ForumsWEPRAn Introduction to Logic

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Moegreche
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Moegreche
3,819 posts
Duke

Part 1: ARGUMENTS

The entire purpose of the logic we're looking at is assessing arguments. So it'll be helpful to understand exactly what an argument is. Here's a working definition:

ARGUMENT: A series of propositions consisting of premises which are purported to support a conclusion.

Of course, to understand this definition, we need to know what a proposition, premise, and conclusion are.
A proposition is a special kind of statement - it's one that can be true or false. Obviously, questions like 'What time is it?' can't be true or false. And neither can commands like 'Shut the door.'
A proposition says something about the world. Here are some examples of propositions:

1) It is sunny outside.
2) Mercury is the closest planet to the sun.
3) All mammals lay eggs.

Notice that 3 is false. But that's okay, it's still a proposition. Remember, these are statements that can be true OR false. As it turns out, there are some philosophers who have some strong arguments about what is and isn't a meaningful proposition. But that discussion is for an Analytic Philosophy class. We don't really care about these things in logic. If it's something to make sense to say it's true or false, then it's a proposition.

So an argument consists of premises and a conclusion. The premises are propositions that give you a reason to accept the conclusion, which is also a proposition. The conclusion is what you're supposed to, well, conclude! Here's as example:

1) All men are mortal.
2) Socrates is a man.
3) Therefore, Socrates is mortal.

In this argument, 1 and 2 are the premises which support 3, the conclusion. You can usually tell the conclusion by keywords like 'therefore' 'so' and 'thus.'
Look back at the definition of an argument - notice it says that the premises are PURPORTED to support the conclusion. That just means that they are intended to give support - but they may fail miserably. The result would be a bad argument, but it's still an argument. Here's an example:

1) Monkeys like bananas.
2) I like bananas.
3) Therefore, I'm a monkey.

In this argument, the premises lend very little support to the conclusion. You may even have an argument where the premises have nothing at all to do with the conclusion. But these are still arguments - just really bad ones!

So now you know what an argument is. Up next, we'll go over the basics of how to assess an argument. This, remember, is the central goal of logic (at least, the logic we're talking about).
It's worth noting here that the kind of logic we'll be talking about is called PROPOSITIONAL LOGIC. This logic deals with, you guessed it, propositions. Overall, it's very weak - there are many arguments it can't assess.
More powerful logical systems like predicate logic and modal logic can handle more arguments. But you have to walk before you can run, and this kind of logic is a very good place to start. If you can understand this, you'll have a much easier time learning more powerful logical systems.

These are the basics, so if there are any questions, please post them. It's vital that you understand these definitions so that the next part will make sense.

  • 114 Replies
pangtongshu
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pangtongshu
9,823 posts
Jester

If you need any logic or ways to win an argument, come talk to me.


i think I'll stick to this guide...but if I really need to talk to someone I'd much rather talk to Kasic
pangtongshu
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pangtongshu
9,823 posts
Jester

Sorry didn't get to finish

MGW, nich, Von, or partydevil

mbbs112
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mbbs112
198 posts
Peasant

This was an excellent guide Moegreche ^_^.

jorgedbp96
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jorgedbp96
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Shepherd

There is not life in the universe
The Earth is in the universo
Therefore there is not life in Earth


There is life in the Earth
Earth is a planet
Therefore there is life in other planets

Moegreche
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Moegreche
3,819 posts
Duke

There is not life in the universe
The Earth is in the universo
Therefore there is not life in Earth


A valid argument, but it's not sound (notice premise 1 is false). But there's an interesting sort of argument that hinges on the ambiguity of the 'in' relation. Consider the following:

1) There is a pain in my finger.
2) My finger is in my mouth.
C) Therefore, there is a pain in my mouth.

We might say there's a fallacy of equivocation going on here. Notice that the use of the word 'in' in (1) is subtly different from (2).

There is life in the Earth
Earth is a planet
Therefore there is life in other planets


An invalid argument, though close to being valid and sound. If we changed 'in other planets' in the conclusion to 'on a planet', we'd have a valid and sound argument
aknerd
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aknerd
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Peasant

Hmm... I'm going to try to be tricky here.

Suppose (Ie, assume these statements are true):
1) The universe is infinite in size, and
2) All non-earth locations do not contain life.

Facts:
1) Life forms must have mass, that is, all living things must be made of matter
2) Matter has a finite, maximum density

Therefore:
1) The maximum volume of all living things cannot exceed the volume of Earth, since we are assuming that there are no living things not on Earth.

2) Then the mass of all living things must be at most finite, since they occupy a finite volume (from fact 2)

3) Then, the density (mass per volume) of living things in the universe is 0, since the volume of the universe is infinite.

4) Then, from fact 1, there are no life forms in the universe as the total mass of all life forms is zero.

I know where I went wrong, do you?

pangtongshu
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pangtongshu
9,823 posts
Jester

4) Then, from fact 1, there are no life forms in the universe as the total mass of all life forms is zero.


This seems to be contradictory to one of your first statements
2) All non-earth locations do not contain life.
But, then I would be assuming the converse is correct, which you never established.
jorgedbp96
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jorgedbp96
4 posts
Shepherd

In a way this is similar to the inequations:

3 > 2 ( 3 higher than 2)
2 > 1 ( 2 higher than 1)
3 > 1 ( 3 higher than 1)

and i understand validity but not sound and how is valid and sound an argument where nothing is related?

aknerd
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aknerd
1,416 posts
Peasant

This seems to be contradictory to one of your first statements
2) All non-earth locations do not contain life.
But, then I would be assuming the converse is correct, which you never established.


Its not that- I was careful not to say "Earth is the only planet that contains life" for a reason. My assumption was merely that Earth is the only planet capable of containing life- it doesn't necessarily have to.
aknerd
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aknerd
1,416 posts
Peasant

Actually, I kinda left out some steps here- perhaps this will make it easier to follow my "logic":

3) Then, the density (mass per volume) of living things in the universe is 0, since the volume of the universe is infinite.


4) Then, on average every finite subsection of the universe contains exactly zero life. (A "finite subjection" here is like a bounded region of the universe)

5) Then NO subsection of the universe contains any life.

6) Then there is no life in the universe.
pangtongshu
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pangtongshu
9,823 posts
Jester

and i understand validity but not sound and how is valid and sound an argument where nothing is related?


A valid argument would be an argument one could make and could essentially back up, doesn't necessarily mean an argument that is correct.

A sound argument is a reasonable argument that is logical.

-----

As for Aknerd, I didn't have much time my last post, nor do I have much time this post, but are you working out a paradox? Would that be the answer to what is wrong?
Moegreche
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Moegreche
3,819 posts
Duke

and i understand validity but not sound and how is valid and sound an argument where nothing is related?


Just keep in mind the definition of validity I've provided. An argument is valid if and only if it's impossible for the premises to be true and the conclusion false. Valid arguments don't have to be convincing or even make sense - it's just a feature of the structure of the argument, not its content.
So just consider this argument structure:

1) P
2) Q
Therefore, P

We can tell this is a valid argument without even knowing what P and Q stand for. This is because there's no way for the premises to be true and the conclusion false.

4) Then, on average every finite subsection of the universe contains exactly zero life. (A "finite subjection" here is like a bounded region of the universe)


This looks like the problem move to me. This claim is supported by the density of life over a finite subsection. But your fact (1) talks about the mass of life, not its density over some area. Thus (4) is an unsupported subconclusion.
But isn't it the case that once we start making finite subsections, the density would be greater than 0? This worry doesn't apply to your previous formulation (which seems stronger to me), I'm just curious about it in general.
aknerd
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aknerd
1,416 posts
Peasant

But your fact (1) talks about the mass of life, not its density over some area. Thus (4) is an unsupported subconclusion.


