Modal Logic (Basics)
ฝัง
- เผยแพร่เมื่อ 17 เม.ย. 2014
- A explanation of the basics of Modal Logic, including the difference between the K, T, B, S4 and S5 systems of modal logic (100 Days of Logic).
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!
These naming conventions are the worst thing since vitamin names
You should see axiomatic set theory.
Lol
Really great explanation despite the complicated topic, thank you
Nice to see standard logic proof methods applied to a modal logic situation. In the books, just reading, it looks a mess and difficult and intimidating to follow, but seeing it laid out step by step has helped make it seem a lot more logical to me, cheers.
Thanks! Glad the proofs are helpful.
22:00 Do all proofs by contradiction assume the system's consistency? 'Cause we never established that for this one.
Thanks for this video! It is a nice wrapup of what I could not instantly grasp in the lecture. And our lectures are not recorded :-(
The example was also pretty nice! :-)
Hey! A minor thing in the example @ 23:07, but in P3, how did the double negation get inside the first parentheses via DeMorgan's Rule? It should be ~~(A>B)&~(A>B), right?
I think you're right. It didn't end up effecting anything, since either way the ~~cancels out, but yeah.
I don't understand why people think this is a great introduction to modal logical without explaining the semantics and underlying meaning of accessibility relations it just a bunch of letters and axioms.
I'm glad to discover you Chanel. Thank you for this video. It will be great if English subtitles were provided to understand better.
Hmm. Odd they usually auto-generate. Perhaps this video is too old to have them. Most of the newer videos should have subtitles, and in most of the newer ones most of the words are written on screen too. Thanks!
Great video
Great video. I was wondering if you could do a video on the semantics of simplest quantified modal logic?
+brantbrns1 Once we are done with Tarski, I am going to do a number of videos on Kripke (in my series on Truth (th-cam.com/video/Cpl5jQpLomE/w-d-xo.html)). Depending on what direction that series does in, I may or may not cover Kripkean semantics and the applications of those semantics to various modal logics.
are p3 not need pharenthesis?
i like your videos. learned a lot from you. thank you.
+Eva Clark I'm glad you like them. Thanks for watching!
When you refer to SEF to explain the problems in axiom B, do you mean the Garson's article, right? Thanks for the lessons.
Gregory Carneiro Yep. That's the one.
Which Modality do you accept?
Interesting video. Thank you for putting this together.
Glad you enjoyed. Thanks for watching!
Carneades.org AntiCitizenX There has to be something very wrong with System B in S5.
In the corollary, if X is Possible it is Necessarily Possible, X is Possible therefore *It Is The Case*.
We could say it's possible that Thor exists, therefore it is the case that Thor exists. This doesn't work.
Potter Suppositionalist
Not quite. Let's take a look. If X is possible, then X is necessarily possible. This is the basis of Axiom 5. However, there is nothing that says if something is necessarily possible, then it is the case. Rather Axiom B gives us if something is possibly necessary, then it is the case. You cannot switch the modal quantifiers.
Think of it like this. Something being necessarily possible means that it must exist in some possible world. There is no way for it not to exist in some possible world. However something being possibly necessary, would mean that it is possible that something exists in all possible worlds. This is hard to explain, as modal quantifiers stop making intuitive sense after the first level, which is why there is controversy as to what is meant by necessarily possible or possibly necessary.
The point is that you can't go all the way from 'possible' to 'actual' as you did in your Thor example in any system as this seem patently false to most intuitions (Fatalists are a notable exception).
Carneades.org Ah, thank you for clearing that up. But there are aspects of this that still seems to defy intuition. If you have commentary, it would be appreciated.
If we defined a _possible world_ as some possible configuration of reality, including the actual reality, how can we quantify these other possibilities? Is it be merely any configuration of reality that isn't a logical contradiction? In this case, we could conceive of different possible worlds with necessary but mutually exclusive elements. This is what leads to the parodies of the MOA (contradictions).
As I understand, a _statement_ is necessary if denying it would lead to a contradiction, and it's contingent if it could have been untrue. However, it's controversial to use _existence_ as a predicate (Kant). The systems in modal logic appear to allow statements about possible configurations of reality to conclude that the subject of those statements exist in the actual world. This is fishy.
There has to be something wrong.
Potter Suppositionalist
Let's take these points one by one. First note that the question is no longer about Modal Logic, it is about modality. We've moved from syntax to semantics.
*"If we defined a possible world as some possible configuration of reality, including the actual reality, how can we quantify these other possibilities? Is it be merely any configuration of reality that isn't a logical contradiction?"*
This depends on what you believe about modality. Some people believe that possible worlds are just all of the possible configurations of the parts of the actual world (combinatorialism). Some people believe that all the possible worlds actually exist and the actual world is just an indexical like 'here' or 'now' or 'I'
*"In this case, we could conceive of different possible worlds with necessary but mutually exclusive elements. This is what leads to the parodies of the MOA (contradictions)."*
Correct. If a MGB is possible, why are not contradictory but also possibly necessary beings possible?
*"As I understand, a statement is necessary if denying it would lead to a contradiction, and it's contingent if it could have been untrue."*
Incorrect, in both category and content. Statements are not necessary. Truths are necessary. Necessity is a metaphysical notion about what must be the case. Statements are not necessary or contingent, they are analytic or synthetic. And the denial of a necessary truth does not have to lead to a contradiction, it is just something that is not the case in any possible world. Similarly contingent is just something that is true in some but not all possible worlds.
*"However, it's controversial to use existence as a predicate (Kant)."*
This is exactly what the MOA avoids. It specifically works to avoid Kan't criticism of some of the other ontological arguments (I have a whole series on them Seven Ontological Arguments (for the Existence of God)) because in this case existence is not a predicate, it is a consequence of being necessary.
