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An Introduction to Logic

Posted Jan 22, '11 at 7:40pm

Einfach

Einfach

1,433 posts

It's a good thread iff Moegreche posts again

 

Posted Jan 22, '11 at 8:52pm

Moegreche

Moegreche

2,777 posts

Moderator

Can you make them even more complex? If you could, than it opens up even more possibilities, like an "and....but" it would allow for contradictions in the same sentence, while still having to use only one symbol. Or did I miss something, can you do that right now?

Heh, it gets much, much more complex. Unfortunately, so does the symbolism, which can't be represented on this site. Although I guess we could use other symbols in their place. But we will see a greater complexity unfold we we turn to proofs.

As for the "and ... but" possibility, those are actually the same in logic.

So the following two propositions:
Bill went to the store and Tim did not.
Bill went to the store but Tim did not.

Can be represented by:
(B ^ ~T)

As for representing a contradiction, we can already do that:

B ^ ~B : Bill went to the store and Bill did not go to the store.

In logic, contradictions do have a use. If you can show the denial of a premise results in a contradiction, you can show that premise to be necessarily true.
There is also a method of proof called an Indirect Proof that tries to generate a contradiction by denying the argument's conclusion. But we'll get to that.

Good questions, though. I'll have the next part up soon, just want to make sure there are no more questions before proceeding.

 

Posted Jan 23, '11 at 2:50am

Impiety

Impiety

18 posts

Just curious, does there exist an XOR operator in this kind of logic?

 

Posted Jan 23, '11 at 12:09pm

Moegreche

Moegreche

2,777 posts

Moderator

Just curious, does there exist an XOR operator in this kind of logic?

Great question. The 'or' operator in this system is not an exclusive one. It can be true if both disjuncts are true.
There are two ways around this. You can still get the logical force of an exclusive or:

(B v T) ^ ~(B ^T)

This would read Bob and Tom went to the store but not both. Thus you can get the same thing as an XOR operator - it's just longer.

Alternatively, we could introduce a logical operator to handle exclusive or stuff. The reason there isn't is because a) it's not necessary and b) we don't do a whole lot of exclusive or stuff.
But as long as you're clear about the truth conditions for the operator (it's false just in case only 1 disjunct is true) then you can certainly introduce it to this logic system.

 

Posted Jan 24, '11 at 7:54pm

Moegreche

Moegreche

2,777 posts

Moderator

But as long as you're clear about the truth conditions for the operator (it's false just in case only 1 disjunct is true) then you can certainly introduce it to this logic system.

Sorry, that should've read "it's TRUE just in case only 1 disjunct is true." Sorry about that.

I'm getting ready to post the next part on inferences, but I thought I'd give you guys a few to practice. If any questions come up, just post them here.

Determine if the following are valid or invalid.
*Remember the definition for validity: An argument is valid just in case it's impossible for the premises to be true and the conclusion false. So if you can imagine a world where the premises for the argument are true and the conclusion is false, then it's invalid.

Mars is closer to the sun than Venus.
Earth is closer to the sun than Mars.
So, Earth is closer to the sun than Venus.

Jim is taller than Bob.
Sam is taller than Bob.
So, Jim is taller than Sam.

Monkeys eat bananas.
Old people eat grapefruit.
So, 2+2=4.

The sun is hot.
The sun is not hot.
Therefore, a square is a circle.

If you go outside when it's raining, you'll get wet.
You go outside when it's raining.
So, you'll get wet.

If I get a haircut, chicks will be all over me.
Chicks are all over me.
So I got a haircut.

These are meant to be tricky and a bit weird, but they should adequately test your understanding of validity. If any of them don't make sense or you want to verify your answer, please just post here.

 

Posted Jan 24, '11 at 8:38pm

driejen

driejen

427 posts

Mars is closer to the sun than Venus.
Earth is closer to the sun than Mars.
So, Earth is closer to the sun than Venus.

Valid

Jim is taller than Bob.
Sam is taller than Bob.
So, Jim is taller than Sam.

Invalid since Sam can be the tallest of the three based on the premises.

Monkeys eat bananas.
Old people eat grapefruit.
So, 2+2=4.

Valid simply because the conclusion can't be false.

The sun is hot.
The sun is not hot.
Therefore, a square is a circle.

Invalid because the premises can't be both true and the conclusion can't be true.

If you go outside when it's raining, you'll get wet.
You go outside when it's raining.
So, you'll get wet.

Valid

If I get a haircut, chicks will be all over me.
Chicks are all over me.
So I got a haircut.

Invalid since it's possible that chicks are all over you for some other reason, like hypnotism.

I hope I did that right...

(~B v ~T) iff ~J

Woo some symbolism, always makes things simpler. Just as a personal test, is the quoted line logically equivalent to;
(B ^ T) iff J
(B ^ T) -> J

 

Posted Jan 24, '11 at 10:57pm

Moegreche

Moegreche

2,777 posts

Moderator

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.

 

Posted Jan 24, '11 at 11:50pm

driejen

driejen

427 posts

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 :)

 

Posted Jan 25, '11 at 3:57am

driejen

driejen

427 posts

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

 

Posted Jan 26, '11 at 3:58am

Pazx

Pazx

4,494 posts

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?

 
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