Le 17-juil.-05, à 11:12, Stathis Papaioannou a écrit :


Yes, it's obvious. Bruno, if only everything you said were so obvious!


(I was asking if the proposition: <For all number x, if x is bigger than 2 then x is bigger than 1.> is obvious). Bigger = strictly bigger, to be sure.

I am pleased you find the propositions obvious. Thanks to Stephen assessing that obviousness too. Logicians and mathematicians are trying hard to make their reasoning decomposable in steps as obvious as this one, at least in principle.

Now, Logicians like puzzles, paradoxes and traps, and you could have guessed that if I ask a such "obvious" question, perhaps it is not so obvious after all.

And, actually, you illustrate very well my point. You will see why.

I guess you agree that if I say <For all number x, the property P(x) is true> where x is supposed to be a natural number, it means that P(0) is true, P(1) is true, P(2) is true, P(3) is true, P(4) is true, P(5) is true, etc.

Now *you* tell me that <For all number x, if x is bigger than 2 then x is bigger than 1> is obvious, and thus this means you agree that:

if 0 is bigger than 2 then 0 is bigger than 1, (line 1)
if 1 is bigger than 2 then 1 is bigger than 1, (line 2)
if 2 is bigger than 2 then 2 is bigger than 1, (line 3)
if 3 is bigger than 2 then 3 is bigger than 1, (line 4)
etc.

So apparently you find obvious that if 0 is bigger than 2 then 0 is bigger than 1.

But then, why do you ask me in your other post? :

Bruno:
I suppose you know some classical logic:
(p & q) is true if both p and q is true, else it is false
(p v q) is true if at least one among p, q is true, else it is false
(~p) is true if and only if p is false
(p -> q) is true if p is false or q is true
(to be sure this last one is tricky. "->" has nothing to do with causality: the following is a tautology (((p & q) -> r) -> ((p -> r) v (q -> r))) although it is false with "->" interpreted as "causality", (wet & cold) -> ice would imply ((wet -> ice) or (cold -> ice)). Someday I will show you that the material implication "->" (as Bertrand Russell called it) is arguably the "IF ... THEN ..." of the mathematician working in Platonia.

Stathis: That last one always got me: a false proposition can imply any proposition. All the rest seem like a formalisation of what most people intuitively understand by the term "logic", but not that one. Why the difference?

You see my point? What you found obvious in my "just a question" post, presupposes what you don't find obvious in the second "RE: Stathis, Lee and the "NEAR DEATH LOGIC"


So, the obviousness of <For all number x, if x is bigger than 2 then x is bigger than 1> presupposes all the correct truth value of "p -> q":

the truth of "false -> false" was implicit in you assessment of
"if 0 is bigger than 2 then 0 is bigger than 1", (line 1).
the truth of "false -> true" was implicit in your assessment of
"if 2 is bigger than 2 then 2 is bigger than 1", (line 3)

"true -> false" will never appear in the list, and I guess that is the reason why you find the question obvious: it is is "trivially" true for x smaller or equal to 2, because those conditions are never met.

So, accepting the "obvious" <For all number x, if x is bigger than 2 then x is bigger than 1>
really means you take as obvious all the (infinity many) propositions:

if 0 is bigger than 2 then 0 is bigger than 1, (line 1)
if 1 is bigger than 2 then 1 is bigger than 1, (line 2)
if 2 is bigger than 2 then 2 is bigger than 1, (line 3)
if 3 is bigger than 2 then 3 is bigger than 1, (line 4)
if 4 is bigger than 2 then 4 is bigger than 1, (line 5)
if 5 is bigger than 2 then 5 is bigger than 1, (line 6)
if 6 is bigger than 2 then 6 is bigger than 1, (line 7)
if 7 is bigger than 2 then 7 is bigger than 1, (line 8)
...
if 79876574327891 is bigger than 2 then 79876574327891 is bigger than 1, (line 79876574327892)
...
etc.

and the first lines confirm the whole truth table of "p -> q".

So if you ask me why I take the truth value of "p->q" as always being true when p is false, the answer is that I want to make "obvious" statements like the question I asked.

Tell me if you see the point. If you see the point I will be able to explain why Bf (and Bt aswell) needs to be true in the cul-de-sac worlds. Probably you could try to figure it out by yourself.

Bruno




http://iridia.ulb.ac.be/~marchal/

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