*"The systems in modal logic appear to allow statements about possible configurations of reality to conclude that the subject of those statements exist in the actual world. This is fishy."*
Well let's see, if something is true in a possible world, that makes it possible in the actual world. That's fundamental to modal logic, so it's not that possible worlds have any effect on what is happening in the actual world, it is that they have an effect on what is actual in the actual world. The problem is that while you may have the intuition that this is a problem, others may not. How are you to demonstrate that one system is worse than another? Beyond just saying that it goes against your intuition?
The necessitation rule confuses me.
Does it apply to theorems or axioms? Does it apply to the axioms of epistemic logic, so if you accept that K(s)(P) -> P then is it necessarily true, as an axiom of that system?
The Necessitation Rule generally does not apply to doxastic or epistemic logic. It would imply that anything that is a logical axiom is something that we know. For a comparison of all the axioms, check out this video: th-cam.com/video/sQ-M4zJ7574/w-d-xo.html
@@CarneadesOfCyrene
What I'm thinking about, though, is the necessitation rule of alethic modal logic applying to axioms of epistemic logic. So a necessitation rule from one logic applying to the axioms of another, or are we not allowed to mix logics together, e.g.:
K(s)(P) -> P (Axiom of epistemic logic)
Therefore, □K(s)(P) -> P (Alethic modal logic necessitation rule)
As for S5, can we prove this by first taking say A is possibly necessary and substituting possibly by expressing possibly in terms of necessary? So instead of saying it is possibly necessary we would say that it is not necessarily not necessary. That would say that it is necessary.
+Jerad Clark Note that this is equivalent, but it does not prove it in any way, unless you are convinced that not necessary not necessary implies it is necessary. This seems to me that this is far from intuitive, so I would still be skeptical.
How do you derive the corollaries?
+3DMint Which ones? Most of them can come pretty easily out of the laws of logic. Take B.
B) (Ax)(x>[]x)
1)x>[]x (B, Universal Instantiation)
2)x>[]~[]~x (1, Definition of Possibility)
3)~[]~[]~x>~x (2, Transposition)
4)[]~x>~x (3, Definition of Possibility)
5)(Ax)([]~x>~x) (4, Universal Generalization)
6)[]~~y>~~y (5, Universal Instantiation (instantiating x as ~y))
7)[]y>y (6, Double Negation)
8)(Ay)[]y>y (7, Universal Generalization)
QED.
Can Modal logic be used in congunction with temporal ?
It certainly can. You should check out my series on temporal logic, particularly the video on branching temporal logic. th-cam.com/play/PLz0n_SjOttTca6krPm5TKsYDaBO6qjOP3.html
Certainly some of the most worthwhile video sites on youtube.
+lomertamahon1 I'm glad you enjoy! Thanks for watching!
Wow this was magical. I have so much to learn but some possible problems that result in "logical wizardry", or highly counterintuitive results definitely arise from the implications of 5, AND B.
You weren't kidding about how it relates to the Ontological problem. I wish I was better at processing multiple stages like this. It always feels like a rhetorical shell game if I miss a detail. "Watch the queen of hearts..." So I see what you mean that some use a rule set to prove or justify a pet conclusion.
I may need to watch this 20 more times.
it helps me instead of using abstract terms of "necessary" and "possible", to think of it as "provable" and "non-disprovable". then the axioms transform into those (and the problems with 5 and B are obvious):
Necessitation Rule: whatever can be inferred from the laws of logic is true.
Change of Modal Quantifier: if something is non-disprovable you can not prove that it is wrong.
Distribution Axiom: if it is provable that from A follows B then from A being provable follows that B is provable. (A and B) is provable if and only if (A is provable) and (B is provable). if A is provable or B is provable then (A or B) is provable.
Axiom M: if A is provable, A is true. if A is true A is not disprovable.
Axiom 4: if A is provable then it is provable that A is provable. if it is non-disprovable that A is non-disprovable then A is non-disprovable.
Axiom 5: if A is non-disprovable then it is provable that A is non-disprovable. if it is not disprovable that A is provable then A is provable.
Axiom B: if A is true, it is provable that A is non-disprovable. if it is non-disprovable that A is provable then A is true.
@@user-wv1in4pz2w impressive.
@@ravissary79 thank you
@@ravissary79 another similar way of thinking is that there are three states of confidence: "i know that yes", "i don't know" and "i know that no"
"necessary" is asserting the first one, "possible" is asserting that one of the first two is correct
@@user-wv1in4pz2w I don't think necessary is exactly like being certain a thing is true.
In a modal sense, necessary tings are not contingent. They aren't the sane as actual for the purpose of modal logic, even though if the logic is correct it's treated as if they are in a manner of fashion, because of their level of ontological grounding... I think I said that right.
Thanks
+Gabriela Duarte No problem, thanks for watching!
how do I benefit from learning about this modal logic stuff? (serious question)
its like learning a new language, increases brain connections and most likely intelligence.
Modal Logic aims to be the shortest description possible of how we relate and equate situations and things in our lives. By understanding these things deeply, you may come to understand why you think the things you think. You could become a more consistent and reliable thinker.
Supposedly it's used in system validation
It can also be used to formalize philosphical reasoning
You can also use it to formalize ethical rules, or create logics that try to capture the notion of "belief" rather than truth, and various other stuff. Google modalities. The best known and most commonly used modalities with modal logic are of course "possible" and "necessary", but you can express other modalities, even temporal modalities.
In that sense, it can also be applied in artificial intelligence.
Dynamic logic is a kind of modal logic used in computer science. If you go deeper, you realize that modal logic can describe possible world which have a similar structure to states in computer science.
@@nukeeverything1802 exactly, a system of rules which can be explored consistently within a possible world is functionally a logical simulation of a reality of some kind.
Virtually reality is a kind of computer processed version of modal logic teasing out the functions within a possible world.
Why are the axioms here presented as material conditionals and not entailments? Shouldn't these be entailments? Thank you.
+cheryllayne I chose not to make this arguably small but confusing distinction in this video since the focus is on Modal Logic. However I will make this distinction more clear when we talk about classical and non-classical logics, particularly when dealing with connexive logic.
