Is it used in AI or is it just a philosophy thing

>>16683319 (OP)

When done right, it's the internal logic of presheaves, so it's pretty much pervasive in topos theory. It's true that philosophers also tend to utter nonsense when they pretend to do logic, so you have to take care.

>>16683322
Can you explain more

>>16683319 (OP)
a modality in general is just a way to qualify the truth of a proposition. Examples are:

necessity:
>[math]P[/math] is possibly true (typically rendered as [math]\Box P [/math])
>[math]P[/math] is necessarily true (typically rendered as [math]\diamond P[/math])
>[math]P[/math] is known to be true

Traditional logic only allows you to evaluate expressions as "true" or "false" but not the qualifications used above. Modal logics add modal operators and semantics that allow you to express and perform inference on expressions like [eqn]\Diamond P \rightarrow \Box Q[/eqn] I.e.: (If P is possibly true, then Q must be necessarily true).

It had its use in AI in the sense that was used in symbolic AI approaches that were most relevant around the 80s and 00s. The approach of symbolic AI is to encode as much domain knowledge in formal logic and allow logical program solvers to draw conclusions. That being said, this form of AI stands orthogonal to the current statistics based approaches of AI that you are probably thinking of when you talk about AI. ChatGPT does not explicitly encode modal logic for instance.

>>16683322
ellaborate for someone who knows modal logic but nothing about topology

>>16683319 (OP)
a way to obfuscate, nothing more.

>>16683319 (OP)
Its the following system of abreviations: Let $L$ be a first-order language. We add to $L$ two unary predicate $W$,a binary predicate $M$, a binary predicate $\leq$, and for every n-place relation symbol $R(x_1,...,x_n)$, a $(n+1)$-place relation symbol $R^*(y, x_1,...,x_n)$. We pick a new set of variables $\Omega:= \{w_1,w_2,w_3...\}$ and define:
1°) $w \Vdash R(t_1,...t_n):= R^*(w,t_1...,t_n)$ for every $w \in \Omega$, every relation symbol $R$ and every terms of the original language $t_1,...,t_n$ (not containing letters from $\Omega$)
2°) $w \Vdash \perp$:= \perp
3°) $w \Vdash A \to B:= (w \Vdash A) \to (W \Vdash B)$
4°) $w \Vdash \forall x C:= \forall x, M(w,x) \to (w \Vdash C)$
5°) $w \Vdash \Box D:= \forall v, W(v) \to w \leq v \to v \Vdash D$

If you add the axioms saying that $\le$ id a preorder and $M$ is increasing (i.e. $\forall x, a, b, W(a) \to W(b) \to M(a,x) \to a \leq b \to M(b,x)$) then the set of theorems of modal logic S4 can be interpreted in the system above (with classical logic).

>>16683400
What's the difference between P and diamond P?

>>16683766
damn i just realized i messed up: [math]\Diamond[/math] is "possibly", [math]\Box[/math] is "necessarily", anyways:

truth in logic is always relative to some model of the world. e.g. in propositional logic: if your model is [eqn]\vdash P,Q,\neg R[/eqn] (i.e. [math]P,Q[/math] are true and [math]R[/math] is not).

the expression [math]P\land Q \land \neg R[/math] would be true, but there are other models for which it is not true (e.g. [math]\vdash P,Q,R[/math]) (i.e. all are true), so the expression is not necessarily true.

An expression is necessarily true if it is true under all models: e.g. [math]P\lor\neg P[/math] is necessarily true cause no matter the truth value of [math]P[/math], this expression evaluates to true, so we can say [math]\Box (P\lor\neg P)[/math].

[math]\Diamond P[/math] just means that there is some model in which [math]P[/math] is true.

>>16683766
Necessary P does not imply that P exists, in the same way that a conditional with a false antecedent is always true.

>>16683319 (OP)
Moral logic is when you extend “truth” and “false” with things like “possible”, “probable”, or “unlikely”. You can also add other “modes” you just have to extend your truth table to accommodate how they work with conjunctives and negations.

>>16684070
that does not explain the difference between P and necessarily P, midwit freak

>>16684080
*modal logic

model jazz rendered as chalk diagrams
https://www.youtube.com/watch?v=rJBWdX9P__g "Modal Jazz Music for Relaxation and Focus | Best Modal Jazz Playlist"

>>16683355

>>16684711
that s something totally different

philosophy thing fucking retard

>>16683322
>>16683400
>>16683506
for fuck's sake. no wonder people like OP are confused about this. philfags are DOGSHIT at explaining this, likely because they just want to peddle bullshit.
>>16683319 (OP)
>>16683355
it's very simple. logic is binary (true or false). things are either true or false, full stop. if something is not true, it is false.

modal logic splits this binary into possibilities. there are things that are true, false, maybe true, and maybe false. if something is not true, it doesn't imply it's false... rather it's just possibly false. or possibly true. i don't fucking know - that's where i stopped caring about their faggotry.

>>16686057
that's got to be the worst explanation ever

>>16687902
i accept your concession

>>16684063
So diamond P means P is satisfiable? Would you say that modal logic is a generalization of satisfiability?

>>16683319 (OP)
It's an extension of regular logic that adds operators that give additional meanings to propositions. Instead of a proposition just simply being true it can be true at a particular moment in time or in a different world.

>>16686057
>things are either true or false, full stop.
is there a set with a cardinality between that of integers and that of reals?

>>16694439
that's a situation in which either case is fine/doesn't break shit

>>16694815
but it's neither true nor false. you can play with its assigned truth value, but viewed from the direction of the rest of regular math it has no intrinsic truth value.

>>16683319 (OP)
>Blocks your philosophically ignorant path

>>16694924
Of course, it might be seen as the ultimate metaphysical pipe-hit to declare that every logically consistent possible world equally exists: at the same time you can't uphold logic and deny this possibility. This is also the only position that does not make quantum physics seem like a schizophrenic episode.

Mathematicians will unironically champion Gödel's incompleteness theorem (which rejects the law of excluded middle) while championing all of their proof by contradictions (which rely on the law of the excluded middle).

>>16695080
I would not say it rejects excluded middle, it's more like it turns any complex enough system + excluded middle into an inconsistent system. but as long as this kind of activity is possible, we'll continue doing it.