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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
Well done, driejen!
Here's the only one you missed:
The sun is hot.
The sun is not hot.
Therefore, a square is a circle.
You said it was invalid and correctly noted that the premises can't both be true and that the conclusion can never be true. But remember the definition of validity: you would have to show that the argument can have all true premises and a false conclusion.
Now, this conclusion will always be false, you're right. But the premises can never both be true, so the argument is valid.
This is why ANYTHING can follow from a contradiction. And that's why contradictions make really bad arguments
As for your equivalency question, we have:
(~B v ~T) iff ~J
Your first one is spot on, the second one should be:
[(B * T) -> J] ^ [J -> (B ^ T)]
The reason you need that stuff after the 'and' has to do with the biconditional (iff).
The statement:
p iff q
is equivalent to:
(p -> q) ^ (q -> p)
That would read "If p then and if q then p"
Does that make sense?
And for the rest of you, definitely read his explanations, they're spot on for what you need to look for when assessing validity.
Perhaps it's just the definition of a valid argument, but I don't really understand why two contradictory premises can be valid when if one premise might support a conclusion, the other would support the opposite. Perhaps it's just that I must take each premise individually and assess wether each can be true while the conclusion false to determine its validity and ignore the contradiction? But then how can one construct arguments from multiple non-related premises? For example;
If I get a haircut, chicks will be all over me.
Chicks are all over me.
So I got a haircut.
I cannot consider the relationship between the premise, "Chicks are all over me" and the conclusion, "So I got a haircut" without taking into account the premise, "If I get a haircut, chicks will be all over me". What I'm trying to say is if for an argument I need to evaluate all the premises leading to a conclusion, how can I evaluate two contradictory premises?
As for your equivalency question, we have:...
Thanks for the explanation, I think I fully understand now. My second line is implied and true for the statement (~B v ~T) iff ~J but not equivalent to it as it does not give the whole relationship between the letters, just an oversight really
Another equivalent of the expression; [(B ^ T)-> J]^[(~B v ~T)-> ~J]
Btw, nice work with this thread. It's certainly informative and I'm looking forward to learning more about proper logic from you
Ah I think I get the validity thing now. It's exploiting the fact that the two premises can never be true at the same time (didn't read doh!). It seemed illogical to me how two contradictory statements could be valid so I didn't reread until now, seems like a major flaw to me that you can use two contradictory premises to construct an argument :/
All ravens are black
I have a white raven
God exists
All ravens are black
I have a white raven
God exists
Not only did I lol at that, my brain reformatted it into a haiku. And, since the second line of my haiku hurt my brain I figured I'd post it here. Only now do I realize that it has 8 syllables but eh. /random
All ravens are black
I have a white raven named black
(Therefore God exists)
Does that make the argument invalid as the two premises make sense (to an extent?) but the conclusion isn't supported by them?
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.
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.
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?
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?
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)
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
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?
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.
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.
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.)
This sounds a lot like C.S. Lewis... One crazy mathematician, I tell you. Logical fallacies? Anyone done TOK in IB? rofl...
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