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Forums: Philosophy, Math, Logic, Proof, Difficult
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This is the last one I really need help with

1. (A • B) v (C • D)
2. ~A / C

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@agarry13,

What rules are available to you?

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@Ding an Sich,

All 8 rules of implication and the 10 rules of replacement

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Spell it out for us. List them.

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@agarry13,

agarry13 wrote:

This is the last one I really need help with

1. (A • B) v (C • D)
2. ~A / C

I'll give you a hint: take advantage of distribution rules. That's the easiest way to attack this.

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@Kolyo,

modus ponens, modus tollens, pure hypothetical syllogism, disjunctive syllogism, constructive dilemma, simplification, conjunction, addition, De Morgan's, commutativity, associativity, distribution, double negation, transpotition, material implication, material equivalence, exportation, and tautology

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@agarry13,

I have a hunch Ding an Sich already knew them all.

As for me, I'll try to find a page on the Internet that has those 18 rules written out in symbols.

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@agarry13,

agarry13 wrote:

modus ponens, modus tollens, pure hypothetical syllogism, disjunctive syllogism, constructive dilemma, simplification, conjunction, addition, De Morgan's, commutativity, associativity, distribution, double negation, transpotition, material implication, material equivalence, exportation, and tautology

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@agarry13,

Note: I don't really have any formal training in this, but I'm used to proofs, because I was a math major.

agarry13 wrote:

This is the last one I really need help with

1. (A • B) v (C • D)
2. ~A

3. ~A v ~B (addition, from #2)
4. ~(A • B) (DeMorgan, from #3)
5. (C • D) (disjunctive syllogism, with #1 and #4 for premises)
6. C (simplification, from #5)

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@agarry13,

The proof here is quite simple using the method of 'backward fell swoop'.
Choosing C to be false makes the truth of the disjunction in premise 1. dependent on the truth of both A and B. But premise 2. negates the possible truth of A, therefore C false makes the premises false, hence the argument is valid.

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