After reading other similar questions I got even more confused:
(1) “Every student in this class has visited Mexico”
$\forall x(S(x) \rightarrow M(x))$
For (example 1) we could either get:
-
T-T (Is a student and has visited Mexico)
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F-T (Not a student and has visited Mexico)
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F-F (Not a student and hasn't visited Mexico)
And I was told this makes sense because example 1 can be re-written as "For every person x, if that person is a student then he has visited Mexico", and we can have the cases F-T, and F-F because they don't go against the evidence of our statement since we didn't say anything about when they're neither a student or went to Mexico.
(2) “Some student in this class has visited Mexico”
$\exists x(S(x) \land M(x))$ is right.
$\exists x(S(x) \rightarrow M(x))$ is wrong
- If person is a student, then he went to Mexico
- If a person is not a student, then he went to Mexico
- If a person is not a student, then he didn't go to Mexico.
In example 1 we came to the conclusion that we can have cases 2 and 3 because we didn't say anything about when they're neither a student or went to Mexico, but now for the existential quantifier with implication, we're not allowed to do that because we're not precisely capturing the desired values, since "there exists a person who is not a student" makes the statement true.
I really don't understand why implication works for Universal but doesn't work for Existential
Best Answer
It's because, as they say, if there is an object in the domain that is not a student, then $S(x)$ would be false, and given the way that we have defined the truth-functional implication, the $S(x) \rightarrow M(x)$ part would then be automatically true. Hence, the statement $\exists x (S(x) \rightarrow M(x))$ can be made true by simply some object in the domain that is not a student. In fact, if there isn’t a single student in the domain, the statement is automatically true! This is of course not what we want. We do want to say that there is a student, and that they visited Mexico, i.e. $\exists x (S(x) \land M(x))$
For the universal, things are different. If we were to use $\forall x (S(x) \land M(x))$, then we are saying that everything in the domain is a student and went to Mexico ... so if we would have objects in the doamin other than student, e.g. bananas, then we end up pointing to those bananas and say: "that is a student, and it visited Mexico" ... which is clearly not what we want ... We are merely saying that all students visited Mexico, not that everything visited Mexico. That is, we should say that out of all objects: if it is a student, then 'it' went to Mexico ... i.e. $\forall x (S(x) \rightarrow M(x))$