Language Proof & Logic 8.31 Fitch Proof

Question

Been working on this one question for the past hours and I can't ever seem to get the last step working. Any help would be appreciated!

Answer 1

Note that the first premise is a conjunction. So, you can just get the two conjuncts using Conjunction Elimination. You have the two conjuncts as the respective assumptions of the two subproofs you have, which would make sense if the conjunction was a disjunction ... but it is not. So again, you can get the two conjuncts at 'ground' level ... and hence you can get the Mythical(a)->Magical(a) and the ~Mythical(a)->Magical(a) sentences at ground level as well. And from there, you should be able to prove Magical(a) without too much effort.

Answer 2

This is as close as I got:

I think I'm stuck on the last step. Please update if you end up finding out how to do it.

Cheers!

Answer 3

Lines 6 and 7 should not be assumptions. They should be derived from Premise 1 by conjunction elimination; and thus do not raise contexts. The disjunction you should be eliminating is ~Mythical(c) v Mythical(c), derived by LEM.

If you cannot use a TautCon to derive ~Mythical(c) v Mythical(c) you can use nested proof by contradiction.

   |_ :
   |  |_ ~Magical(c)       Assumption
   |  |  |_ Mythical(c)    Assumption
   |  |  |  :
   |  |  |  :
   |  |  |  Magical(c)
   |  |  |  _|_            Negation Elimination
   |  |  ~Mythical(c)      Negation Introduction
   |  |  :
   |  |  :
   |  |  Magical(c)
   |  |  _|_               Negation Elimination
   |  ~~Magical(c)         Negation Introduction
   |  Magical(c)           Double Negation Elimination

Answer 4

I know it’s a formal logic question with a mechanical proof, but maybe taking a semantic approach might help you see what’s going on.

Firstly, we’re talking about a constant object C. (In practice C might be a parameter for any object, but let’s think of it in specific terms first)

We have three premises about C to start with.

Firstly, the last one is the simplest; if C (specifically) has horns then it is Magical. One example of this might be if we were assessing C as a horse - if it had horns then magic would be at work!

Next, if either C is not mortal or it is a mammal, then it has horns. Perhaps given what we know about C, by way of evidence of some action C has taken, we have narrowed it down to the possibility that the only way C could possibly not have horns is if it were some mortal non-mammal animal.

Last, some principles of Cs; if C is mythical then it is not mortal, and if C is not mythical then it is a mammal. I don’t know for sure if for real, non-mythical Centaurs existed, but I can tell you one thing about them for sure; they’d be mammals, rather than reptiles or plants.

Detective groundwork laid out about what we know about C - so, what conclusions can we draw? (I leave that to you and others)

This shift to semantics, over syntax, helps us watch out for situations where our intuitions struggle with general principles, and identifies possible remedies; often, it’s that we’re not being specific enough with our symbols, or hiding their specificity through the form our reasoning takes. Obviously this example is a bit silly but it should teach you something important about Logic as a nuanced formal tool.