I just was told about modal logic so thought I would see what it is about and how I could use it. I am completely lost here, seriously I am... is there a pre-basics? To maybe learn what these terms means?
Start with the 100 days of logic, then you can build up to this video.
Is the reason that people (like myself) get confused about Axiom 5 due to this: that when the corollary of Axiom 5 says "possibly necessarily A" means "necessarily A", the Axiom is referring to a situation where A is known to be necessary (and has thus been identified as "necessarily A") and saying "possible" is just referring to the fact that what is necessary is possible?
Or is Axiom 5 literally saying that whatever might possibly exist by virtue of its own nature (i.e. be necessary) - but which we do not know if it exists by virtue of its own nature or not - definitely *does* exist by virtue of its own nature?
Yusuf1187 Note that this is all metaphysical. There should be nothing about knowledge at all. The axiom simply says that if in some possible world, there is some A that exists in all possible worlds, that A does exist in all possible worlds. The problem here is that mixing different modal operators means that our semantics cannot catch up with our syntax. Or, in other words, there is no way to properly describe what it would look like in therms of possible world for something to be necessarily possible or possibly necessary. That's why people's intuitions differ on whether or not to include the axiom. There's not a clear way to understand what something being possibly necessary is. If you believe the axiom, then that can just be boiled down to necessary.
Thank you, I think my wording was poor (saying "known" was misleading). So I guess it falls under my first interpretation, which makes sense to me. I don't see any problem with the axiom in that sense.
However in regard to the Modal Ontological Argument, I think I would still disagree with it on the basis of 1) some of the points you mentioned in your series on it, and 2) also because I don't think that it makes any logical sense to define any entity as including "necessity" since being "contingent" or "necessary" are never really independent properties in and of themselves, but rather are descriptions of the nature of the other combined properties that define a thing. i.e. Would it be inconceivable for that combination of properties (omnipotence, omniscience, and perfect goodness in this case) to *not* be instantiated in all possible worlds?
In other words, a thing is contingent or necessary as a *result* of its other properties, not by necessity of contingency being a property within the entity's definition. So if that view is correct, including necessity or contingency into an entity's definition would simply be nonsensical, so we'd have to reject the MOA's definition of God as being incoherent before the argument even begins.
Do you think that objection is valid? You are more familiar with modal logic and the MOA than myself.
Yusuf1187 That looks like an interesting objection. It seems to me that it will boil down to the original objection of equivocation. In the sense that the theist will claim that the God that they are thinking is possible has necessity as an essential property, while the atheist will claim that the possible God that they are thinking of would only have it as an accidental property if he did in fact exist in all possible worlds, but does not and therefore only exists in some but not all worlds.
Yusuf1187: "Or is Axiom 5 literally saying that whatever might possibly exist by virtue of its own nature (i.e. be necessary) - but which we do not know if it exists by virtue of its own nature or not - definitely does exist by virtue of its own nature?"
One of the confusing features of formal logic is that even though the propositions "If P then Q," "P only if Q," and "If not Q then not P," have different connotations in English, they are all considered to be logically equivalent and essentially interchangeable within formal logic; they all mean that it is not possible (in some sense) that it is both the case that P is true and Q is false. So the following are all logically equivalent characterizations of the S5 axiom.
"If it is possible A is necessarily true then A is necessarily true"
"A cannot possibly be necessarily true unless A is necessarily true."
"If A is not necessarily true, then it is not possible A is necessarily true."
So yes, working in S5, if you establish it's possible A is necessarily true, then you may conclude A is necessarily true. But this is because working in S5, in order to establish A is possibly necessarily true, you must show it's necessarily true. There's no such thing as a free lunch in formal logic.
Once one understands this, the modal ontological argument quickly falls apart.
S5 is meant to model metaphysical possibility and necessity. As Carneades pointed out, when you say "but which we do not know if it exists by it's own nature. . ." you're appealing to epistemic possibility (what we can in principle know) which is best modeled in S4 rather than S5. When one mixes S5 with an epistemic interpretation of possibility, one gets nonsense.
"In other words, a thing is contingent or necessary as a result of its other properties, not by necessity of contingency being a property within the entity's definition."
I share your distaste at making necessity or contingency a defining property of an entity. The machinery of formal modal logic can handle it, but we shouldn't be surprised when we get queer results.One way of looking at it (this was the position of Frege, others including Plantinga I think, would disagree) is that necessary existence is not a property of an entity, but a property of the definition of an entity; that it is necessarily instantiated.
So for axiom 5, []A, Since we can express A as ~[]~A then we can substitute this into axiom 5 giving us, ~[]~[]A => []A =>A.
+Jerad Clark That sounds correct. Yes.
+Carneades.org For the corollary of axiom 4, isn't it A => A because A=~[]~A and so if we negate this equality and substitute A=~B it is
~~B=[]B and then using axiom 4 []B => [][]B we get
~~B => ~~(~~B) which simplifies, getting rid of double negatives to ~~B => ~~B and so if we carry out the substitution.
~A => ~A... Oh I see, you can't negate this because that'd be denying the antecedent, but the contrapositive is A => A. Well that was accidental but Q.E.D.
I don't think of S4 and S5 as being controversial. They just model different notions of possibility. S4 models apodictic modality, the modality of provability. Something is necessarily false if one can derive a contradiction from it, or if it is obviously impossible due to the meaning of the terms making it up, and is possibly true otherwise. S5 models alethic modality, where necessity and provability are not always the same thing. I would agree that S4 better matches my intuitive notions of possibility, but that doesn't mean S5 is "wrong." In fact the idea of a formal axiomatic system being right or wrong doesn't really make sense.
Working in S4, it's reasonable to grant the premise of the modal ontological argument, at least provisionally, but the argument is not valid. When people argue that we should accept it's possible a MGB exists because no one has proven it hasn't, or because the idea appears coherent, they are appealing to intuitions that are valid in S4, to possibility in terms of provability. Working in S5, the argument is valid, but it would only be rational to grant the premise of the argument if one already agreed with the conclusion.
