Logic

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Danoff

Danoff

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How well do you know your logic? Figure out if the conclusions to these premises are logically correct.

Use an exclusive "or" (xor) whenever you find an "or" statment, and remember that anything follows a contradition.

(1)
I have seen many swans in my lifetime and they have all been white, therefore all swans are white.
(this first case is the basic logical foundation of all of science)

(2)
premise
If A then B.
B
conclusion
A

(hint: the premise here is that if A then B and that we know that B is the case. Can we draw the conclusion that A is the case?)

(3)
premise
If A then B
If B then C
not C
conclusion
not A

(4)
premise
not a or not b
not b or not c
if a then d
d
conclusion
c (note: or is not the same thing as and)

(5)
premise
not a or not b
not b or not c
if a then d
c
conclusion
d

(6)
premise
not a or not b
not b or not c
if a then d
not a or not c
conclusion
x

(7)
premise
if b then c
if a then d
if d then c
not d or not a
conclusion
c

which ones are right and which ones are wrong?
 
Oh Hell No you didn't... :irked: it's freakin summer you dumb ass... j/k but no seriously that wasn't cool because I just added a class to take Philosophy. :banghead:
 
1 is no, thats all i know, haha.
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Let's work on number 2 then.

Let's say A stands for apple and B stands for banana

here's what we know.

If we have an apple, we have a banana

we also know that we have a banana

given this information can we conclude that we must have an apple?

edit: fixed banana spelling. Thanks M5.
 
no, cause u didnt say if we have a banana then we must have an apple



EDIT: no M5 we are not :p 👍
 
Are we going on the danoff suggested spelling of 'banana?'

I think 'danoff' = 'Dan Quayle.'
 
Something tells me this would be inspired by the lack of logic you see in others, in some debates... or is it not?

1 - False
2 - False
3 - True
4 - False
5 - False (unless you consider not b or not c = b or c ?)
6 - What does x stand for impossibe? we can only imply "not b", so it would be false.
7 - False

How many points? :dopey:
 
jpmontoya

you got 4 points.

To answer your question. X is another variable. You might ask, how could we possibly get X if it isn't in our premise? The answer to that is that (according to the rules of logic) if you have a contradiction in your premise you can prove anything.

So if there is a contradiction in 6, you can show X.
 
danoff
jpmontoya
The answer to that is that (according to the rules of logic) if you have a contradiction in your premise you can prove anything.
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I do not remember this - and don't understand this rule of logic. I guess it's the cause for the 3 misses, but I always thought that if there is a contradiction in the premise, you can't prove anything from it.
 
If the premise is contradictory, anything follows.

I don't really like that rule either, but just use X to indicate whether you should look for a contradiction.

Also, if you can find a contradiction in any of the others than those are true as well.

This is not the cause for all three of your misses.
 
following this rule I could say

My dog is purple
Sky is blue
Dogs are not purple.
Conclusion
Put anything here, it's true?

I just saw my error on #3. doh. :banghead:
 
My dog is purple
Sky is blue
Dogs are not purple.
Conclusion
Put anything here, it's true?
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Yea. There's some kind of way to prove anything from contradictory premises but I can't remember how they proved that.
 
danoff
Yea. There's some kind of way to prove anything from contradictory premises but I can't remember how they proved that.
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Neither do I ...but it might make sense, looking at how much it's used in practice.
 
jp,

Either you changed your answer or I graded you incorrectly. From what I'm seeing in your post you got 5 points. Sorry about the mix-up.
 
Answer Time!!! (don't read this if you haven't taken the quiz)


(1)
Number 1 is an example of inductive reasoning and it is not sound. Just because you've never seen something before or there have been no cases where something has happened doesn't mean that it is logical fact. Inductive reasoning is the basis of science and is very useful, but it is not ironclad.

Edit: :mad: Stupid deductive, inductive mixing up grrr.... ok I fixed it. The above is correct now.

Just because the sun has come up every day doesn't mean it will come up tomorrow. Just because F has equaled ma every day so far doesn't mean it will tomorrow (of course, that would put me out of a job so I'm betting F=ma tomorrow). Anyway the answer to number 1 is false and, in fact, there are black swans (in Australia I believe).

(2)
If A then B
B
therefore A

In order to get A from B you have to have a statement that says If B then A which is not the same thing as if A then B.

To rephrase this let's replace A with rain and B with the ground being wet.

The premises say if it rains then the ground is wet. They also say that we know the ground is wet. Can you conclude that it rained? The answer is no because there are other reasons why the ground could be wet. So (2) is false.

more answers later.
 
Good Job Nick.

(3)
premise
If A then B
If B then C
not C
conclusion
not A

The first two yeild the result of If A then C because A implies B which implies C. So the question is,

If A then C
not C
not A?

