Re: modal logic and possible worlds
Title: Re: modal logic and possible worlds George Levy wrote: I have been following the latest very scholarly exchange involving different logical models in relation to the MWI, however I fail to see how it relates to my own perception of the world and my own consciousness unless I think according to those formal systems which I think is unlikely. Using different logical models to describe possible worlds is interesting but isn't it true that if the problem of consciousness (as an observer, and definer, for these worlds) is to be addressed, then the only logic that matters is the one in my, or in your, own head? Of all these logical models which one is the right one? Are all of them right? One thing is sure. Those modal logics are right for the consistent machines. This is essentially what Godel (followed by Lob, ...) proves. The rest is a matter of definition. Can we find those logic by pure introspection? The S4 first person: yes (it happens all the time from Heraclite to toposes ... The G and G* logics? Not at all, they formalize by construction the most counter-intuitive feature of the computationalist hypothesis. When Copernicus formulated the heliocentric system, he didn't go around saying that a new logic had to be used to explain the central position of the sun. He simply used a physical model. People just had to accept the new paradigm that the Earth moves even though they do not feel the Earth move. Can't we just accept the fact that the world - and our consciousness - split or merge even though we do not feel them split and merge? It seems to me that if we define a good physical model, then classical probability could do the job of formulating the decision theory desired by Wei. I think all the problem relies in the question what is a physical model? Even what is a physical reality? Does that exists in some absolute sense? With the comp hyp, that's doubtful, as I am used to argue. I think this is coherent with your relativism, as we concluded before, independently of comp! Bruno
Re: modal logic and possible worlds
Wei Dai wrote: Thank you for the explanation on S4, IL, and CL. I'm interested in more details, but rather than bombarding you with endless questions, can you suggest a book on this topic? Something that talks about what you just did, but in more detail? BM: Try perhaps the book by Van Dalen at Springer Verlag. It is good for CL and IL. For modal logic there is the excellent book by Chellas. CHELLAS Brian F., Modal Logic an introduction, Cambridge University Press 1980. But Boolos 1979 or 1993 introduces well both modal logic and provability theory (the G, G* logics). WD: Unfortunately I'm still not able to understand much of your second post yesterday. I continue to hope you will put all of your ideas into English in an organized form, whether as a paper or a book. Why don't you translate your Ph.D. thesis into English, expand it into a book, and publish it? (Like Nick Bostrom did with his thesis, except it was already in English.) BM: My PhD Thesis presupposes ALL the logical and computer-science stuff and presupposes the physicist motivations. So a translation of my thesis would not help more than my english paper (CCQ) and probably less than our conversation on the net. Perhaps I should translate my brussels thesis/technical report. It is more self-contained (yet not completely) but then it makes 800 pages, and my experience is that people reading it loose the track or the line of the reasoning. I am hoping making some technical progress to be able to write some intermediate presentation. In any case, thank you very much for what I take as encouragement. Bruno
Re: modal logic and possible worlds
Dear George, I was missing your input lately, I like this one a lot. 2 remarks: 1./ Logic in 'your', 'my', or anyody else's mind may be different. Does it allow to restrict it from being "any"? Any may be right in their own rite. We may not like 'some'. 2./ The world just HAD to accept Copernicus and his conclusions But was Copernicus right? (Partially: yes, of course). (A step forward does not make it a complete novelty. Important and salutable, but also debatable - especially when even newer ideas coincide). Thanks for the words of reason. John MIkes - Original Message - From: George Levy To: [EMAIL PROTECTED] Sent: Friday, August 16, 2002 7:39 PM Subject: Re: modal logic and possible worlds I have been following the latest very scholarly exchange involving different logical models in relation to the MWI, however I fail to see how it relates to my own perception of the world and my own consciousness unless I think according to those formal systems which I think is unlikely.Using different logical models to describe possible worlds is interesting but isn't it true that if the problem of consciousness (as an observer, and definer, for these worlds) is to be addressed, then the only logic that matters is the one in my, or in your, own head? Of all these logical models which one is the "right" one? Are all of them "right?"When Copernicus formulated the heliocentric system, he didn't go around saying that a "new" logic had to be used to explain the central position of the sun. He simply used a physical model. People just had to accept the new paradigm that the Earth "moves" even though they do not feel the Earth move. Can't we just accept the fact that the world - and our consciousness - "split" or "merge" even though we do not feel them "split" and "merge?" It seems to me that if we define a good physical model, then classical probability could do the job of formulating the decision theory desired by Wei.George
Re: modal logic and possible worlds
Thank you for the explanation on S4, IL, and CL. I'm interested in more details, but rather than bombarding you with endless questions, can you suggest a book on this topic? Something that talks about what you just did, but in more detail? Unfortunately I'm still not able to understand much of your second post yesterday. I continue to hope you will put all of your ideas into English in an organized form, whether as a paper or a book. Why don't you translate your Ph.D. thesis into English, expand it into a book, and publish it? (Like Nick Bostrom did with his thesis, except it was already in English.)
