Bay 12 Games Forum
Finally... => General Discussion => Topic started by: Roc CURIOUSBEAST_EATER on March 08, 2015, 05:53:18 pm
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A couple days ago, I watched about the Modal ontological argument when I Started to think about the fact that axiom S5 (that is used for 3rd premise of the Modal argument) can be used to support a number of self-contradictory positions including an irresistible (or unstoppable) force and unmovable object existing in the same world (what happens if a irresistible force hits the unmovable object?). I spend a couple creating this argument, although most of it was wasted trying figure out how premise 13 should stated.
Axiom S5 states that if it exists in some possible worlds then it exists in all possible worlds. The Logic I am talking about of course is Modal logic, I don't know much about it apart from the axiom I trying to refute.
This argument has premises and a conclusion. it is a Reductio ad absurdum argument you will see in the argument present below.
A Reductio ad absurdum argument works by assuming what your trying to disprove to be true then push that assumption to it's logical (and absurd) conclusion to expose contradictories that the assumption creates. since everything that creates contradictories are false, it logically follows that the assumption the assumption is false.
The argument goes as follows.
1. Axiom s5 (X in some worlds = X in all worlds) is true.
2. It is possible that a unmovable objects exists.
3. If it is possible that a unmovable objects exists, then it exists in some possible worlds. (from 2)
4. If a unmovable object exists in some possible worlds, then it exists in all possible worlds. (from 1 and 3)
5. if a unmovable object exists in all possible worlds, then it exists in the real world. (from 4)
6. It is possible that a irresistible force exists.
7. If it is possible that a irresistible force exists, then it exists in some possible worlds. (from 6)
8. If a irresistible force exists in some worlds, then it exists in all possible worlds. (from 1 and 7)
9. If a irresistible force exists in all possible worlds, then it exists in the real world. (from 8)
10. The idea of a unmovable object and irresistible force existing in the same world is self-contradictory. (from 5 and 9)
11. all self-contradictory ideas are false.
12. any axiom that creates self-contradictory ideas are incoherent. (from 11)
13. Axiom S5 created this self-contradictory idea. (from 10 and 11).
Conclusion: Premise 1 cannot be true. (from 12 and 13)
I want this argument to be critiqued, since I want to see if the argument is logically valid, the logic is sound and the premises are true. The best way to do this to have other try expose the flaws in the argument. I will be surprised if the argument survives the critiquing without any changes to it.
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That's a weird-as-hell axiom. Wouldn't it basically mean that all possible worlds are absolutely equivalent to each other?
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1. Axiom s5 (X in some worlds = X in all worlds) is true.
I am unfamiliar with axiom S5, but as you've presented it, it does not appear to imply that any worlds besides the real one exist. They might, or they might not. 'All worlds' could be a set with only one member: {the real world}. If so, then your logic chain falls apart.
2. It is possible that a unmovable objects exists.
3. If it is possible that a unmovable objects exists, then it exists in some possible worlds. (from 2)
6. It is possible that a irresistible force exists.
7. If it is possible that a irresistible force exists, then it exists in some possible worlds
You appear to be making assertions other than the one you're attempting to disprove. 2 and 6 don't appear to follow from 1, and if they're removed, your logic chain falls apart.
10. The idea of a unmovable object and irresistible force existing in the same world is self-contradictory. (from 5 and 9)
There may be reasonable definitions for 'unmovable object' and 'irresistible force' for which 10 is not implied by 5 and 9.
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PTW. Modal logic makes no damn sense.
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PTW. Modal logic makes no damn sense.
QFT
That axiom is strangely arbitrary and metaphysical: Any possible proposition is declared necessarily possible simply because the logic requires it, and a certain mystical force ensures that its terms have the same referents, come what may, in all possible worlds. "End of Discussion." Your attempt to "refute" such an axiom is admirable, although I'm not entirely sure why you would have accepted it in the first place.
Can you give us a reasonable definition of an 'unmovable object?'
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Can you give us a reasonable definition of an 'unmovable object?'
Some such definitions are used in this video (https://www.youtube.com/watch?v=9eKc5kgPVrA).
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Yes, objects with infinite mass are very reasonable. :D
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Yes, objects with infinite mass are very reasonable. :D
:P
I didn't really think that one through.
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Can you give us a reasonable definition of an 'unmovable object?'
A definition? Not going to step into that minefield.
However, it's very easy to give an example of one. For example: your frame of reference.
Some such definitions are used in this video (https://www.youtube.com/watch?v=9eKc5kgPVrA).
Which demonstrates another flaw with the OP: the immovable object and irresistible force could be the same phenomenon, in which case step 10 is not valid.
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All you've done is shown that 2 and 6 are contradictory. There's no reason to accept that either of them is necessarily possible anyway.
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All you've done is shown that 2 and 6 are contradictory.
But they're not. We have two examples already:
1) they're the same phenomenon
2) they pass through each other without interacting
Here's a third:
3) they never come into contact with each other.
Though 2 and 3 are basically the same thing: they don't interact.
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All of those things are semantic quibbles that don't address the core argument in any meaningful way though.
