Modus Tollens: Valid or Invalid

[Gary Childress]

While Googling around this evening, I stumbled upon this website. I was rather shocked to hear that someone thinks Modus Tollens is an invalid argument. What do you think? Is Modus Tollens an invalid argument? Why or why not?

http://changingminds.org/disciplines/argument/syllogisms/modus_tollens.htm

For those unfamiliar with Modus Tollens it's basic form is:

If X, then Y
Not the case Y
Therefore not the case X

My answer: Modus Tollens is a valid argument.

 

[xchen08]

Modus Tollens is valid, but the argument given in the link is also correct. These two things are both possible because the link gives an incorrect definition of Modus Tollens, which replaced "it is (not) the case that X" with "X is true (false)."

 

[Atticus]

There are some good reasons to deny modus tollens.

Kurt Gödel proved in the 30s that there are propositions in maths that can not be proven to be true or false.

From this point of view, if you know that A => B and that B is not true, you could conclude that A is false or it's one of those propositions which can not be proven to be true or false.

 

[Till]

The author obviously doesn't understand the material implication.

 

[tycoonist]

modus tollens is, in principle, ok. however the way it is used isn't, as the first statement has the habit of often confusing sufficient and necessary conditions, which can affect the answer.

 

[Rashiminos]

Modus Tollens is appropriate when the "middle" is excluded (the only possibilities are {true, false}).

 

[Defiant47]

If X, then Y
Not the case Y
Therefore not the case X

By mathematical logic, this is wrong. Y has no bearing on X and does not tell us whether the case X is true or not, because something false can imply anything. (speaking in technical mathematical terms)

(X => Y) =/> ((NOT Y) => (NOT X))

X => Y
True -> true
False -> true or false

 

[Rashiminos]

By mathematical logic, this is wrong. Y has no bearing on X and does not tell us whether the case X is true or not, because something false can imply anything. (speaking in technical mathematical terms)

(X => Y) =/> ((NOT Y) => (NOT X))

X => Y
True -> true
False -> true or false

Houston, we have a problem. In mathematics this concept is known as the contrapositive.


X, Y can be either true or false (in mathematical/formal logic), but it has been given here that when X is true, so is Y.

Now,

Assume Y is false, and let's ignore the wiki for just a moment.

Either X is true, which by X => Y implies Y is true and leads to a contradiction,

or,

X is false, and there is no contradiction,

so

not Y => not X whenever X => Y.

In mathematical logic, the statements:

X => Y
not Y => not X

are logically equivalent.

Maybe it would help if I reverse the arrow:

X => Y
not X <= not Y

 

[Gary Childress]

By mathematical logic, this is wrong. Y has no bearing on X and does not tell us whether the case X is true or not, because something false can imply anything. (speaking in technical mathematical terms)

(X => Y) =/> ((NOT Y) => (NOT X))

X => Y
True -> true
False -> true or false

Not true. If Y is not the case, then it follows X is not the case either.

If X, then Y.
Not the case Y but X being the case would contradict the meaning of "If X, then Y" because we would have an example of "If X, then not the case Y" which is the opposite in meaning of "If X, then Y."

A premise being false does not necessarily make an argument invalid. For instance, you can say:

All men are immortal
Socrates is a man
Therefore Socrates is immortal

Notice the argument above is valid but the first premise is false. The fact that the argument draws a false conclusion is based upon the falsity of the first premise, not upon an invalidness of the argument. The argument is still valid, even though the conclusion is false.

 

[Fifty]

By mathematical logic, this is wrong. Y has no bearing on X and does not tell us whether the case X is true or not, because something false can imply anything. (speaking in technical mathematical terms)

(X => Y) =/> ((NOT Y) => (NOT X))

X => Y
True -> true
False -> true or false

You have no idea what you're talking about.

There are some good reasons to deny modus tollens.

Kurt Gödel proved in the 30s that there are propositions in maths that can not be proven to be true or false.

From this point of view, if you know that A => B and that B is not true, you could conclude that A is false or it's one of those propositions which can not be proven to be true or false.

I ask this because I really don't know... does this come from a non-superficial understanding of Goedel? I don't know much about the Incompleteness Theorems, but this intuitively strikes me as one of those instances where the caricature of Goedel says something different than what Goedel's theorems really say. This is a notorious issue with the Incompleteness Theorems.

Perhaps Till can enlighten us?

The author obviously doesn't understand the material implication.

indeed

modus tollens is, in principle, ok. however the way it is used isn't, as the first statement has the habit of often confusing sufficient and necessary conditions, which can affect the answer.

what the crap are you talking about?

 

[Rashiminos]

I ask this because I really don't know... does this come from a non-superficial understanding of Goedel? I don't know much about the Incompleteness Theorems, but this intuitively strikes me as one of those instances where the caricature of Goedel says something different than what Goedel's theorems really say. This is a notorious issue with the Incompleteness Theorems.

