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If black is white then red is green
By Angelina Lai on Saturday, December 07, 2002 - 05:12 pm:

Hi, just a q that we were confronted in lesson and had a big debate (and still didnt come off with any agreement)...


Quote:

Bob says: 'If 5<2 then I am a Dutchman'. Bod isn't a Dutchman. Is the statement true/false?



Now the intuition is that 5 is NOT less than 2 therefore this statement is false. However, imagine that in some distant alien world that 5<2 is true and
Let A represent the statement '5<2'
Let B represent the statement 'I am a Dutchman'

Now the statement basically says A Þ B. My initial argument was that since the arrow is pointing to B then knowing whether or not B is true does not tell us anything about A. And then I realised the question asks whether this statement is true...

Would the fact that B is false invalidate this whole statement?

Thanks!

Angelina
By Kerwin Hui on Saturday, December 07, 2002 - 05:37 pm:

How about this:

F(father) said to S(son) "If you had got full mark in the exam I would have bought you a new game console."
S didn't get full mark in the exam.
F didn't buy S anything.

Is F a liar?

This is essentially the same thing as you describe. The statement A is replaced by the statement 'S gets full mark in the exam' and B is 'F will buy S a new game console'.

Of course in this case we would say F isn't a liar. The only possibility that will lead us to the conclusion 'F is a liar' is when both 'S gets full mark in the exam' and 'F didn't buy S a new game console' are true, but this is not the case here. There may be some parallel universe where S did in fact get full mark in the exam, but that is not in our system and you wouldn't know whether F actually buy the new game console for his son in that particular parallel universe.

The important thing is you can say anything as a consequence of a false claim, and the statement is still true as long as we work inside this particular system. You cannot jump to another system and expect things to continue to behave themselves.

Kerwin

By Angelina Lai on Saturday, December 07, 2002 - 06:01 pm:

So are we saying that the statement 'If 5<2 then I am a Dutchman' is false?

By Brad Rodgers on Sunday, December 08, 2002 - 02:04 am:

No; The statement "If 5 < 2 then I am a Dutchman" is perfectly true.

A statement is false if there is a counter-example to it. In this instance, there is no counter-example to "If 5 < 2 then I am a Dutchman," because there are no instances where 5 < 2 (and there are certainly no such instances when the person speaking is not a Dutchman). Strictly speaking, in fact, the statement is true even when the person speaking is a Dutchman.

Sorry if that's more confusing than helpful,

Brad

By Stephen Burgess on Sunday, December 08, 2002 - 11:35 pm:

Professor Tim Gowers (Field's Medallist) showed us a similar example in a lecture, namely

If x is both odd and even, then x=17

and said that that this makes logical sense, so I'm going to side with him :-)

By Richard Thomas on Monday, December 09, 2002 - 02:49 am:

On Stephen's note of Tim Gowers above, I refer to this URL (on his website) which may help explain anything that hasn't already been explain in the above conversation.

By Vicky Neale on Tuesday, December 10, 2002 - 08:53 pm:

He's Prof actually, but I'm sure he won't mind.

I think the point is that one can deduce anything from a false premise. If x is both odd and even, then x is 17 is fine. The fact that there exists no x that is both odd and even means that it is a completely vacuous statement, but that's okay. I could just as well say "if all sheep are green then I can fly" - just because not all sheep are green does not mean that we cannot make the statement. So "if 5<2 then I am a Dutchman" is fine, regardless of whether or not I come from Holland.

Vicky