Re: modal logic and possible worlds
jamikes wrote: 007f01c24609$8a1cfa00$5e76d03f@default"> I was missing your input lately Yes, I am very busy preparing for a patent bar. But I still read the list. I don't have too much time to dig deep into the references so I can't comment intelligently when the going gets too technical. 007f01c24609$8a1cfa00$5e76d03f@default"> 2 remarks: 1./ Logic in 'your', 'my', or anyody else's mind may be different. Does it allow to restrict it from being "any"? Any may be right in their own rite. We may not like 'some'. The arbitrariness of "my," "your" or anybody's own mind point to the need for the relativistic approach which I have been advocating. The frame of reference here is the logical system residing in the observer's mind. It may not be the type of formal system which has been discussed in the list. There may be a need to develop some kind of "fuzzy" logical system for human mental processes corresponding to the formal systems already in existence. As far as I know Fuzzy Logic has not been developped to the same extent as the branches of logic that have been discussed in the list. In any case, a totally different approach involves using physical models, just like Copernicus and Einstein did. An interesting conjecture is that the "physical model" approach and the "logical model" approach will converge or even will be proven to be equivalent. 007f01c24609$8a1cfa00$5e76d03f@default"> 2./ The world just HAD to accept Copernicus and his conclusions But was Copernicus right? (Partially: yes, of course). (A step forward does not make it a complete novelty. Important and salutable, but also debatable - especially when even newer ideas coincide). I agree with you here. I have been somewhat imprecise. George 007f01c24609$8a1cfa00$5e76d03f@default"> - Original Message - From:George Levy To: [EMAIL PROTECTED] Sent: Friday, August 16, 2002 7:39PM Subject: Re: modal logic and possibleworlds I have been following the latest very scholarly exchangeinvolving different logical models in relation to the MWI, however I fail tosee how it relates to my own perception of the world and my own consciousnessunless I think according to those formal systems which I think isunlikely. Using different logical models to describe possible worlds isinteresting but isn't it true that if the problem of consciousness (as anobserver, and definer, for these worlds) is to be addressed,then the only logic that matters is the one in my, or in your, own head?Of all these logical models which one is the "right" one? Are all of them "right?" When Copernicus formulated the heliocentric system, he didn'tgo around saying that a "new" logic had to be used to explain the centralposition of the sun. He simply used a physical model. People just had toaccept the new paradigm that the Earth "moves" even though they do not feelthe Earth move. Can't we just accept the fact that the world - and ourconsciousness - "split" or "merge" even though we do not feel them "split" and"merge?" It seems to me that if we define a good physical model, thenclassical probability could do the job of formulating the decision theorydesired by Wei. George
Re: modal logic and possible worlds
On Saturday, August 17, 2002, at 08:06 PM, George Levy wrote: The arbitrariness of my, your or anybody's own mind point to the need for the relativistic approach which I have been advocating. The frame of reference here is the logical system residing in the observer's mind. It may not be the type of formal system which has been discussed in the list. There may be a need to develop some kind of fuzzy logical system for human mental processes corresponding to the formal systems already in existence. As far as I know Fuzzy Logic has not been developped to the same extent as the branches of logic that have been discussed in the list. Well, count me as skeptical that the hype about fuzzy set theory and fuzzy logic has ever, or will ever, live up to some of the claims made by Bart Kosko, Lofti Zadeh, and others. Most of what passes for fuzzy logic just looks like ordinary Bayesian probability. Here's a comment from Saunders Mac Lane in his book Mathematics: Form and Function, 1986: Not all outside influences are really fruitful. For example, one engineer came up with the notion of a _fuzzy_ set--a set X where a statement x elementof X of membership may be neither true nor false but lies somewhere in between, say between 0 and 1. It was hoped that this ingenious notion would lead to all sorts of fruitful applications, to fuzzy automata, fuzzy decision theory and elsewhere. However, as yet most of the intended applications turn out to be just extensive exercises, not actually applicable; there has been a spate of such exercises. (. pp 439-40). While maybe Mac Lane is a little too snippily dismissive, here we are more than 15 years later and what do we have? Fuzzy rice cookers which look like nothing more than rice cookers with various algorithms Newton could have calculated, fuzzy-logic elevators which are simply implementing similar acceleration algorithms, and not much else. Certainly fuzzy logic has not been significantly in the foundations of mathematics. Logicians have not been using fuzzy sets and fuzzy logic in any significant way, judging by the books and articles I've seen. I agree that formal logic is not easily applied to minds. Logicians would agree. A mind is weighing large numbers of inputs, far beyond what would normally fill an entire page of First Order Logic equationssurvival has made the ability to reason with uncertainty (a better core concept that calling it fuzzy logic, in my opinion) a survival trait. Those minds which can find solutions in the midst of noise and uncertainty tend to reproduce more than those minds which are paralyzed or too slow in reaching survival-enhancing conclusions. What we have talked about here in this sub-thread on _modal logic and possible worlds_ is an idealization of logic, just a snapshot or facet of things, in much the same way a line or a plane is a facet of the world around us (and understandable at some level by birds and reptiles even). --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
At 10:29 -0700 13/08/2002, Wei Dai wrote:
Does it mean anything that S4 and intuitionistic propositional
calculus (= 0-order intutionistic logic, right?) ...
Right.
have the same kind of models, or is
it just a coincidence? I guess Tim is saying that it does mean something,
but I don't understand what.
I almost missed this fair question. It is not a coincidence.
I choose S4 for not frightening Tim with irreflexive or non
transitive accessibility relation :-).