Perhaps Till can enlighten us?

Perhaps a good place to start would be here: http://en.wikipedia.org/wiki/Decidability_(logic)

Now if someone can find something about determining all non-abelian (algebraic) groups of finite order n, that person may get bonus points.

 

[Munch]

Can't we settle this peacefully?

X => Y
True -> true
False -> true or false

By that argument the only case where Y is false is when X is also false.

 

[Atticus]

I ask this because I really don't know... does this come from a non-superficial understanding of Goedel? I don't know much about the Incompleteness Theorems, but this intuitively strikes me as one of those instances where the caricature of Goedel says something different than what Goedel's theorems really say. This is a notorious issue with the Incompleteness Theorems.

Yes, I know. My post was more about intuitionism than Gödel's proof. This part is Gödel:

Kurt Gödel proved in the 30s that there are propositions in maths that can not be proven to be true or false.

I suppose that's not controversial. (My knowledge of it is superficial by mathematicians standards, but not by lay man's).

This part is intuitionism:

From this point of view, if you know that A => B and that B is not true, you could conclude that A is false or it's one of those propositions which can not be proven to be true or false.

Of intuitionism I have superficial knowledge. As far as I know, intuitionists don't accept modus tollens, and one reason for that is Gödel's proof.

Support for intuitionism is extremly small (I don't even know if there are any living intuitionists). So it would be quite correct to say "Gödel's theorem has no bearings on modus tollens". That is the generally held view. However you can say that some people think it has.

 

[pau17]

Kurt Gödel proved in the 30s that there are propositions in maths that can not be proven to be true or false.

From this point of view, if you know that A => B and that B is not true, you could conclude that A is false or it's one of those propositions which can not be proven to be true or false.

I don't think this is quite right. The propositions that Godel identified as being problematic in this way are particular self-referencing sentences of the type "this sentence is false," and the issue is that they cannot be proven true or false within a given system, not that they are unprovable period. The idea is that within any given axiomatic system of reasoning, you will have such problematic propositions, and you can add axioms to conclusively deal with a given proposition, but you will always have more, i.e. every consistent axiomatic system is by definition incomplete. (And if you think about it, it makes sense that it would be this way, because while there are infinite mathematical truths, only in a self-contradictory system of reasoning can you derive them all at one go without expanding or revising the system.) You can also certainly prove them outside of that particular system. It's not to say that there are mysterious "unprovable" statements floating about that give rise to skepticism in basic logic or anything like that.

You might be interested in this book, which discusses this in great detail and most of it is available as a preview:

http://books.google.com/books?id=71...nzen&hl=es#v=onepage&q=torkel franzen&f=false

 

[Defiant47]

Awesome, get rusty in some mathematics I did a few years ago and never touched again, and get crucified. Not surprised.

*crawls head in sand*

 

[Atticus]

Thanks for clarification, pau17! I guess I knew that, but I never think about other logical systems, I've never been particularly into logic, so the real numbers is the axiomatic system to me.

However, do you by this:

It's not to say that there are mysterious "unprovable" statements floating about that give rise to skepticism in basic logic or anything like that.

mean that Gödel isn't one of the reasons why intuitionists abandoned modus tollens?

 

[pau17]

Thanks for clarification, pau17! I guess I knew that, but I never think about other logical systems, I've never been particularly into logic, so the real numbers is the axiomatic system to me.

However, do you by this:

mean that Gödel isn't one of the reasons why intuitionists abandoned modus tollens?

I don't know enough to say. I was merely commenting that saying

Kurt Gödel proved in the 30s that there are propositions in maths that can not be proven to be true or false

wasn't telling the whole story as far as I know, since I think the idea of unprovability was a property of self-referentiality in a given system and not a general statement that "there are certain unprovable statements, period" and that any particular statement could somehow turn out to be one of "those" unprovable statements.

If you meant to say something else, let me know, and in the meantime I'll try to read into Intuitionism a bit.

 

[Flying Pig]

Not true; I explain it thus:

If I was happy, I would be smiling
I am not smiling
Therefore I cannot be happy

That doesn't follow, does it?

 

[Bill3000]

Not true; I explain it thus:

If I was happy, I would be smiling
I am not smiling
Therefore I cannot be happy

That doesn't follow, does it?

That just means that the material implication ("If I was happy, I would be smiling") is false. Nothing wrong with the structure of the argument; it's just that the first premise is false.

 

[pau17]

Not true; I explain it thus:

If I was happy, I would be smiling
I am not smiling
Therefore I cannot be happy

That doesn't follow, does it?

It is logical if the first premise is true; the thing is that the first premise is usually not true in real life. Of course, in real life you can be happy and not smile; just the way you wrote the first sentence makes it such that you would have to smile if you were happy.

You can flip the first premise around and say, "If I am not happy, I am not smiling" to the same effect.