DIR Return Create A Forum - Home
---------------------------------------------------------
EC901 Micro
HTML https://warwickec901micro.createaforum.com
---------------------------------------------------------
*****************************************************
DIR Return to: PS1
*****************************************************
#Post#: 2--------------------------------------------------
Q1(a)
DIR By: Francesco_Longo
Date: October 22, 2012, 11:20 am
---------------------------------------------------------
Hi,
I'd like to ask if the way I've used to demonstrate that A2
holds in case of strict preference is ok or not. Similarly to
the indifference case I exploited the definition of strict
preference:
if x R y and NOT y R x -> x P y
so:
if y R z and NOT z R y -> y P z
then:
if x R y and y R z -> x R z
and
if NOT y R x and NOT z R y -> NOT z R x
hence:
if x R z and NOT z R x -> x P z
Is this plausible?
Thanks
#Post#: 3--------------------------------------------------
Re: Q1(a)
DIR By: Ariel_Li
Date: October 22, 2012, 3:32 pm
---------------------------------------------------------
I have the exactly the same proof! I think it is okay overall.
But I am not sure about the part "if NOT y R x and NOT z R y ->
NOT z R x". See, it's not a converse-negative proposition of
transitivity. It's logically right, but... not based on a
certain theory...
#Post#: 4--------------------------------------------------
Re: Q1(a)
DIR By: Francesco_Longo
Date: October 23, 2012, 12:49 am
---------------------------------------------------------
--- Quote from: Ariel_Li link ---
>
> I have the exactly the same proof! I think it is okay overall.
But I am not sure about the part "if NOT y R x and NOT z R y ->
NOT z R x". See, it's not a converse-negative proposition of
transitivity. It's logically right, but... not based on a
certain theory...
>
--- End Quote ---
From my point of view (but I'm not 100% sure) we're still
talking about transitivity, because it should hlod for the
negative reasoning as well. It would be the same of saying, you
can do this :
a>=b and b>=c -> a>=c
and this too:
a<b and b<c -> a<c
#Post#: 5--------------------------------------------------
Re: Q1(a)
DIR By: Julia.Wirtz
Date: October 23, 2012, 5:51 am
---------------------------------------------------------
I would say this is not strictly correct.
"If A then B" is not logically equivalent to "if not A then not
B"
On the other hand, "If A then B" is indeed logically equivalent
to "if not B then not A"
Example:
"If it rains, the street gets wet".
You cannot logically conclude: "If it doesn't rain, the street
doesn't get wet." (The street could get wet because I washed my
bike or something)
But you can conclude: "If the street is not wet, it doesn't
rain."
I hope this makes sense.
You can only make the reverse inference for "if and only if"
statements.
#Post#: 6--------------------------------------------------
Re: Q1(a)
DIR By: Francesco_Longo
Date: October 23, 2012, 6:10 am
---------------------------------------------------------
--- Quote from: Julia.Wirtz link ---
>
> I would say this is not strictly correct.
>
> "If A then B" is not logically equivalent to "if not A then
not B"
>
>
> On the other hand, "If A then B" is indeed logically
equivalent to "if not B then not A"
>
> Example:
> "If it rains, the street gets wet".
> You cannot logically conclude: "If it doesn't rain, the street
doesn't get wet." (The street could get wet because I washed my
bike or something)
> But you can conclude: "If the street is not wet, it doesn't
rain."
>
> I hope this makes sense.
>
> You can only make the reverse inference for "if and only if"
statements.
>
--- End Quote ---
Thanks for your reply. The example is very clear, but can you
help me to apply it to this specific case?
Just to be sure, the statement "if NOT y R x and NOT z R y ->
NOT z R x" is not (completely) logically correct, is it?
Thanks
#Post#: 7--------------------------------------------------
Re: Q1(a)
DIR By: Julia.Wirtz
Date: October 23, 2012, 11:53 am
---------------------------------------------------------
Yes, I think it is not logically correct. I would say a proof by
contradiction as in Ben's model solution is easier than a
constructive proof here.
*****************************************************
Page 1 of 1