Arguments in sentential logic and contradictions/tautologies.

[pandaBee]

From what I understand, and please correct me if I'm wrong:

An "Argument" in sentential logic is a set of propositions, or premises, which logically lead to a conclusion. I took this definition from http://en.wikipedia.org/wiki/Propositional_calculus

However, in my text that I am currently studying, How to Prove It A Structured Approach by Velleman, he defines arguments as the following: "We will say an argument is valid if the premises cannot all be true without the conclusion being true as well." -pg.9

These definitions aren't quite the same thing, for example in the following argument:

It will either rain tomorrow or not rain tomorrow (premise)
Jason (a human being) is either alive, or dead. (conclusion)

It is clear that the if the premise is true, then the conclusion is also true in the case where the conclusion is a tautology, my example above being that a human being is either alive or dead.

Is this Argument considered valid? It should be, according to Velleman's definition, but I'm not so sure it fulfills the wikipedia definition since the premise does not 'logically' lead to the conclusion.

In that case, how do you even show formally that the premises don't 'logically' lead to the conclusion(s)?

Also, the problem with the 2nd definition is also apparent because it introduces the possibility of redundancies in the premises, since I could simply make all of the premises tautologies and make any true conclusion.

So I guess my first major question is which of these definitions should I follow? Or am I missing something here? Please correct me if I'm mistaken about something.

 

[micromass]

The wikipedia definition is correct, but not so useful. Indeed, you have already remarked that you do not know what it means to "lead logically to a conclusion". The wikipedia definition should clarify this matter and define precisely what it means. If they do clarify this matter, then they will likely take Velleman's definition.

So I would say to follow Velleman's definition.

Now, you have already made some good remarks about the definition. You have given an example of an argument which certainly satisfies Velleman's definition but which does not really looks like an argument to us! Indeed, the premises and the conclusions are completely unrelated! If a lawyer would make this kind of argument in a court of law, the jurors would not take him very seriously!

So what we want is somehow that the conclusion is related to the premises. This is an argument in "real life". But it is quite impossible to formalize this to a math setting. The definition of an argument would be either vague (like wikipedia) or would be horribly complicated!

On the other hand, Velleman's definition is simple, it is easy to check and it does give useful results in mathematics. So this is why we adopt Velleman's definition. True, it does not always correspond to our intuition of what an argument is. But that is a small price that we pay.

 

[Nick O]

Edit: Disregard this. You said "if the premises are true", which covers your bases.

Note that your example argument isn't complete without axioms. That is, unless we assume that being alive and raining are binary (either true or false) and that Jason exists, we cannot assign truth values to those propositions.

Stating assumptions can be very important in mathematical proofs.

 

[pandaBee]

So in other words I should approach situations like this in math with a bit more intuition then. I asked this question because of a problem in the book brought up these concerns while I was thinking of the problem, the problem was as follows:

Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument? What if the conclusion is a contradiction? What if one of the premises is either a tautology or a contradiction?

And thinking about including tautologies/contradictions in the conclusion and/or premises brought up the aforementioned concerns I had with both definitions.

Then isn't it the case that an argument with a tautology/contradiction as it's conclusion is a fallacy and that using a tautology as a premise in an argument is redundant since all it does is create a closed circle of logic - something of the form (A or not A) or (B and not B) are obviously self-contained and have no bearing in an argument other than demonstrating a law of logic.

Using a tautology as a conclusion would bring up a problem like in the example I stated above, and using a contradiction would obviously mean that the conclusion is always false.

So in other words, finally, if the conclusion has nothing to do with the premises I can safely outright state that the argument is invalid, correct? This is of course independent of whether the conclusion is a tautology.

 

[Nick O]

Valid conclusions are tautologies; they just aren't obvious ones.

That is to say, a conclusion will usually have the form "if P1 and P2 and ... and PN are true, then Q is true." If the conclusion is valid, then that statement is a tautology. If you assume Q is false, then you'll find that necessarily one of the premises is false and the conclusion still holds.

The problem is proving that it is a tautology.

 

[micromass]

So in other words I should approach situations like this in math with a bit more intuition then. I asked this question because of a problem in the book brought up these concerns while I was thinking of the problem, the problem was as follows:

Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument? What if the conclusion is a contradiction? What if one of the premises is either a tautology or a contradiction?

And thinking about including tautologies/contradictions in the conclusion and/or premises brought up the aforementioned concerns I had with both definitions.

Then isn't it the case that an argument with a tautology/contradiction as it's conclusion is a fallacy and that using a tautology as a premise in an argument is redundant since all it does is create a closed circle of logic - something of the form (A or not A) or (B and not B) are obviously self-contained and have no bearing in an argument other than demonstrating a law of logic.

Using a tautology as a conclusion would bring up a problem like in the example I stated above, and using a contradiction would obviously mean that the conclusion is always false.

So in other words, finally, if the conclusion has nothing to do with the premises I can safely outright state that the argument is invalid, correct? This is of course independent of whether the conclusion is a tautology.

The only way an argument can be invalid is if the premises are all true and the conclusion is false. Any other argument is considered valid.

