I know you can prove multiple ways, but I actually don't know what each means.
Is semantics where you provide a paragraph proof and use objects and deduction where you do subproofs/elimations/intros to get to the conclusion?
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Semantics is to do with meaning, and syntax is to do with form. Its quite possible to set up a deductive system and then formally prove something that is wrong; this because the semantics of the situation hasn't been taken into account.
To give a crude example one could formally develop a theory of happiness by scoring peoples happiness out of ten; and then you can ask questions about the total amount of happiness, or its maximum and so on; but the semantics of this situation, when thought about, is can you measure a qualitive thing such as happiness quantitively? If this is even possible, how to take account of people lying, or not being aware of the true state of their happiness: it isn't for nothing that 'know thyself' was inscribed in the forecourt of the Temple of Apollo at Delphi. All these are common objection to theory of Utilitarianism as developed by Bentham.