Language Proof and Logic 13.4 13.5

Language Proof and Logic 13.4 13.5
yvanrc51
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Posted Apr 26, 2011 - 12:44 AM:

13.4

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yvanrc51
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Posted Apr 26, 2011 - 12:46 AM:

13.5

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yvanrc51
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Posted Apr 26, 2011 - 12:46 AM:

Now if some could help me out with 13.7, that would awesome. Anyone?
bartolmarin
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Posted Apr 26, 2011 - 5:59 AM:

First try from (P & Q) ->R deduce (P->(Q->R)). The procedure is same.



In 13.5:

step 3. and 4. can be one step (ab - side by side), step 6. and 7. can be one step, 9. and 10.

You can input universal generalization and implication together in one step (rule is: universal generalization )

Edited by bartolmarin on Apr 26, 2011 - 6:05 AM
Spoonwood
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Posted Apr 26, 2011 - 10:46 PM:

deseraimp wrote:


HEY ASSHOLE. No one cares that you have your PhD in first-order logic. Don't be a condescending prick. If someone needs help, help him. I promise you do not need an egoboost.


Interesting enough he had a vacuous quantifier in step 5 also. So easy? Then why isn't every line of your proof perfect?
Axiomatica
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Posted Apr 21, 2012 - 9:17 PM:

You can do the proof for 13.5 in 8 steps instead of 14 steps. I think this is worthy of "resurrecting" the thread for. Here's why: The solutions to this problem turns out to be very easy if one remembers that when one introduces one or more individual constants (two in the case of these two problems), these constants are representative of arbitrary individuals in the domain of discourse. They are not tied to particular variables.
Thus, in 13.5, having introduced a subproof with boxed c and d, the first premise can be instantiated as ‘(Cube(c) ˄ Dodec (d)) →Larger (d,c)’ and the second premise can be instantiated as ‘Larger (d,c) ↔ LeftOf (d,c)’ . It doesn’t have to be instantiated as ‘Larger (c,d) ↔ LeftOf (c,d)’. The proof follows immediately (and easily).


1. ∀x ∀y ((Cube(x) ∧ Dodec(y)) → Larger(y, x))
2. ∀x ∀y (Larger(x, y) ↔ LeftOf(x, y))
3. Assume Cube(c) ^ Dodec(d)
4. ((Cube(c) ^ Dodec(d)) → Larger(d,c)) ∀Elim 1
5. (Larger(d,c) ↔ LeftOf(d,c)) ∀Elim 2
6. Larger(d,c) -> Elim 3,4
7. LeftOf(d,c) <-> Elim 6,5 End assumption
8. ∀x ∀y ((Cube(x) ^ Dodec(y)) → LeftOf(y, x)) ∀Intro 3-7

Hope this helps.
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