04:34
@PrithuBiswas I believe the author wanted the reader to wave with both hands vigorously rather than write with one hand rigorously.
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And this is not even counting the numerous issues with the cited text, such as the failure to recognize that the stated recursion theorem is not directly applicable to defining summation on a function whose domain is not all of ℕ⁺, and the failure to realize that the logical basis for being able to drop brackets in a chained sum cannot be glossed over without making the use of the recursion theorem pointless.
I mean, just imagine.. the justification for dropping brackets may go something like this: Using associativity, we can iteratively drop the outermost bracket, until there are none left, where lack of bracket is assumed to mean left-associativity. Sure, but as you definitely can tell, "iteratively" and "drop" and "outermost" are not just hard to define rigorously but in fact harder to define rigorously than defining summation via recursion!
So if one wants to go to the extent of using the recursion theorem to justify the recursive definition of summation, one definitely ought to do all the other things equally rigorously, including what the cited text failed to do.
@NavBhatthal Yup. I had made a rudimentary version in high-school, then in my first year of undergraduate my professor taught me a Fitch-style system, so I realized that the style is not new. However, there are hundreds of possible design choices, and my system is purposely designed to be practical, in contrast with essentially all other systems.
@NavBhatthal In English, "If A then B" and "B if A" and "B given A" usually mean the same thing, as long as A,B are both boolean statements. soupless has given a great simple example with both "if" and "given" in my system, where these words have only exactly the meaning that the system wants you to think of them. It's always best to try to avoid using the same word for too many things; this is what makes natural language ambiguous in the first place.
@soupless "Given x∈S:" creates a subcontext in which we have been given an object of a specified type, and in which we can reason about that object. Yes you are right that your sample doesn't actually reason about that object, and I know that you know that it doesn't matter; "we can reason" not "we must reason".
@EE18 I am unfamiliar with Ebbinghaus but I know at least one person who studied it but didn't learn much from it.
It's best not to read any textbooks on logic before learning how to use logic. I emphasize this only because students always do not realize that using logic is completely different from studying logic.
I believe Prithu bolded "learn" for the same reason.
