Give a formal proof of the sentence p from the single premise ¬¬p using only Modus Ponens and the standard axiom schemata. Warning: This is surprisingly difficult. Though it takes no more than about ten steps, the proof is non-obvious.
Give a formal proof of the sentence p from the single premise ¬¬p using only Modus Ponens and the standard axiom schemata. Warning: This is surprisingly difficult. Though it takes no more than about ten steps, the proof is non-obvious.
I am interested in how you solved it. When a derivation involves the third axiom (not B -> not A) -> (A -> B), which is responsible for double-negation elimination and classical logic, I know of only one systematic way of doing this. One can construct a derivation using the Deduction Theorem, which is usually much simpler. Since the proof of the theorem is constructive, one can then emulate it in this concrete instance to get a complete derivation.
I got the question from internet—I don't remember where, but it's quite interesting. I was taken by the warning, but it turned out to be a bluff.
As I understood it, only in the execution of a new line, Modus Ponens is required. The negation of an atomic formula is not an issue, particularly when used in the assumptions, so I did it as follows:
Given Argument: ~~P |- P
Proof:
1. P.................................Hypotheses
2. ~ ~ P...........................Premise
3. P.................................Hypotheses
4....... ~ P........................Assumption
5...... ~~P -->~~P............Hypotheses
6..............~~P.................Assumption
7. ......~~P.......................2,5, Modus ponens
Explanations:
Line 6 satisfies the hypotheses on line 5—cross out line 6. Line 5 becomes a premise.
Line 4 and 7 satisfy the hypotheses on line 3—cross out lines 4,5, and 7. Line 3 becomes a premise.
Line 3 satisfies the hypotheses on line 1. Lines 1 and 2 make up a valid argument.
If you don't like the contradiction for an inference, you can try this:
1. P.................................Hypotheses
2. ~ ~ P...........................Premise
3. P-->~~P.......................Hypotheses
4. P..................................Assumption
5...... ~~P -->~~P............Hypotheses
6..............~~P.................Assumption
7. ......~~P.......................2,4, Modus ponens
Line 6 satisfies line 5.
7 satisfies 3
4 satisfies 1
I am now more confused than before. What deductive system are you using: natural deduction, Hilbert system, or something else? In my usage, hypothesis, assumption and premise are all synonyms.
Sorry, brother. I don't have the slightest idea.
Will this help now?
1.Show P
..|------------------------------|
2|. ~ ~ P...........................|
3|. Show P........................|
..|.|-------------------------|....|
4|.|~ P.........................|....|
5|.|Show ~~P -->~~P.|....|
..|.|...|--------|...............|....|
6|.|.. |.~~P..|...............|....|
..|.|...|--------|...............|....|
7|.|.~~P......................|....|
..|.|------------------------|....|
..|------------------------------|
This looks like Fitch variant of Natural Deduction, but I still can't recognize it. For one, one does not write what one needs to prove (like "Show P") in natural deduction derivations; all formulas are either assumptions or have already been proven.
Bit that's OK if the solution works for you.
I have learned the method you mentioned, but with only one given it will not take you any where.
This method I have just begun learning is more powerful.
A free book for you:
Philosophy 110 on-line text