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So, my questions are these: Why do 2 and 6 follow from 1? Also, why would 1 be true? What does "world" mean?
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All you've done is shown that 2 and 6 are contradictory.
But they're not. We have two examples already:
1) they're the same phenomenon
2) they pass through each other without interacting
Here's a third:
3) they never come into contact with each other.
Though 2 and 3 are basically the same thing: they don't interact.
I can give an example of an (apparently) unstoppable force at least-- Hubble expansion. Current projections indicate that it will overpower all other forces in the universe in the far distant future.
Black hole singularity is not an immovable object. While it has infinite density, it has finite mass, and therefore only requires finite energy to move. If these objects were unmovable, they could not be in the rotating reference frame surrounding sagitarius A, the center of our galaxy-- yet they are. Moreover, these objects would not move in accordance with hubble expansion, yet they do. Black hole singularities are not immovable objects. About the only immovable object I can think of would be the universe itself, but that is a contrivance; it is logically unable to move, because there is no coordinate system outside of itself in which it could move. If there is a coordinate system outside the universe, we cant know about it.
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All you've done is shown that 2 and 6 are contradictory. There's no reason to accept that either of them is necessarily possible anyway.
Since the axiom I am trying to refute requires the existence of the 2 objects since they are possible, despite the fact that the 2 things existing in the same world is self-contradictory.
The reason why I say those 2 things are possible is because the existence of either thing is not logically impossible (unlike square circles).
Anyway, The argument needs to be changed as it is currently too easy to miss the core argument.
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I've done more reading and I think you have just misunderstood the nature of S5, or possibly muddled two parts of it together. What it actually says is
"If A is possible, then A is necessarily possible"
"Necessarily possible" is not equivalent to "true". It just means that A being true in one world implies it must also be possible in all other worlds. So yes, if we accept there could be a world with an immovable object and a different world with an unstoppable force, and that doing so implies that both are possible in this world. It does not, however, imply that both (or indeed either) of them exist in reality.
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Really? I thought rule 1 is that basic identity of something must exist in all worlds.
So X doesn't become a B in a different world. What is X in one world is X in all worlds.
But your logic fails for one reason... Zebras
Zebras exist on earth. How many Zebras live on mars?
But Axiom S5 isn't that...
It states that a Zebra on earth is the same as a Zebra on mars.
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Under that interpretation, the logic breaks with black holes, and with general relativity in general.
A second in one part of the universe, is not the same as a second in another part of the universe, as measured from a 3rd reference frame.
Let's say we are observing the difference between "one second" between extra-galactic space (where there is little matter, and thus very little acceleration fields in effect) and behind the event horizon of a black hole, as measured from the reference frame of the Earth.
In the first, the second happens very quickly, as time in our reference suffers from local acceleration fields from the sun, the galaxy as a whole around us, and even our planet, which causes dilation.
In the second instance, the second seems to take an infinite amount of time to pass, due to the infinite acceleration field found behind the black hole's horizon.
Since the flow of time is not constant between observational frames, the axiom is false.
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The somewhat strange conclusion that arises from S5 is
"If A is possibly necessary, A is necessary" (=> A is true in reality)
If you try to parse this in day-to-day language it sounds pretty strange, but it's basically saying "If there's a world where A is necessarily true, then A is necessarily true in all worlds". Note that you can simply reject the notion that it's possible for there to be a world where, say, God necessarily exists, so this isn't simply saying "everything you can imagine is true".Under that interpretation, the logic breaks with black holes, and with general relativity in general.
A second in one part of the universe, is not the same as a second in another part of the universe, as measured from a 3rd reference frame.
These extra worlds do not have to actually exist, they can just be metaphorical devices to represent a phase space of different possible universes.
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The somewhat strange conclusion that arises from S5 is
"If A is possibly necessary, A is necessary" (=> A is true in reality)
If you try to parse this in day-to-day language it sounds pretty strange, but it's basically saying "If there's a world where A is necessarily true, then A is necessarily true in all worlds". Note that you can simply reject the notion that it's possible for there to be a world where, say, God necessarily exists, so this isn't simply saying "everything you can imagine is true".Under that interpretation, the logic breaks with black holes, and with general relativity in general.
A second in one part of the universe, is not the same as a second in another part of the universe, as measured from a 3rd reference frame.
These extra worlds do not have to actually exist, they can just be metaphorical devices to represent a phase space of different possible universes.
What is the observational difference between a locality where time is unmeasurable (because it is a coordinate singularity), and one where time does not exist? How can one tell the difference?
This is an important question, because a place in the universe where time ceases to exist would break the axiom-- while one with time as a coordinate singularity narrowly escapes.
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That's a very interesting physics question but I don't see what it has to do with logical frameworks?
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The axiom, roughly translated, says "If A is possible, then A is necessarily possible", per your own interpretation.
So, we have regions of space where time can be said to no longer be possible, due to infinite spacial curvature, where outside of those regions, time is a measurable and clearly required quanta.