Kripke knew a 1933 result(*) by Godel according to which, (with IL
standing for Intuitionist logic):
Theorem: S4 proves T(A) if and only if IL proves A
where T is a function from the propositional language to the modal
propositional logic given by
T(A) = A if A is a propositional letter = A belongs to {p, q, r ...}
T(-A) = []-T(A)
T(A-B) = [](T(A) - T(B)
Kripke first discovered his semantics for general normal modal logic.
By Godel 1933 this provides him (and us!) the S4 possible world
semantics of IL.
Note that Beth developed a more awkward but similar semantics before.
The world of the S4 models (= of the IL) models are sometimes
interpreted as *state of knowledge*. This fit well with the fact
that the S4 logic in my thesis describes a pure first person knower.
(But I get S4Grz, its antisymmetrical extension, good for subjective
irreversible time).
The following remarks may help.
S4, which is a *classical* extension of CL
(Classical Logic), is capable of simulating IL. This is not an argument
that CL is better, for Godel (again) found that IL can simulate CL:
basically IL proves (- - A) when CL proves A. It's a key of IL that
IL does not prove (- - A) - A. (but IL proves A - (- - A))
Like CL admits an algebraic semantics in term of Boolean Algebra, ...
parenthesis:
(where the propositions A, B, C ... are interpreted by subsets of
a set W, the and by intersection, the or by union, the - by the
complementary, the constant f by the empty set and the constant t by
the whole set W. You can even (in our modal context interpret the element
of W as worlds verifying the formula, for example a tautology being
true in all world you see a tautology, like the constant t, is
interpreted by W). For example [A union -A] = W, i.e the exclude
middle principle is universally valid, in CL.
end of parenthesis
... IL admits a topological interpretation, where the propositions
are interpreted by open sets in a topological space. The and by
intersection, the or by union, the not by ... the interior of
the complementary. For exemple take as topological space the real
line. Interpret A by the open set (-infinity 0) then -A is (O infinity)
you see [A union -A] does not give the whole space, and this
shows the exclude middle is not universally valid, in IL.
Quantum Logic (QL) like IL, is a syntactically weaker logic than CL.
(And thus IL and QL are semantically richer). Algebraic semantics
for QL is given by the lattice of subspace of vector spaces (or Hilbert
spaces).
A good shape to remember is the following drawing (Arrows from bottom
to up = syntactically extend:
S4 B
\/
\ /
CL
/ \
/ \
IL QL
S4 gives a classical representation of IL, and B gives (thanks to
Goldblatt result) classical representation of QL.
(*) Godel was so famous that he did'nt need no more to prove his
affirmation/conjecture. It is a two pages paper without proof.
The Godel's result will be proved by McKinsey and Tarski in 1948.
Bruno
Re: modal logic and possible worlds
At 10:11 -0700 14/08/2002, Wei Dai wrote: Let me generalize my question then. Is it true that for any modal logic that has a semantics, any sentence in that logic has a corresponding sentence in non-modal quantificational logic with the same meaning? It depends of the semantics. It depends of the order of quantification. It depends what you want to do. At 10:11 -0700 14/08/2002, Wei Dai wrote: Before the invention of possible world semantics, people had to reason about modalities on a purely syntactical basis. Are there still modal logics for which no semantics is known? There are the modal logics of your servitor :-) Z, Z1, Z*, Z1* and X, X1, X*, X1* Those are well-defined modal logics. It is easy to prove them even decidable. I have partial soundness theorem for Z and Z1. No completeness! None of those logics have not even been axiomatized. (Open problems!). Remember the high constraint due to the fact that I interview sounds universal machines. The mother box corresponds to the Goedelian beweisbar predicate. If we interpret the propositional letters p, q, r ... by arithmetical sentences, and []p is interpreted by beweisbar(godel-number-of(p)) then a natural question, which curiously did not appear in Godel 1933, is: which modal logic, if any, does the beweisbar box obey? Thanks to the work of Lob, Magari, Boolos, Solovay and others we know such a logic exists and it is G. Here is a presentation of G: AXIOMS: [](p - q) - ([]p - []q) []([]p - p) - []p []p - [][]p (this one can been shown redundant) RULES:p p-q p ,--- q []p A class of Kripke frames (there exists others!) is set of finite, irreflexive, transitive frames. Actually G gives the part of that logic which remains provable by the machine itself. Solovay showed much more: the following *non normal* (indeed: NON necessetation rules) logic gives the whole propositional logic, including what is true but such that the machine cannot prove: AXIOMS: All theorems of G []p - p RULES:p p-q q In particular G* minus G gives the set of all 0-order modal propositions corresponding to true unprovable (by the sound machines), but bettable self-referential sentences. G* has no Kripke semantics, but it can be shown G* has some natural semantics in term of sequences of Kripke models. All the logics defined in my work are defined syntactically *from* G and G*, so that it is non trivial at all to find semantics. Roughly speaking the knower is defined by ([]p p), the observer/bet-ter by []p p, the observer/better-embedded-in-UD*-with-comp-true is []p p with p interpretation restricted on \Sigma_1 sentences (if they are true there are provable). The sensible observer = []p p p , with p \Sigma_1) At 10:11 -0700 14/08/2002, Wei Dai wrote: We know that in general syntactical formulas and rules are not powerful enough to always let us reason without meaning, because the set of mathematical truths that are derivable syntactically from a fixed set of axioms is just a subset of all mathematical truths. Yes but self-transforming theories or machine/brains can learn to makes bets and change themselves. That never gives the whole truth, a priori, but can help a machine to progress or just survive. At 10:11 -0700 14/08/2002, Wei Dai wrote: The rest can only be obtained by considering the semantic consequences of the axioms. I think the point of syntax is just to give us a way to obtain at least some of the truths through syntactical manipulation - a way to grab the low-hanging fruit. Yes. And it is a point of the brain/body too. Bruno
