Complexity Theory’s 50-Year Journey to the Limits of Knowledge
"How hard is it to prove that problems are hard to solve? Meta-complexity theorists have been asking questions like this for decades. A string of recent results has started to deliver answers.
"Logic" is used to carry many different, but associated, meanings. Its formal meaning usually refers to the discipline that allows us to examine an argument, and conclude that its structure is such that it can deliver a deduced and reliable conclusion when provided with sound premises.
In everyday usage, we often say "logic" when we are actually referring to Reason. I blame Mr Spock for that... 😐
Reminder: logic and reasoning is not the determination of the truth. It is about defining the relationships between the truthfulness of separate statements. That is all.
"Reminder: logic and reasoning is not the determination of the truth. It is about defining the relationships between the truthfulness of separate statements. That is all."
Dear Dr. Freemo: I would argue that people who have attained adequate and sufficient competent seasoned logic and reasoning skills would be able to reasonably employ or utilize such mental tools to be able to attempt to define the relationship between the truthfulness of separate statements via learned psychological mental disciplined meticulous examinations of such statements in question. Defining the relationships between the truthfulness of separate statements must be an evidenced based probative inquiry process.
I would further argue that a trained mind would be more adept at unearthing the truth. There must be reliable accurate authoritative intersections of correlated reference points in which to rely upon in order to definitively factually determine the relationships between the truthfulness of separate statements in question.
Toshiyasu Arai just put a preprint with a pretty big result in arxiv: "In this paper we give an ordinal analysis of a set theory with (\Pi_N)-Collection."
If right, he found the proof-theoretical ordinal of second-order arithmetic.
More than argument, #logic is the very structure of reality
“Philosophers have sometimes fallen into that trap, thinking that logic had nothing left to discover. But it is now known that logic can never complete its task. Whatever problems logicians solve, there will always be new problems for them to tackle, which cannot be reduced to the problems already solved.” #philosophy#knowledge#scientificmethod
I've wanted to be able to extract the Notes field (usually where lyrics, lead sheets, chords, etc. are) from a Logic Pro session file WITHOUT opening it for a looooong time.
Thanks to a helpful article, I was able to kludge a shell script together that finally does it!
#AmReading this 1854 book by George #Boole , which summarises his thoughts (first published a few years earlier) on #logic , as well as probability. Boole built the world I live in as a logician (and to extent, the world we all live in in the age of computers) but this is the first time I've read him in the original, so I thought I might make a thread with a few notes in it as I read it over the next few weeks.
Before I get to the book itself, note that while Boole was revolutionary, the revolution didn't catch on quickly; this precious family heirloom, a logic textbook signed by my great-great grandfather in 1885, mostly focuses on logic as Aristotle would have understood it. It does, in fact, cover Boole, but mostly to complain that his work is obscure and unnecessarily mathematical! #logic
#Boole 's propositions do not range merely across 0 and 1, as often presented today, but across subsets of all objects in the universe (or some agreed upon universe of discourse). If this sounds like Boolean Algebra, you're half right; conjunction is indeed intersection, but disjunction (which he writes +) is disjoint union, so x+y is not meaningfully defined in general, as with x/y in arithmetic (as y might be 0). This strikes me as something which might cause trouble later. #logic
#Boole develops #logic by close analogy with arithmetic, though he is at pains to say this is mere analogy and there is no a priori reason the rules should be the same. So while we usually think of logic as being about entailment, Boole virtually ignores it in the early going and makes equality primary; see the attached proof of the principle of contradiction (here 1 stands for the whole universe, and x - y, defined only if y is a subset of x, is set difference), with its arithmetical flavour.
This footnote is an example of the arithmetical approach to #logic making life terrible for #Boole ; given that x² (i.e. x and x) = x is an axiom, shouldn't x³ = x hold? Apparently not, as x³ - x = 0 'factorises' into gibberish terms like 1 + x (we can't add new things to the universe), or -1, which has no meaning at all (not to be confused with the negation of 1, which is 1 - 1 = 0). I must admit to my doubts about the well-definedness of this whole enterprise!
#Apple, you screwed the pooch in this instance not just slightly, but very, very damn badly indeed. What kind of quality control systems aren't in-place to allow a build of #Logic (specifically Logic 10.8) to go out of the door where something as simple as recording MIDI notes duplicates every single one of them? A new shiny release and this is how you fail?
Glad I don't have clients in today or I'd be losing business because of you. Livid does not even begin to cover how I feel today. Not a damn bit of it.
TLDR; there are least 4 problems with the claim that "Argument mapping is about twice as effective at improving student critical thinking as other methods".