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.

= New attempt at Definition:Interderivable =

Definition
If two statements $p$ and $q$ are such that:


 * $p \vdash q$, i.e. $p$ therefore $q$
 * $q \vdash p$, i.e. $q$ therefore $p$

then $p$ and $q$ are said to be logically equivalent or interderivable.

That is:
 * $p \dashv \vdash q$

means:
 * $p \vdash q \ \text {and} \ q \vdash p$.

Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign.

Also see

 * Provably Equivalent, an interpretation of this logical concept in proof theory
 * Semantically Equivalent, a related concept in model theory

Discussion
In the present case, I think a "master" page Definition:Equivalent Statements pointing to or transcluding the three specific types of "equivalent" in logic and symbolic logic. Perhaps these three can also already be mentioned on the disambig Definition:Equivalent in a sublist below the link to Definition:Equivalent Statements.

Naturally, this will require a subpage set-up (so don't mind the awkward headings). I would however rather wait a few more days for the completion of the extension (yes, I've actually picked up on that again) so as to be able to do two paradigm shifts at once. In the mean time, comments on the proposed approach are most welcome. --Lord_Farin (talk) 21:26, 19 February 2013 (UTC)

To be clear: Definition:Provably Equivalent will contain something like "In $\mathcal D$, $p \vdash_{\mathcal D} q$ and $q \vdash_{\mathcal D} p$" for a deductive apparatus $\mathcal D$. Definition:Semantically Equivalent will contain "$\mathcal M \models p$ iff $\mathcal M \models q$ for all models $\mathcal M$". --Lord_Farin (talk) 21:30, 19 February 2013 (UTC)


 * No responses? :( Viewpoint should be that the "logical" definitions have two interpretations: proof- and model-theoretic. It may be possible to indicate this by an appropriate transclusion scheme. --Lord_Farin (talk) 18:16, 22 February 2013 (UTC)


 * Sorry, I completely missed this when it first came out.


 * I've been uncertain about the treatment of this for a while. We need a) a short-and-sweet page defining "logical equivalence" for those who only need a fairly intuitive overview of what it means, b) an in-depth analysis of the biconditional, c) a rigorous proof that the two halves of a biconditional are logically equivalent statements and vice versa, and d) a page that can be linked to (possibly the same as a) above) when invoking the clause "logically equivalent" in a mathematical proof. Oh, and e) the model-theoretic approach that you allude to above. We also need to be able to treat logical equivalence as a relation so that Biconditional is Reflexive makes semantic sense.


 * Every time I come this way I give it another tweak, but I've been here and I'm moving on again, and so if you want to take this on, then feel free. --prime mover (talk) 22:51, 22 February 2013 (UTC)