## Contradictory Conclusions - Both cannot be valid

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Q: Consider the following two propositions: Oswald shot Kennedy; Oswald did not shoot Kennedy. Suppose that we have two pieces of reasoning, one with the first as conclusion and the other with the second as conclusion, and each with a single premise that is true. Explain why the pieces of reasoning cannot both be valid.

A: They both cannot be valid as they are contradictions

If P = True – Conclusion = True :. O shot K

If P = True – Conclusion = True :. O did not shoot K

Both conclusions cannot be true and valid as it’s not possible to do and not do the same thing.

The above is my answer, which I have been told is wrong. Could someone please advise me?

Just to clarify with reference to what you have put down for "A", was that your answer, or is it the apparently correct answer? – Five σ – 2015-03-24T23:07:28.507

Sorry - that was my answer. I don't know the correct answer (the answer the Lecturer is wanting). I am apparently on the right track.... – Trish – 2015-03-24T23:20:00.910

Hmm, this is interesting. My thoughts would have been in line with yours. Let's look at argument 1: Premise: A, Conclusion: X. Argument two: Premise: B, Conclusion: ~X (the negation of the previous conclusion). I would say that X and ~X both cannot be simultaneously true, as it leads to a contradiction. It's like me saying "I am answering and not answering your question". This is known as the "law of contradiction" in classical logic. – Five σ – 2015-03-24T23:34:23.843

Thanks - I think your wording is a lot clearer (I am new to this). When you say "simultaneously true" can it be substitute for 'simultaneously valid'? or would this not be correct terminology? – Trish – 2015-03-24T23:53:44.247

"True" and "Valid" are different concepts. An argument can be valid whilst having premises that are not true. With reference to your original question, I have offered my thoughts as an answer, rather than continuously adding comments under your post. – Five σ – 2015-03-25T00:42:36.453

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A valid argument with true premises has a true conclusion.

Take modus ponens for example:

If A, then B

A

Therefore B

If the antecedent and consequent are both true, then the conclusion "Therefore B" also has to be true.

Now, let's take a look at your question. For starters, let's assume that both arguments are valid and have true premises, which should imply that they both have true conclusions. Now we note that both conclusions cannot be simultaneously true , because that would lead to a contradiction. This means that (at least) one of the conclusions has to be false.

We know from your question that both premises of both arguments have to be true. We have also noted that both conclusions can't be true at the same time. As aforementioned, a valid argument with true premises has a true conclusion.

For (at least) one of the arguments, the conclusion is not true. Since a valid argument with true premises has a true conclusion, and this argument has a true premise but a false conclusion, this argument is not valid, so both arguments cannot be valid as (at least) one argument cannot be valid.

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Both conclusions cannot be true and valid as it’s not possible to do and not do the same thing

If I'm understanding, your sentence correctly you seem to be saying:

1. both conclusions cannot be true

2. both conclusions cannot be valid

It might just be a fault of wording, but the second claim is incorrect. Validity is a concept that applies to arguments -- not conclusions. It's possible you're getting the answer wrong because of this somewhat nit-picking problem.

Now on to the answer as to why at least one of these arguments must be invalid.

First, we need a definition for reference: validity means that if the premises were to be true, then the conclusion must also be true.

Second, I'm not quite sure what "pieces of reasoning" are. I'm going to assume that means arguments.

Third, it is possible for contradictory conclusions to be reached by valid arguments. This is true because validity only looks at the logical relationship between the premises and conclusion. Thus, we could conceivably have two valid arguments with the conclusions Oswald shot Kennedy and Oswald did not shoot Kennedy. (Thus, the explanation you provide is wrong (or at least horribly unclear) -- because contradictory conclusions means nothing).

What matters here is what happens when we apply the definition of validity to each argument in conjunction with the stipulation that the sets of premises for each are true. Let's say the shot argument is argument A. The did not shoot argument is argument B.

We can construct the following argument:

1. If A is valid and A's premise is true, then Oswald did shoot
2. If B is valid and B's premise is true, then Oswald did not shoot
3. A's premise is true [given]
4. B's premise is true [given]
5. | A is valid (assumption of sub-argument)
6. | A is valid and A's premise is true [&I5,3]
7. | Oswald did shoot [MP 1,6]
8. | ~ [B is valid and B's premise is true] [MT 2,7]
9. | Either B is not valid or B's premise is not true [DeM 8]
10. | B is not valid [DS 9,4]
11. Ergo, if A is valid, B is not valid

Then repeat 5-10 for the assumption B is valid as a proof that if B is valid, then A is not valid.

This may be what you meant to say with your sentence, but it's not clear enough. A clearer wording is that their conclusions are incompossible and the premises are true, so at least one argument must not be valid since if both were valid both conclusions would follow necessarily.

1Thank you Virmaior. Your detailed explanation has helped greatly. Thanks for also pointing out my wrong / unclear statements - this helps the learning process – Trish – 2015-03-25T02:14:06.403