Tuesday, May 05, 2009

Some basic information about logic

A redated post.

Argumentation about controversial issues is at the heart of philosophy, and logic is the science of studying arguments. The lecture material (which I didn’t write, just so you know), says some things about logic. The fundamental idea that founded the science of logic was discovered by Aristotle, who realized that arguments could you could distinguish the question of the internal logic of an argument from the question of the truth of the premises. He discovered that some arguments had the characteristic of being such that, if the premises are true, the conclusion must be true, while others do not. The key idea is the idea of validity. There is a common use of the term “valid,” which just means legitimate. So, for example, the you ask me “If I told you the dog ate my homework, would that be a valid excuse” I might well say no that is not valid. But when logicians talk about the concept of validity, what they mean is that the argument is structured in such a way that if the premises are true, the conclusion must be true. Conversely, if an argument is invalid, that means that it is possible that the premises are true and the conclusion is false.

Let’s take this argument:
1) If I say my dog ate my homework, then I have a valid excuse.
2) I say that my dog ate my homework.
3) Therefore, I have a valid excuse.

Even though the excuse isn’t valid in the informal, common-sense sense, the argument is valid in the logic al sense.

Aristotle worked with the logic of classes, so the arguments he considered were arguments like:

1. All dogs are animals.
2. All beagles are dogs.
3. Therefore, all beagles are animals.

This argument is valid, in that if the premises are true, the conclusion must be true. Now the premises are true, so the argument is sound. A sound argument is a valid argument with true premises, and when we have a valid argument with true premises, then the conclusion must be true.

But an argument can be valid if it has false premises. For example:

1. All angels are mortal beings.
2. All persons identical to Bill Clinton are angels.
3. Therefore, Bill Clinton is a mortal being.

In this case the premises are both false and the conclusion is true. But if the argument had had the same structure, you could never have and argument with the premises true and the conclusion false. That is the possibility that validity leaves out. One the other hand, some arguments are invalid, in that the premises can be true and the conclusion false. You have encountered this is what you have heard. There are some statements made, and then there is a leap to a conclusion that is not warranted by the prior statements.
1. All beagles are mammals.
2. All dogs are mammals.
3. Therefore, all dogs are beagles.

This argument has a true conclusion, but the same argument form could be used to prove a very different conclusion;

1. All cats are mammals
2. All dogs are mammals.
3. Therefore, all dog are cats.

Obviously a false conclusion. The conclusion doesn’t follow from the premises.

Now let’s look at the logic of conditional statements. That is what modus ponens and modus tollens are all about. A conditional statement is a statement of the type “If A then B.” That statements means that if A is true, B must also be true.

Modus ponens is:

1. If A then B.
2. A
3. Therefore B.

Let’s take a look at this:

1. If you study, you will pass.
2. You study.
3. Therefore, you will pass.

This arguments is a valid argument. You can’t accept both of these premises and reject the conclusion. The statements is about what will happen if you do study, namely, that you will pass.

But let’s try this argument.

1. If you study, you will pass.
2. You passed.
3. Therefore, you studied.

Uh, no. Just because studying means that you pass doesn’t mean that if you don’t study, you still won’t pass. Remember the irritating kid from high school who used to ace all his classes without studying? This is an argument form called affirming the consequent. It’s not valid. The premises can be true, and nevertheless the conclusion could be false. The structure of the argument is as follows:

1. If A then B
2. B
3. Therefore A.

Here’s another argument form:
1. If you study, you will pass.
2. You didn’t study.
3. Therefore, you didn’t pass.
Again, no. You could be that annoying kid from high school, in which case the premises of the argument are true but the conclusion is false. The structure here is

1. If A then B
2. Not A
3. Therefore not B.

This is denying the antecedent, and it’s invalid.

The last argument form is modus tollens, and it is valid.

1. If A then B
2. Not B
3. Therefore A.

Let’s try the same concepts again.
1. If you study, you pass.
2. You didn’t pass.
3. Therefore, you didn’t study.

This does follow. If studying is sufficient to make you pass, and you don’t pass, the only conclusion we can draw is that you just didn’t study. The structure is:

So two of these argument-forms are valid, and the conclusion follows from the premises. Two of these argument-forms are not valid. These argument forms involve a leap in logic that isn’t justified. The premise could be true and the conclusion not be true. The conclusion doesn’t follow from the premises.

Most arguments can be presented in a valid argument form. In fact, you can present a valid argument for an insane conclusion.

If the moon is made of green cheese, the moon is made of yellow cheese.
The moon is made of green cheese.
Therefore, the moon is made of yellow cheese.

That’s a valid argument, but the conclusion is, well, a little cheesy.

Now let’s take a look a controversial issue.

1. If fetuses are persons, then it is wrong in virtually all cases to abort them.
2. Fetuses are persons.
3. Therefore, it is wrong in virtually all cases to abort them.

