# Where to make a conjunction instead of an implication?

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.

• What you quote your teacher as saying is not right. Dec 15, 2015 at 17:56
• It might depend on the domain, but I think your teacher is wrong : his formula implies that everyone is a politician Dec 15, 2015 at 17:56
• @KevinQuirin: Well, isn't this true in some philosophical sense? :) But Benjamin must indeed be a very angry person.
– user98602
Dec 15, 2015 at 18:18
• 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. Dec 15, 2015 at 18:30

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).

• 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. Dec 15, 2015 at 21:16
• 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. Dec 15, 2015 at 21:18
• @BrianO thanks, edited. Also, saying "Benjamin hates everyone" I mean everyone from the set $X$ belongs to. Dec 15, 2015 at 21:26
• 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 Dec 15, 2015 at 22:53