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"If I do not approve of this subject, my girlfriend will be sad," says Juan. Later we know that Juan has approved this subject, but that his girlfriend is sad. Was John's statement true or false? Why?

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    Is this homework? Regardless, we ask users to describe their thought process and specific issue that obstructs them. – Conifold Jan 27 '17 at 21:17
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There's not much evidence to determine whether the statement is true or not. It might be easiest to explain with a truth table:

  • A = Juan approves
  • S = His girlfried is sad
A  S  |  ~  A  →  S
--------------------
0  0  |  1  0  0  0  
0  1  |  1  0  1  1  
1  0  |  0  1  1  0  
1  1  |  0  1  1  1  

The column under the arrow indicates the truth value of the statement under all the possible scenarios. It's true under all conditions except for when both A and S are false.

Knowing that Juan has approved and that his girlfriend is sad only tells us that A and S are true, corresponding with the last row of the table. That doesn't falsify the statement, but it also isn't enough to establish its truth. There are other scenarios which could possibly falsify it.

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The instant Juan/John approves of the subject, his statement becomes a counterfactual statement, which means its truth depends on the truth-functionality of the particular modal logic and on the particular metaphysic of concern. Regardless of whether or not Juan approves of the subject, his statement cannot be made false by his girlfriend being sad. His statement could be made true if, in some possible world (that is not the actual world), Juan does not approve of the subject and Juan's girlfriend is sad. Likewise, his statement could be made false if, in some possible world (that is not the actual world), Juan does not approve of the subject and his girlfriend is not sad.

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