How to prove $(A \rightarrow ( B\vee C) ) \rightarrow ((A \rightarrow B) \vee(A \rightarrow C))$ when $\vee$ means or, using natural deduction?
It is easy to prove the converse , but I didn't success to prove that.
Help me please. Thanks.
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(Note that I changed ○ to $\vee$ [\vee].) – Christopher Carl Heckman Apr 01 '16 at 08:47
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@CarlHeckman thank you. – user115608 Apr 01 '16 at 08:50
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You also need to clarify what you mean by "natural deduction"; are you allowed to use rules of inference? – Christopher Carl Heckman Apr 01 '16 at 08:51
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@CarlHeckman yes I think. – user115608 Apr 01 '16 at 08:53
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This can be done easiest by contradiction; proof by contrapositive is also possible. – Christopher Carl Heckman Apr 01 '16 at 08:55
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You missed a bracket, which I added. – user21820 Apr 01 '16 at 10:28
1 Answers
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Hint: Use law of excluded middle to split cases cleverly. $\def\imp{\rightarrow}$
Sketch:
If $A \imp B \lor C$:
$B \lor \neg B$. [Put the proof of LEM for $B$ before this.]
If $B$:
...
$A \imp B$.
$( A \imp B ) \lor ( A \imp C )$.
If $\neg B$:
If $A$:
$B \lor C$.
...
$C$.
$A \imp C$.
$( A \imp B ) \lor ( A \imp C )$.
...
user21820
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thanks,but I didn't understand the case which B is false. Explain please. – user115608 Apr 01 '16 at 10:52
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@user115608: Do specify which line you cannot figure out how to derive. – user21820 Apr 01 '16 at 11:02
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@user115608: From "$\neg B$" and "$B \lor C$" it is a short proof to derive $C$. Can you see how? Hint: If $\neg C$ then ... – user21820 Apr 01 '16 at 14:24
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In Polish notation one might write 1. CCpAqrACpqCpr. 2. CCpAqrACpqCsr and 3. CCpAqrACsqCpr have the same bracket type as 1. and are tautologies also. A double check on that is here: http://www.codeskulptor.org/#user42_BY5vtccr9k_0.py Neither 2. nor 3. has a converse like 1. If we assume the disjunction of the first variable and the negation of the first variable instead of that of the second variable, with some more work, all three of those tautologies can proved with a similar plan for each of them. But, I believe any of those proofs longer than the idea sketched above. – Doug Spoonwood Oct 07 '16 at 02:07
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@user115608: http://meta.math.stackexchange.com/questions/2831/troll-or-skeptic/2867. – user21820 Oct 31 '16 at 11:29