We use the logical system that we know from observations (empirical data) holds true in the world we live in (please correct me if I am wrong). Hence the axioms of logic we choose are themselves dependent on our observations. Does this mean that logic is also limited to observations, and is neither the absolute or eternal truth nor fundamental?
I have currently learned that we have various types of logic. The two-valued logic teaches us for example:
1.The pot is red (A)
2.The pot is not red. (~A)
These are the only two cases possible in classical logic. But the logic used in the East before colonization was the multi-valued. In Buddhist Tradition Dīgha Nikāya provides an example. As the Buddha explains in the Brahmajāla Sutta, there are four alternatives:
(1) The world is finite, this is one case.
(2) The world is not finite, this is another case.
(3) The world is both finite and infinite, this is the third case.
(4) The world is neither finite nor infinite, this is the fourth case.
(5) There are no other cases.
The quantum logic has already shown that
(p and a) or (p and b) is not equal to
p and (a or b). The distributive law fails in quantum logic. Now if you say that we have to pick a suitable logical system for the particular area we are working in, then how can mathematics be the same everywhere, it will also be empirical then.
In quantum logic an electron can be both at Position A and B at the same time. Classical Logic does not permit it. When we prove something by contradiction we take advantage of the above condition. What I mean to say is we prove that if root of 2 is not rational then it can be irrational, or if root 2 is irrational it cannot be rational. But in quantum logic such proofs will fall flat.
Please See This Question : Is Logic Subjective
What I am not able to understand is: If logic can vary, how can Mathematics be universal?
Why Do Not We Allow Empirical Proof In Mathematics which gradually become more precise with each observation (The way it is in physics) if both the Logical System and the Axioms are themselves are dependent on our observation, they are based on our empirical observation?
EDIT : How the Distributive Law Fails ? (source)
p and (q or r) = (p and q) or (p and r), where the symbols
rare propositional variables.
To illustrate why the distributive law fails, consider a particle moving on a line and let
p= "the particle has momentum in the interval [0, +1/6]"
q= "the particle is in the interval [−1, 1]"
r= "the particle is in the interval [1, 3]"
(using some system of units where the reduced Planck's constant is 1) then we might observe that:
p and (q or r) = truein other words, that the particle's momentum is between
0 and +1/6, and its position is between
−1 and +3. On the other hand, the propositions
"p and q"and
"p and r"are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So,
(p and q) or (p and r) = false. Thus, the distributive law fails.