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I was going to post this in the "something a little lighter..." thread, but they aren't really jokes or riddles as such, more a serious subject which i am fascinated by.
Logic has always amazed me from the moment i studied it at uni, and i find myself constantly analysing statements made by people and determining the truth values of them.
Here are a few questions you may wish to try and answer using logic. If you have any of your own i would love to hear them!
1) The statement "this statement is false" is a paradox. If someone says to you "everything i say is a lie" is this also a paradox?
2) Consider the statement "I break some eggs if i make an omelette" This means that making an omelette implies breaking some eggs.
Now remove the word "if" from the statement and put in another word(s) which make the statement mean that breaking some eggs implies making an omelette.
3) Does "2+2=5" imply that mickey mouse was the archbishop of canterbury?
Hope you have fun with these! Again if you know of any yourself then i would love to hear them!
1. no, he simply lied. That has no implication that everything he says is a lie thus meaning that the statement is not a paradox.
Ouch that hurt my brain, I probably got it hopelessly wrong and made an idiot of myself but thats happened enough now that I'm used to it.
I dont know the other 2, I'm interested in #3 though. Triumph Of Man thinks its because both statements are false. #2 requires far to much grammatical logical effort then I'm willing to put in right now. :rolleyes:
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Yup(Y) Many people make the mistake of thinking that the negation of "everything i say is a lie" is "everything i say is the truth" which of course it's not.Originally Posted by LordLink
Don't wanna give the others away just yet...
Here's something else which is fun to look at. Does changing the order of the words in a statement alter the truth values of it? Look at these examples
"For all n there exists an m such that m is greater than n"
"There exists an m for all n such that m is greater than n"
Simply by changing the order of the words we have changed the truth value of the statement! Am i the only one who finds this fascinating?
No. The initial statment is of the form "If P, then Q", which is a logical method which most of science depends on, and is very prevalent in computer programming. Remove the conditional element and you then have the form "P, therefore Q", which never necessarily holds. It may do in a specific case, such as "I can see, therefore I am not blind", but is never a general rule, which is why things like Descartes' famous "I think therefore I am" can be logically questioned.Originally Posted by Stonehambey
On another level entirely, if you break eggs and are thus making an omelette, you are replacing the normal definition of omelette (A dish consisting of beaten eggs cooked until set and folded over, often around a filling, dictionary.com) with the definition "the state which eggs become when they are broken" or words to that effect. If that is indeed the definition of omelette (which is generally not the case), then you have no problem. But if it is not, then the statement is not a deductive one, merely an inductive one.
To illustrate the difference between the two:
Premise 1: The Eiffel Tower is in the city of Paris
Premise 2: The city of Paris is in France
Premise 3: Therefore, the Eiffel Tower is in France
Premise 1: An omelette can be generated from broken eggs
Premise 2: Eggs are broken
Premise 3: Therefore, an omelette has been generated
The difference between the two is that the premises of an inductive argument do not necessarily lead to the conclusion. Yes, an omelette is generated from broken eggs, and yes eggs have been broken, but there is no necessary connection between the two.
Hope that makes sense!
You made the mistake of assuming that i wanted the statement to be true, i never said that. Of course breaking an egg doesn't imply making an omelette!
The original statement was in the form P implies Q, now the question i pose is what can you replace the word "if" with in that statement in order to reverse the implication of the statement (i.e. Q implies P) without changing the order of any of the other words? The final statement need not be true.
A noble attempt though
Erm...everything. I was simply wording a statement which took the form P implies Q (which you correctly pointed out)Originally Posted by Xerxes
You seemed to grasp the idea the first time round, so why the sudden change of heart? Please don't get huffy
The simplest answer is "only if"
In a mathematical proof you are sometimes asked to show something if and only if something else. Which means that you need to prove the statement both ways (i.e. P implies Q and Q implies P)
There are other wordings commonly used, such as "is necessary for" is the implicative opposite of "is sufficient for"
"Breaking an egg is necessary for making an omelette"
"An omelette is sufficient for breaking some eggs"
These two statements are equivalent. The fact that they are also true is irrelevant to the point i am trying to make. It's just that its easier for people to visualise a true statement.
That is to say if we have an omelette, we can safely say some eggs have been broken.
Fair enough. And sorry if I sounded huffy, I wasn't trying to be. It was just that the answer that I gave seemed ludicrously simple (something which I was suprised at, given then nature of the question).
And a slight flaw here; although in the example you give, necessary and sufficient condition are the same thing, they are not always. For example, necessary conditions do not always result in the thing they are necessary for. Fertilisation is necessary for a new organism, but it is not the be all and end all, you've then got cell division etc etc. It is necessary, but it is not sufficient. Although it doesn't appear to be the same the other way round, ie. all sufficient causes are also necessary ones.
On the contrary, "is necessary for" and "is sufficient for" mean the opposite of each other (see in my omelette example i switched the P and Q round, as i said before the order of a statement is very important). So if P is necessary for Q then this tells us nothing about whether P is sufficient for Q. It does however, tell us that Q is sufficient for P.Originally Posted by XerxesWhat you say is indeed correct, necessary conditions are not always sufficient ones, this is only true if the implication works both ways (back to the old "if and only if" argument). The example i put forward shows how they mean the opposite, not the same.Originally Posted by Xerxes
I really like your example of fertilisation, so i'll stick with that for a moment if i may
Let's define our P and Q.
Let P be the event of a new organism
Let Q be the event of fertilisation
So if we have a new organism then there must have been fertilisation somewhere along the line. So we can say that
P implies Q
So say we have a new organism, and we wish to know whether fertilisation has taken place. Is the fact that we have a new organism sufficient evidence to say that fertilisation took place? Yes, thus we can say that
P is sufficient for Q
Now i said earlier that "is sufficient for" and "is necessary for" are implicative opposites, therefore the following must be true (assuming the above is true, which i think it's safe to say is)
Q is necessary for P
So here we are saying that fertilisation is necessary for a new organism, which of course holds.
Now consider the following two statements
Let P be the fact that x is an even number
Let Q be the fact that 2 divides x
Now here we can say that P is necessary and sufficient for Q
A great response from Xerxes and Lordlink so far Has anyone any ideas about #3?
Im sorry, this belongs in GD, not Enhanced.
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