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An Introduction to Logic

Posted Jan 26, '11 at 9:25am

Moegreche

Moegreche

3,385 posts

Moderator

Does that make the argument invalid as the two premises make sense (to an extent?) but the conclusion isn't supported by them?


Ah, very good question. What's you've done is used the fallacy of equivocation for your benefit
With equivocation, you use a word in two different ways. Here's an example:

Joe takes his money to a bank.
A bank is the side of a river.
So, Joe takes his money to the side of a river.

Here we have a valid argument, but only because we're using the word 'bank' to mean: 1) a place where you put money, and 2) the side of a river. We have to be consistent with our terms.
In your raven example, you're using black as a color and as a name, so you are equivocating on the word 'black.'
The raven argument, though, is still invalid because you do try to make it clear how you're using the word. The raven named black is still white, so it contradicts the first premise.

And 1 quick but very important point. When you said
but the conclusion isn't supported by them?

it's very important to look at validity in the form of the argument. The premises could have nothing at all to do with the conclusion, but the argument could be valid. It's still a bad argument, but not because it's invalid.
Some logic books teach validity as the premises guarantee the truth of the conclusion. But this definition isn't just misleading, it's flat wrong. Sometimes you have a conclusion that can't be false, which is going to make for a valid argument no matter what the premises are.

(premise 1)
(premise 2)
Therefore, 2+2=4

The conclusion of this argument can't be false, so it's a valid argument. You could put whatever you wanted in the premises and it would still be valid.
 

Posted Jan 26, '11 at 10:01am

Asherlee

Asherlee

5,331 posts

Knight

It's like a surge of Intro to Logic from my sophomore year! I LOVE IT! I would participate in a truth table topic...as nerdy as that sounds. But I'm going to leave a note with my favorite logical paradox. I think it goes like this:

All Cretans are liars,
I am Cretan.
Therefore, I am a liar

----
It's hard to put in modus ponens form, but the point is Epimenides was talking about Zeus to the Cretans who believed he was dead.

They fashioned a tomb for thee, O holy and high one
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.
 

Posted Jan 26, '11 at 11:39am

aknerd

aknerd

1,431 posts

it's very important to look at validity in the form of the argument. The premises could have nothing at all to do with the conclusion, but the argument could be valid. It's still a bad argument, but not because it's invalid.
Some logic books teach validity as the premises guarantee the truth of the conclusion. But this definition isn't just misleading, it's flat wrong. Sometimes you have a conclusion that can't be false, which is going to make for a valid argument no matter what the premises are.

(premise 1)
(premise 2)
Therefore, 2+2=4

The conclusion of this argument can't be false, so it's a valid argument. You could put whatever you wanted in the premises and it would still be valid.


I'm going to have to disagree with you, Moegreche, because your argument only works on the assumption that I already know that 2+2=4. I personally believe that a proof should work without using any outside knowledge. In my math class, we aren't allowed to use anything that we haven't already proved in class. This may seem ridiculous when applied to basic arithmetic, but with more complex math...

Roses are red
Violets are Blue
There is no rational number that squares to two

Pretend that you did not already that the square root of two is a rational number. Could you possibly say that this is a valid argument?

Okay, maybe something more complex:

Batman is okay
Superman is my hero
e^(i*pi) + 1 = 0

Yeah. Also an "always true" conclusion. But how can this be a valid argument when the conclusion isn't even argued for?
 

Posted Jan 26, '11 at 2:11pm

Asherlee

Asherlee

5,331 posts

Knight

aknerd, I can't speak for Moegreche, as he is the master logician, but it might be possible that something like 2 + 2 = 4 is a tautology. But, is there a case that it would not equal 4?

 

Posted Jan 26, '11 at 4:06pm

aknerd

aknerd

1,431 posts

but it might be possible that something like 2 + 2 = 4 is a tautology.


Are things tautologies before they are proven to be tautologies? Well, yes. And, in a way, no. e^(i*pi) + 1 = 0 is a tautology. But no one would believe you until it you prove it to them. It would still be true, however. In fact, pretty much all of mathematics consists of manipulating tautologies.

But my question remains: supposing that I do not know that 2+2=4, how do I assess the truth value of Moegreche's argument?

(Note: There is actually a proof for why 2+2=4. The one I know of is amazingly complex A full proof requires almost 26,000 steps, if you start by assuming nothing and then slowly prove the existence of numbers)
 

Posted Jan 26, '11 at 8:16pm

Moegreche

Moegreche

3,385 posts

Moderator

Good question, aknerd. As Ash suggested, I was stipulating 2+2=4 to be a tautology. You're concerns about its status are right, I was just being lazy. But you've called me out, and rightly so.
I really should've used something more like:
(p v ~p)
which is a proper tautology. I just didn't want to further confuse the issue by bringing in symbolization that I hadn't yet explained.

The statement 2+2=4 is actually really problematic to classify. I consider it to be an analytic statement, but even this classification needs defending.
But even to understand this would require one to understand the meanings of '2' the + function, the = function, and the term '4'.
In the future, I'll just stick with proper tautologies. Thanks for keeping me on my toes

 

Posted Jan 27, '11 at 12:35am

Einfach

Einfach

1,502 posts

Really, if 2+2=4 is acceptable, then e^i*pi should be, too.

You can prove it with (materials needed):
Function of addition
Function of multiplication
Function of exponents
Function of subtraction
The Binomial Theorem
Taylor Series
Function of derivatives
The sin function
The cos function (it's how you get the pi part)
Meaning of e, i, and pi
The concept of a limit (for e)
Concept of a circle (for pi and sin and cos)
Understanding of combinations (for the Binomial Theorem)
The concept of the infinite sum (for e)
Complex numbers.
Understanding of variables
Understanding of square roots (for i)
Concept of negative numbers
Understanding of factorials

Missing anything? If 2+2=4 is a tautology, then this is too (I almost put the number "2" instead of "too"). If this is not, then why not? But then, isn't any true mathematical function a tautology?

 

Posted Jan 27, '11 at 8:20am

Moegreche

Moegreche

3,385 posts

Moderator

As I said, I had presented 2+2=4 as a tautology out of laziness. It's not one, however. I may be analytically true or even necessarily true, but it's not a tautology. I apologize.

 

Posted Feb 6, '11 at 8:02am

jacksonghuntington

jacksonghuntington

352 posts

Math makes up everything. Logic is something. Therefor, logic is math and 2+2 does infact equal 4? im just saying and im sorry if i dont belong with all these mods and high rep people.

 

Posted Feb 7, '11 at 9:02pm

Einfach

Einfach

1,502 posts

Math makes up everything. Logic is something. Therefor, logic is math and 2+2 does infact equal 4? im just saying and im sorry if i dont belong with all these mods and high rep people.

OK - your premises:
Math makes up everything...explain our consciousness. This can prove difficult. Can consciousness really be reduced to physics and mathematical equations? What about our perception of free will? You might want to read up on physicalism and reductionism.
Logic is something ... how does logic exist? It's not a material object, so what IS it? It doesn't exist physically, does it? Or does it exist in our minds, which can be explained using physics, and therefore it does "exist" in some form, physically.

(Personally, I sympathise with physicalism and reductionism, but I'd like to hear you defend these points.)
 
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