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proof of functional completeness
editThe supposed proof is only a `proof' that such a single operator system is not necessarily incomplete, that is, it fails to satisfy a sufficient condition for incompleteness. In short the argument establishes nothing of interest (towards the supposed aim of demonstrating functional completeness). Pushing the negation in: "using only nor *might* be functionally complete" is what the mini-section actually makes an argument for.
The first principles method for a corrected argument would be an explicit construction of all 16 binary boolean functions as expressions using only "nor".
Of course it would suffice to demonstrate just the reduction of `and', `or', and any form of `not' (the obvious one being unary negation, but it suffices to to consider either of the binary forms that ignore either the first or second argument); any reader would already recognize, indeed, prefer, this definition of what makes a Boolean algebra complete.
Formally this looks like, perhaps going the extra mile to define false (i.e., as (false p) := (nor p (not p)), so that one truly needs just one logically constant symbol "nor"), in prefix notation:
(not p) := (nor p p), (or p q) := (not (nor p q)), and finally (and p q) := (nor (not p) (not q)).
With strict formality the parentheses can be everywhere dropped, and so for example "(and p q)" expands to:
nor nor p p nor q q.
A fully detailed proof then finishes by actually backing up the implicit claims of the definitions, say: "(true-not (nor p q))" is, by the definition of "nor", equivalent to "(true-or p q)" ergo "(true-not (not p))" expands to "(true-or p p)" and hence "(true-not (not p)) = (true-identity p)". It is an elementary property that true-not is its only inverse, therefore both "not = true-not" and "or=true-or" are shown. Finally "and=true-and" follows by applying De Morgan's law.
So far as the article is concerned though merely the definitions should suffice, or at least, such would go just as far as the current mini-section goes, but, in the `right' direction.
--
However the present section actually serves to point to far more advanced topics on formal results pertaining to abstract boolean algebras; a truly correct edit must figure out the right way to phrase why the observation+reference is interesting rather than merely replacing the content with a correct argument towards the wrong goal. — Preceding unsigned comment added by 72.223.117.66 (talk) 16:43, 7 March 2012 (UTC)
emphasis
editShouldn't the articles for Logical NAND and Logical NOR both be treated equally? Now the NOR article emphasizes the actual operation while the NAND article emphasizes the denotational symbol (Sheffer stroke).
Proposed project
editI have been working on all of the logical operators recently. I would like to see a consistent format for them. There is a wikiproject proposal for this at: Wikipedia:WikiProject_Council/Proposals#Logical_Operators. Also see Talk:Logical connective.
I would like to see the logical, grammatical, mathematical, and computer science applications of all of the operators on the single page for each of those concepts.
Ampheck
edit"Ampheck" is listed under "see also" at the bottom of this "Logical NOR" page, but that link merely redirects back to the "Logical NOR" page. Perhaps some mention of the term "Ampheck" should be included in the text. I'd add it myself except I don't know what it means (which is why I clicked on it).
I agree this is confusing. --68.0.124.33 (talk) 07:40, 19 February 2008 (UTC)
- "Ampheck" once had its own article, started by a somewhat problematic editor. Apparently this is a term coined by Charles Peirce that can refer to NAND as well as to NOR. I don't think we have to cover all of Peirce's idiosyncracies here, so I just removed the word from the "see also" section. See also Jon Awbrey's ampheck article on PlanetMath. --Hans Adler (talk) 18:06, 19 February 2008 (UTC)
- Other articles link to "Ampheck", which redirects here. In that case one would expect to see at the very least a short history of the original term whose definition is sought. --Nekiko (talk) 22:29, 20 September 2012 (UTC)
Er, what?
