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We have this statement in natural language:

Benjamin hates all politicians.

I suggested this formula:

$$ \forall X ( politician(X) \Rightarrow hate(benjamin,X)) $$

But our teacher has written this as a solution:

$$ \forall X ( politician(X) \land hate(benjamin,X)) $$

And I quite can't understand the difference between conjunction and implication in cases like this.

Maybe I need more clarification on implication in the real world to get this straight.

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    $\begingroup$ What you quote your teacher as saying is not right. $\endgroup$ Dec 15, 2015 at 17:56
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    $\begingroup$ It might depend on the domain, but I think your teacher is wrong : his formula implies that everyone is a politician $\endgroup$ Dec 15, 2015 at 17:56
  • $\begingroup$ @KevinQuirin: Well, isn't this true in some philosophical sense? :) But Benjamin must indeed be a very angry person. $\endgroup$
    – user98602
    Dec 15, 2015 at 18:18
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    $\begingroup$ That's just wrong. It says, everything $X$ is a politician and Ben hates it (= him). You understood the difference between conjunction and implication just fine before you saw your teacher's solution ;) Stick to that. $\endgroup$
    – BrianO
    Dec 15, 2015 at 18:30

1 Answer 1

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Your teacher is right if and only if $X$ is iterating over a set of politicians (i.e. $X\in \rm{Politicians}$). The mathematical formula $$\forall X \quad\rm{politician}(X) \implies \rm{hate}(B,X)$$

is translated into

For all $X$, if $X$ is a politician, $B$ hates it.

Which is true. The second formula $$\forall X \quad\rm{politician}(X) \land \rm{hate}(B,X)$$

is translated into

For all $X$, $X$ is politician and $B$ hates it.

It means not only all $X$ are politicians but also that $B$ hates everyone (from the set $X$ is iterating over).

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    $\begingroup$ Nevertheless, this is not an argument that "Benjamin hates all politicians" should be rendered with a conjunction. If everything in a model is a politician, then for truth in that model you don't have to qualify that anything is or isn't a politician, because the statement is equivalent to "B hates everyone", including himself, as B must be a politician in such a model. Anyway, that's just one weird model which you had to cook up to show that the claim is even possible somewhere. The teacher is still wrong. $\endgroup$
    – BrianO
    Dec 15, 2015 at 21:16
  • $\begingroup$ PS "Your teacher is right if and only if X is a set of all politicians". Just to be a stickler for correctness in these matters: $X$ is not a set of anything, it's a variable, ranging over elements of models. $\endgroup$
    – BrianO
    Dec 15, 2015 at 21:18
  • $\begingroup$ @BrianO thanks, edited. Also, saying "Benjamin hates everyone" I mean everyone from the set $X$ belongs to. $\endgroup$ Dec 15, 2015 at 21:26
  • $\begingroup$ I understood. But if "everyone" in a model has to be a politician for the conjunctive rendering to be correct, then in that model Benjamin is a politician who hates himself. :P $\endgroup$
    – BrianO
    Dec 15, 2015 at 22:53

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