Arising_uk wrote: ↑Thu Oct 18, 2018 11:59 am
I guess what I'm trying to understand is what's the difference between a 'procedure", a set of logical axioms and an algorithm? As I'm pretty sure I could recreate the procedure for finding primes in Prolog and that would be a set of logical axioms?
The distinction is technical rather than semantic, so let me switch gears. An axiom is something that cannot be doubted. You accept it or you reject it - for reasons that are your own. 1 + 1 = 2. Axiom. You could just as well accept an axiom which says 1+1 = 3. What you are accepting isn't the axiom as such, but the RULES which come married to the axiom. The RULES, which WHEN violated indicate that a contradiction has occurred.
And so if you do some calculation and you get a result that says 1+1 = 4, but the AXIOM says 1+1 = 3 then that is against the rules of the system.
That is all that a contradiction means: rule violation.
And the question of "what AXIOMATIC RULES should I accept?" is rather difficult to answer without appealing to a false authority
Lets take the black box as a fundamental building block or all systems we speak about:
https://en.wikipedia.org/wiki/Black_box
The generic model is: input->MAGIC->output
Even logic itself is a system so logic can be modeled as a black box.
You can think of logic as:
input->MAGIC->(True,False)
Or you can write a system which determines if a number is odd or even:
Input: N
Magic:
if reminder of (N/2) = 0 then output(even)
else output(odd)
Output: (even, odd)
or if the number is prime:
Input: N
Magic:
(some algorithm for testing prime numbers here)
Output: Prime, Not-prime
The point is that every one of those algorithms produces a BOOLEAN response.
True/False
Even/odd
Prime/Not-prime
And the statistical name for any magical box that classifies N inputs into 2 categories is a Binary classifier:
https://en.wikipedia.org/wiki/Binary_classification
You can have a black box with 1 in put and 1 output (tautology)
You can have a black box with 1 in put and 2 outputs (ambiguity)
You can have a black box with M in puts and N inputs
The point is that the black box maps the inputs to a set of outputs.
Which is why it is commonly known as a transfer function:
https://en.wikipedia.org/wiki/Transfer_function