editWhy on Earth does a search for "pierce arrow" redirect to this page? -- 64.223.230.30 01:39, 25 May 2008 (UTC)
- I suppose you mean Peirce arrow. The redirect seems to be due to an obsolete(?) notation of NOR as an arrow pointing downwards. This should probably be added to the article, as a minor detail of notation. --Hans Adler (talk) 11:40, 25 May 2008 (UTC)
No, I mean exactly what I said. The Pierce Arrow was an automobile from the early 20th century (later, with a hyphen added, it became the name of the marque), and a search for "pierce arrow", spelled with the I before the E but without the hyphen, redirects to the Logical NOR page instead, which is... mildly baffling. -- 64.223.230.30 16:00, 29 May 2008 (UTC) —Preceding unsigned comment added by 64.223.230.30 (talk)
- Someone fixed the problem. --Hans Adler (talk) 22:51, 23 November 2008 (UTC)
2 questions about this article
edit1) why does logical NAND redirect to Sheffer stroke, while logical NOR doesn't redirect (and is itself redirect from Pierce Arrow?)
2)what the hell means "worldwide view" in this article? you can't talk in "worldwide view" about logical operation ... I mean, it's the same in Canada and Tanzania..
--89.24.85.79 (talk) 19:25, 23 November 2008 (UTC)
- 1) I don't know the answer, but it may have to do with a very idiosyncratic editor who had to leave the project. He was a great fan of Pierce. You could be bold and try to solve this problem.
- 2) An anonymous user from Slovenia added this tag with the edit remark "Jan Łukasiewicz should be equally credited" [1]. Unfortunately he didn't bother to tell us what Łukasiewicz has to do with this trivial article, and I am not sufficiently interested in it to do something.
- In my opinion, all this mini-articles should be merged anyway. --Hans Adler (talk) 22:20, 23 November 2008 (UTC)
- I think he was confusing this with Peirce's law; see [2]. Tijfo098 (talk) 19:06, 12 April 2011 (UTC)
NOR and Material Conditional
editI noticed that the material conditional is written to be equivalent to NOR as:
p → q ≡ ((p NOR q) NOR q) NOR ((p NOR q) NOR q)
However, when I tried to verify this with a truth table, I determined that the truth value of this equivalence does not match that of the material conditional, but a switching of the last p with the q in the both the left and right terms fixes this:
| p | q | (p NOR q) | (p NOR q) NOR p | ((p NOR q) NOR p) NOR ((p NOR q) NOR p) | p → q |
|---|---|---|---|---|---|
| T | T | F | F | T | T |
| T | F | F | F | T | F |
| F | T | F | T | F | T |
| F | F | T | F | T | T |
| p | q | (p NOR p) | (p NOR p) NOR q | ((p NOR p) NOR q) NOR ((p NOR p) NOR q) | p → q |
|---|---|---|---|---|---|
| T | T | F | F | T | T |
| T | F | F | T | F | F |
| F | T | T | F | T | T |
| F | F | T | F | T | T |
Have I made a mistake, or is this article's equivalence incorrect? If I am wrong, where did I err in my table? --BBUCommander (talk) 02:34, 8 July 2009 (UTC)
Here is a small proof I made that my proposed alternative equivalence is correct. It makes use of the equivalences ~~p ≡ p, ~p ≡ (p ↓ p), and (p ∧ q) ≡ ((p ↓ p) ↓ (q ↓ q))
p → q ≡ ~(p ∧ ~q) ≡ ~((p ↓ p) ↓ (~q ↓ ~q)) ≡
≡ ~((p ↓ p) ↓ (~~q)) ≡ ~((p ↓ p) ↓ q) ≡ ~(~p ↓ q) ≡
≡ (~p ↓ q) ↓ (~p ↓ q) ≡ ((p ↓ p) ↓ q) ↓ ((p ↓ p) ↓ q)
--BBUCommander (talk) 16:31, 8 July 2009 (UTC)
- I agree with your table as well as with your proof. Good catch. Remember kids: don't blindly trust Wikipedia when you write your cheat sheets. Adrianwn (talk) 19:52, 1 August 2009 (UTC)