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 Definitions/Logic 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)

Addressing WFFs
We have the following ways to address "formulae":


 * Definition:WFF of Propositional Logic
 * Definition:Logical Formula Different scope
 * Definition:Statement Form Disambig, WIP
 * Definition:Propositional Formula
 * Definition:Logical Form Disambig

This multitude seems to arise from generality being sought with only one application in mind. We conclude that this needs to be uniformised. &mdash; Lord_Farin (talk) 11:41, 1 October 2013 (UTC)


 * The current pair seems to be adequate and clear. Job done. &mdash; Lord_Farin (talk) 23:13, 5 December 2013 (UTC)

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

Refactoring/rewriting

 * Definition:Model (Logic) Only the "PredCalc" subpage needs to be addressed.
 * Definition:Statement Form
 * Models for Propositional Calculus
 * Model Defined by Implication
 * Definition:Substitution Instance
 * Definition:Main Connective
 * Method of Truth Tables and subpages Adapted to new scheme.
 * Definition:Tableau Extension Rules
 * Model of Branch of Propositional Tableau

In PropLog
In due time, we will need to set up the following pages:

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.
 * 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.
 * This includes noting that Huth-Ryan call the "Labeled Tree WFF" associated to a "normal WFF" a parse tree. This can be used to replace the source entry for that book on Definition:Language of Propositional Logic/Labeled Tree.
 * 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.

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)


 * To separate results about strings/words from the Formal Systems category.
 * To graft the Bourbaki pages as just another branch onto the Formal Language/Predicate Calculus definitions tree.
 * To separate any symbolic treatment of logic (such as present in Definition:Converse) from the Definitions/Logic category.
 * To move computability theory stuff out of the Category:Definitions/Mathematical Logic category as it doesn't belong there.

What needs to be contemplated:


 * A way of rigorous distinction between PropLog/PredLog and PropCalc/PredCalc, or an identification of the relevant pairs.

I have refrained from going into things that come properly after this mostly metalinguistic cq. formal-symbolic treatments. In particular, model theory has been left untouched, both in boolean and relational form.