Hmmm. I would claim that (4) is actually logically valid. Its like this: Imagine you have an infite grid made up of unit squares (ie the side length of each square is 1). And you have a ball of putty of some sort of finite volume/mass. If you squished the putty evenly over a given number of squares, found the mass of the putty in just ONE of the squares, and then multiplied that mass by how many squares you squished over, you would have the original mass of the entire ball of putty. Right?

BUT if you squished the putty over all of the squares (of which there are infinite), the mass over any single square would be zero. This is really easy and intuitive to prove.

So, lets prove it, using my most favorite kind of proof ever, a proof by contradiction! (skip this if you don't really care about this aspect)
__________________________________

Suppose that after you squished the putty evenly over the squares, the mass over a single square is greater than zero.

Then, because you squished the putty perfectly evenly, every square must contain the same amount of putty.

Then every square contains an amount of putty with mass greater than zero.

But there are infinite squares, so the mass of the putty over all the squares must be infinite as well.

But I initially stated that the putty had finite mass, so this is a contradiction. Therefore, the mass over a single square must be zero.

-----------------------------------------------------------------------


I claim that the error in my proof is actually between steps 4 and 5.

BUT to keep this from getting too mathematical, I actually would rather talk about the implications of my proof. Above, I just demonstrated how a proof by contradiction works. Basically, if you arrive in a logical contradiction in your proof, one of your initial assumptions must be incorrect.

If I add the initial assumption to my original proof "I am alive", then clearly the conclusion of my proof (there is no life) is a contradiction. Which means at least one of my assumptions must be false.

So, Either:
1) There is infinite life in the universe*
2) The universe is not infinite
or
3) I am not alive

*This isn't what I directly assumed, but it is an equivalent assumption as the choice of Earth was arbitrary

But there is no reason why any of those assumptions should have to be false for there to be no life in the universe. Which means that I MUST have messed up the proof somehow- that is, my contradiction is not actually a contradiction.
Moegreche
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Moegreche
3,819 posts
Duke

Your reductio makes perfect sense to me. But then again, one could use a reductio against a variety of arguments that incorporate claims about infinity. But this doesn't mean that there is an actual contradiction in the premises.
I suppose we could try to formalise the argument, but I'm thinking that this will be beyond my limited abilities as a logician. Trying to formalise the notion of an infinite universe, for example, is going to be very problematic. I'm not even sure how it would be done. We would need something more than a mere property statement (it would seem), but I'm just not sure what that 'something more' would amount to. At any rate, it's something fun to think about!

aknerd
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aknerd
1,416 posts
Peasant

But this doesn't mean that there is an actual contradiction in the premises.


But, in this case, there is! Let's look at my first outline, going from steps three to four:

3) Then, the density (mass per volume) of living things in the universe is 0, since the volume of the universe is infinite.

4) Then, from fact 1, there are no life forms in the universe as the total mass of all life forms is zero.


Basically, what I am saying in (4) is that since the denisity of life throughout the entire universe is zero, the mass must be zero as well. In step three, I defined density as mass per volume.

So, equivalently, what I am saying in (4) is that if mass/volume = zero, then mass must equal 0. Or, mass/volume = 0 only if mass = 0.

HOWEVER.

In step three, I also state that mass/volume = 0 if the volume is infinite. Therefore, the statement: mass/volume = 0 only if mass = 0 is actually false! It is possible to have zero density but positive mass in an infinite area. This means that step four does not logically follow from step 3.

I said this proof was sneaky, and it is. Most people know from basic algebra that if x/y = 0, then x must be 0, and therefore don't catch onto the logical contradiction. But, within my proof I explicitly state that there is another way for a quotient to be zero. And then in the very next step I just ignore that fact.

For those who are interested:
Mathematically, you could see that this proof contained a critical error more easily than if you were just following the logic (though I claim it is still possible to find the error with little knowledge of math). This is because the division involving infinity actually uses a different operator than regular division. So, in my proof, I used the infinite kind of division (which involves limits and such) in step three, but pretended like I had used the regular kind of division in step four.
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