The argument is perfectly valid. If the premises are true, then the conclusion must be true. However, both of the premises are controversial. Defenders of abortion often argue that fetuses are nonpersons, so they are abortable. They also maintain that even if fetuses are persons, there are a significant number of cases in which abortion is still justified. However, the valid argument clarifies the debate. An abortion defender has to attack one or both of the premises, in order to defend their position.

22 comments:

One Brow said...

That's a very nice beginning primer. Do you cover that in your class?

Victor Reppert said...

Yes.

Ilíon said...

I'm thinking you didn't actually mean everything as actually written (i.e. I suggest editing it).

For instance: "There are some statements made, and then there is a leap to a conclusion that is not warranted by the prior statements.

All beagles are mammals.
All dogs are mammals.
Therefore, all dogs are beagles.

This argument has a true conclusion, but the same argument form could be used to prove a very different conclusion;
"

One Brow said...

ilion, I will let Dr. Reppert defend himself in detail. However, the statement you quote seems to mean exactly what it says, and is a very good example on his part. There is no error or mischaracteriazaiton and no reason for him to edit the post.

Maybe you could give an example of how you think it could be improved?

Ilíon said...

"This argument has a true conclusion,"

Really?

Victor Reppert said...

Ilion: Thanks. I made the correction.

Ilíon said...

Hey, your fans have to give you criticism. Heavens knows that most of your "critics" are incapable of giving you real criticism.

One Brow said...

Hey, your fans have to give you criticism. Heavens knows that most of your "critics" are incapable of giving you real criticism.

Well, you certainly saw something I missed there. Congratulations.

Ilíon said...

One Brow,
I explicitly quoted a typo which ended up saying something quite different from the intent, which, in fact made the whole section incoherent, and you *still* tried to play the "yer ignert/stoopid" card.

Do you ever wonder why I intentionally do my best to ignore your posts, both here and at AS? I'm not a patient man, in general; and I have no patience at all behavior that comes across to me as based in willfull ignorance, as yours so often does. But since you (personally) don't seem to be so totally hopeless as most of the 'atheists' I encounter, it's better that I leave you to those who are not so impatient as I.

lurk-to anon-back to lurk said...

ilion,
wow.
you took the words right out of my mouth.

Ilíon said...

Lurk-to,
I do hope that wasn't too unpleasant an experience.

going back to lurking again said...

ilion,
it wasn't.
I don't like debating because i'm not very good at it. I get nervous and kind of shook up with it. i just like to read the article and comments. WHat you said toward ob is how I was feeling after reading his posts over the past couple of months.

Ilíon said...

"I don't like debating because i'm not very good at it. I get nervous and kind of shook up with it."

I know exactly what you mean. Actually, I don't so much think of myself as debating as trying to discuss important topics in almost impossible situations. ;)

But however we characterize it, here's how I do it:

First, I consciously try to never say anything I don't have good reason to believe is true. But I also don't worry whether anyone or even everyone calls me stupid (and this is such a boringly common reaction) or arrogant (also boringly common ... like what? am I supposed to be a doormat?).

There are many things I have good reason to believe are true that I don't say, because it's my intention to be always ready to back up what I say with references. At the same time, in situations where it seems clear to me that the person(s) disputing what I've said are not actually willing to reason or accept evidence, I don't waste my time providing it.

At the other same time, if what I've said follows from reasoning, rather than from mere facts, and someone is disputing it, than there is no evidence other than more reasoning with which to back it up. If a person is determined to refuse to reason, there is nothing anyone can do to force the situation to be otherwise.

Second, I generally write out and re-read and re-edit something before I post it. It frequently takes me hours to compose a single post ... consequently, I often have to ignore some of the responses; there just isn't enough time.

Third, choose your "battles" wisely (I still need to work on this one) and don't allow yourself to be thrown off-balance by (frequently intentional) distractions.


"WHat you said toward ob is how I was feeling after reading his posts over the past couple of months."

Well, I really don't *enjoy* having such a belief about anyone; but I also refuse to lie to myself about what I'm actually seeing due to some vacuous duty to politeness.

Still, and quite unlike so many of the 'atheists' I've encountered, I think One Brow does try, at least upon occasion, to approach these issues with an open and critical mind. But *I* can't deal with off-and-on: so it seems to me better that I try to ignore him; others don't have the same limitations that I have.

Ilíon said...

Also, this isn't my blog; this is Mr Reppert's blog, and One Brow is as much his guest as I am.

One Brow said...

One Brow,
I explicitly quoted a typo which ended up saying something quite different from the intent, which, in fact made the whole section incoherent,


Thank you for the replay.

and you *still* tried to play the "yer ignert/stoopid" card.

Except, I did no such thing. I said that it "seems to mean exactly what it says" because I overlooked the typo as well, and asked you to point out the precise error. I know it is not at the level of regualrly claiming that all atheists are intellectually dishonest, still it certainly does not claim anything at all about you.

Do you ever wonder why I intentionally do my best to ignore your posts, both here and at AS?

Not especially. Most of the time when you do, the discussion continues until it appears you have no valid response, then you stop. I don't bother to fill in the "why".

I'm not a patient man, in general; and I have no patience at all behavior that comes across to me as based in willfull ignorance, as yours so often does.

Your perception is your own problem, of course.

But since you (personally) don't seem to be so totally hopeless as most of the 'atheists' I encounter, it's better that I leave you to those who are not so impatient as I.

Whatever works for you. I'll be fine whether you respond or not.

One Brow said...

going back to lurking again:

I don't like debating because i'm not very good at it. I get nervous and kind of shook up with it.

Well, should you change your mind, I think you'll find I'm cofortable discussing things at whatever level you choose. It certainly doesn't have to be antagonistic. For example, you might compare how my responses to Dr. Reppert differ from the responses to more aggressive people.

One Brow said...

First, I consciously try to never say anything I don't have good reason to believe is true. But I also don't worry whether anyone or even everyone calls me stupid (and this is such a boringly common reaction) or arrogant (also boringly common ... like what? am I supposed to be a doormat?).

Well, we all reap what we sow. People who write broad, sweeping condemnations are frequently read as being categorical and close-minded, and so get treated that way. It is quite possible to be firm, or even assertive, without being arrogant. You might study Dr. Reppert's style more while you are here.

There are many things I have good reason to believe are true that I don't say, because it's my intention to be always ready to back up what I say with references.

Although at times when you are asked for more information, and nothing else, you choose no to provide it.

At the same time, in situations where it seems clear to me that the person(s) disputing what I've said are not actually willing to reason or accept evidence, I don't waste my time providing it.

Sometimes when you don't cast pearls, you make it look like you have none, regardless of the swine.

Second, I generally write out and re-read and re-edit something before I post it. It frequently takes me hours to compose a single post ... consequently, I often have to ignore some of the responses; there just isn't enough time.

Well, we all reap what we sow. That is one thing we have in common.

Well, I really don't *enjoy* having such a belief about anyone; but I also refuse to lie to myself about what I'm actually seeing due to some vacuous duty to politeness.

Do you feel any obligation to see if your personal perception is invalid, or even allow for the possibility?

But *I* can't deal with off-and-on: so it seems to me better that I try to ignore him; others don't have the same limitations that I have.

Perhaps if put out more honey and less vinegar, you'll find the flies more to your liking.

Ilíon said...

One Brow: "... For example, you might compare how my responses to Dr. Reppert differ from the responses to more aggressive people. ..."

One Brow: "... Do you feel any obligation to see if your personal perception is invalid, or even allow for the possibility? ..."

You're so self-delusional.

I'm pretty sure that this blog/thread is where I first became aware of you ... and decided to do my best to ignore you (and specifically due to this post).


You (the general reader) see, it is One Brow's "style" to constantly accuse those who disagree with him of making unsubstantiated (unsubstantiatable?) claims ... even as he himself does the very thing.

One Brow said...

You're so self-delusional.

How kind of you to tell me.

You (the general reader) see, it is One Brow's "style" to constantly accuse those who disagree with him of making unsubstantiated (unsubstantiatable?) claims ... even as he himself does the very thing.

Well, I thought the strongest claim I was making was that we don't know human reasoning is qualitatively different from computers. Let's look at the unsubstantiated claims again.

Unsustantiated claim:
*We* don’t actually use that formal system when we add 2 + 2 -- because, as Mr Reppert says, we *understand* the meanings and the relationships between the meanings.

Offered without evidence of what was we understand that computers don't, outside of us being better at it. After all, we start off learning about "2" the same way computers see it, by your description. Computers are quite capable of looking at the number of bytes in a file, and deciding if that number matches "2", just as we do. We just understand the breakdown of what happens in the computer CPU in much more detail that the breakdown of how that is processed by humans. We don't know that this means the human breakdown is of a different quality.

Unsustantiated claim:
Computers don’t add (or subtract or multiply or divide), they count, just as an abacus “adds” by counting up -- counting is what that formal system is about. That is, whereas we add “2 + 2,” a computer counts “(1 & 1) & (1 & 1)”

The fact that we use a look-up table, rather than repeated counting, is feature of human memory storage. If it were faster on a computer, and space was not an issue, computers could also use a look-up table.


Unsustantiated claim:
LOL. Please! By definition, all “difference in meaning is qualitative instead of quantitative”

Again, is there a qualitative difference in meaning between "gram" and "kilogram"?

Yes, I can see where the unsubstantiated claims would be annoying.

One Brow said...

Also, this isn't my blog; this is Mr Reppert's blog, and One Brow is as much his guest as I am.

If this is really a constraint for you, come on over to my blog, and please do not in any way feel constrained by being a guest there. You can unleach both barrels freely.

Ilíon said...

This post is off-topic in many ways, but on topic in a very important way: Christ is Lord of all (including of reason and logic).


Islam’s ‘Public Enemy #1’

Ilíon said...

I see that the post still contains the miswording I explicitly pointed out, along with other infelicities which I did not explicitly point out.