Recall that in formal modal logic, If P then Q is logically equivalent to P only if Q. Classical logic has no notion of cause and effect, and makes no distinction between the truth of P somehow forcing the truth of Q, or the truth of P requiring the truth of Q.
Now consider the B axiom: If it's possible something is necessarily true, then it's true. That's logically equivalent to, something can be possibly necessarily true only if it's true. Those two statements sound different because we naturally think in terms of cause and effect, but from the point of view of the formal logic we are using, they are indistinguishable.
Now, suppose someone wishes to show it is possible a MGB exists. What do they need to do? Well, they need to establish it's possible it's necessarily true a MGB exists, because they've defined a MGB so that it must necessarily exist in order to exist at all, and therefore it won't be possible for it to exist unless it's possibly it necessarily exists. So, what do they need to do to establish it's possible it's necessarily true a MGB exists? Well, at the very least they need to establish a MGB exists; that's what the axiom from B tells us; a minimal requirement for it to be possible for it to be necessarily true a MGB exists is that a MGB at least exist! And here it becomes evident the argument is worthless as a tool of persuasion.
But of course advocates of the MOA typically ignore the B axiom when considering what they need to show to establish the major premise of the argument. They behave as though we're working in S4, and that we may provisionally grant something to be possible if it seems coherent. And then they turn around an invoke the axiom to establish the conclusion of the argument. That's what's called, trying to have your cake and eat it too.
I'm working on a video series on all this, but my progress has been slow. ^^
That's roughly how I think about it as well. S5 is perfectly intuitive as long as people understand the ideas of possible and necessary employed in the axiom. It seems to me that most objections to it are rooted in the fact that they have esthetic objections to the definitions, which is fine, but doesn't really make S5 controversial any more than Base 10 arithmetic becomes controversial because someone prefers Roman numerals.
I agree with most of what you are saying. I think it is important to point out that there is a very live debate in modal logic right now around which is correct, or in your schema which is truly apodictic modality. There are many theists that hold that when we are talking about S5 we are actually talking about real, metaphysical necessity.
*"In fact the idea of a formal axiomatic system being right or wrong doesn't really make sense."*
Wait, wrong being similar to false. False being not corresponding to reality (by most definitions). It seems that most would say that many axiomatic systems do not correspond to reality. Such as the axiomatic system that includes both p and not p, or something more complicated like Frege's Naive set theory. Perhaps it is the claim that these axiomatic theories correspond to reality that is not the case (not the theories themselves) but that is exactly what the proponent of the MOA is doing.
I would love a video offering more explanation of your definitions of apodictic and alethic modalities as they are dissimilar to the standard Kantian or Aristotelian definitions of those words (as it seems to me we have discussed before).
Gnomefro
Be careful. When S5 is claiming that there are specific notions of necessity and possibility employed in it, it is talking about full throated metaphysical necessity. If S5 and the first premise of the MOA are the case, the consequence is God. That is not disputed by any philosopher that does not dispute logic itself. If you claim that Axiom B implies some other notion of necessity and possibility then it is just false because the symbols used there are talking about metaphysical necessity and possibility. An apposite comparison to math would be how to define zero over zero. This is something that is about what is similar to reality, not simply what is consistent within a system.
I'm not sure how it would be possible to deny that S5 is valid. You could get the same results using (what is to my knowledge) completely uncontroversial things in modal logic:
1. ◊p
2. □p ∨ □~p
3. ◊p ↔ ~□~p
4. ~□~p (1 and 3)
5. □p (2 and 4)
So, as long as we're talking about something that is analytic in nature (premise 2), then possibility will always entail necessity.
I assume by analytic you mean either necessarily true or necessarily false?
S5 allows to conclude p is necessarily true given that it is possibly necessarily true, without the additional assumption it is analytic. Another way of expressing S5 is to say, if it is possible a proposition is analytic, then it is analytic.
If X is a necessary condition for Y but Y is not necessarily true, can it be said like "(Y>[]X)" ?
No. Necessity in this formation is about being true in all possible worlds, not necessary conditions for something. That is a different meaning of the word. th-cam.com/video/ibjL90iY1d0/w-d-xo.html
Thanks for the response. I've seen contingency depicted as "A>(B&~B)"; is that correct? So, my situation is that I'm trying to describe a necessary condition for knowing _x_ is that _x_ must be the case, which will then be compared to contingency. Would it be correct to use dedicated predicates for this notion of necessity/possibility? For example: "Ksx>Nx" for necessity and "Fs>(Psa&Psb)" for contingency?
by that logic the multiverse is necessary since it possibly necessary right?
+0cards0 By S5 if something is possibly necessary, then it is necessary. But be careful that you don't mix up your modalities too much. There is a difference between nomic modality that is usually used for multiverses (limited by the laws of physics) and metaphysical modality (limited by the laws of logic). This video is talking about the latter, which is generally seen as casting a wider net than the former. th-cam.com/video/ilDowk0DvMw/w-d-xo.html
+Carneades.org
so are you saying that by s5 a multiverse is not necessary because its not metaphysical?
btw can you tell me whats wrong with the s5 argument? i mean its not valid right?
+0cards0 S5 is not an argument, it is an axiom of logic. Axioms are not valid or invalid. They must be assumed without argument. The question is, do we want to assume S5 without argument. The point here is that there are some compelling reasons not to. With or without S5, a multiverse is only necessary if it is a logical truth that one exists, like either p or not p is the case.
+Carneades.org
sorry but i cant find the logic in this axiom, Possibly necessarily X --> Necessarily X, that does not follow, dont you agree?
+0cards0 To be clear, axioms are not the kind of things that follow. They are the kinds of things that map onto reality or not. It does not seem to me that S5 maps onto reality []X>[]X, but that does not make it invalid or illogical in some way. Reason is a slave to the passions, logic can only function with axioms that we irrationally choose.
