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Hi I'm currently working through the book "How to Prove It" by Daniel Velleman and I'm having some trouble choosing between the conjunction and conditional statement when constructing logical statements. The most recent issue I had is when solving a problem where we have to convert a statement in English to one written using logical connectives and quantifiers.

The statement is "Everyone has a roommate who dislikes everyone". Using M(x) to stand for "x is a math major" and L(x,y) to stand for "x likes y" I came up with the following

$\forall x \exists y \space(\space R(x,y) \rightarrow \neg \exists z \space L(y,z) )$

which I think fits the bill. But after searching online I found another persons answer which was

$\forall x \exists y \space(\space R(x,y) \wedge \forall z \neg \space L(y,z) ).$

Now I know that $\neg\exists$ is the same as saying $\forall\neg$ so I don't have an issue with that. But why would you use the conjunction instead of the conditional statement? Reading them "in English" they seem to imply the same meaning but my mind just immediately jumped to using the conditional and it seems more "natural" to me in this case because of it.

So I guess my question is how do you decide between using the conjunction and the conditional statement. I know that, of course, these two are not interchangeable and represent different "operations" since they don't have the same truth table.

Maybe this comes down to my tenuous understanding of the conditional statement. Perhaps the reason I am misusing the conditional statement here and in other occasions is because it's definition is a bit weird to me.

Like the conditional statement is basically only ever false when the hypothesis is true but the conclusion is false. Yet if the hypothesis is false then it doesn't matter whether the conclusion is true or false, we just say the conditional statement is true. And it's the part where the hypothesis being false automatically makes the conditional statement true that kinda trips me up. I've tried to reason why that's the case and I've kind of just accepted it on faith but it still doesn't sit right with me. If anyone could shine some light on this then maybe I'd be able to stop misusing the conditional. But if this is too tangential and worth it's own post then let me know and I'll edit this bit out.

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Consider the similar but simpler case: "there is a man who likes books".

The formalization will be:

$\exists x (\text {Man}(x) \land \text {Likes-Books}(x))$.

Due to the presence of $\land$, the quantified formula: $\text {Man}(x) \land \text {Likes-Books}(x)$, is false if we replace $x$ with the name of my dog: "Fido" (Fido is not a man).

If we write instead: $\exists x (\text {Man}(x) \to \text {Likes-Books}(x))$, now we have that the formula $\text {Man}(x) \to \text {Likes-Books}(x)$ will be true for Fido, because a conditional with false antecedent is true.

Thus, in every "universe" without men, the sentence $\exists x (\text {Man}(x) \to \text {Likes-Books}(x))$ will come out true.

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  • $\begingroup$ Ok thanks for your response. The example you gave is helpful, but I just had a quick question. So is the point of your example to suggest that a conjunctive statement is more fundamentally linked to the underlying universe of discourse than a conditional statement? Like in your example the conjunctive statement only has the possibility of being true if the UOD includes men, while the conditional will be true even if the UOD has no men. $\endgroup$ – skippy130 Feb 6 '18 at 2:43
  • $\begingroup$ @skippy130 - NO; it is simply an efefct of the truth table for the conncetives. $\endgroup$ – Mauro ALLEGRANZA Feb 6 '18 at 6:58

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