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I am a graduate teaching assistant at a university and I teach recitations for a discrete mathematics course for computer science majors. Right now they are studying propositional logic, and the recitation included the following question:

If $x$ represents an element of the set containing all people, and $y$ represents a certain time, and $P(x)$ is the statement that "$x$ is pleased by you", and $T(y)$ is the statement "the time is $y$", translate the following sentence into a logical statement using quantifiers and logical operations: "Everyone is pleased by you some of the time".

The solution manual I was given stated that the solution is

$$\forall x \; \exists y \; s.t. \; P(x)\wedge T(y)$$

which I didn't initially see a problem with. But then in class students came up with the solution

$$\forall x \; \exists y \; s.t. \; T(y)\Rightarrow P(x)$$

which also seems correct to me. Is it possible for an English sentence to have two logical representations that are not equivalent or is one of these logical statements not equivalent to the English sentence "Everyone is pleased by you some of the time"?

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  • $\begingroup$ A weird question for propositional logic ... $\endgroup$
    – Bram28
    Oct 11, 2019 at 19:29

3 Answers 3

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The solution should use a two place predicate, $P(x,y)$ to mean $x$ is pleased by you at time $y$. Separating the two loses the idea that someone can be pleased by you at some times and not at other times. Using $T(y)$ to mean "the time is $y$" seems to involve the time when the sentence is said. Using that predicate the solution is $\forall x \exists y P(x,y)$

It is certainly possible for one English sentence to have more than one representation because English sentences can be ambiguous. A simple example is "A or B". In math it is clear that or is inclusive, but in English it is not, so this could be $A \vee B$ or it could be $(A \vee B) \wedge \lnot (A \wedge B)$

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The manual is right, and the students' answer is wrong.

The two formulas are equivalent if we add an axiom that all times exist. However, this was not stated in the task.

Indeed, assume that some time does not exist, such as negative values of time (measured from the moment of the Bing Bang, the origin of the Universe), or complex values of time $a+ib$, etc. Taking $t=-1$, we have $\forall x: T(-1)\Rightarrow P(x)$: either the time is not $-1$ (which is always true for all possible times) or $x$ is pleased with you (which might not be true for any point in time).

However, with the solution provided by the manual this is not a problem: it requires that both the moment of time exist (say, be non-negative) and at this time $x$ be pleased with you.

More precisely, the task presupposes a binary function on the Cartesian product of some set of persons $X$ and some set of time moments $T$: whether a person $x$ is pleased by you at the moment $t$. Unless you presuppose that $T$ includes all numbers $y$, the students' solution is wrong.

Or, you can see this task as ambiguous: yes, this sentence's meaning can be different depending on how much knowledge of the real world you assume to be presupposed. Say, whether you assume it to be granted that all moments of time exist. Or, if you know a person who never likes you, then the required logical expression is just "FALSE" :)

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The general problem with logical statements of the kind:

$$\forall x \; \exists y \; s.t. [bla bla(x,y)] \Rightarrow [thisorthat(x,y)]$$

is that it is often easy to make such a statement true, without capturing what one tried to capture. For example, if for every $x$ you can simply point to any $y$ for which the antecedent is false, then the conditional becomes true, and hence the whole statement is true ... but you never ended up saying anything about there being any $y$ for which the antecedent is true ... and for which the consequent is (therefore) also true. To do that, you have to use the $\land$

Now, what complicates the issue here, is that this is a really horribly posed question.

First of all, to interpret $T(y)$ as "The time is $y$" is really weird. ... do they mean: $y$ is the same time that it is now? But what is now? Maybe they simply meant $y$ is a point in time? But $y$ was already (somehow) restricted to points in time, in which case $T(y)$ is trivially true ... And by the way, if you do want to restrict your use of a variable to a certain domain, you'll have to do something like $\exists y \in T$ ... with $T$ being a set of times. Without that, you really cannot just say that $y$ is restricted ...

Second, as RossMillikan points out, the question really wants to declare that everyone is pleased by you at some time. So, the 'please' predicate will need to make reference to the time, i.e. you really need a $P(x,y)$ kind of predicate.

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