Comments on: The Law of Excluded Middle http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/ Mathematics for computers Wed, 20 Aug 2008 12:18:55 +0000 http://wordpress.org/?v=2.3.3 By: pa97001569 http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-5739 pa97001569 Mon, 16 Jul 2007 06:56:47 +0000 http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-5739 Hi Andrej. I once told you that my brain can handle only classical logic (to which you of course disagreed). But now I have come across a real-life phenomenon that I simply cannot grasp; so maybe you can make non-classical sense of it. In the old days I often used to go cycling but now, with family and kids, find it difficult to get around to it. Specificially each time I want to go, my wife says: you can cycle whenever you like, just not now. I tried to formalize this as the statement forall t: forall s: ( st P(s) ) on the predicate P(t) := "I can go cycling at instant t" and found it (at least classically) paradoxical. So can NON-classical logic help understanding women ? (Notice that, if so, this would make an ultimate selling point!) Hi Andrej.

I once told you that my brain can handle only classical
logic (to which you of course disagreed). But now I have
come across a real-life phenomenon that I simply cannot
grasp; so maybe you can make non-classical sense of it.

In the old days I often used to go cycling but now, with
family and kids, find it difficult to get around to it.
Specificially each time I want to go, my wife says:
you can cycle whenever you like, just not now.

I tried to formalize this as the statement
forall t: forall s: ( st P(s) )
on the predicate P(t) := “I can go cycling at instant t”
and found it (at least classically) paradoxical.

So can NON-classical logic help understanding women ?
(Notice that, if so, this would make an ultimate selling point!)

]]> By: roconnor http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-27 roconnor Sun, 12 Jun 2005 12:02:24 +0000 http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-27 Nice explaination. When I explain my construtivism I say that I study what is provable, while classical mathematicians study what is true. Thus P ∨ ¬P states that either P is provable or ¬P is provable, and every knows that isn’t provable (it isn’t even true). My explaination is not incompatable with yours. Nice explaination.

When I explain my construtivism I say that I study what is provable, while classical mathematicians study what is true. Thus P ∨ ¬P states that either P is provable or ¬P is provable, and every knows that isn’t provable (it isn’t even true).

My explaination is not incompatable with yours.

]]> By: PeterMcB http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-19 PeterMcB Fri, 20 May 2005 09:05:05 +0000 http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-19 In response to Thad, a great deal of attention in AI over the last 30 years has been on logics for defeasible reasoning -- ie, logics in which one can reach conclusions which can be subsequently defeated in the light of new information, or on the basis of further reasoning, or due to some change in the environment. This work has led AI researchers to look at argumentation, the theories of argument and debate first developed in philosophy. In this area, a key figure has been Aristotle, so it is nice to see his ideas finally being applied in contemporary AI. In response to Thad, a great deal of attention in AI over the last 30 years has been on logics for defeasible reasoning — ie, logics in which one can reach conclusions which can be subsequently defeated in the light of new information, or on the basis of further reasoning, or due to some change in the environment. This work has led AI researchers to look at argumentation, the theories of argument and debate first developed in philosophy. In this area, a key figure has been Aristotle, so it is nice to see his ideas finally being applied in contemporary AI.

]]> By: Thad Coons http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-13 Thad Coons Mon, 16 May 2005 21:50:01 +0000 http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-13 It appears that mathematicians as well as philosophers are perpetually tip-toeing around the excluded middle, trying to weaken it without rejecting it entirely. Perhaps a logic that can deal directly and consistently with equivocation would clarify matters. It appears that mathematicians as well as philosophers are perpetually tip-toeing around the excluded middle, trying to weaken it without rejecting it entirely. Perhaps a logic that can deal directly and consistently with equivocation would clarify matters.

]]> By: PeterMcB http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-12 PeterMcB Mon, 16 May 2005 19:36:45 +0000 http://math.andrej.com/2005/05/13/the-law-of-excluded-middle/#comment-12 Thanks, Andrej. Call your final question Q1: Q1: "If a program does not run forever, does it terminate?" Certainly, if I potentially had money at stake in some online marketplace, I would want an affirmative answer to a second question: Q2(X): "If a program does not run forever, does it terminate within time X?" (eg, X = my expected lifetime). If all you can tell me is that the answer to question Q1 is YES, but you cannot give me an answer to Q2(X), then I would not participate in your marketplace. An answer of YES to Q1 is not necessarily an answer of YES to Q2(X). And, for any given X, a non-constructive proof-by-contradiction of Q1=YES will provide no information on the answer of Q(X). (I believe this; it should be possible to prove it, I imagine.) A constructive proof of Q1=YES, on the other hand, may well provide information on the answer to Q2(X), at least for some values of X. From this I conclude that all venture capitalists should be constructivists. :-) Thanks, Andrej.

Call your final question Q1:

Q1: “If a program does not run forever, does it terminate?”

Certainly, if I potentially had money at stake in some online marketplace, I would want an affirmative answer to a second question:

Q2(X): “If a program does not run forever, does it terminate within time X?” (eg, X = my expected lifetime).

If all you can tell me is that the answer to question Q1 is YES, but you cannot give me an answer to Q2(X), then I would not participate in your marketplace. An answer of YES to Q1 is not necessarily an answer of YES to Q2(X). And, for any given X, a non-constructive proof-by-contradiction of Q1=YES will provide no information on the answer of
Q(X). (I believe this; it should be possible to prove it, I imagine.)

A constructive proof of Q1=YES, on the other hand, may well provide information on the answer to Q2(X), at least for some values of X.

From this I conclude that all venture capitalists should be constructivists. :-)

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