Re: modal logic and possible worlds
I have been following the latest very scholarly exchange involving different logical models in relation to the MWI, however I fail to see how it relates to my own perception of the world and my own consciousness unless I think according to those formal systems which I think is unlikely. Using different logical models to describe possible worlds is interesting but isn't it true that if the problem of consciousness (as an observer, and definer, for these worlds) is to be addressed, then the only logic that matters is the one in my, or in your, own head? Of all these logical models which one is the "right" one? Are all of them "right?" When Copernicus formulated the heliocentric system, he didn't go around saying that a "new" logic had to be used to explain the central position of the sun. He simply used a physical model. People just had to accept the new paradigm that the Earth "moves" even though they do not feel the Earth move. Can't we just accept the fact that the world - and our consciousness - "split" or "merge" even though we do not feel them "split" and "merge?" It seems to me that if we define a good physical model, then classical probability could do the job of formulating the decision theory desired by Wei. George
Re: modal logic and possible worlds
Wei Dai wrote:
Thanks for your answers. They are very helpful.
Y're welcome. I want just add something.
Your general question was Why using modal logic when
quantifying on worlds is enough. My basic answer was
that Kripke's possible world semantics works only on a
subset of the possible modal logics.
You can do modal logics without semantics. In fact modal
logic appeared because of apparent existence of modalities.
The main one is possible and necessary. But others
occurred like permitted and obligatory; provable and
consistent, believable and imaginable, etc.
The fundamental motivation of a logician is to give purely
syntactical formula and rules for manipulating formula so that
we can reason and communicating reasoning *without* any
meaning. The traditional joke is that a logician does not
want understand what he talk about!
When you do that you fall automatically on the following sort
of problem:
Take a formal theory like S4, again:
(I suppose a language with the usual logical symbol
including the propositional constant f and t, + the [])
AXIOMS: axioms of classical propositional logic
[](p - q) - ([]p - []q)
[]p - p
[]p - [][]p
RULES:p p-q p
,---
q []p
This gives at once an infinity of formula: those derivable
from the axioms by using finitely many times the rules.
I recall that -p is an abbreviation for (p-f), and p is
an abbreviation for -[]-p.
The question is: is the formula []p - []p derivable in S4?
Now that question is not so hard and you can solve it *syntactically*.
Just find a pathway from the axioms to the formula by using the
rules of inference. (That is: just prove []p - []p in S4).
But I can say this because I know the solution!
Now logician want not only prove theorems in their system, they
want also know if the system is consistent, that is, if the system
does not prove f, and question like that.
For example, and this is a *very* difficult exercise, try to prove
that S4 does *not* prove the formula p - []p.
Before the rise of semantics such question was almost not answerable
in general. You cannot solve them by searching all the proofs
because you have an infinity of proofs.
Something like Kripke semantics makes such an exercise very easy,
once you have soundness and completeness metatheorem relating your
logic (here S4) with the semantics.
Now it not very difficult to prove such completeness and soundness
theorem for system as simple as S4 (see also ref in the archive
below). I give you only the two main metatheorems we can use here.
I recall that a Kripke frame is just a set (of worlds) with some
binary relation among them (the accessibility relation).
A model is a frame with, for each world w, a function from L in {0,1}.
L = our set of propositional letters {p, q, r, ...}.
(That is: a model assigns truth to the proposition in each world).
I recall also that classical logic is verify in each world, that is:
if p is true in world w and q is true in world w, then p-q is
true in world w, etc. Now the S4/Kripke-semantics soundness
and completeness (SC) metatheorem is:
SC theorem: S4 proves A if and only if A is true in any
world of any model based on a reflexive and transitive frame.
Now, if p - []p was derivable in S4 it would follow from
the SC metatheorem that p - []p would be true in any world
of any model based on a reflexive and transitive frame.
So, to prove that p - []p is not a theorem of S4 it is
enough to find a model based on a reflexive and transitive frame
in which p - []p is false in some world.
Let us build that counterexample. For having p - []p false
in a world w1, by classical logic, you need a world with p true
in it, and []p false in it, i.e. -[]p true in it.
But -[]p is equivalent with []-p, so it is enough to join
a unique world w2 with -p true in it.
Our counterexample is:
The frame = {w1, w2}
The accessibility relation R is given by w1Rw1, w2Rw2, w1Rw2
(if you prefer: R = {(w1 w1) (w2 w2) (w1 w2)}
The model is given by making p true in w1, and false in w2.
R being reflexive and transitive, and p - []p being false at
w1, p - []p has been shown not derivable in S4.
