I'm an undergraduate student and I have take a Mathematical Logic course this semester. I just read about the theorem of generalization in constants and I have (maybe a silly one) question. Why the hypothesis that symbol c does not occur in any wffs in Γ is essential? Could someone give me an example that the theorem will not work without it? Thanks you!
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$\begingroup$Suppose I know that Charlie is a dog, and so I have as a premise $Dog(c)$.
Now, if the universal generalization rule would not mandate that the symbol from which we generalize is a 'new' symbol, then we can use the premise to conclude $\forall x Dog(x)$, i.e. everything is a dog! Clearly we should not be able to infer that just because Charlie is a dog.
This is why we mandate the constant to be a 'new' constant. It should be a temporary name that we use to denote "some arbitrary object from our domain ... let's call it $c$" ... but that of course means that $c$ should not denote some specific individual as was the case with Charlie.
$\endgroup$ 0 $\begingroup$(Rule EI - Existential Instantiation) If Γ;αxc⊢β where the constant symbol c does not occur in any wffs in Γ , α or β , then Γ;∃xα⊢β (and there is a deduction of β from Γ;∃xα that does not use c ).
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