How is A>possibly A a corollary of M?
+Barton Smith-Johnson
P1) (Ap)([]p>p) (Axiom T/M)
P2) []~q>~q (P1, UI) (th-cam.com/video/eU3JmbrgmSc/w-d-xo.html)
P3) ~~q>~[]~q (P2 Trans) (th-cam.com/video/63e482YP29A/w-d-xo.html)
P4) q>~[]~q (P3, DN) (th-cam.com/video/NR6E6ZajOqg/w-d-xo.html)
P5) q>q (P4, CMQ)
P6) (Ap)(p>p)
Hope that helps!
My head hurts!!! 😯
I feel like this is the level of logic where it's just theory for situations that never occur. To the point where even students of philosophy just tune out lol.
Some of these axioms seem dubious - I'm not sure I understand this correctly, but by necessary I understand probability equal to 1, by possible - probability (0,1>. Axiom @10:39 says: if it is possible that A is necessarily the case, then A is true. There's a chance that there's a 100% probability that A is true, therefore A must be true? What is it I don't understand here? May God bless you for these videos!
[]P > P
P=1 => Pe(0,1] - the other axioms should be fine as well. Just experimenting, really :)
Now:
[] [] = 1*1 = 1 = [] (axiom 5)
~ = ~(0,1] = 0 = []~
= (0,1] * (0,1] = (0,1] = (axiom 4)
Also, the difference between []A and A would be that the first is a calculated probability, and A is the verified state. Now we can see why []A>A but A does not > []A.
So the whole modal logic could be obtained by applying the rules of logic to probabilistic statements.
+Pawel Wysocki Forgot what the point was:
[] A > A
1*(0,1] (A is true) => A is true
does not seem to obtain, because 1*(0,1] I understand to be equal to (0,1]. That's just reformulating the doubts you mention in the video, sure. So really:
1*(0,1] (A is true) (0,1] (A is true) *=> A is true
And sure, that's supposed to be an axiom, but at least we can see why people don't like it.
+Pawel Wysocki
One last thought:
~ = ~[] = [0,1)
I haven't checked it, but if we take into account the scope of ~, the other axioms should be fine as well, except for the corollary of axiom B.
+Pawel Wysocki Those are some really interesting ideas. My intuition is that it may be possible for use to express probability in some type of modal logic, but it does not seem to map onto Alethic Modal Logic, which is what we are discussing here (th-cam.com/video/NRMf-4PDljY/w-d-xo.html).
The reason that it is not going to map on is that if p is the case in the actual world, it seems that most would say that it has a probability of 1. However if some p is the case in the actual world, that does not make it necessary, because it could be different in another possible world. It seems that once you have verified some state, when you then calculate the probability you will arrive at 1 because you have verified that state. So p will imply []p.
Furthermore there could be something with a probability of 0, which is not logically impossible. It could be the case that according to the physical laws of the actual universe there is no way to create a perpetual motion machine. So the probability of creating such a machine is 0. However this does not mean that this is logically impossible, merely that it is nomically impossible (th-cam.com/video/ilDowk0DvMw/w-d-xo.html).
It would be possible for you to simply define probability in line with our understanding of logical modality, allowing for things which are nomically impossible to have a probability higher than 0 and claim that things that have already happened can still be characterized as having a probability of less than 1 (though I would bet this would grind against many people's intuitions about probability). Or, conversely you could use this as a basis for a completely separate type of modal logic. It seems to have many of the characteristics (th-cam.com/video/JHyfy0Chcs4/w-d-xo.html).
Finally, it sounds like you would be very interested in Bayesian Epistemology, if you have not studied it before, you should really check it out(th-cam.com/video/YRz8deiJ57E/w-d-xo.html)!
+Carneades.org Hi, thanks for your comment, I was going to reply earlier but I'm still trying to find my own answers to some of the questions to do with modal logic.
By the way, it seems logically impossible to create a physical machine that defies the laws of physics.
In formal logic, P=>Q means that it cannot be the case that P is true and Q is false. P=>Q is used to model both the idea of Q following P and the idea of P requiring Q to be true.
For example, suppose I want to say, if I jump into the pool I will get wet. If P=I jump into the pool, and Q=I get wet, I can say this by writing P=>Q, it's not possible I'll jump into the pool and not get wet.
Suppose I want to say, in order to be able to join the army, I must be at least 18. If P=I am able to join the army and Q=I am at least 18, then I can say this by writing P=>Q, it's not possible I am both able to join the army and not at least 18.
Formal logic doesn't distinguish between these two uses of P=>Q, because there is no sense of causality or time within formal logic.
Bearing this in mind may make it easier to understand the axiom from B, If it is possible P is necessarily true, then P is true. Put this way, it sounds odd; how can the mere possibility of P being necessarily true somehow cause it to be true? This doesn't make sense. But suppose we interpret it as saying, in order for it to be possible P is necessarily true, it must at least be true. This at least makes sense. It means we have a high requirement for admitting it's possible something is necessarily true. If something is false, it might possibly be true, but it cannot . . possibly. . be necessarily true, at least if we're working in B or S5.
Understanding it this way explains the problem with the MOA. The advocate of the MOA hears that if it's possible a MGB exists then a MGB exists (which is a direct application of the B axiom), and thinks, oh, if I can show it's possible a MGB exists, then I can conclude a MGB exists. But interpreting the B axiom as written above, what this says is, In order for it to be possible a MGB exists, a MGB must exist. Of course the advocate of the MOA typically ignores this interpretation, in the belief that there is some way of showing it's possible a MGB exist short of showing a MGB exists. This results in appeals to intuitions about possibility that are appropriate to epistemic rather than metaphysical possibility, and which are modeled in S4; not S5.