Another use of Kripke CS result is to show that S4, for example,
is decidable (and then write a theorem prover for S4). This is easy
if you succeed in refining the completeness part of the CS theorem
above with finite frame instead of any frame. In that case
you know that if a counterexample exist you can find it.
But Kripke semantics is useless with a non normal
logic, for example a modal logic without the necessitation rule.
Chellas excellent book has a chapter on Scott-Montague semantics
(also known as minimal model) which can be used in the same
way for weaker modal logic. The Scott-Montague semantics gives
topological or quasi-topological structure on the set of worlds.
In Kripke []p is true at world w if p is true at all worlds x such
that wRx. In Scott-Montague
Re: modal logic and possible worlds
Hi Tim, just some quick comments. On Tue, Aug 13, 2002 at 10:08:50AM -0700, Tim May wrote: * Because toposes are essentially mathematical universes in which various bits and pieces of mathematics can be assumed. A topos in which Euclid's Fifth Postulate is true, and many in which it is not. A topos where all functions are differentiable. A topos in which the Axiom of Choice is assumed--and ones where it is not assumed. In other words, as all of the major thinkers have realized over the past 30 years, topos theory is the natural theory of possible worlds. Frankly I think you exaggerate :) I could be very long on this even without mentioning my thesis. But I want to be short, and in my thesis, toposes could only be used for first person semantics (and even this is still an exaggeration). Toposes are just (enlarged) S4 model, or (classical) model for intuitionist logic. At 11:15 -0700 13/08/2002, Tim May wrote: Worlds _are_ propositions. This can be misleading. In modal context we have a duality: we can define world by set of propositions (the proposition true in the world), and we can define dually proposition by set of worlds (the world in which p is true). At 15:51 -0700 13/08/2002, Tim May wrote: (You might also want to take a look at the paper by Guts, a Russian, on a Topos-Theoretic Model of the Deutsch Multiverse. Available at the usual xxx.lanl.gov site.) Thank you very much for this interesting reference (and the reference therein, including a Russian website on Everett!). At 15:51 -0700 13/08/2002, Tim May wrote: As far as the math of nonstandard logic goes, I think the most interesting application within our lifetimes will come with AI. I agree. Perhaps Wei Dai should look at the non monotonic logics and to the logics of relevance. Especially if he want escape the problem of omniscience. At 21:29 -0700 13/08/2002, Tim May wrote: Nor do I take Schmidhuber's all running programs notion very seriously. Interesting ideas to play with, and to use some tools on. [...] At 21:29 -0700 13/08/2002, Tim May wrote: Lack of even the slightest piece of evidence for all possible mathematical universes actually exist and/or the all runnable computer programs.' I also don't believe there are gods or other supernatural beings, for the same reason. If and when I see an experiment that points to there being other universes which have tangible existence, then I'll start to believe. Then I urge you to read my thesis (which results, btw, has been published about ten years before Tegmark and Schmidhuber and which results goes far away beyond, ... :) Why. Because even *without experiment*, but with just a small amount of platonism in arithmetic and computationalism in the cognitive science, you will understand that the many computations are unavoidable, and that the physical laws necessarily emerges from simple elementary relation between integers ... I am more skeptical than you, I don't believe in a *physical* universe. Actually I show that with comp physics cannot be fundamental, but must emerge from numbers and numbers as seen by numbers ... Physicalism and materialism is *just incompatible* with mechanism. Perhaps read just my Computation, Consciousness and the Quantum loadable from my URL below. I will say more in a post which I am writing to you and where I make a comment on Yetter's Functorial Knot theory. Bruno -- http://iridia.ulb.ac.be/~marchal/
Re: modal logic and possible worlds
On Wed, Aug 14, 2002 at 04:38:45PM +0200, Bruno Marchal wrote: Your general question was Why using modal logic when quantifying on worlds is enough. My basic answer was that Kripke's possible world semantics works only on a subset of the possible modal logics. Let me generalize my question then. Is it true that for any modal logic that has a semantics, any sentence in that logic has a corresponding sentence in non-modal quantificational logic with the same meaning? In other words, are there any modal sentences whose meaning cannot be expressed by quantifying directly on the appropriate objects? You can do modal logics without semantics. In fact modal logic appeared because of apparent existence of modalities. The main one is possible and necessary. But others occurred like permitted and obligatory; provable and consistent, believable and imaginable, etc. The fundamental motivation of a logician is to give purely syntactical formula and rules for manipulating formula so that we can reason and communicating reasoning *without* any meaning. The traditional joke is that a logician does not want understand what he talk about! Before the invention of possible world semantics, people had to reason about modalities on a purely syntactical basis. Are there still modal logics for which no semantics is known? We know that in general syntactical formulas and rules are not powerful enough to always let us reason without meaning, because the set of mathematical truths that are derivable syntactically from a fixed set of axioms is just a subset of all mathematical truths. The rest can only be obtained by considering the semantic consequences of the axioms. I think the point of syntax is just to give us a way to obtain at least some of the truths through syntactical manipulation - a way to grab the low-hanging fruit.
