User:Lord Farin/Sandbox/Proof Rules

Refactoring of Proof Rules
The pages in Category:Proof Rules are currently focused on PropLog. As I cover more ground in PredLog, most of the proof rules transfer.

Not all systems have equivalent rules of inference and derived rules.

To accommodate for further coverage of these subjects we ought to design these pages in a way that we can easily add the references to new proof systems having these rules.

This includes the proof of being a derived rule if necessary. Those pages will look like Deduction Theorem and will be interlinked in a standard way.

Applications will mostly lie in deriving stuff like Completeness Theorems, but once we get to deeper parts of proof theory it is going to be very convenient to have this solid foundation.

Considerations
The current state of affairs is reminiscent to the conflation of first order model theory and formal semantics. In a similar manner the generic and specific are intertwined and need to be extracted. In this case, the specific being the coverage of PropLog.

Aim is to restructure the definition/exposition, and not affect the content and/or the proofs based on that. Just give them a more specific foundation in PropLog, or PredLog as applicable.

There will be references in multiple ways:
 * The PR page will link to the systems having these rules, and will link to the proofs of these statements where it concerns derived rules
 * The systems will link to the PR page for the standard rules (as done now)
 * The systems will link to the derived rules in a separate section for derived rules
 * The proofs will by definition link to PR and PS, but aim for doing so in a standard manner

Exposition of relevant pages
Definitions:


 * Definition:Proof Rule (subpage of Definition:Natural Deduction)
 * Definition:Rule of Inference
 * Definition:Derived Rule

Lists and categories:


 * Definition:Natural Deduction
 * Definition:Elementary Valid Argument Form
 * Definition:Natural Deduction System
 * Category:Proof Rules

Decisions made

 * Definition:Proof Rule and Definition:Rule of Inference will merge.
 * The PR pages will be refactored to generalise them from Definition:Natural Deduction to a general proof system
 * There will be a clear distinction made between a standard rule and a derived rule. As in "the following PS have this as a rule of inference: list. the following PS have this as a derived rule: list (link to proof)".
 * There will be a separate page/separate pages listing out the specific PRs for the various types of PropLog. These will be appropriately referenced.
 * Main focus of this project will be to come up with a crisp and clear presentation of these concepts and their occurrences for various types of logic and proof systems.
 * There will remain a separate branch for coverage of PropLog/PredLog on . Just like we have a "canonical" Language of PropLog and PredLog.
 * Proofs of equivalence of the various proof systems will be out of scope of this project to keep it manageable.
 * PR and RoI will remain as interchangeable names on due to prohibitive and unwarranted refactoring costs associated.
 * There will be a category Category:Derived Rules of Inference or Category:Derived Proof Rules or just Category:Derived Rules to contain the derived rules proofs.
 * Definition:Natural Deduction will be phrased as a proof system, with reference to a Definition:Gentzen Proof System. However, due to the presentation in e.g. Huth-Ryan being informal, it will be necessary to give a warning about the informality. Obviously, the informality does not influence the everyday applicability of the system on, and it is exactly because of this informality that it is easily used and properly readable.
 * The rules stated on Definition:Natural Deduction/Proof Rule need work. They do not carry over to predicate logic and are substandardly handwaving. One reason for these rules is that in PropLog it can be common to define the axioms over letters, rather than over WFFs. The rules of substitution then govern that actually all sensible formulae can be reached. But actually this is just a poor choice of axioms / rules of inference. Therefore on these will be made into derived rules.

Chunks of work: Preparation

 * $\checkmark$ User:Lord Farin/Sandbox/Proof Rules/Definition:Natural Deduction: Rewrite of Definition:Natural Deduction
 * $\checkmark$ User:Lord Farin/Sandbox/Proof Rules/Definition:Natural Deduction/Rules of Inference: This will absorb Definition:Elementary Valid Argument Form.
 * $\checkmark$ User:Lord Farin/Sandbox/Proof Rules/Definition:Natural Deduction/Derived Rules: Per decision above, the rules stated on Definition:Natural Deduction/Proof Rule will be derived rules on this page.
 * $\checkmark$ Integrate Definition:Natural Deduction/Proof Rule into Definition:Rule of Inference
 * $\checkmark$ Prior to that, make sure that Definition:Rule of Inference is sufficiently accessible. At least Modus Ponens should be referenced. Additionally make sure that RoI defines the notion of "proof rule" as equivalent to RoI
 * $\checkmark$ Additionally, Definition:Rule of Inference needs to become standalone and not a subpage of Definition:Proof System
 * $\checkmark$ Conclusion, sandbox preparation needed. User:Lord Farin/Sandbox/Proof Rules/Definition:Rule of Inference
 * $\checkmark$ A separate subpage is to be prepared for the "structure of a proof rule" section. User:Lord Farin/Sandbox/Proof Rules/Definition:Rule of Inference/Structure
 * In preparation to the next step, refactor the live proof rule pages to expand their scope (currently they say they pertain to PropLog)
 * $\checkmark$ Rule of Assumption/Proof Rule
 * $\checkmark$ Rule of Conjunction/Proof Rule
 * $\checkmark$ Rule of Simplification/Proof Rule
 * $\checkmark$ Rule of Addition/Proof Rule
 * $\checkmark$ Proof by Cases/Proof Rule
 * $\checkmark$ Modus Ponendo Ponens/Proof Rule
 * $\checkmark$ Modus Tollendo Tollens/Proof Rule
 * $\checkmark$ Modus Tollendo Ponens/Proof Rule
 * $\checkmark$ Modus Ponendo Tollens/Proof Rule
 * Rule of Implication/Proof Rule
 * Double Negation Introduction/Proof Rule
 * Double Negation Elimination/Proof Rule
 * Biconditional Introduction/Proof Rule
 * Biconditional Elimination/Proof Rule
 * Principle of Non-Contradiction/Proof Rule
 * Proof by Contradiction/Proof Rule
 * Rule of Explosion/Proof Rule
 * Law of Excluded Middle/Proof Rule
 * Reductio ad Absurdum/Proof Rule
 * Refactor the proof rules pages themselves to include the references to natural deduction and other systems. Sandbox to follow.
 * Preserve any relevant talk page contributions in a sensible place if the parent page gets refactored.

Chunks of work: Execution

 * The links to Definition:Natural Deduction do NOT need to be updated as the page remains applicable
 * Definition:Elementary Valid Argument Form becomes a redirect to Definition:Natural Deduction/Rules of Inference
 * Any direct links to Definition:Natural Deduction/Elementary Valid Argument Forms need to be addressed. Upon first inspection, it would seem that EVAF and Definition:Elementary Valid Argument can be subsumed in the treatment with appropriate AKA sections. This would clean up idiosyncratic terminology.

Follow-up organisation

 * Create Category:Definitions/Examples of Proof Systems and possibly associated result category. For now, no separate subcats for PropLog and PredLog will be entertained.
 * Create categories surrounding proof rules, "proof rules of natural deduction", and related results.
 * Extracted refactor template from Definition:Natural Deduction: For each of the logical connectives, include a link on the relevant page the appropriate "introduction" and "elimination" proof rules.

Source review activities
The following sources are affected: