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 Posted: Mon Aug 10, 2009 7:00 pm
This is logic if you don't know. sorry the implies symbol didn't come out right.
The problem is said "if p then q and if q then r logically implies if p then r"
tell if it is a valid argument.
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Posted: Mon Aug 10, 2009 7:35 pm
Valid in which system? Hypothetical syllogism is derivable in many of the common systems of logic, and certainly so in classical logic, but there are systems where it is not always true.
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Posted: Mon Aug 10, 2009 9:43 pm
I'm going to ignore Layra's prudence and bull ahead with some sentential logic that's probably wrong.
We have one premise
1. (If p then q) and (If q then r)
---
2. If p then q [1 Conjunction Elimination]
3. If q then r [1 Conjunction Elimination]
Let's assume p
4. p
---
5. q [2,4 Conditional Elimination]
6. r [3,5 Conditional Elimination]
7. If p then r [4-6, Conditional Introduction]
ninja
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Posted: Tue Aug 11, 2009 2:01 pm
I haven't yet taken logic; I've only had glimpses of the workings. But I know that if A means B and B means C then A means C. So yeah, that's valid logic.
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Posted: Tue Aug 11, 2009 2:21 pm
Jerba I haven't yet taken logic; I've only had glimpses of the workings. But I know that if A means B and B means C then A means C. So yeah, that's valid logic. MOAR CIRCUMLOCUTION!
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Posted: Tue Aug 11, 2009 9:20 pm
The hypothetical syllogism is almost an axiom in itself. For example, in the Russel-Whitehead axioms:
[RW4] [q→r]→[p∨q → p∨r]
Which under p→q ≡ ¬p∨q used as a definition in that system can be rewritten as [q→r]→[(p→q)→(p→r)], and the hypothetical syllogism follows immediately. There is also an alternative axiom:
[Tsuf] [p→q]→[(q→r)→(p→r)]
that does the work even more directly. Tpre, prepending r instead of appending it, is similar.
Just thought I'd add an axiom-based system perspective to the more rule-based one Morberticus used.
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Posted: Fri Aug 14, 2009 10:06 am
Because I'm bored, here's my shot at showing that RW4 can be a derived inference rule.
1. q → r [Assumption]
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2. p ∨ q [Assumption]
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3. q [Assumption]
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4. r [1,3, Conjunction Elimination]
5. p ∨ r [4 Disjunction Introduction]
6. q → p ∨ r [3-5 Conditional Introduction]
7. p [Assumption]
---
8. p ∨ r [5 Disjunction Introduction]
9. p → p ∨ r [5-6 Conditional Introduction]
10. p ∨ r [2,6,9 Disjunction Elimination]
11. p ∨ q → p ∨ r [2-10 Conditional Introduction]
12. (q→r) → (p ∨ q → p ∨ r) [1-11, Conditional Introduction]
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Posted: Fri Aug 14, 2009 3:16 pm
I think by 'conjunction elimination', you meant modus ponens (aka conditional elimination, detachment, etc.).
For reference, in that system, the full axioms are:
[RW1] p∨p → p
[RW2] q → p∨q
[RW3] (p∨q) → (q∨p)
[RW4] (q→r) → [(p∨q)→(p∨r)]
And the rules:
-- Uniform substitution.
-- Modus ponens (from ⊦p and ⊦p→q, infer ⊦q).
Although something like conditional introduction would make things a lot easier, I'm not sure whether it's strictly necessary. Still, I'd rather have it as a third rule.
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Posted: Fri Aug 14, 2009 4:02 pm
VorpalNeko I think by 'conjunction elimination', you meant modus ponens (aka conditional elimination, detachment, etc.). Ah yes. Wasn't paying attention when I was typing there. Thanks
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