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[wierd]

That is not exactly true.

You DO exist, statistically, in the entire universe.

There is a non-zero probability that if I checked for the existence of the atoms that comprise your body, in the habitable zone of bernard's star, that I would find them there. (The probability is ABSURDLY low, but non-zero. This just means I have to keep looking until I find you.)

Subatomic particles exist as both particles and wavefunctions. Wave functions taper off to infinity. This means that you DO exist in the whole universe, but a statistically significant proportion of you exists within a much tighter locality.

As for time's necessity, we can prove it is necessary in our part of the universe at the very least, since we have experimentally proven that General Relativity and Special Relativity are totally features of our universe. (If they werent, the sun would not shine, we would not exist, we wouldnt be able to detect muons from cosmic particle collisions with our atmosphere at the surface of our planet, the GPS network would not work, gravitational lensing would not be a thing, and a whole bunch of other things.) 

Time and space are functions of each other, and are intrinsically related.  Inside the event horizon of a black hole, one of thge sides of this function becomes infinite, making the other into an unquantifiable property.  The rules of the universe go batshit crazy. You can trace the transition going in smoothly, but then you can't resolve a mathematical way OUT again. (Start from infinity, and trace the trajectory back out to the point of origin outside the horizon)

Time is VERY MUCH "required" for our part of the universe.

[Bohandas]

Does modal logic by any chance use some strange definition of the terms "possibly" and "necessarily" that most people would not be aware of?

EDIT:
also

I'm gonna try and say this as nicely as I can: you don't understand what we're talking about here at all.  This is about mathematically building up the fundamental ideas of logic and reasoning, it makes absolutely no statements about the real world or how it operates.

What good is it then? It sounds like you're saying that this field of philosophy is nothing more than a massive circlejerk with no practical application.

[wierd]

It uses the logic definitions of those terms.

For a thing to be "Possible", it must exist in at least one of the computable manifestations of the statement.

EG, "X=1" is "Possible" given the set of

X=1, Y=0
X=0, Y=0
X=0, Y=1
X=1, Y=1

In at least one of those relationships between X and Y, X is equal to 1, thus X=1 is "possible."

If however, the set is this:

X=0, Y=0
X=0, Y=1

Then X=1 is not possible.

A "Necessary" condition happens with X=1 given the following set

X=1, Y=1
X=1, Y=0

Because all members of the set have X=1 as truth.


"Necessarily possible" means that there is not a condition between multiple sets, where a value fails to occur.

EG:  X=1 is necessarily possible between these two sets:

X=1, Y=0
X=1, Y=1
X=0, Y=1
X=0, Y=0

AND

X=1, Z=0
X=1, Z=1
X=0, Z=1
X=0, Z=0

In both sets, it is necessary for X to possibly be equal to 1.


The problem with the black hole and time's existence, is that the rules of reality go absolutely apeshit. The person watching somebody fall into the black hole will never see them enter. They seem to get "Stuck", then smeared all over the surface of the event horizon.

The person falling into the black hole (assuming they dont get roasted to a crisp, have their atoms torn apart by powerful magnetic fields, or undergo spontaneous fission as space rips apart underneath them) would not experience anything unusual at all about the passage of time, except noting that the universe seems to be suddenly speeding up all around them, as they watch every star in the galaxy supernova, every new star get born, etc-- in order, until the heat death of the universe.  They will fall away from everything in the universe at an absurd rate as well.

So, Time both exists, and does not exist-- SIMULTANEOUSLY.

This means that instead of a discrete X=1 being "possible", you get a state of "X=1 AND 0"

That's why it breaks the axiom.

The axiom RELIES on X being discrete.  X being undefined or undefinable takes the whole idea of being true or false, wads it up, and throws it out the window.

To make matters more confusing, recent theoretical research based on high energy physics experiments suggests It can be quite literally stated that on the other side of the event horizon, is a completely different universe.

[Sergarr]

It seems that "X=1 is possible" can be replaced with "There exists at least one X, for which X=1" and "X=1 is necessary" with "For all X, X=1".

It sounds much cleaner this way, with less baggage associated with words "possibility" and "necessity".

[Leafsnail]

I'm not using "special pleading".  I am pointing out that your physics-based objections fundamentally are not in any way relevant to the logical statements that S5 makes.