(Edit: inserted a "possibly" I should have put in originally)
It seems to me that most of the unintuitive results come from an ambiguity in what "possible" means. For example, I might say, "It is possible that the Reimann zeta hypothesis is true," meaning that I don't know that it is false. However this is different from what possible means in modal logic. In modal logic, for something to be possible, it must not be necessary that it is false. In the case of the Reimann hypothesis, it might not be the case that the Reimann hypothesis is possible. If the Reimann zeta hypothesis is false, then there is no possible world in which it is true, i.e. if it is false, it is necessarily false and thus not possibly true. By this definition of possible, it is NOT possible that the Reimann hypothesis is true if it is actually false, regardless of whether or not I know that it is false. Thus we can't say for sure that the Reimann zeta hypothesis is possibly true. By this definition of possible, S5 seems obviously true: If A is not necessarily false, then it is necessary that A is not necessarily false, or, if you want to frame it in terms of possible worlds: if A is true in at least one possible world, it is true in all possible worlds that A is true in at least one possible world.
You seem to be pointing out the distinction between epistemic modal logic and alethic modal logic. The former is talking about what it is possible to believe or know based on your current beliefs. The latter is dealing with what is actually possible (i.e. true in some possible world) irrespective of your knowledge or beliefs. It might be epistemically possible for the Reimann zeta hypothesis to be true (it is not contradictory with any of your current beliefs) but not possible in terms of alethic modal logic (it is not true in any possible world). Check out my series on the three months of modal logic for more on alternative versions of modal logic.
it helps me instead of using abstract terms of "necessary" and "possible", to think of it as "provable" and "non-disprovable". then the axioms transform into those:
Necessitation Rule: whatever can be inferred from the laws of logic is true.
Change of Modal Quantifier: if something is non-disprovable you can not prove that it is wrong.
Distribution Axiom: if it is provable that from A follows B then from A being provable follows that B is provable. (A and B) is provable if and only if (A is provable) and (B is provable). if A is provable or B is provable then (A or B) is provable.
Axiom M: if A is provable, A is true. if A is true A is not disprovable.
Axiom 4: if A is provable then it is provable that A is provable. if it is non-disprovable that A is non-disprovable then A is non-disprovable.
Axiom 5: if A is non-disprovable then it is provable that A is non-disprovable. if it is not disprovable that A is provable then A is provable.
Axiom B: if A is true, it is provable that A is non-disprovable. if it is non-disprovable that A is provable then A is true.
You seem to be the type to truly wholeheartedly believe that logic truly is the heart/engine of philosophy, not wonder, not the ethics of the Other, not metaphysics, and not epistemology. Logic comes before all of the previous.
What is the correct order to watch this videos...?
The 100 days of logic is here: th-cam.com/play/PLz0n_SjOttTcjHsuebLrl0fjab5fdToui.html. Once you have made it through that, check out the three months of modal logic th-cam.com/play/PLz0n_SjOttTfP_liEHPNCzvESZsh5eirP.html
It sounds like you are just uncomfortable with S5.
What gave you that idea?
S5 IS GARBAGE LOGIC.
@@dooganchode9447 You sound very passionate.
@@dooganchode9447 why?
If you do not accept system T they you are saying it's ( [necessary] A does not imply A ). However, the distribution law in K logic would then break under a change of variables definition (A > B --> C), as you would have ( [nec] A > [nec] B ) > [nec] C giving (A > B) > C using system T. If it is possible that this is false (system T doesn't hold) then it is possible that the very transitivity of truth if false; viz. possible that ( A > B ) > C is not true even if it is a definition (seems to me to be proof by contradiction).
Can anyone help me out as to how a position where system T does not hold but system K does is tenable? I assume this must be a provably valid position, and therefore that my musing above are flawed.
To be clear, if you do not accept T, it does not imply that you deny it. I am unclear what you thing that the argument that would lead to a contradiction is. A denial of T would constitute an affirmation of
~T) ~[]([]A>A), (K does not allow us to distribute necessity under the scope of a negation) so we then can conclude
P1) ~([]A>A) (~T, CMQ)
P2) ~(~[]AvA) (P1, Impl)
P3) (~~[]A&~A) (P2, DM)
P4) ([]A&~A) (P3, DN)
This may seem like a contradiction, but it is not, remember that Alethic modal logic is not the only type of modal logic there is (th-cam.com/video/NRMf-4PDljY/w-d-xo.html). It makes complete sense for Deontic Modal Logic (th-cam.com/video/sQ-M4zJ7574/w-d-xo.html). Check out the full three months of modal logic for more: th-cam.com/play/PLz0n_SjOttTfP_liEHPNCzvESZsh5eirP.html
Thank you very much for your reply. I think I should clarify that I did not, in my attempt above, claim (or use) that [not accept]T > [deny]T. I tried to express not accepting T as the expression that it is possible (not necessary) that T is false; ( [necessary] A does not imply A ). Following that through as above, if its *possible* that T is false then its possible that (A > B) > C is false even if it is a definition of C ( and therefore []((A>B)>C) ).
Your expression of denial of T makes sense (a very uneasy sense to me, you have given me much to read and think about), but it does not show me where my thinking went wrong (if it did). Is my expression of doubting (not accepting) T not adequate/accurate and if so, why? Or is is that I need to clarify the argument from that through to the contradiction?
Hello, would like to reply to your reply to my comment posted 4 days prior but I am responding from my phone and am unable to do so. I posted that if []a, then we can substitute a for~[]~a and therefore arrive at ~[]~[]a=> [][]a=>a. It seems that I am treating this as one would in algebra where ~ acts as a negative sign for integers. This led to my premature conclusion. Sorry, I am new to logic. Anyways love your channel and enjoy conversing. Maybe some day we could debate your stance on gnostic atheism IE dogmatic atheism.
Can one formally reduce the entirety of Modal Logic into ordinary logic using the "in possible worlds" idea?