This is WITHIN A SINGLE UNIVERSE--
Since we can observe this break inside our own universe, we can conclude that an axiom that necessitates other universes to conform must be false.
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I think you've misunderstood what "possible" means in this context. If statement A could be true then A is possible - A doesn't actually have to be true in every world. "Time" is not a meaningful statement in any case so you cannot say whether it's possible or not. If you tried to make a statement like "time is the same everywhere" then it might be possible, but it would be false in our universe due to the things like the example you pointed out.
This statement is not saying that the universe is homogenous everywhere nor that any other world need be the same, it's actually not really saying anything too exciting at all when you break it down. It's unlikely to be relevant to you unless you're trying to create long logical proofs.
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This seems like special pleading to me.
Time is possible in our universe. Time is also IMPOSSIBLE in our universe. (as I pointed out.)
For the axiom to be true, then time must be possible inside the singularity, because it is inside our universe. But it is not.
Within the world domain of "Our universe", a tautology involving this axiom exists.
As for the assertion of "cant prove time impossible", The relationship between space and time has been well established experimentally. There is infinite space inside the black hole's volume, making time either a coordinate singularity, or making it a non-existant property.
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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.
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I see it like this:
Modal logic is about the how the qualifiers possible and necessary work in logic. Under the system S5 those are equivalent across possible worlds.
So if X is possibly true in world A then X is possibly true in all worlds.
If Y is necessarily true in world A then X necessarily true in all worlds.
Also, if Z is not possibly true in world A, then Z is not possibly true in all worlds.
So if it is not possibly true that an immovable object and an unstoppable force both exist in world A, then it is not possibly true in any world.
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And that is the very definition of special pleading. (http://en.wikipedia.org/wiki/Special_pleading)
Your personal definition of what constitutes logic, is not congruent with more established sources.
Wikipedia (yes, I see your eyes rolling) has this to say.
Logic (from the Ancient Greek: λογική, logike)[1] is the use and study of valid reasoning.[2][3] The study of logic features most prominently in the subjects of philosophy, mathematics, and computer science.
Logic was studied in several ancient civilizations, including India,[4] China,[5] Persia and Greece. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Buddhists and Jains.
Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.
Since we are discussing an axiom of modal logic, let's get the established definition of what it is.
Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. Modals—words that express modalities—qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").[1] Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),[2][3] deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")[4] and doxastic modalities, or modalities of belief ("It is believed that p").[5]
A formal modal logic represents modalities using modal operators. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, Possibly, attached to the sentence "It will rain today".
So in essence, Modal Logic is a structured and rigorous exploration of possible states in formal logic.
As pointed out eariler, formal logic is the foundation for Modal logic. Formal logic was created as a basis to establish reason, and one of the foundational types of proof that can be given for logic is a physical one. I just provided a physical proof.
Specifially, I used deductive reasoning to arrive at an uncertainty about this axiom. Since there is uncertainty, the axiom is not necessarily true, and thus cannot be called true.
Deductive reasoning, also deductive logic or logical deduction or, informally, "top-down" logic,[1] is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.[2] It differs from inductive reasoning or abductive reasoning.
Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.
Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an open domain, one where there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.
I noted a closed domain-- "Our universe"-- and a feature-- "Time". Time is both possible and impossible within that closed domain, as I pointed out. For the axiom expressed to be true, then this event cannot occur. It DOES occur, therefore the axiom must be false.
You responded with special pleading.
Basically, I am showing that there is uncertainty about the necessity of the axiom, and thus showing that the axiom can be false, which removes it from candidacy as an axiom.
One could twist this around in a knot, and say that because time can be shown to be possible and also impossible in the same universe, then time must be possible and impossible in all universes, but that is a useless statement. It just means that black holes must exist in all universes, or at least, that all universes must have this kind of inconsistency. This is not true, as mathematically cogent models of universes without these features exist-- which creates a contradiction with this axiom.
If we use the other form you quoted:
"If A is possibly necessary, A is necessary" (=> A is true in reality)
We can say that singuarities are possibly necessary, Singularities ARE necessary ==> Singularities exist
But we also have things like Godel spacetime models that fully satisfy the same mathematical fundaments of our spacetime, which are not able to have these kinds of features.
BOOM. Contradiction.
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If time exists in the universe, then it is possible that time exists in the universe.
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That's not what the axiom says.
The axiom says that if time is potentially necessary in the universe, that it *IS* necessary in the universe, and thus exists.
There are places in the universe where it (seemingly) does not.
A better counter-argument would be that I have argued myself into a corner with something like this:
"You have just shown that in places where time is necessary, time exists, and in places where it is not necessary, it does not, which satisfies the axiom"
However, both areas are essentially "The same" mathematically.
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No-one has stated that time potentially necessarily exists in any possible universe, let alone this one. So it doesn't necessarily have to exist in this one, as we stand.
There are places in the universe where I don't exist, that doesn't mean that I cannot possibly exist in this universe.
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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.
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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.
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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. (http://www.gizmag.com/quantum-black-holes-singularity/29242/)
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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".
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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.
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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"?
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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.
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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.
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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.
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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.
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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.
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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.