You seem to think that a statement has to somehow apply all over the universe under this system - it doesn't.  This is like saying that the statement "air exists" is contradicted by the fact that there are places in the universe where there is no air - it is not.  Air still exists, it just doesn't exist everywhere, in the same way that there are regions of space where the conventional laws of time apply (I assume this is what you mean when you make the very ill-defined statement "time exists"), they're just not everywhere.

So the statement "there are regions of space where the conventional laws of time apply" is true.  The statement "the conventional laws of time apply in all regions of space" is false, but it is in in absolutely no way implied by S5, and if you think it is then you have fundamentally misunderstood it.  All S5 implies is "In all universes it is possible that there are regions of space where the conventional laws of time apply" (although note that it doesn't require all other universes to have such regions, it's just saying that it's possible).

You may have noticed this is not a very interesting implication from a practical standpoint, and that's because it isn't - it's a logical axiom, not a statement about reality.

[Sergarr]

I'm not using "special pleading".  I am pointing out that your physics-based objections fundamentally are not in any way relevant to the logical statements that S5 makes.

You seem to think that a statement has to somehow apply all over the universe under this system - it doesn't.  This is like saying that the statement "air exists" is contradicted by the fact that there are places in the universe where there is no air - it is not.  Air still exists, it just doesn't exist everywhere, in the same way that there are regions of space where the conventional laws of time apply (I assume this is what you mean when you make the very ill-defined statement "time exists"), they're just not everywhere.

So the statement "there are regions of space where the conventional laws of time apply" is true.  The statement "the conventional laws of time apply in all regions of space" is false, but it is in in absolutely no way implied by S5, and if you think it is then you have fundamentally misunderstood it.  All S5 implies is "In all universes it is possible that there are regions of space where the conventional laws of time apply" (although note that it doesn't require all other universes to have such regions, it's just saying that it's possible).

You may have noticed this is not a very interesting implication from a practical standpoint, and that's because it isn't.  It's really just a tool that helps people create logical proofs far more easily.

Is it "possible" in a sense that "it exists somewhere/somewhen inside of any universe" or in a sense that "it exists somewhere/somewhen inside of at least one universe"?

[Leafsnail]

Is it "possible" in a sense that "it exists somewhere/somewhen inside of any universe" or in a sense that "it exists somewhere/somewhen inside of at least one universe"?

The second one, it's definitely not saying it has to be true in any other universe.  It's kindof weird but I think it's basically trying to define what we mean by "possible" in the first place.

[Urist Arrhenius]

I'm with leafsnail. Think of it this way: you are using world to mean a particular space separated from other spaces by arbitrary boundaries. There are bound to be contradictions. We don't have to get into time stuff. You could just say: this space is black, this other space is white, there's a contradiction.

What the axiom actually relates to are metaphysically separate universes, where the boundary is defined by an inability for matter to exchange between the two.

[Neonivek]

Under that interpretation, the logic breaks with black holes, and with general relativity in general.

Well no... A Black Hole is a Black Hole no matter where it is.

A Zebra is lighter on mars than it is on earth. Does that mean it isn't a Zebra anymore? No.

Axiom S5 is basic identity.

[Sergarr]

Is it "possible" in a sense that "it exists somewhere/somewhen inside of any universe" or in a sense that "it exists somewhere/somewhen inside of at least one universe"?

The second one, it's definitely not saying it has to be true in any other universe.  It's kindof weird but I think it's basically trying to define what we mean by "possible" in the first place.

So essentially it means that "for any sentence, there's at least one system of logic, objects and meanings, under which this sentence is true", right?

Sounds like an axiom all right.

[Leafsnail]

No that's not what it means at all.  The whole point of this logical system is to try and distinguish between things that just happen to be false in our world and things that must be false in every world.

[MagmaMcFry]

After giving modal logic some of my attention, it has come to said attention that OP's representation of S5 is technically incorrect.
Here's the actual S5:

If X is possibly necessarily true, then X is necessarily true.

In other words (assuming the ontological representation of "necessarily" and "possibly" and a complete accessibility relation): If there exists a world such that in all worlds X is true, then in all worlds X is true. Yeah, this is pretty much a tautology, and it is usually introduced to ensure that the accessibility relation is component-complete ("euclidean") in the first place. So S5 is perfectly okay in ontological modal logic.

Of course there are other applications of modal logic where this axiom is not appropriate, but OP seems to use the ontological representation.