B is pretty much beside the point on this one.

So if A implies C, and we don't have C then how could we have A? If we had A we would have C. This one is True, not A can be concluded from the premises.




(4)
premise
not a or not b
not b or not c
if a then d
d
conclusion
c

So we have d. But all we know is that if we have a we have d, we don't know that if we had d we have a. So we don't know if we have anything besides d. We either have c or b (that is... not b or not c) but we don't know which one we have.

So this one is false, we can't conclude c.
 
Half guessing


1.
False.

2.
No

3.
False

4.
True

5.
False

6.
True

7.
False
 
2 right; 2 wrong.
 
I just realized that these last two require that my or statements are exclusive or's. Sorry if that causes problems - I'll post it on the first post. Maybe that's why you missed these last two jp.


(6)
premise
not a or not b
not b or not c
if a then d
not a or not c
conclusion
x

So keeping in mind that anything follows contradictory premises, the only way to prove x (since it isn't in the premises at all) is to show a contradiction.

if a then d is totally beside the point because d doesn't show up anywhwere else.

So we're left with
not a or not b
not b or not c
not a or not c

the first two can be combined to look like the following:

(not a and not c) or (a and c)

the reason is because if you just look at the first two you know that you have two options.

option 1
not b is the case. that means that you have a and you have c.

option 2
not b is not the case. that means that you have not a and not c.

those two options combine in into
(not a and not c) or (a and c)

but the third premise is:
not a or not c

which means you can't have both but that you have to have one. This contradicts the earlier result that says you have to either have neither or both. So there is a contradiction and anything follows. So x is true.

(7)
premise
if b then c
if a then d
if d then c
not d or not a
conclusion
c


Either you don't have d or you don't have a is the same thing as saying you either have d or you have a. The reason is because you can't not have both, it says one or the other. It also says you can't have both.

So you either have d or you have a. But if you have a then you have d (premise 3), so you can't have a, so you must have d (premise 5). But if you have d then you have c (premise 4). So c is true.

We didn't need premise 1.
 
Not that any of you want to learn more about this but just in case, here are the real logical operations that should be used in each of these derivations (I’m reading from my book). When you see an arrow read "if then".

Modus Ponens
Q, Q->W implies W
Modus Tollens
not Q, W-> Q implies not W
Disjunctive Syllogism
not Q, Q or W implies W
Contradictory Premises (<- jp’s favorite)
not Q, Q implies anything

I’ll stop naming them now
A and B implies B
A and B implies A
A, B infer A and B
some of these are pretty obvious
A->B, B->C implies A->C
here’s one I had to use for #7
not A -> A implies A
A or B, not A or Q implies B or Q
not (A or Q) = (not A and not Q)
not (A and Q) = (not A or not Q)
not not X = X
not (X or Q) = (not X or not Q)

the list goes on… that’s probably 3/4 of them. The proper way to do this is to list which of these rules you use in a series of steps until you get the conclusion. It’s pretty good stuff, but the amazing thing is that when you’re moving around all of these letters you could be moving sentences or ideas around (rather than just numbers like algebra) so it’s pretty powerful stuff. Definitely something worth studying.
 
I just realized that these last two require that my or statements are exclusive or's. Sorry if that causes problems - I'll post it on the first post. Maybe that's why you missed these last two jp.
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Yes, it is. The conventions that I learned were that with a or b you can have either a, b or both for the statement to be true unless you specified it's an exclusive or [xor].

For the contradiction, unless it is only for the purpose of sarcasm I don't get it. Maybe I mix these with concepts learned in my demonstration maths course. You can only demonstrate that the premises don't work by showing a contradiction, but that wouldn't imply (or demonstrate) anything else.

I made a mistake on #3, which was very easy though. (corrected and edited on the first post).

Yep, logics are quite useful. I guess the main limitations is that in many cases, the empirircal (as opposed to logical) premises are either subjective, not universally accepted, or may contains errors, which will often invalidate the conclusions. Otherwise the threads in the opinion forums would be quite shorter...
 
For the contradiction, unless it is only for the purpose of sarcasm I don't get it.
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I'll check my book for a proof of the contradictory premises rule.

I learned "or" the same way you did, but when I set up the quiz I did so thinking of xor rather than just or. I should have been aware that I was making that assumption and included it in the instructions.
 
I was just going back through this thread because I brought it up in the opinions forum and I realized I never got back to Carl about the contradictory premises.

I checked it in the reference material on logic, sets and recursion. It says clearly in there that anything follows from contradictory premises. Think of it this way. If your world includes dogs that are purple but also the rule that no dogs are purple, your world includes anything you can dream up - because it doesn't make sense.
 
I can divide :)
 
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