How to translate SAME and DIFFERENT in predicate logic?

Question

Betty told every secretary a lie.

I was told that this sentence is ambigous as it can be intrepreted in 2 different ways.

These are my 2 interpretations:

• "Betty told every secretary the same lie"
• "Betty told every secretary a different lie".

I know how to translate "Betty told every secretary a lie" in predicate logic. But I don't understand how to translate the words SAME and DIFFERENT.

Answer 1

Hint:

• For the general case (possibly different, possibly the same, possibly some mixture), you want to say that for each secretary there was a lie that Betty told that secretary.

• For the "same lie", you want to say that there was a lie which Betty told every secretary.

• For exclusive "different lies", you want to say that for each secretary there was a lie that Betty told that secretary and did not tell any other secretary.

Answer 2

The sentence "Betty told every secretary a lie" does not specify whether the lie was determined for each person or whether every person received the same lie. Formally, it does not specify the order of the quantifiers. As you put it, we don't know whether it is the same lie or potentially a different lie.

First, we define our predicates and constants. Let b = Betty, L(x) mean "x is a lie", let S(x) mean "x is a secretary", and let T(x,y,z) mean "x told y to z."

For same lie, we want the lie to be picked on the outset. That means the lie quantifier should come first. Thus we have

$\exists x ~ \left[L(x) \: \land \:\forall y ~ \big(S(y) \to T(b,x,y)\big) \right]$

This translates back to English as "There exists an x such that x is a lie and for all y, if y is a secretary, Betty told x to y." Less formally, "There exists a lie x such that for any secretary named y, Betty told the lie x to y." This sounds good!

For a different lie, we want Betty to find a secretary first, and then pick the lie afterwards. This means the existential quantifier comes after the universal quantifier on the secretary. Therefore, we have,

$\forall y \left[S(y) \to \exists x ~ \big( \:L(x) \land T(b,x,y)\: \big) \right]$

This translates back to English as "For all y, if y is a secretary, then there exists an x such that x is a lie and Betty told x to y." Less formally, "For any secretary named y, there exists a lie x that Betty told to y." This sounds good. Note that this sentence doesn't imply Betty told every secretary a different lie; that is possible, but it is possible some lies were the same. If we wanted the lie to be unique, we would have to add

$\forall y \left\{S(y) \to \exists x ~ \left[ \:L(x) \: \land \: T(b,x,y)\: \land \: \forall z ~ \big((S(z) \land z \neq y) \to \lnot T(b,x,z) \big) \:\right] \right\}.$

This translates to, "For any secretary y, there exists a lie x such that Betty told that lie x to y and for any other secretary z, Betty did not tell that lie x to z."

However, I believe the question didn't intend for you to specify that it was a unique lie, just that it could be a different lie. So I believe you can ignore this third version.

Hope this helps!

Answer 3

Betty told every secretary a lie.

I was told that this sentence is ambigous as it can be intrepreted in two different ways.

Yes; due to hanging quantifiers—ambiguity due to mixed quantifiers not all specified at the beginning of the sentence—the above sentence matches both these symbolisations:

1. $\exists l{\in}L \;\,\forall s{\in}S \quad T(s,l)$
2. $\forall s{\in}S \;\,\exists l{\in}L \quad T(s,l).$

These are my two interpretations:

• Betty told every secretary the same lie.

This reading corresponds to Translation 1 above.

On the other hand, Translation 2 above means that every secretary gets a lie, not necessarily the same one, that is,

• To every secrerary, Betty told some lie.

• Betty told every secretary a different lie.

While it may be true that each secretary gets a distinct lie, the given sentence however you read it is not actually claiming this. This reading is at odds with Translation 1, and stronger than—though isn't precluded by—Translation 2.

(It is symbolised as $\forall s{\in}S \;\exists l{\in}L \;\big(T(s,l)∧\forall r{\in}S \;(T(r,l)→r=s)\big)$ or, equivalently, $\forall s{\in}S \;\exists l{\in}L \;\big(T(s,l)∧\forall p{\in}S \;\forall q{\in}S \;(T(p,l)∧T(q,l)→p=q)\big).$)

On a similar note, observe that with the given sentence, every secretary getting multiple lies is also a possibility, but asserting this would be over-reading what it actually says (reading more than what it says).

I know how to translate "Betty told every secretary a lie" in predicate logic.

Emphasising the last two words "a lie" doesn't actually disambiguate the given sentence; the question remains whether the secretaries are being fed a common lie; after all, this suggestion can be understood as "Betty told every secretary a lie (as opposed to a joke? or a half-truth?)".