For example:
Premise: X is a square
Condlusion: X is a rectangle

This is a true argument. The only way to make this false is if you can find a square that is not a rectangle, but that cannot exist. So if X is in reality a triangle, then the premises are false, but so is the conclusion. The argument still remains valid. So it if the premises and conclusions are both false, then the argument is valid.

It can also be that the premises are false and the conclusion is true. For example, if X is in reality a nonsquare rectangle, then the premise is false, but the conclusion is true. The argument is still valid.

So if you have a tautology as conclusion, then the argument is always valid. No matter what the premises are. So something like "If 1+1=3, then 1=1" is a valid argument (but a rather useless one).

If you have a contradiction as conclusion, then the only way this can be a valid argument is if the premises are false. If the premises are true, then you would have true premises but a false conclusion. This would be an invalid argument.

 

[WWGD]

Maybe this : http://en.wikipedia.org/wiki/Material_inference will be related.

 

[twoslit]

So in other words I should approach situations like this in math with a bit more intuition then. I asked this question because of a problem in the book brought up these concerns while I was thinking of the problem, the problem was as follows:

Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument? What if the conclusion is a contradiction? What if one of the premises is either a tautology or a contradiction?

And thinking about including tautologies/contradictions in the conclusion and/or premises brought up the aforementioned concerns I had with both definitions.

Then isn't it the case that an argument with a tautology/contradiction as it's conclusion is a fallacy and that using a tautology as a premise in an argument is redundant since all it does is create a closed circle of logic - something of the form (A or not A) or (B and not B) are obviously self-contained and have no bearing in an argument other than demonstrating a law of logic.

Using a tautology as a conclusion would bring up a problem like in the example I stated above, and using a contradiction would obviously mean that the conclusion is always false.

So in other words, finally, if the conclusion has nothing to do with the premises I can safely outright state that the argument is invalid, correct? This is of course independent of whether the conclusion is a tautology.

Several points need to be made here.

First, logical notions like validity, tautology, contradiction etc. are all defined purely in terms of truth, not usefulness or non-circularity or redundancy. So while it might be somewhat useless to derive one tautology from another, this is not logically incorrect. Anyway, sometimes it is useful to demonstrate a law of logic, especially if that law is complicated. Some laws of logic are named for their discoverers, such as deMorgan's law (that (not (A and B)) is equivalent to (not A) or (not B) ). The correct definition of validity of an argument is: an argument (or proof) with premises P1...Pn and conclusion C is valid just when the truth of the premises guarantees the truth of the conclusion; that is, when it is impossible for the premises to be true and conclusion to be false. (Notice by the way that the conclusion is C, not ((P1 &...&Pn) implies C). That longer formula is logically valid - is a tautology - when the proof is valid, but it is not the conclusion of the proof; unless, of course, there are no premises, ie when n=0, in which case the conclusion is itself a tautology.) It follows that an argument is valid when the conclusion is a tautology (always true) and also when any premis is a contradiction (always false). (And this is so even when the premis is not actually used in the argument, by the way.) But it does not follow that having a tautology as a premis makes an argument invalid. Invalid arguments may have true conclusions, eg the argument: Men are mortal, therefore fish swim, is invalid, even though the conclusion is true, because the conclusion could be false without making the premis false. Strangely, the argument Men are both mortal and immortal, therefore fish fly is valid, since its premis is contradictory.

Second, the two ways of characterizing correctness of an argument - in terms of truth, or in terms of proofs - are both correct and, when properly stated and made precise, they are equivalent. That was first proved by Goedel in 1929, and is called the completeness theorem. Proving it requires having a mathematically exact theory of truth (called model theory) and a precise definition of what counts as a proof. There are many varieties of formal proof - just about any textbook in modern logic will have one system in it - but they all involve being very exact about applying rules. Here for example is a proof of (P or not P) using one common proof system:

1. assume not (P or not P)
2. assume P
3. P or not P (from 2 by or-weakening)
4. Contradiction (from 1 and 3)
5. not P (by reductio from 2-4, discharging line 2)
6. P or not P (from 5 by or-weakening)
7. Contradiction (from 1 and 6)
8. not ( not ( P or not P)) (by reductio from 1-7, discharging line 1)
9. P or not P (from 8 by double negation)

This uses rules like or-weakening (derive (A or B) from A) and reductio (derive not-A from a proof of contradiction with A as an assumption). It may strike you as a very baroque kind of an argument for such an obvious conclusion, and you would be right, but this kind of extreme exactness in proof rules is necessary when the idea of proof gets formalized.

This whole topic, of rule systems for logical proofs and the relationships between them, is called proof theory. It is a large and (IMO) fascinating research area, with many connections to the theory of computation and computational complexity. But if one is primarily interested in getting ordinary arguments stated and proved, using your intuition is usually more useful than actually trying to follow one of these very exact rule systems.

If the conclusion seems to have nothing to do with the premises, can you say the argument is invalid? No, you can't. In any case, it is very hard to know whether one statement has nothing to do with another. Bertrand Russell was once, famously, asked how it was that 1=0 implied that he was the Pope, and he immediately replied: if one equals zero then, adding one to each, two equals one. The Pope and I are two; therefore, the Pope and I are one.