Aloitin helposta päästä eli mahdottomista maailmoista. Tuo Vacekin IEP-entry oli lähinnä raapaisu, mutta muistin sitä kautta opiskelleeni joskus parakonsistenttia logiikkaa SEP:stä, ja löytyi kevyempi IEP-tekstikin. Kertailen nyt niitä ja annan välillä ajatuksen harhailla. Parasta iltapuhdetta. 🤗

https://plato.stanford.edu/entries/logic-paraconsistent/

https://iep.utm.edu/para-log/

Relevanttia aluetta kaikille, joita johdonmukaisuus sitoo, mutta ei voi välttyä sisäisiltä ristiriidoilta.

#logic #paraconsistent #contradiction

@ryk047 @andrejbauer

I've written some here (search for #RM3) https://mathstodon.xyz/@CubeRootOfTrue/110775995709463374 about the naturalness of RM3, it is essentially the "complex logic" you get by solving the equation \( A \wedge \lnot A = \top \), akin to \( x^2 + 1 = 0 \) in the reals. In fact it's \( x^2 + x + 1 = 0.\)

Gödel said that blah blah either inconsistent or incomplete. 20th century mathematicians were so horrified at the thought of inconsistency it's been effectively banished (the "law" of excluded middle). We're happy, apparently, with incompleteness. But what about the opposite case!? Gödel himself developed a 3-valued logic, because he obviously understood that if you allow inconsistency, you can have completeness.

Normally inconsistency can't be tolerated because \( (A \wedge \lnot A) \supset B \), you can prove anything from an inconsistency, aka the principle of explosion. Hence the horror.

In a 3-valued logic, there are statements that are inconsistent, but the logic doesn't allow explosion, so everything's under control.

So yes, #paraconsistent and #relevant #logic have very much to do with foundations.

And yes, it's possibly the simplest example of a symmetric closed monoidal category (symmetry is optional), and maybe a useful teaching tool, not to mention that it's a superior logic than 2-valued logic, as it can handle vagueness.

@andrejbauer (I am assuming classical logic, but nothing really changes if we switch to intuitionistic logic, just replace Boolean algebras with Heyting algebras.)

This is a bit like saying "I am assuming euclidean geometry but nothing really changes if we switch to spherical." Because there are more than 2 kinds of geometry, and intuitionistic logic is not the only other choice here. Is RH "neither true nor false" or perhaps RH is "both true and false"? You need a dialetheic logic, a #paraconsistent logic. And in that case, these are both valid truth values, and RH would be formally undecidable. But more than that, asking for it to have no truth value at all is different from just not being true and also not being false. It might be similar to Graham Priest's "ineffable" logical value. #logic #dialetheism #twothingscanbetrue

@logicbot (¬a∨((b∧¬b)⇾c)) is valid in binary logic but not in #paraconsistent #logic because (b∧¬b) sets off little alarm bells that are built into closed monoidal categories

I honestly wonder why the relevance #logic #RM3 is not taught in high school. Not only is it an obvious extension, and still relatively simple, but it works BETTER than regular binary logic!

It is a myth, by the way, that modern computers use binary logic. They don't. Deep within their electrical circuits are logical states like X for Don't Care and N for No Connection. Some have many more than 2 states.

You don't need non-binary logic everywhere! It's pretty useless in an adder, for example. Binary logic is good at stuff like that. As long as your inputs are consistent.

Where non-binary logic really comes in handy is things like the law, or politics, where disentangling the validity of assertions can be hard, because natural language is full of ambiguities and vagueness. You need a #paraconsistent logic.

#MariaDB, the database formerly known as #MySQL, is one of the only popular computing environments on the planet to implement #paraconsistent logic. It's quite natural, of course, for a database system which accepts inputs from the real world to have a mechanism to deal with inconsistent or missing data. That mechanism is the paraconsistent logic LP.

It is thus possible to build a fully relevant #RM3 implication in an SQL Select statement. The three values of the logic are 0, 1, and None. The table t, here, is just a list of all pairs (a, b).

select a, b, (not a) or b, a is false or b,
(a is false or b) and ((not b) is false or not a) from t

a b ~a|b a->b a=>b
----------------------------------------
0 0 1 1 1
0 None 1 1 1
0 1 1 1 1
None 0 None 0 0
None None None None None
None 1 1 1 1
1 0 0 0 0
1 None None None 0
1 1 1 1 1

The last three columns show the ordinary (non-paraconsistent) conditional, the paraconsistent conditional, and the relevant conditional, resp., the latter being implemented by applying the contrapositive to the paraconsistent conditional.

Validity is expressed by "not false". This is an important concept. For the binary conditional, validity is ALSO "not false", but the lack of any other choice makes this concept invisible. Graham Priest has written extensively about this topic, and the semantics of LP.

In particular, \[ (a \wedge \lnot a) \rightarrow b \]is not valid, because None -> 0 is false, and your database won't explode when it hits a None.

Note to the geeks out there who want to add some robustness to their databases: you can usually create a function in SQL like Imp(a, b) or something, so you don't have to constantly write the whole expression out every time.