Re: modal logic and possible worlds
Bruno probably does, but I'll put my spin on it. Each distinguishable world is a description*, which is a conjunction of propositions I have green eyes _and_ I live in Sydney _and_ the twin towers were destroyed by airliners on 11/9/2002 _and_ ..., and as such is a proposition. I'm not completely convinced that one can simply apply modal logic to the set of all descriptions in this way, but it does have some plausibility. Cheers * This is the case in the Schmidhuber and Tegmark ensembles, but not so obviously true of Deutsch's Multiverse. Wei Dai wrote: Now I'm lost again. Again A is a world not a proposition so what would A or not-A mean even if A and B are comparable? If anyone else understand the point Tim is making please help me out... A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
Re: modal logic and possible worlds
Wei Dai wrote: According to possible world semantics, it's necessary that P means that P is true in all worlds accessible from this one. Different modal logics correspond to different restrictions on the accessibility relation. Before the invention of possible world semantics, people argued about which modal logic is the correct one, but now philosophers realize that different notions of accessibility (and the corresponding notions of modality) are useful at different times, so there is no single correct modal logic. That's my one paragraph summary of possible world semantics. Please correct me if I'm wrong, or read these articles if you're not familiar with this topic: http://www.xrefer.com/entry.jsp?xrefid=552831 http://www.xrefer.com/entry.jsp?xrefid=553229 My questions is, why not just quantify over the possible worlds and refer to the accessibility relation directly? This way you can talk about multiple accessibility relations simultaneously, and you don't have to introduce new logical symbols (i.e. the box and the diamond). Is modality just a syntactic shorthand now? BM: Each time you can reduce a theory in another you can considered it as a syntactic shorthand. In fact you could just as well throw all math in the basket keeping numbers, right at the start. It reminds me early programmers saying that FORTRAN was just a toy language for those stupid guy unable to manage binary code. Of course your question is not *that* stupid, for sure, :-) The first answer I would give is that Kripke Possible world semantics just works for a portion of the possible modal logics. (Sometimes called the classical normal modal logics). So, if modal logician would have defined the modal logic by the accessibility relations, they would have missed the whole forest. Just in my thesis, among G, G*, S4Grz, Z, X, Z*, X*, Z1, Z1*, X1, X1*, only G and S4Grz have Kripke semantics. (Z, X, Z1, X1 have Scott- Montague semantics, the others ... are more difficult...). Note that in first order logic, the quantifier For all and it exists are sort of modal connector and are used in your sense: the variable x denoting a sort of abtract world). But even for modal logic with possible world semantics, it is important (for a logician) to distinguish 0-order and first-order complexity, and this in a a priori, non semantical way. Also first order modal logic, would be syntactically awkward if everything was done by quantifying on the worlds. Last answer: if you take a simple modal logic like S4, that is: AXIOMS: [](p - q) - ([]p - []q) []p - p []p - [][]p RULES:p p-q p ,--- q []p You can define the accessibility relation in a first order formula. The accessibility relation, indeed, is just a transtive and reflexive relation, making the frame of world a partially ordered set (the model of intutionnistic propositional calculus). But take the famous G Lob formula: []([]p-p)-[]p, there is *no* semantics for it describable by a first order formula. The frame is an inverse of a well founded set. This means that to continue your semantical use of modal logic in a synctatical well defined way, you should use second order logic. Here, the fact that there is a simple modal formula making all the heavy work for you is *very* nice. It really simplify things a lot. To sum up, my main answer is that Kripke semantics has been invented to handle better some--not all--modalities (invented by Aristotle!). Modalities has not been invented for shortand description of Kripke worlds. It is the other way round. In the real life (like in comp!) Kripke semantics can only be used exceptionally. Bruno
Re: modal logic and possible worlds
On Monday, August 12, 2002, at 11:41 PM, Russell Standish wrote: Bruno probably does, but I'll put my spin on it. Each distinguishable world is a description*, which is a conjunction of propositions I have green eyes _and_ I live in Sydney _and_ the twin towers were destroyed by airliners on 11/9/2002 _and_ ..., and as such is a proposition. I'm not completely convinced that one can simply apply modal logic to the set of all descriptions in this way, but it does have some plausibility. I think small. Attempting to reason about entire worlds with huge amounts of state (put various ways: long description, high logical depth, high algorithmic complexity, big) is not useful...to me. So I use A and B for two possible worlds. The outcome of a coin toss, for example. The click of a geiger counter or not. Schrodinger's cat alive, or dead. These states are, as Russell notes, propositions. Or sets of propositions (or huge sets of propositions, for entire worlds). I prefer at this time to ignore the implied complexity of an entire world and just call them A and B. Two outcomes, two branches in the MWI sense, two possible worlds, two points in a lattice, two points in a pre-ordered set (see below), two points in a partially-ordered set (poset, see below). I picked WWIII happens this year (or doesn't) to illustrate the general point that modal logic applies, that classical logic cannot apply to find and implication from A to B or B to A, as they represent contradictory to each other worlds. I didn't mean it to imply that modal logic is going to somehow tell us how likely such a world is, or what life might be like in either of those worlds, etc. I just wanted to make A and B more tangible to MWI sorts of folks. (Goldblatt, in his book Topoi, uses Fermat's last theorem is true or false as the two contradictory possible worlds. At the time he wrote his book, 1979, the truth or falsity of FLT was unknown. These were two possible worlds, visualizable by mathematicians and others, each having a kind of tangible reality. In fact, something that was shown to be equivalent to FLT was the Taniyama-Shimura