Logic of Given

[Posina Venkata Rayudu]

The science of logic in the ancient philosophical sense means the study of the general laws of the development of thinking.

Thinking

(1) reflects reality (i.e., has a content)

but also

(2) is itself part of reality and so has some motions that are oblivious to content.

Therefore the science of logic finds two aspects of thought’s motion:

(1) the struggle to form a conceptual image of reality that is ever more refined, whose laws we may call objective logic,

and

(2) the motion of thought in itself (for example the inference of statements from statements), whose laws we may call subjective logic.

The long chains of correct reasonings and calculations of which subjective logic is justly proud are only possible within a precisely defined universe of discourse. Since there are many such universes of discourse, thinking necessarily involves many transformations between universes of discourse as well as transformations of one universe of discourse into another. The results of applying logic in the narrow sense (i.e. the inference of statements from statements by means dependent on their form rather than on their content) to the laws of these objective transformations are necessarily inadequate; for example, such attempts have led to phrases such as "let X be a set in which there exists a group structure," which are essentially meaningless. Rather than using "there exists" in such contexts, one needs instead a logic of "given."

http://conceptualmathematics.wordpres...

Sets for Mathematics