+Arya Pourtabatabaie Kripkean Semantics does just that. I have not covered it officially, but it is hiding behind all of my videos on Modal Logics th-cam.com/play/PLz0n_SjOttTfP_liEHPNCzvESZsh5eirP.html
According to the Stanford Encyclopedia of Philosophy:
"It has been shown that S5 is sound and complete for 5-validity (hence our use of the symbol ‘5’). The 5-valid arguments are exactly the arguments provable in S5. This result suggests that S5 is the correct way to formulate a logic of necessity."
Source: Garson, James, "Modal Logic", The Stanford Encyclopedia of Philosophy (Spring 2013 Edition), Edward N. Zalta (ed.), URL = .
So is it fair to interpret this as S5 not being so controversial after all?
Not at all. In the very same SEP article they go into why B and S5 are controversial (see the quote below). A logical system may be complete and sound without in any way representing the way that the world works. The question is whether you accept that if something is possibly necessary then it is the case (not if the system that contains such an axiom can prove all of the claims in that system). Some people claim that it is. Some people claim that it is not. One of the reasons that I didn't offer this point in my MOA objections is that it does seem to be more of a notion of preference around your understanding of modality and the MOA proponent can easily bite the bullet and say that they are committed to S5. Then it is on the interlocutor to show that this commits them to something outlandish. Perhaps in a future video I will speak on why the axioms of S5 end up committing one to strange beliefs about modality. For now it should suffice to say that most people will balk at the claim that "it is possible that it is necessary that A implies that A is the case in the actual world" which is implied by S5. It's soundness or completeness cannot make such a claim more palatable.
"It is interesting to note that S5 can be formulated equivalently by adding (B) to S4. The axiom (B) raises an important point about the interpretation of modal formulas. (B) says that if A is the case, then A is necessarily possible. One might argue that (B) should always be adopted in any modal logic, for surely if A is the case, then it is necessary that A is possible. However, there is a problem with this claim that can be exposed by noting that ◊□A→A is provable from (B). So ◊□A→A should be acceptable if (B) is. However, ◊□A→A says that if A is possibly necessary, then A is the case, and this is far from obvious. Why does (B) seem obvious, while one of the things it entails seems not obvious at all? The answer is that there is a dangerous ambiguity in the English interpretation of A→□◊A. We often use the expression ‘If A then necessarily B’ to express that the conditional ‘if A then B’ is necessary. This interpretation corresponds to □(A→B). On other occasions, we mean that if A, then B is necessary: A→□B. In English, ‘necessarily’ is an adverb, and since adverbs are usually placed near verbs, we have no natural way to indicate whether the modal operator applies to the whole conditional, or to its consequent. For these reasons, there is a tendency to confuse (B): A→□◊A with □(A→◊A). But □(A→◊A) is not the same as (B), for □(A→◊A) is already a theorem of M, and (B) is not. One must take special care that our positive reaction to □(A→◊A) does not infect our evaluation of (B). One simple way to protect ourselves is to formulate B in an equivalent way using the axiom: ◊□A→A, where these ambiguities of scope do not arise."
Source: Ibid.
Carneades.org Hmmm well it doesn't seem like too much damage has been done here to S5. It primarily seems like people balk at the claim but then again lots of people balked in regards to the heliocentric model. I guess we'll just wait and see where the conversation takes us.
GainingUnderstanding
In the end if you feel that if something is possibly necessary then it is the case, you can help yourself to S5. I have never seen a convincing argument for such a claim so I'll refrain from assenting. Axiomatic systems like that always worry me as the basic principles are always completely unjustified. But like the Modal Ontological Argument that rests on it, the central premise is simply taken on faith (unless you have an argument for B). You should talk to Roderic Taylor (that also commented here) as he has some strong opinions on S4 and S5 and what they should be used for.
Carneades.org Well, with all due respect you are a radical skeptic. Apparently you haven't seen a convincing argument for any claims whatsoever, whether its the affirmation or negation of S5 lol
What in the world does "necessity" contribute to a discussion? If a proposition is the case then it is in fact necessarily the case, yes? And if it is not the case, what would we be doing using it in the affirmative anyway? Only when we dont know the truth value of a proposition can we say its possible, yes? Im just not seeing the point to the modal logic. It seems we're adding symbols that complicate our system, obfuscate our arguments, without contributing anything meaningful. I know Im probably being very naive but I at least understood the point of predicate logic; I just dont see a point in distinguishing that a statement is necessary from a statement in general. To me P implies necessity of P, and not P implies the impossibility of P; if a statement is true its already known to be true and so its necessarily true, and if a statement is false its known to be false and so cannot possibly be true, yes?
Simply because a statement is true, that does not make it necessary. Necessity means being true in ALL possible worlds. Just because a statement is true in the actual world does not mean it is true in any and all hypothetical scenarios.
@@Nicoder6884 All possible worlds? This world is the only one that matters. I can imagine up a world where contradictions are perfectly allowed, therefore modal logic is nonsense.
@@leesweets4110 Think back to the part about truth tables; each row of the truth table is a possible world. If you write out the truth table for something like "(P v Q) v ~(P v Q)", you will notice that it is false in all rows; in other words, it is false in all possible worlds.
@@Nicoder6884 Sounds like you just reduced modal logit to propositional logic.
@@leesweets4110 Well, not quite. Regular propositional logic doesn't have a symbol that represents "in every row of the truth table". Additionally, you can also add the modal quantifiers onto predicate logic statements.
If it is possible that it's necessary that 1+1=1, then it is necessary that 1+1=1. I'm not sure about those corollaries.
The problem is that I don't know what you mean by "It is possible that it is necessary" the standard possible world semantics break down. And, the problem with corollaries is that they are saying the exact same thing, and if you accept one, you are committed to them all.
I'm talking about the corollary to axiom 5, and the fist sentence I wrote was an example of that corollary where I replaced A with (1+1=1). Maybe I should've made it more clear what I meant by that statement.
*first
I wanted to know if those corollaries could fit even with situations like the one I wrote in the original comment.
