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Piggy backing off of this post here I want to make sure I understand this point clearly.

Are valid arguments content blind?

Even if the conclusions they make contradict?

For example, the following 3 arguments are all valid (as they use modus tollens) but the conclusions contradict.

Argument 1:

  1. If God does not exist, there are no objective moral values.
  2. There are objective moral values.
  3. God exists.

Argument 2:

  1. If 2 Gods do not exist, there are no objective moral values.
  2. There are objective moral values.
  3. 2 Gods exists.

Argument 3:

  1. If God exists, there are no objective moral values.
  2. There are objective moral values.
  3. God does not exist.
fgharo91
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    If by "valid" you mean formally valid then yes, they only depend on the form and blind to content/matter (hence "formal"). However, most real world arguments are informal and rely on inference warrants that are content dependent. For instance, "if one sample of compound X is soluble in water then all of them are" is a materially valid inference in chemistry, but "if one human is a liar then all of them are" is not in sociology, even though it has the same form. See argumentation theory – Conifold Oct 10 '17 at 19:33
  • Your premises (statement 1) in Arguments 1 and 3 are inverses (or converses) of each other. They are not contrapositives and are not logically equivalent. Therefore, you would expect to get different theorems from these premises/axioms. However, I suspect you already knew this, and your question is deeper? –  Oct 11 '17 at 16:53
  • @barrycarter Statement 1 in argument 1 and statement 1 in argument 3 are neither inverses, converses, or contrapositives of each other. I simply wanted to know whether 3 different arguments which have a contradicting conclusion can still be valid or in other words content blind. The 3 different conclusions here are that 1 god exists, 2 gods exist and no god exists. converse-inverse-contrapositive – fgharo91 Oct 11 '17 at 19:07
  • Yup, you're right. My mistake. –  Oct 11 '17 at 19:20

2 Answers2

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The simple answer is: yes.

Your examples above use modus tollens, one of the most commonly known valid argument formats. Deductive validity is about abstract format exclusively, not about the content of the premises.

The reason arguments involving conditional statements (as the ones above do) are valid or invalid here may help. What the first premise states is in the abstract format "If A, then B." What A and B are really expressing are necessary and sufficient conditions (hence calling it a "conditional"). What "If A, then B" asserts is actually a relationship between A and B. This is so no no matter what you're using for either (true, false, unknown, or nonsense).

In formal logic, the argument would run like this: P1. A -> B (If A is true, then B is true) P2. ~B ("Not" B - B is not true) Therefore, ~A. (Not A - A is not true)

What's happened? A and B are asserted in (1) to have a certain relationship to each other in which A is sufficient for concluding B (A is enough by itself to show B's truth), and at the same time, B is necessary for A (B must be true, otherwise A is false, but B may not be sufficient). Since the second premise goes on to state that we "don't have" B - it's false - and the first premise asserts, in part, that we have to have B to have A... but we don't have B... it follows that we don't have A either.

So using one of your examples above: P1. If God does not exist, there are no objective moral values. P2. There are objective moral values. Therefore, God exists.

A = God does not exist. B = There are no objective moral values. (These are both actually negated, and thus P2 and the conclusion would wind up being double negation, but that's complicating matters unnecessarily for an explanation, since ~~P (not not-P) is equivalent in truth to P.)

What P1 is asserting is that God's non-existence is sufficient for concluding that there are no objective moral values. At the same time, it's also asserting that the non-existence of objective moral values is a necessary condition for God's non-existence.

I highly doubt that P1 is true, because I don't think the relationship between the two holds, but that does not make the argument invalid. It makes it unsound.

Dolfyn
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  • Your opinion about the truth of P1 is obviously not based on any substantial reason. A little reflection would reveal that the logically independent nature of normative principles is due to them having an independent source, i.e. a norm which is often at odds not only with real matters of fact but also with the inclinations of the person concerned. The theory of evolution, which only claims to explain things as they, in fact, are, is hardly equipped to account for any counterfactual standards as to how things should be; nor, for that matter, how one might wish* them to be. –  Oct 12 '17 at 12:11
  • God, on the other hand, has authority to legislate, and His laws are normative by their very nature. Because of this, there is a clear and logical relation between God's existence and the normative nature of mankind's sense of morality. –  Oct 12 '17 at 12:12
  • @PédeLeão - The opinion to which you are referring is quite "obviously" just an assertion of my belief that was used to perform a clear function in the explanation above as an explanation. I assure you I have done much more than just "a little reflection," and I have quite a bit of "substantial reason" for believing what I do. However, your comments are vastly off-topic and as such, inappropriate. – Dolfyn Oct 13 '17 at 17:42
  • @PédeLeão - Actually, your discussion is extremely theory-laden, and I suspect you are working from conceptual analyses quite different from mine. Let's leave it at that, shall we? – Dolfyn Oct 13 '17 at 17:44
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Valid arguments preserve the truth of their premises. Given false premises, valid arguments are not sound, and do not prove anything. (This is the point of all the arguments in Alice in Wonderland. A single false premise can make nonsense out of anything.)

Your first 'step' in each case above is a different premise, so your result is different. Given that we do not know the truth value of the second premise, none of these arguments, even though they are formally valid, are the least bit helpful.

Pure logic cannot indicate the truth value of premises except via identifying internal contradictions. If you combine a premise with another premise which may be false, logic does not even help clarify the premise itself.