as a half-way house, using Loglish as a bridge between FOL and English. But it is only a half-way house. You now need to ask: how would you say much the same thing (as far as truth-conditions are concerned) in normal English? The book answer gets that right.