And if you're wondering whether there is a world where it is possible that 1+1=1, there is such a world, and it's called Boolean algebra.
Doesnt that hole debate confuse metaphysical and epistomological necessety. This is al metaphysical... Not epistomological.
This is a logical framework that can be applied to various types of modality. My full series "3 Months of Modal Logic" explains more. th-cam.com/video/JHyfy0Chcs4/w-d-xo.html
Yes! I'm acctualy watching that one now. Its realy good.
You shouldnt use words you dont understand. Use thw word simple when you mean simple. Simplistic does not mean simple.
😂😂😂 !!! Lol
Try to spell the correctly next time or consider actually looking over what you type instead of vomiting it out because your feelings are hurt
The corollary to S5 simply makes no sense. Even if it were true that the possibility of necessity implies necessity, it certainly isn't obvious and therefore would require argumentation, so how can such a controversial statement possibly serve as an axiom?
Well the idea is a little simpler given kripke semantics it just says that what is necessary doesn't change from one world to another. That requires argumentation but its not as insane as plain language would suggest
@@jhonjacson798
Sorry, 1 year on and I just saw your response lol!
I guess thinking about it in possible worlds semantics actually makes more sense intuitively. If it's possible that something is necessary then it must exist necessarily in some possible worlds, but from this it follows that it must exist in all possible worlds otherwise it would be contingent rather than necessary.
Hmmm, I guess it makes sense but it still seems wrong somehow!
I think it comes down to the fact that that reasoning works only if we are talking about metaphysical possibility, whereas often when we talk about possibility we are not talking about metaphysics, but rather about the limits of our knowledge and saying that there may be a possibility due to our incomplete knowledge.
When I say that something is possible it is due to intellectual humility as I do not want to assert something as being impossible if I don't know, but I'm not actually saying it is a metaphysical possibility either - I just don't know.
I think theists can sometimes conflate the two to create a disingenuous argument.
@@Oners82 that's a good point actually. In Kripke semantics then you'd be saying that maybe it's possible, since possible is now speaking about a positive state of existence.
One argument against S5 is to say that a human notion of what is possible is dictated by what is conceivable to us. Like a dog could not conceive of a world with certain colors but we can. So in that sense the dog's multiverse of possible worlds is smaller than a human's multiverse. To state otherwise would be to posit possibilities that we would have no means of conceiving or believing in just because some alien says it's possible and we silly hoomans can't sense.
check out wittgensteinian combinatorialism, it's a compelling case to reject the axiom of S5 modal logic.
@@jhonjacson798
Yeah, I mean if a theist asks me, "do you think it is at least possible that god exists as a metaphysically necessary being" I would probably answer, "sure, I guess it's at least possible".
Then they use this argument to show how god must necessarily exist.
This is of course a completely disingenuous argument though because I don't know if such a being is possible, maybe such a being is metaphysically impossible but obviously I can't prove that so taking on such a burden of proof is rather silly.
It's a slick but dishonest argument which commits a fallacy of ambiguity regarding the word "possible".
"One argument against S5 is to say that a human notion of what is possible is dictated by what is conceivable to us."
I'm not sure that is a very good argument. For example it is impossible to conceive of 4 dimensional curved spacetime so we have to use 3 dimensional analogies but we still accept it is possible.
Same with much of the weirdess of quantum mechanics - we cannot conceive of such things, we only believe it because the math produces predictions that agree with experimental results.
Unless I'm not understanding what you mean by "conceive", but your dog analogy suggests you mean visualise, or intuitively understand.
@@Oners82 We can understand that certain things bounce back certain frequencies of light waves after light touches it but if you don't have functioning eyes the object wouldn't look any different. If you don't see color you can't conceive of a world where something looks red because you can't see what the color red looks like. So to someone who doesn't see red there isn't a possible world where something "looks" red, even if it bounces back light waves that fit what we see as red.
We can understand 4d space and much like a person who sees in monochrome can understand the dynamics of color and what it is and can even acknowledge it exists. But they can't think of a blue ball and a red ball looking different (aside from it looking like a different shade of black or white).
I am probably not explaining the position as best as the paper I read on this did though. I'd suggest googling witgensteinian combinatorial is, there is a specific paper that I read that expounds on the point better than I could.
I dislike the sharp attack on each utterance. Makes it a pain to hear. Still, good clarifications in substance.
the fuck
If you are confused, start the 100 days of logic series at the beginning. By the time you get here it should all make sense.
27 minutes :) Better get a notebook.
And this is just the basics! I'll probably do another series in the future covering more. Thanks for watching!
The axioms in S5 and B have serious fallacies in their conception.
I imagine tree diagram of epistemically spheres.. and all becomes clear. I should write a paper on it.
Why did I leave philosophy after 1 year of it at Cambridge?! I was made for it!
It’s necessary that it is possible had no meaning., due to the infinite regress possible in the ‘modality’ tree where one can always create a universe of possibilities to the side of another.. and thereby create another modality level back up one step of the tree.. it’s all just epistemology.. layers of considering possibility.
Also to say something is the case is just to rule out bubbles / universes from the final layer of the tree.. it says nothing about the previous epistemically structure of the tree.. and once you settled on a universe.. you’re not really talking about the epistemically tree at all. I think..
S5 is obviously unacceptable
Why do you think so?
@@chrismathew2295 I mean to alethic. Because a corolary of S5 is:
⯁⯀A→⯀A
if it is possible that A is necessary, then A is necessary.
Just because something is possible, doesn't make it so, much less necessarily so.
In general axiom S5 is too strong in most situations.
@@Alkis05 take it like this. Possible means that "there exists at least one world such that...." and Necessary means that "A is true in all worlds".
Thus, corollary of ax 5 would mean: "If there is at least one world wherein the fact that A is true in all worlds is true, then A is true in all worlds."
Applying this to every axiom, I dont even know why axiom S4 and S5 are incompatiblr, but Im leaning towards S4 being more correct lol idk why.