User:Lord Farin/Sandbox

This page exists for me to be able to test features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Subpages of this one may exist; they are listed at this PW special page.

= Restructuring of the logic department =

This is in progress. The Definitions/Formal Systems category has been reworked for the directly relevant part. (I've let Bourbaki and some pages on Definition:String resp. Definition:Word (Formal Systems) rest for now.)

I will have to do a lot of reading in the category and its subcategories to try and bring order in the fragmentation. &mdash; Lord_Farin (talk) 12:27, 8 September 2013 (UTC)

User:Lord_Farin/Sandbox/LogicCategories will contain an attempt to classify and categorise the pages. Hopefully, it will largely overlap with the existing system. &mdash; Lord_Farin (talk) 12:24, 9 September 2013 (UTC)

The below two radical proposals seem to clear major problems with categorisation and clarity of the site. &mdash; Lord_Farin (talk) 18:09, 9 September 2013 (UTC)

Radical Proposal #1
Ditch the Category:Mathematical Logic and Category:Definitions/Mathematical Logic categories. For, most, if not all, of the material currently there is one of:


 * 1) Actually more general, and applicable to either the Symbolic Logic or the Formal Systems categories;
 * 2) Part of computability/recursion theory;
 * 3) Applicable only to predicate logic/model theory

Now, although we have at least historical and sourced merit for pages like Definition:Mathematical Logic, the grand scheme of universal coverage that we strive for on will cause us to move material presented by authors in the context of mathematical logic according to one of the three rules (particularly the first and last). We thus do not need to cripple ourselves and artificially fragment the site by forcing this mould onto our system. Please post your thoughts. &mdash; Lord_Farin (talk) 15:03, 9 September 2013 (UTC)


 * I understand "mathematical logic" as being a more specific category than symbolic logic but more general than formal systems. It is the general framework into which formal systems goes, and requires as its starting-point a working model of number theory. It is the category in which Godel's Theorem lies. SO I would say: keep it, make Formal Systems a subcategory of it, and make it a subcategory of Logic (or Symbolic Logic, or both), but it definitely needs to be in there as there is a wealth of literature that refers to "mathematical logic".


 * It may turn out, on revisiting everything in that category in turn, that everything belongs in "formal systems" or "model theory" or whatever other categories we identify as a subcategory of "mathematical logic", but this will not invalidate the existence of the category itself. --prime mover (talk) 18:34, 9 September 2013 (UTC)


 * I read formal systems as being more or less the most basic category. I think it could be applied to defining programming languages, e.g. letting the theorems correspond to properly compiling programs.


 * But since the precise (or perhaps it's better to say "intuitive") definitions of, and interplay between, "mathematical logic", "symbolic logic", and "formal system" are not universal, it's at least clear that we need to do something to separate them (if appropriate). I'll get cracking on the other radical proposal first, then conduct a literature search. &mdash; Lord_Farin (talk) 21:41, 9 September 2013 (UTC)

Radical Proposal #2
Merge Category:Propositional Logic and Category:Propositional Calculus, similarly for Category:Predicate Logic and Category:Predicate Calculus.

I have not been able to find an explanation for any distinction between these terms. They also seem to be used interchangeably, both on and in sources. At least, their distinction is not clear to the reader, and I deem it not useful. We might want to rename Definition:Propositional Calculus to Definition:Language of Propositional Logic or something like that. Please also post your thoughts on this one. &mdash; Lord_Farin (talk) 18:09, 9 September 2013 (UTC)


 * I think I'm inclined to agree. "Language of Propositional Logic" is a propositional calculus, and the latter term can remain in, but, instead of a category, as an appropriately written definition page.


 * Bear in mind that some of this material dates from the very early days of and we hadn't got to grips with how versatile we could structure it, so we (or I did, anyway) were constrained by what we thought were limitations of the medium. So I'm happy for that to happen. --prime mover (talk) 18:34, 9 September 2013 (UTC)


 * Progress at User:Lord_Farin/Sandbox/PropLog. &mdash; Lord_Farin (talk) 22:11, 9 September 2013 (UTC)

Proposal #3
Ditch Category:Boolean Algebra and Category:Definitions/Boolean Algebra. We don't have a clear explanation of what "Boolean algebra" is, except for "the field of study of Boolean algebras" -- in which sense it isn't used.

It seems as a kind of intermediate mess between that general context and the PropLog context of the BA two: $\{0,1\}$. In particular, I propose the following renaming scheme:


 * Definition:Boolean Domain $\to$ Definition:Set of Truth Values
 * Definition:Boolean Function and Definition:Boolean Function/Truth Function $\to$ Definition:Truth Function
 * Definition:Basic Proposition $\to$ delete
 * Definition:Boolean Variable $\to$ delete
 * Definition:Logical Complement $\to$ the BA part can be deleted

Definition:Complementary Pair, Definition:Contradiction, Definition:Model (Logic)+subpages, Definition:Tautology, Definition:Truth Table, Definition:Sheffer Operator can be treated in one of the logic categories.

It may make sense to create a category Category:Truth Functions which can take the logical counterpart of things like Definition:Sheffer Operator and the theorems now in Category:Boolean Algebra. Thoughts? &mdash; Lord_Farin (talk) 14:27, 16 September 2013 (UTC)


 * While at it, is there anyone with a good alternative for $\mathbb B$? Not calling it "Boolean domain" anymore makes picking another letter sensible, perhaps $\mathbb T$ or even $\mathbb 2$, $\mathbf 2$, or another variant? &mdash; Lord_Farin (talk) 21:58, 16 September 2013 (UTC)


 * I think in general I'm in favour of this move, as long as we add "also known as" sections in whatever pages we doctor. As for alternatives to "$\Bbb B$, I quite like it. As for $2$ and its variants, despite its use in category theory and its clarity therein, I'm not sure about how it will go over with people who have not studied cat theory, set theory and ZFC in which $2 := \{0, 1\}$ - neat though it is to use it, it's just too esoteric. That leaves $\Bbb T$, which I have used betimes as a "set of sets whose subsets are $T_i$", which would need to be revisited. --prime mover (talk) 16:48, 17 September 2013 (UTC)