Conjecture about some curious relationships between elliptic functions and modular forms. And for many years before Taniyama was proved, papers would start with this perfect example of modal logic: Assuming Taniyama-Shimura is true, then People _believed_ T-S was probably true, but it hadn't been proved formally until Andrew Wiles did so, thus proving Fermat's Last Theorem as almost a trivial afternote.) A series of moments or events is drawn as a graph, with vertices linked with edges, with some events clearly coming after others, because they are causally-dependent on earlier events. But also some events _independent_ of other events, with no known (and perhaps no _possible_ causal relationship, e.g., events outside each other's light cones, i.e., spacelike intervals). This graph, this set of vertices and edges, is a per-ordered set. More than just a set, any category with the property that between any two objects p and q there is AT MOST one arrow p -- q is said to be pre-ordered. There are lots of examples of this: the integers (and the real numbers) are pre-ordered under the operation greater than or equal to or less than or equal to. Moments in time are pre-ordered. Containment of sets is pre-ordered. Following Goldblatt, I'll call the arrow R. So the p -- q example above is written as pRq. Here are some properties of pre-orders: 1. Reflexive: for every p, pRp. Example: For every p, p implies p. Example: For every real number, that real number is less than or equal (LTE) to itself. And also greater than or equal (GTE) to itself. Example: For every event, that event occurs before or at the same time as that event. (Here I'm using time, because time is the most interesting pre-order for our discussion of worlds, MWI, causality, etc.) Example: Every set contains itself (where containment is contains or is equal to). (This may say like a tautological definition. Draw pictures of sets as blobs. The motivation for this example will become clearer with later properties.) 2. Transitive: Whenever pRq and qRs, then pRs. Example: If p implies q and q implies s, then p implies s. Example: if p is less than or equal (LTE) to q and q is LTE to s, then p is LTE to s. Example: if event A happens before (or at the same time as) event B and event B happens before (or at the same times as) event C, then even A happens before (or at the same time as) event C. Example: (short version--you know the drill by now): If A contains B and B contains C, then A contains C. Discussion: These are all simple points to make. Obvious even. But they tell us some important things about the ontological structure of many familiar things. I encourage anyone not familiar with these ideas to think about the points and think about how many things around us are pre-ordered.
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 10:08 AM, Tim May wrote: This graph, this set of vertices and edges, is a per-ordered set. More than just a set, any category with the property that between any two objects p and q there is AT MOST one arrow p -- q is said to be pre-ordered. I meant to type pre-ordered in the first line above. I don't normally worry overmuch about minor typos, especially when I used the correct spelling right after the typo, but I wouldn't want anyone thinking there's some kind of per-ordered set! --Tim May
Re: modal logic and possible worlds
On Monday, August 12, 2002, at 11:18 PM, Wei Dai wrote: Tim, I'm afraid I still don't understand you. On Mon, Aug 12, 2002 at 06:00:26PM -0700, Tim May wrote: It is possible that WWIII will happen before the end of this year. In one possible world, A, many things are one way...burned, melted, destroyed, etc. In another possible world, B, things are dramatically different. Ok, but what about my point that you can state this by explicit quantification over possible worlds rather than using modal operators? I.e., There exist a world accessible from this one where WWIII happens before the end of this year. instead of It is possible that WWIII will happen before the end of this year.? That is indeed saying just the same thing (though the language is slightly different). The important part of modal logic is not in the accessible from this one or it is possible language. Rather, the forking paths (a la Borges) picture that is described by posets and lattices. There can be no implication from one world to the other. That is, we can't say A implies B or B implies A. What does that have to do with my question? Anyway A and B are supposed to be worlds here, not propositions, so of course you can't say A implies B. I don't know what point you're trying to make here. Worlds _are_ propositions. And the causal operator (time) is the same as implication. With some important caveats that I can't easily explain without drawing a picture. In conventional logic, implication is fully-contained or defined from some event A (or perhaps some combination of events A, B, C, etc., all causally contributing to a later event). There are two interesting cases to consider where implication does not follow so easily from A: 1. Possible worlds. The event A forks down two (or more) possible paths. A future where war occurs, a future where war does not. A future where Fermat's Last Theorem is proved to be true. A future where it is not. A future of heads, a future of tails. 2. Quantum mechanics. Schrodinger's cat. (It was Einstein and Podolsky's belief that classical logic must apply that led to their belief that there _must_ be some other cause, some hidden variable, that makes the outcome follow classical logic. Bohm, too. But we know from Bell's Theorem and the Kochen-Specker no-go theorems that, basically, these hidden variables are not extant.) (By the way, the book Interpreting the Quantum World, by Jeffrey Bub, has an interesting section on how modal logic applies to QM.) Bruno is much more of a logician than I am, but the various terms of logic, lattices, and set theory are analogous (probably a very efficient category theory metaview, but I don't yet know it). 