Fallacy of the distributed middle

[Underseer]

For whatever reason, some people have decided that rather than using the Logic forum to ask questions about logic, that we should use the Logic forum to talk about puzzles.

Uh, anyway, no matter how many times someone explains the fallacy of the distributed middle to me, I keep re-forgetting it. So can any of you explain it in a way that I won't immediately forget again?

 

[-RRH-]

Is the first problem that you say distributed middle instead of undistributed middle, or is this some other fallacy with a confusingly similar name?

 

[Loren Pechtel]

For whatever reason, some people have decided that rather than using the Logic forum to ask questions about logic, that we should use the Logic forum to talk about puzzles.

to talk about logic puzzles.

 

[Rilx]

First, a proper syllogism:

1. All chickens have 2 legs.
2. This animal is a chicken.
3. Therefore, this animal has 2 legs.

'Chicken' is the distributed middle term that connects the two premises.

Then the fallacy, without a distributed middle term:

1. All chickens have 2 legs.
2. Underseer has 2 legs.
3. Therefore, Underseer is a chicken.

 

[Underseer]

First, a proper syllogism:

1. All chickens have 2 legs.
2. This animal is a chicken.
3. Therefore, this animal has 2 legs.

'Chicken' is the distributed middle term that connects the two premises.

Then the fallacy, without a distributed middle term:

1. All chickens have 2 legs.
2. Underseer has 2 legs.
3. Therefore, Underseer is a chicken.

No wonder I never remember that. I bet I wouldn't spot that in an actual argument. Why am I so dense?

Let me try:

1. All teabaggers are morons.
2. I am a moron.
3. Therefore, I am a teabagger.

Now the next trick is remembering that one. Maybe I need to learn more about syllogisms?

- - - Updated - - -

to talk about logic puzzles.

It's still puzzles. Don't we have a games forum around here for that?

 

[-RRH-]

The typical form of a syllogism is

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

If you think of the "are" and "is" verbs as the centre for each premise, see that "man" is on the right in one premise and on the left in the other premise. So you can think of it as it needs to be distributed to both the left and the right, if you want to use it to connect the two premises. In your teabaggers example, "moron" is on the right in both premises.

 

[Wiploc]

Working on a mnemonic.

Undistributed middles ruin syllogisms.
This syllogism lacks an undistributed middle.
Therefore, this syllogism isn't ruined.

Well, that sucks. Clearly invalid, despite the fact that I tried to construct it using what I've learned from the posts above.

Does anybody know what is meant by "middle"? All I've got so far is that it comes first in P1 and last in P2. And I assume that that rule wouldn't stand inspection. For instance, in "P1: syllogisms are ruined by undistributed middles," the middle comes at the end of P1.

Unless I'm way off base, which is entirely possible.

 

[Rilx]

Something is ruined in the state of Denmark.

Wiploc. Syllogisms are constructed from categorical propositions, not from any kind of sentences which set some properties to some objects. Except formal logic, practical sense is also necessary. The major premise P1 defines a category ("all men are mortal") and the minor premise P2 uses that category ("Socrates is a man").

"Undistributed middles ruin syllogisms" is not categorical: for instance, "general stupidity" can also ruin syllogisms. If you replace P1 by "Um and only um ruin syllogisms" the rest of the syllogism seems at least logically correct. I don't mind analyzing it further though ...

 

[Speakpigeon]

Does anybody know what is meant by "middle"? All I've got so far is that it comes first in P1 and last in P2. And I assume that that rule wouldn't stand inspection. For instance, in "P1: syllogisms are ruined by undistributed middles," the middle comes at the end of P1.

The middle term is indeed the term occuring in the two premises (and not in the conclusion), hence undistributed middle. The rule would stand inspection.
EB

 

[ficino]

Let me try:

1. All teabaggers are morons.
2. I am a moron.
3. Therefore, I am a teabagger.

Underseer, did you mean to give an example of the fallacy of affirming the consequent? 'Coz that's what you did.

for all x, if x is a teabagger, x is a moron.
I am a moron.

It doesn't follow that I am a teabagger. In this argument, morons can include other classes of people besides teabaggers.

 

[Perspicuo]

I have never really understood it either. I just go with common sense.

But after thinking of a good rule of thumb this very moment, the following has occurred to me:

P2 has to invoke the subject of P1 and not its predicate, given that P1 is universally quantified and P2 is an example of an individual (which it attempts to induce into the group represented by the predicate of P1).

I still have a problem with what "distributed" means. Wiki says:

The fallacy of the undistributed middle is a formal fallacy, that is committed when the middle term in a categorical syllogism is not distributed in either the minor premise or the major premise. It is thus a syllogistic fallacy.

 

[beero1000]

I have never really understood it either. I just go with common sense.

But after thinking of a good rule of thumb this very moment, the following has occurred to me:

P2 has to invoke the subject of P1 and not its predicate, given that P1 is universally quantified and P2 is an example of an individual (which it attempts to induce into the group represented by the predicate of P1).

I still have a problem with what "distributed" means. Wiki says:

In a categorical proposition, we assert that all (or some) members of the subject category are in (or not in) the predicate category.

For example: All S are P. Here S is the subject term and P is the predicate term.

There are four classes of categorical propositions:

All S are P
All S are not P
Some S are P
Some S are not P

Each category can be distributed or undistributed. If the proposition asserts something about every member in the category, then we say it is distributed, and otherwise it is undistributed.

So for the four classes:

All S are P: S is distributed. P is undistributed (as we are not claiming something about all members of P).
All S are not P: S is distributed. P is distributed (as we are saying that All P are not some S).
Some S are P: S in undistributed. P is undistributed.
Some S are not P. S is undistributed. P is distributed (as we are saying that that all P are not all S).

A syllogism connects two premises with three categories, generally of the form A -> B and B -> C. But if the B term is undistributed in both premises, then we can't guarantee that the subset of B in the first premise overlaps the subset of B in the second premise. Therefore, the conclusion may be false, and we have the fallacy of the undistributed middle.

So for example:

All teabaggers are morons.
I am a moron.
Therefore, I am a teabagger.

If S = class of teabaggers, P = class of morons, Q = class of Underseer. The syllogism is

All S are P
Some Q are P (or All Q are P, it doesn't matter)
Therefore, some/all Q are S.

Here, the goal is to link Q -> S by saying Q -> P and P -> S, so P is the middle. However, P is undistributed in both premises so we don't know that the morons that are teabaggers are guaranteed to be the same morons that are Underseers (sorry Underseer ). That is the fallacy of the undistributed middle.

 

[Wiploc]

Thank you so much, Beero.

 

[steve_bnk]

Learned something new.


If it were an engineering problem wewould say the stated variables are incomplete or insufficient inorder to completely bound the problem or draw a conclusion..


http://en.wikipedia.org/wiki/Fallacy_of_the_undistributed_middle




Forexample:

All students carry backpacks. My grandfather carries a backpack. Therefore, my grandfather is a student.

All students carry backpacks. My grandfather carries a backpack. Everyone who carries a backpack is a student. Therefore, my grandfather is a student