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I would like to convert the following English sentence to a Horn clause.

For every man that likes a woman, if she likes him back, then he is happy.

I tried the two following conversions, but I'm not sure if they are correct. 

man(x) ∧ woman(y) ∧ (likes(x,y) ∧ likes(y,x)) → happy(x)

man(x) ∧ woman(y) ∧ (likes(x,y) ↔ likes(y,x)) → happy(x)
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    Is the second formulation a Horn clause? What about the first? In which case(s) do they disagree? And, finally, which one is right? – 2017-01-10

1 Answers 1

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Let's first symbolize it, and then we'll worry about putting it into a Horn clause.

OK, the straughtforward symbolization is:

$\forall x \forall y ((Man(x) \land Woman(x,y) \land Likes(x,y)) \rightarrow (Likes(y,x) \rightarrow Happy(x)))$

Using Exportation ($P \rightarrow (Q \rightarrow R) \Leftrightarrow (P\land Q) \rightarrow R$) we can rewrite this as:

$\forall x \forall y ((Man(x) \land Woman(y) \land Likes(x,y) \land Likes(y,x)) \rightarrow Happy(x))$

Drop the quantifiers, and this is your 1)

Your 2) is not correct, because first of all it is not the correct symbolization (note that $Likes(x,y) \leftrightarrow Likes(y,x)$ would be true if both $Likes(x,y)$ and $Likes(y,x)$ are false, which would certainly not make the man happy), and second, it is not even a Horn clause.