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)


 * Definitions are practically taken care of. Theorems remain. &mdash; Lord_Farin (talk) 22:46, 25 September 2013 (UTC)


 * All of the existing and directly related material has been prepared, ordered and modified as needed. In principle, the operation can commence. &mdash; Lord_Farin (talk) 12:56, 26 September 2013 (UTC)

Loose Ends
The follwing pages will have to be revisited once most of the rubble is out of the way:

Refactoring/rewriting

 * Definition:Model (Logic)
 * Definition:Statement Form
 * Models for Propositional Calculus
 * Model Defined by Implication
 * Definition:Substitution Instance
 * Definition:Propositional Calculus/Formal Grammar/Labeled Tree Specification
 * Definition:Main Connective
 * Method of Truth Tables and subpages
 * Definition:Tableau Extension Rules
 * Model of Branch of Propositional Tableau

Deletion

 * Category:Definitions/Propositional Calculus
 * Definition:Alphabet of Propositional Calculus
 * Definition:Connective of Propositional Calculus
 * Definition:WFF of Propositional Calculus
 * Definition:Propositional WFF
 * Definition:Sign of Propositional Calculus
 * Definition:BNF Specification of Propositional Calculus
 * Definition:Bottom-Up Specification of Propositional Calculus
 * Definition:Main Connective (Propositional Calculus)
 * Definition:Precedence (Propositional Calculus)

Archiving

 * Definition:Abbreviation of WFFs of PropCalc and subpages
 * Definition:Propositional Calculus and subpages
 * Definition:Binding Priority/Propositional Calculus
 * Definition:Classical Propositional Calculus
 * Definition:Main Connective/Propositional Calculus

In PropLog
Cover the non-standard approaches on subpages of Definition:Language of Propositional Logic, e.g. Ben-Ari's trees and Bourbaki's links and Polish notation (although the latter could justifiably be put under PredLog alone). &mdash; Lord_Farin (talk) 08:27, 11 September 2013 (UTC)


 * There's a case for making Polish notation categorically distinct from PropLog. Don't know where it does fit, but in my mind it transcends PropLog as it applies to all linguistic construction. Conceptually it fits into Formal Systems, and demands a category for itself in which it is proven a) that it is isomorphic to standard infix, and b) that you can express a statement in Polish without using brackets, both of which are important in that context. --prime mover (talk) 08:36, 11 September 2013 (UTC)


 * I meant "Bourbaki's (links and Polish notation)". Of course, Polish notation is itself more general than PropLog. The other suggestions for Polish notation are good ones. It will take time to get there, though. &mdash; Lord_Farin (talk) 08:44, 11 September 2013 (UTC)


 * A first example of a non-official approach is here: User:Lord_Farin/Sandbox/PropLog/Definition:Language of Propositional Logic/Keisler-Robbin. Thoughts and suggestions appreciated. &mdash; Lord_Farin (talk) 13:12, 11 September 2013 (UTC)


 * That's sort of how I envisaged it. In theory we can document as many approaches as there are published, but my desire to acquire more and more books on the same subject merely for the purpose of documenting them on pw is limited. --prime mover (talk) 17:18, 11 September 2013 (UTC)


 * I'm glad you like it. Now in due time, we will need to set up the following pages:


 * Definition:Translation Scheme for Language of Propositional Logic, in which back-and-forth techniques to transliterate the wffs of respective formal languages into "our" FL, and back.
 * Definitions for any type of semantics set out in a source using a non-official FL.
 * Equivalence proofs to "intertwine" the semantics in the non-official FL and the transliteration.
 * This will ensure complete rigour, but it's also quite... boring, to say the least. Alas, I don't see a way around it. It's kind of a basic version of what is known as "definitional equivalence" or "interdefinability" in proper model theory (where we can rely safely on a unique Prop/PredLog base): that two theories are strong enough to define each others' operations, relations, and constants.
 * Prolonging my thought, there may be a call for similar things in any proof-theoretical context. The nice thing is that this approach, once done, gives complete, formal (as opposed to intuitive) justification for using the PW formalisation of Prop/PredLog everywhere. Now it remains to be motivated for this massive undertaking... &mdash; Lord_Farin (talk) 19:11, 11 September 2013 (UTC)

At some point, the unique parsability of the various PropLog languages will need to be established. This will probably need some things like WFF of PropCalc is Balanced. &mdash; Lord_Farin (talk) 13:57, 16 September 2013 (UTC)

We have a disambiguation to settle with Definition:Truth Value and Definition:Boolean Interpretation/Truth Value. &mdash; Lord_Farin (talk) 14:44, 17 September 2013 (UTC)