Does anyone here know the state of the idea that Probability is the foundation for Logic ? That is if A =>B then we know that ~B => ~A the contrapositive. But we only know this iff P [A=>B] = 1 Has Mathematics accepted Probability as the foundation for Logic ? If so, then is it not so that there is no dichotomy between QM and Classical Physics thru GR. Classical physics is just that body of knowledge where we think P = 1
Probability is based on logic, not the other way around. The laws of probability follow from logic and simple definitions, like |P| = P and Sum(P_i) = 1. From those definitions, you can then use logic to reason about probability, including probabilistic logic. P_AB + P_A!B + P_!AB + P_!A!B = 1 (definition of P in the A,B basis) P_[A=>B] = P_AB + P_!AB + P_!A!B = 1 - P_A!B (definition of A=>B) P_[!B=>!A] = P_AB + P_!AB + P_!A!B = 1 - P_A!B (defintion of ~B=>~A) So P_[A=>B] = P_[!B=>!A] = 1 - P_A!B The statement that "the statement that 'B is true whenever A is true' is X% likely to be true" is identical to "the statement that 'A is false whenever B is false' is X% likely to be true". Of course this gets complicated when A and B are statements over sets, and very complicated when A and B are statements over infinite sets. But QM is not just probability writ large. Instead, the definition is simply: Sum( |Q_i|^2 ) = 1 Or as Scott Aaronson claims "Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call 'probabilities' can be negative numbers. As such, the theory could have been invented by mathematicians in the 19th century without any input from experiment. It wasn't, but it could have been." http://www.scottaaronson.com/democritus/lec9.html
Interesting Perhaps I should not have brought QM in to it. I am just curious if it is now accepted that the foundation for Logic is Probability See .......... http://en.wikipedia.org/wiki/Probability_logic http://en.wikipedia.org/wiki/Bayesian_probability http://en.wikipedia.org/wiki/Conditional_probability Recall that P [ A | B ] is equivalent to P [ B => A ] This is what links Conditional Probability to Propositional Logic Thus Logic is merely the subset of conditions when P = 1
Not so. In the finite universe where labels A and B can only have two values, P(A|B) = (P_AB)/(P_AB + P_!AB), and is undefined when P_AB + P_!AB = 0. P(A|B) is read as "the probability of A being true, given that B is true," and "given B is true" restricts our universe being considered to a smaller one that originally, and so you have to rescale the probabilities, so that Sum(P) = 1 in the new universe. A and B in all these posts are sentences with truth values. So one universe shrinks from 4 to 2 possibilities. But, since B=>A is a truth function of the sentence "A is true whenever B is true", then P(B=>A) = 1 - P_!AB because only when A is false and B is true does that sentence evaluate as false. None of your sources use the notation P(B => A) to mean P(A|B) since implication (part of logic) is the basis of probability and therefore probability will not redefine terms used by its predecessor.
Probaability => Logic if P=1 I don't follow you here. Are you disputing that P [B=>A] is equivalent to P [A | B] ? If so, simply go thru the language; B implies A : is the same as : A is true, given that B is true And what problem is there with defining Logic as the subset case for those situations where P=1 That is if P[ B ] = 1 then P[B=>A] becomes B=>A. The Probability Statement reduces to pure Logic
"B => A" = "B implies A" = "A is true whenever B is true", etc. is a truth-valued function. It can be expressed a variety of equivalent ways. You have not provided any reference which supported your viewpoint of very old definitions. Code: A | !A | B | !B | B => A | A or !B | not( !A and B ) --+----+---+----+--------+---------+---------------- T | F | T | F | T | T | T T | F | F | T | T | T | T F | T | T | F | F | F | F F | T | F | T | T | T | T Here's just some references for implication being a truth-function: http://us.metamath.org/mpegif/df-or.html http://us.metamath.org/mpegif/df-an.html http://en.wikipedia.org/wiki/Material_conditional http://en.wikipedia.org/wiki/Logical_implication http://planetmath.org/encyclopedia/VacuouslyTrue.html http://www.informatik.htw-dresden.de/~nestleri/logic/02/index.html
I have no issue with Logic linked to a Truth Table. But I am not clear on what you do object to in connection with my original question ... which is ............. Has Mathematics accepted Probability as the foundation for Logic ? Also I am curious on what grounds you object to my previous statements ..... 1/ Are you disputing that P [B=>A] is equivalent to P [A | B] ? If so, simply go thru the language; B implies A : is the same as : A is true, given that B is true 2/ And what problem is there with defining Logic as the subset case for those situations where P=1 That is if P[ B ] = 1 then P[B=>A] becomes B=>A. The Probability Statement reduces to pure Logic And I did provide references above http://en.wikipedia.org/wiki/Probability_logic http://en.wikipedia.org/wiki/Bayesian_probability http://en.wikipedia.org/wiki/Conditional_probability This area is very interesting to me. Fascinating and important applications of this are http://en.wikipedia.org/wiki/Prosecutors_fallacy http://en.wikipedia.org/wiki/Meadows_law http://en.wikipedia.org/wiki/False_positive_paradox Again, anyone ?? ......... Has the Mathematics community accepted Probability as the foundation for Logic ?
