Translating statments into symbolic logic.

Question

Let $F(x,y)$ be the statement, “$x$ can fool $y$,” where the domain consists of all of the people in the world. Translate this statement into symbolic logic.

No one can fool both Fred and Jerry.

I think this could be: There exists an $x$ or $y$ in $F(x,y)$?

Answer 1

The statement

There exists an $x$ or $y$ in $F(x,y)$,

if it makes any sense at all, is a statement about symbols: there is a symbol $x$ or a symbol $y$ in the $6$-character string of symbols $F(x,y)$. This is a true statement, as it happens, but it has nothing to do with the problem that you’re trying to solve.

Note also the following three excellent reasons to doubt that your suggestion could possibly be right:

1. It doesn’t mention Fred.
2. It doesn’t mention Jerry.
3. It says that something does exist, when the English statement that you’re trying to translate says that something — a person who can fool both Fred and Jerry — does not exist.

Back up. Suppose that there were someone — Pamela, say — who could fool both Fred and Jerry. How would we express symbolically the fact that Pamela can fool Fred? Since $F(x,y)$ means that $x$ can fool $y$, ‘Pamela can fool Fred’ is simply $F(\text{Pamela},\text{Fred})$. Similarly, ‘Pamela can fool Jerry’ is $F(\text{Pamela},\text{Jerry})$. And ‘Pamela can fool both Fred and Jerry’ is

Pamela can fool Fred and Pamela can fool Jerry,

or in symbols

$F(\text{Pamela},\text{Fred})\land F(\text{Pamela},\text{Jerry})$.

Okay, now suppose that we know that someone can fool both Fred and Jerry, but we don’t know who. Now we can’t use a specific name for this clever person, so we’ll call her $x$: the expression $F(x,\text{Fred})\land F(x,\text{Jerry})$ means that $x$ can fool Fred and $x$ can fool Jerry, i.e., that $x$ can fool both Fred and Jerry. But this $x$ is just a tag, a place-holder for a name; the expression $F(x,\text{Fred})\land F(x,\text{Jerry})$ doesn’t actually say anything about the world until we either replace that place-holder with a name or quantify over it. To say that there is such a clever person, we use the existential quantifier $\exists$:

$$\exists x\Big(F(x,\text{Fred})\land F(x,\text{Jerry})\Big)\;.$$

This says that

there exists an $x$ such that $x$ can fool Fred and $x$ can fool Jerry,

or, in more normal English,

there is someone who can fool both Fred and Jerry.

Your statement is that there is no such person. That is, your statement is the negation of this. How would you write that in symbols?

I’ve now looked at your previous question. You need to keep asking yourself exactly what the symbols that you’re trying to use mean. In particular, when you conjecture an answer to one of these problems, you need to back off and ask yourself:

• If someone handed me this string of symbols and asked me to translate it into English, what answer would I give if I’d never seen it before and didn’t know what it was supposed to mean?

In particular you seem to have a tendency to confuse symbols with the things that they represent; you did this in the previous problem when you wrote ‘$F(x,y)$ is in $W$’ and again here when your answer talked about $x$ or $y$ being ‘in $F(x,y)$’. You should probably make a point of asking yourself whether you’re making that particular mistake.

Answer 2

The statement, "No one can fool both Fred and Jerry" can be written as

$$\forall x \neg \bigg(F(x, \operatorname{Fred}) \wedge F(x, \operatorname{Jerry})\bigg)$$

which is equivalent to

$$\forall x \bigg(\neg F(x, \operatorname{Fred}) \vee \neg F(x, \operatorname{Jerry})\bigg)$$

Answer 3

A more general response to this and your previous question. The questions are so very elementary that the fact that you are asking them strongly suggests you need to read (slowly, carefully) a good elementary introduction to the language of first-order logic, which explains clearly how to translate/transcribe/render in and out of logical symbolism.

There are a lot of good elementary logic books which do this sort of thing well, available in any library (including one by P*t*r Sm*th, An Introduction to Formal Logic, CUP). One very good treatment is in Paul Teller's Modern Formal Logic Primer, now freely available online. Look at the opening chapters of Volume II, particularly Chapter 4 on 'Transcription'.