translating one sentence to FOL , an interview question is wrong !?

Question

We ran into an Interview question, writing part 3 days ago. one of the question is as follows: (definition of A(x) and B(x)‌ is not given by OP)

what is the logical interpretation of following sentence:

"each muslems is not privy to god."

$I)‌ \exists x (A(x)‌ \wedge \neg B(x))$

$II)‌ \forall x (A(x)‌ \to \neg B(x))$

$III)‌ \neg \forall x (A(x)‌ \to B(x))$

$IV)‌ \forall x (\neg B(x)‌ \to A(x))$

the answer sheet wrote (2)‌ is the answer. but we think (4) is true, too. and we must have Objection to this question. are we right?

Answer 1

If the formulas $A(x)$ and $B(x)$ were defined as part of the question then there there could be no ambiguity. If not:

1. if you define $A_2(x) =$ "$x$ is a Muslim" and $B_2(x) =$ "$x$ is privy to God", then your sentence translates into $\forall x (A_2(x) \to \neg B_2(x))$;

2. if you define $A_4(x) = \neg B_2(x)$ and $B_4(x) = \neg A_2(x)$, then your sentence translates into $\forall x (\neg B_4(x) \to A_4(x))$.

Please note that for 2. you need to assume "double-negation elimination", i.e.

If $\Gamma \vdash \neg \neg \theta$, then $\Gamma \vdash \theta$.

which may or may not be the case (it is if you are working with classical logic).

Anyway, when translating a sentence from a natural language into a logic formula I would start by the syntax, i.e. by translating the connectives, and what is left (if it makes sense) into atomic formulas. I couldn't find a reference (yet), but if this is the convention, then case 2. cannot arise, because the connective $\neg$ could not be part of, say, $A_4(x)$.

Edit: Gary M. Hardegree writes in his Symbolic Logic: A First Course (see this freely available early edition):

An English statement is said to be in standard form, or to be standard, if all its connectives are standard and it contains no simple statement that is straightforwardly equivalent to a compound statement; otherwise, it is said to be non-standard.

Once a statement is paraphrased into standard form, the only remaining task is to symbolize it [...].

Later on (see page 108 of the linked edition) he explains:

If sentence $S$ is symbolized by the formula $\mathcal{A}$, then the negation of $S$ (standard or idiomatic) is symbolized by the formula $\sim \mathcal{A}$.

where an example of standard and idiomatic negation is (quotation marks mine):

Statement: "it is raining"

Idiomatic negation: "it is not raining"

Standard negation: "it is not true that it is raining"

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In conclusion, the choice (IV) could be correct only if, when translating from English to sentential logic, we could define, say, $B(x)$ as "$x$ is not a Muslim". The quote above states that this by convention is not allowed, i.e. if you find a "not" you must translate it with $\neg$ and you cannot "hide" it in an atomic formula.

In other words: (IV) is a valid answer if and only if you can translate "each Muslim is not privy to God" into $\forall x (\neg B(x) \to A(x))$. If you drop the quantifier, this is the same as translating (in what follows, read '$\leadsto$' as "translates into") $$ \text{"if a person is Muslim, then that person is not privy to God"} \;\leadsto\; \neg B(x) \to A(x) $$ To do so, you need to make the following assignments:

1. Implication: "if ... then" $\;\leadsto\;$ '$\to$'

2. Variable: "person", "that person" $\;\leadsto\;$ '$x$'

3. First atomic formula: "$x$ is Muslim" $\;\leadsto\;$ '$\neg B(x)$': by the law of excluded middle, this implies the assignment "$x$ is not Muslim" $\;\leadsto\;$ '$B(x)$'

4. Second atomic formula: "$x$ is not privy to God" $\;\leadsto\;$ '$A(x)$'

The problem is that you are not allowed to do this, because assignments 3 and 4 are not possible in the conventional translation algorithm: you must translate "$x$ is not privy to God" into $\neg P(x)$, where $P(x)$ is "$x$ is privy to God" (and similarly for the other sentence), as explained in the above quote. In other words, every atomic formula (resulting from a translation of English into sentential logic) is assumed to be "positive".

Answer 2

With $Ax\equiv$ "$x$ is a Muslim" and $Bx\equiv$ "$x$ is privy to God", we have:

I) $\exists x(Ax\land\neg Bx)$

This is the usual existential statement: "there exists something that is Muslim and is not privy to God", or "there is at least one Muslim who is not privy to God". However, you can easily see that this is too weak compared to "each Muslim is not privy to God" (assuming there are more than one Muslims).

II) $\forall x(Ax\to\neg Bx)$

This is the usual universal statement, where the object is passed as the premise of the implication and the property as the conclusion. This reads: "everything that is Muslim is also not privy to God" or "each Muslim is not privy to God". Hence this is the correct solution.

III) $\neg\forall x(Ax\to Bx)$

By using the fact that $\neg\forall x\equiv\exists x\neg$, and then distributing the negation over the implication, we can rewrite the above as: $\exists x(Ax\land\neg Bx)$, which is exactly the same as case I). You could also directly read it as "not all Muslims are privy to God", which again is not strong enough because it implies that there is some Muslim who is not privy to God while the other Muslims might be.

IV) $\forall x(\neg Bx\to Ax)$

This a correct universal statement, saying that "everything that is not privy to God is Muslim", which is very different from the interpretation we're looking for. Indeed, it is quite plausible that there are things that are not privy to God but also are not Muslim.

Note again, as in II), that if you want to express a universal statement of the form "all $y$ are $z$", then $y$ needs to be the premise and $z$ the conclusion, such that, logically, if something is $y$ it must also be $z$ (while the converse might not hold).

Final remark: if both II) and IV) were true, such that you would have both $\forall x(Ax\to\neg Bx)$ and $\forall x(\neg Bx\to Ax)$, then you could conclude $\forall x(Ax\leftrightarrow\neg Bx)$: "being Muslim is equivalent to not being privy to God".