This page is for developing a "simpler" description of Quantification; see Talk:Quantification

(0): At the deepest level of all, the most sophisticated and least understood, is the notion of your personal experience of a "sensory event". For example, you see a shape, and your brain (the physical) processes the input from your eyes, and your mind (what exactly is a "mind"?) experiences " This dog is blue!! ". Some philosophers argue that you are experiencing two "quales" -- the one having to do with the sensory image of "dog" and the other having to do with the sensory image of "blue-ness". Some arugue that having/experiencing/being! "quales" are what it means when someone asks you "What's it like to be you?" (I hurt when I stub my toe). Others argue that you're an automaton and there is no "you", there is "nothing that it is like to be you" (i.e. that which is "the automaton-which-is-me" expresses the learned/inherited/programmed/built-in-at-birth belief that I hurt when I stub my toe.) To proceed, we probably need to accept the first belief, i.e. that you experience/are quales: "there is something that it is like to be you."

(i) The notion, in language, of the "object of immediate sensation" and of the "primitive utterance" that describes the object as perceived by direct physical action on sensory nerves, transmitted to the brain then processed there (seen, felt, heard, tasted, smelled) or perceived indirectly in the mind itself (dreamed, thought). As noted in (0) here's some very serious (contemporary) philosophic issues around this notion of "object of immediate sensation". As a starting point: For an eye-opening discussion in very elegant simple prose see Bertrand Russell 1912/1997 The problems of philosophy, Oxford University Press, NY, ISBN 0-19-511552-N; Russell wrote this during his work on his and Whitehead's Principia Mathematica. You can buy this book at the e.g. Borders, for not very much money ($10.95).

Thus we point at an object directly before us and say, " This dog is blue! ", " That symbol is an X ".

Clearly, to say that the dog is blue requires the notion of "dog" and "blue", more below about this.

(ii) The notion of talking about a "collection", a bunch of, these sensations; to begin with let's work with simple collections that are restricted in the number of "elements" (sense-objects) as in " This collection of symbols ": { ✖, ■, ✸, ■, ✖,✸,✸, ■, ■ } . Or, the collection may come to our senses piecemeal, one after another, spread over time,

(iii) The notion of matching sense-objects (distinguishing the elements of the collection however the collection is obtained and comparing them) to "similarity-templates" with "qualities" quales, i.e. "categorizing", i.e. determining that there are some objects with quality ✖, some with quality ■, some with quality ✸ and teasing these apart, from one another (see iv). Example: The fact that the four-legged thing I know from prior learning has a quality of "doggy-ness" about it, so my mind asserts: " a dog! ", and moreover it seems to have the quality "blueness" i.e. my mind asserts " blue! " that I know from prior experience, as well.

(iv) The notion of dividing the collection of quales (immediate sensations) into equivalence classes i.e. "categories" that are "mutually exclusive" ("disjoint", not-joined, thus " cat " and " dog " are two mutually-exclusive quales associated with 4-legged things) and "exhaustive" (meaning every single immediate quale, however derived, gets thrown into its one and only one quale-category). By doing this we generalize the qualities so that we can speak about them as "concepts" e.g. a symbol having (a "shape", i.e. the quality of "✖-ness") opposed to symbols having quality ■ or quality ✸. Qualities: { doggy-ness, blueness }. Thus a " blue truck " and a " blue cat " share a similar quale (blue-ness).

(v) the notion of generalizing such quales/sensations over the entire universe of such equivalence classes as we know them, i.e. asserting "universality", as in " All dogs (in this entire physical universe) are blue ", all ✖ are Xs ", "all ■ are solid-squares ", "all ✸ are stars ".

(vi) the notion of asserting "existence" of a sensation in a restricted collection, as " At this dog show, there's a blue dog, " or as a logician might say: " At this dog show there exists a (at least one) blue dog ",

(vii) the notion of asserting "existence" without restriction, as " (In this big wide universe as we know it) There exists at least one blue dog" , and finally

(viii) the notion of " denial " i.e. " not ", i.e. "asserting the contrary" as applied to all of the (i) through (vii), for example " This dog is NOT blue" ( " This dog is NOT-blue ", " It is not the case that "this dog is blue " ).

(ix) the notion of "symbol": ∀ and ∃ are distinct (their quales can be put into mutually exclusive categories). Moreover these are distinct from other symbols such as those being used in this line of type, and the sensory-image "blue" and "dog" and etc etc. Thus the sensory images of these symbols are quales, too.

(x) The notion "abbreviation-by-symbol": The two fundamental notions are -- "universality" (generalization) and "existence". These are symbolized (by contemprorary philosophers and logicians) by the easy-to-remember upside-down A, as in "All" i.e. ∀, and the backwards ∃ for "Exists".

(xi) When a contrary or contradiction (i.e. NOT-blue) is applied to an expression when the universal is asserted (spoken as true) e.g. ∀ or ∃, the rules of the game get a bit complicated. This relationship is shown by the diagram immediately above. Clearly the drawing needs great explication (the author Reichenbach required 3-4 pages).

Thus we might say that " It's not true that ' All dogs are blue ' ", or we might say that " Blue dogs do exist: moreover " It's true that ' At least one blue dog exists ' " " (the quote-marks here are not trivial and have to be respected). If done properly these two statements are equivalent (they're saying the same thing).

(xiii) From here we get into the notion of generalization via the notion of function in the mathematical sense. As in " This x is blue" (a simple propositional function where either x is restricted to a "domain" or "universe of discourse" of objects worthy of discussion) and a more complicated " This x is a y ". ∀x:(x is blue) i.e. "for all things x, x is blue ", ∃x: "At least one of these x is blue ". The object x is taken from the so-called domain of discourse (the "set" of equivalence-class concepts) e.g. { trucks, skies, dogs, cats, fruitflies, pumpkins }. Note that a substitution of an "object" into the "function" may result in a so-called "truth value" of "falsity", as in "There exists at least one blue fruitfly."

and here I'm going to stop because this is getting too long and too complicated. Plus "It's raining" (notice the indefiniteness of this expression). Bill Wvbailey (talk) 23:57, 11 April 2011 (UTC)

User:Wvbailey/Quantification

Contents

cc'd over from Talk:Quantification