How do I prove this argument valid?

[Protoman2050]

How do I prove this argument valid using reductio ad absurdum and disjunction elimination?

"I am a Christian iff I have Jesus as my Savior. I have Jesus as my Savior. Therefore, I am a Christian."

1. C<->J
2. J
3. :.C

Thanks!

[boagie]

Hi Protoman,

I am not really into syllogisms but I believe this one is circular, actually as near as I can understand it, its solid, but circular arguments are apparently a no no. We have some gifted people here in the realm of logic I am sure they will show you the error of your ways. Good luck!!

[Protoman2050]

Originally Posted by boagie

Hi Protoman,

I am not really into syllogisms but I believe this one is circular, actually as near as I can understand it, its solid, but circular arguments are apparently a no no. We have some gifted people here in the realm of logic I am sure they will show you the error of your ways. Good luck!!

Circular? How? Biconditionals express equivalence, and they have rules of inference. As you said, I'll wait for a logician to get to this thread.

[boagie]

In circular reasoning, "The definition comes first and then the supposed proof is based on that definition. This is proving something (at the end) by making logical deductions from premises that themselves contain the conclusion. Looping from the end to the beginning that way is called circular reasoning. Circular reasoning often sounds right, but it is invalid nonetheless. ... It is often hard to recognize reasoning as circular because the steps between the first and last may be many." {Logic and Literary Argument by Eric Rabkin}

[VideCorSpoon]

First, your criteria for the proof’s solution. Deductive elimination commonly goes by the name “constructive dilemma.” Same concept, different name. Reductio ad absurdum is a different matter. That is (from what I can tell) an “indirect proof.” Both are indeed inferences, but the Reductio ad absurdum is a little more complex, because it requires nesting an argument inside another argument. But still, a good question.

But Boagie makes a good point. Bi-Conditionals could be considered a tautologies if we examine them a bit closer. But for all intensive purposes, I’m looking only at the logical rules to extrapolate a comparison. A tautology inference states that for any P, PvP or P&P can be inferred. And for a Bi-Conditional, it is true as long as both components have the same truth value. A lot like a tautology as Boagie points out. But you also have a good point in that we must look only at the equivalence factor in a bi-Conditional because we must adhere to inferences. So actually, both you and Boagie are right… and further with really good perspectives on the same problem.

But to get to your proof.

There are a few problems though. If you want to use a constructive dilemma for your argument, you run into a problem setting up your argument when you are required to provide a disjunction (either, or) of the two antecedents of your existing conditionals. The only conditionals you can use are derived from equivalence of the bi-conditional. So you would need to provide, in your given argument before the proof, C v J (basically, I am a Christian or I have Jesus as my Savior). But that does not sound right, even though it could theoretically be done in the proof. You would in a sense need to contradict your faith to further the argument that you are indeed a Christian…oddly enough. And this is where the reductio comes in. This is also a rule which needs you to contradict your belief in order to affirm it. (i.e. ~P, then P by Indirect proof). The use of these inferences are problematic… but not impossible. But still, we want to solve the proof.

This may be a simpler way to solve your proof.

Gotta love Modus Ponens! Hope this helps.

[Protoman2050]

Originally Posted by VideCorSpoon

First, your criteria for the proof’s solution. Deductive elimination commonly goes by the name “constructive dilemma.” Same concept, different name. Reductio ad absurdum is a different matter. That is (from what I can tell) an “indirect proof.” Both are indeed inferences, but the Reductio ad absurdum is a little more complex, because it requires nesting an argument inside another argument. But still, a good question.

But Boagie makes a good point. Bi-Conditionals could be considered a tautologies if we examine them a bit closer. But for all intensive purposes, I’m looking only at the logical rules to extrapolate a comparison. A tautology inference states that for any P, PvP or P&P can be inferred. And for a Bi-Conditional, it is true as long as both components have the same truth value. A lot like a tautology as Boagie points out. But you also have a good point in that we must look only at the equivalence factor in a bi-Conditional because we must adhere to inferences. So actually, both you and Boagie are right… and further with really good perspectives on the same problem.

But to get to your proof.

There are a few problems though. If you want to use a constructive dilemma for your argument, you run into a problem setting up your argument when you are required to provide a disjunction (either, or) of the two antecedents of your existing conditionals. The only conditionals you can use are derived from equivalence of the bi-conditional. So you would need to provide, in your given argument before the proof, C v J (basically, I am a Christian or I have Jesus as my Savior). But that does not sound right, even though it could theoretically be done in the proof. You would in a sense need to contradict your faith to further the argument that you are indeed a Christian…oddly enough. And this is where the reductio comes in. This is also a rule which needs you to contradict your belief in order to affirm it. (i.e. ~P, then P by Indirect proof). The use of these inferences are problematic… but not impossible. But still, we want to solve the proof.

This may be a simpler way to solve your proof.

Gotta love Modus Ponens! Hope this helps.

That's so cool! But I'd love to see the RAA proof...that'd look interesting --I have no problem in denying my faith if it's in the course of a logical proof to prove said faith--, in both propositional form and plain English. As well as the RAA proof of this:

1. God is either unity or triune
2. If God was unity, He could not know Himself
3. God can know Himself
4. Therefore, He must be triune

Thanks!