Validity and Soundness

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

According to the definition of a deductive argument (see the Deduction and Induction), the author of a deductive argument always intends that the premises provide the sort of justification for the conclusion whereby if the premises are true, the conclusion is guaranteed to be true as well. Loosely speaking, if the author's process of reasoning is a good one, if the premises actually do provide this sort of justification for the conclusion, then the argument is valid.

In effect, an argument is valid if the truth of the premises logically guarantees the truth of the conclusion. The following argument is valid, because it is impossible for the premises to be true and the conclusion nevertheless to be false:

Elizabeth owns either a Honda or a Saturn.

Elizabeth does not own a Honda.

Therefore, Elizabeth owns a Saturn.

It is important to stress that the premises of an argument do not have actually to be true in order for the argument to be valid. An argument is valid if the premises and conclusion are related to each other in the right way so that if the premises were true, then the conclusion would have to be true as well. We can recognize in the above case that even if one of the premises is actually false, that if they had been true the conclusion would have been true as well. Consider, then an argument such as the following:

All toasters are items made of gold.

All items made of gold are time-travel devices.

Therefore, all toasters are time-travel devices.

Obviously, the premises in this argument are not true. It may be hard to imagine these premises being true, but it is not hard to see that if they were true, their truth would logically guarantee the conclusion's truth.

[from the

Internet Encyclopedia of Philosophy https://www.iep.utm.edu/val-snd/ ]

Note the instance of a disjunctive syllogism—the first argument. Note also that the premises of an argument do not have to be true in order for the argument to be valid.

Logik wrote: ↑Wed Jan 09, 2019 8:14 pm

Given the structure of your argument, anything you put after the comma in P1 is true.

If you mean that if the structure of an argument is “disjunctive syllogism”, then anything that is put after the comma in the first premise in

P1. Either nothing exists, or Hugh Nose was born on Mars.

P2. Something exists.

C. Therefore, Hugh Nose was born on Mars.,

Is true [it is the conclusion of the argument], you are simply mistaken. In addition to being a disjunctive syllogism, a valid argument form, the premises must be true. So, what is put after the comma in the first premise of an argument of this form will be true only if the first premise [along with the second premise] is true.

Hugh Nose wrote: ↑Wed Jan 09, 2019 7:24 pm

Clearly you can’t mean that one can prove anything with a “sound argument”, since that is obviously false.

I assume you agree that argument P is a valid argument.

I do not.

You do not what? You do not mean that one can prove anything with a sound argument or you do not agree that argument P is a valid argument. If the latter then you are simply mistaken as the excerpt from the IEP shows.

Hugh Nose wrote: ↑Wed Jan 09, 2019 7:24 pm

You cannot believe that the fact that you can produce a valid argument in the form of a disjunctive syllogism, with a false conclusion [or any conclusion you like] shows that there is something wrong with argument P.

I do.

What exactly do you think it shows? The fact that you can produce a valid argument in the form of a disjunctive syllogism, with a false conclusion does not show that the first premise is false. If it were true, then the following argument would have something wrong with it also:

Argument A

Either Hugh Nose is a resident of Delaware or Hugh Nose is a resident of Pennsylvania.

Hugh Nose is not a resident of Delaware.

Therefore, Hugh Nose is a resident of Pennsylvania.

But Argument A is a disjunctive syllogism and a sound one at that, since it is valid and the premises are true. But if you believe, as you say you do, that the fact that you can produce a valid argument in the form of a disjunctive syllogism, with a false conclusion [or any conclusion you like] shows that there is something wrong with argument P, it will show that there is something wrong with Argument A. But there is nothing wrong with Argument A.

Hugh Nose wrote: ↑Wed Jan 09, 2019 7:24 pm

Clearly one cannot prove that some conclusion is true with an unsound argument. But you haven’t done anything to show that argument P is unsound.

OK, Martian. What would convince you that the argument is invalid?

You are asking “What could show that a valid argument is invalid?”. The answer is "Nothing can show that a valid argument is invalid." One could, of course, show that an argument that is valid is unsound by showing that one or more of the premises is false. But you haven’t done that. You haven’t even attempted to do that.

Your P1 is a false dichotomy. If you are to rely on classical logic, and if the law of excluded middle is to be considered, then the options need to be all-exhaustive and mutually exclusive. Why? Decision theory.

Nowhere in classical logic does it say that the disjunctive premise in a disjunctive syllogism has to be a disjunction of all of the possible distinct alternatives. Look again at the excerpt from the IEP. Beyond that, even if P1 is, in some sense, a false dichotomy, being a false dichotomy is not inconsistent with being true. [Argument A, above]

One might urge a complaint such as yours if one thought that the first premise was being offered as a necessary [logical] truth, but it is not, nor does classical logic anywhere say that the disjunct in a disjunctive syllogism must be a necessary truth. I am not saying that you are saying that it is a necessary truth. I am just trying to figure out what point your remark is intended to make.

To say that "Apples are either red or they are yellow" is an invalid syllogism. Your decision-space is incomplete because you have left out green apples.

The first statement baffles me! “Apples are either red or they are yellow” is not a syllogism. Neither is the first premise of argument P, nor is the first premise of argument A, a syllogism.

And so if I said "The apple I am holding is not red" does not allow you to make a conclusive deduction about the color of the apple that I am, in fact, holding. It could be yellow; or it could be green.

You are right here, but I don’t know why you are offering this because it has nothing to do with anything that I have said. Unless…

Unless you are interpreting the first premise of Argument P as equivalent to a material implication,“If something exists, then God exists”, and then viewing the conditional statement as expressing an entailment relation between the antecedent and the consequent. Such an interpretation is completely gratuitous. I am not saying that this is what you are saying. Once again, I am just trying to understand what you are saying, since so much of it, to be charitable, appears to be irrelevant to anything that I have said.

You doubt my ability to choose my own words?

Do you really mean to insinuate you are a mind reader?

No, I do not doubt your ability to choose your own words, nor do I mean to insinuate that I am a mind reader. But given the oddity of your remarks against the background of the basic logic displayed in the IEP excerpt, I thought you might have simply “misspoken” at points in the post. Now it seems clear that you didn’t misspeak. But then… ???

Cheers,

Hugh