Forums

ForumsWorld Events, Politics, Religion, Etc.

An Introduction to Logic

Posted Dec 1, '13 at 7:11pm

jorgedbp96

jorgedbp96

4 posts

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?

 

Posted Dec 1, '13 at 7:14pm

aknerd

aknerd

1,275 posts

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.

 

Posted Dec 1, '13 at 7:27pm

aknerd

aknerd

1,275 posts

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.

 

Posted Dec 1, '13 at 10:01pm

pangtongshu

pangtongshu

8,230 posts

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?

 

Posted Dec 3, '13 at 5:11am

Moegreche

Moegreche

2,760 posts

Moderator

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.

 

Posted Dec 3, '13 at 11:14am

aknerd

aknerd

1,275 posts

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.

 

Posted Dec 5, '13 at 2:23pm

Moegreche

Moegreche

2,760 posts

Moderator

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!

 

Posted Dec 7, '13 at 5:48pm

aknerd

aknerd

1,275 posts

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.

 

Posted Dec 11, '13 at 5:42pm

Moegreche

Moegreche

2,760 posts

Moderator

This is all really interesting. But in the interest of full disclosure - I'm definitely out of my league at this point. I'm happy to talk about lots of different logical systems and the theory behind logic. But I can't meaningfully contribute to this, although I am learning a lot!

 

Posted Apr 20, '14 at 11:53am

09philj

09philj

834 posts

Any equation dealing with infinity causes any constant to approximately equal zero.

 
Reply to An Introduction to Logic

You must be logged in to post a reply!