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Truth table / proof help

 
 
Reply Wed 27 Jul, 2016 06:02 am
I all, I'm pretty new to logic but I've been trying to teach myself from The Art of Reasoning by David Kelley, if you've ever come across it. I thought I was doing fine, but I think I may be misunderstanding something and I hope someone can clear it up for me.

In the book, the author walks through the construction of a proof. I won't go through the proof itself, but the argument is as follows, and the author does finish with a completed proof.

1. A ⊃ B
2. C · ~B
3. (C ∨ D) ⊃ E
4. E ⊃ F / ~A · F

I have been getting a little stuck on proofs, particularly the use of addition, so I thought I would construct a Truth Table around this argument to show its validity. The problem is, I seem to have found a way of making the conclusion false and the premises true. From this I gather either either 1)the author made a mistake in his proof, 2) I have made a mistake in the construction of my truth table, or 3) I am misunderstanding the function of truth tables and/or proofs. I have included my table below (can't seem to get it to format very well I'm afraid - just skip D as that's the only statement for which I haven't specified a truth value), if anyone can shed any light on this I would be very grateful!

A ⊃ B C · ~B (C ∨ D) ⊃ E E ⊃ F / ~A · F
F T F T T T F T T T T T T T T F F T
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fresco
 
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Reply Wed 27 Jul, 2016 09:28 am
@Monkeymox,
Instead of dealing with the whole truth table, choose values for the components of the conclusion which make it false. If those values are inserted into the premises, and any premise turns out false, then the argument is valid.
(Ref: Method of Backward Fell Swoop).
Monkeymox
 
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Reply Wed 27 Jul, 2016 12:38 pm
@fresco,
Unless I'm misunderstanding you, that's what I've done. If you don't mind I'll go through my reasoning (forgive me if I use incorrect terminology in places, I'm new to this!), because I'm not sure if I'm doing this wrong.

The conclusion is a conjunction, which I've made False. As the conjuncts are ~A and F, I have made F true, A false, but the negation true.

To make the conditional in the first premise true, I have made B false, since A is true. This makes the negation in the second premise true, so I have made C in the second premise true, to make the whole premise true.

This makes the disjunctive in the third premise true, as C is true and I don't need the truth value of D, since if either disjunct is true then so is the whole statement. I then made E true, to make the entire statement true.

If E is true, I only need to make F true to make the fourth premise true. So now all four premises are true, and the conclusion is false.... I think....
fresco
 
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Reply Wed 27 Jul, 2016 12:53 pm
@Monkeymox,
You seem to have it the wrong way round. With the conclusion ~A . F you need either A true, F false, or both, to make it false. A false conclusion from false premises is a valid argument.
(If F false then from 4, E is false,
so both C and D are false in 3,
which makes 2 false)
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fresco
 
  1  
Reply Wed 27 Jul, 2016 02:05 pm
@Monkeymox,
Ah...I see your mistake. You are reading the conclusion as ~(A . F ) rather than
(~A) . F. .......the negation only applies to A.
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