1 is True 0 is False lattice infimum or Boolean meet, ^ , is conjunction (AND) lattice supremum or Boolean join, v , is disjunction (OR) lattice or Boolean orthocomplement is negation (NOT) (Understanding this is not essential to my arguments here...I just wanted to make the point that there are mappings between the languages of logic, set theory, and lattices. In a deep sense, they are all the same thing. Definitions do matter, of course, but e-mail is not a great place to lay out long lists of definitions!) This branching future is exactly what I was talking about a week or so ago in terms of partially ordered sets. If the order relationship is occurs before or at the same time as, which is equivalent to less than or equal to, A and B cannot be linearly ordered. In fact, since both A and B are completely different states, neither can be said to be a predecessor or parent of the other. In fact, A and B are not comparable. I'm with you so far in this paragraph. We cannot say A or not-A. Now I'm lost again. Again A is a world not a proposition so what would A or not-A mean even if A and B are comparable? The two forks in the road are given the same truth value weighting in this possible worlds approach. We have _assumed_ A in this fork I described, so not-A is certainly not necessarily the other path. In fact, the meaningful interpretation of not-A in the complement sense is that which precedes A, that is, the events leading up to A in this world. I realize this sounds confusing. Draw a picture. Just have three points in it, arranged in a triangle: A B \ / X Time is in the upward direction. The points/events/states X, A, B form a poset. One arrow between any two points. Pre-ordering (reflexive, transitive) and partial-ordering (reflexive, transitive, antisymmetric). We cannot, however, say X implies A because X has given rise to _both_ A and B. Besides the possible worlds situation, where we assume X could give rise to either of these events, there is also the distinct possibility that this will be the only logic we ever know for quantum mechanics. The situation X gives rise to either the cat being dead or alive at the time we make the measurement,
Re: modal logic and possible worlds
Tim, I think I'm starting to understand what you're saying. However, it still seems that anything you can do with intuitionistic logic, toposes, etc., can also be done with classical logic and set theory. (I'm not confident about this, but see my previous post in reponse to Bruno.) Maybe it's not as convenient or natural in some cases (similar to how modal logic can be more convenient than explicitly quantifying over possible worlds even when they are equivalent), but if one is not already familiar with intuitionistic logic and category theory, is it really worth the trouble to learn them? For example, posets can certainly be studied and understood using classical logic. How much does intuitionistic logic buy you here?
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 02:34 PM, Wei Dai wrote: Tim, I think I'm starting to understand what you're saying. However, it still seems that anything you can do with intuitionistic logic, toposes, etc., can also be done with classical logic and set theory. (I'm not confident about this, but see my previous post in reponse to Bruno.) Maybe it's not as convenient or natural in some cases (similar to how modal logic can be more convenient than explicitly quantifying over possible worlds even when they are equivalent), but if one is not already familiar with intuitionistic logic and category theory, is it really worth the trouble to learn them? I don't know. One learns a field for various reasons. Clearly a lot of people think classical logic with the right exceptions and terminology serves them well. A lot of others, though not as many, think program semantics and possible worlds semantics are best understood in the natural logic of time-varying sets and topos theory. I'm in the latter category (no pun intended). I also don't know what your goals are, despite reading many of your posts. If, for example, you are looking for tools to understand a possible multiverse, or how multiverses in general might be constructed, I'm not at all sure any such tools have ever existed or _will_ ever exist, except insofar as tools for understanding toposes, lattices, etc. exist. This is quite different from understanding, say, general relativity, where the tools of differential geometry and exterior calculus are immediately useful for understanding and for calculations. The MWI/Tegmark/Egan stuff is very far out on the fringes, as we know, and there is unlikely to be anything one can do calculations of. Still, it seems likely that a _lot_ of mathematics is needed...a lot more math than physics, almost certainly. Modal logic seems to me to be _exactly_ the right logic for talking about possible states of existence, for talking about possible worlds, for talking about branching universes. So the issue is not But can't I find a way to do everything in ordinary logic? but is, rather, to think in terms of modal logic offering a more efficient basis (in the conceptual vector space), a basis with a smaller semantic gap between the formalism and the hypothesized world. (You might also want to take a look at the paper by Guts, a Russian, on a Topos-Theoretic Model of the Deutsch Multiverse. Available at the usual xxx.lanl.gov site.) For example, posets can certainly be studied and understood using classical logic. How much does intuitionistic logic buy you here? Posets can certainly be studied with classical logic. However, posets fail the law of trichotomy, that two things when compared by some ordering result in one of three outcomes: A is less than B, A is greater than B, or A is equal to B. This is the common sense comparison of objects, one with a linear or total order. However, posets are partially-ordered precisely because they don't follow the law of trichotomy. Is one more natural than the other? More common? More useful? I have my own beliefs at this point. A book I strongly recommend, though it is difficult, is Paul Taylor's Practical Foundations of Mathematics. 1999. (I buy many books not to read straight through, but to consult, to draw insights and inspirations from, and to let me know what I need to learn more of. This is one of those books. The first 175 pages, which I've been reading from, is making more and more sensethe terms become familiar, I see connections with other areas, and by a process akin to analytic continuation the ensemble of ideas becomes more and more natural. Beyond these pages, though, it's mostly incomprehensible.) I recommend this book for the broad insights I am gaining from it, but not as any kind of manual for tinkering with multiverses! (insert silly smiley as one sees fit) My conclusion from Tegmark's paper, which dovetailed with Egan's treatment of all topologies models in Distress, was that to make progress a lot of math needs to be learned. Which is my current approach. These are not my only inspirations. Indeed, I came to join this list after becoming fascinated (again, after a nearly 28-year absence) in topology, algebra, and the physics of time and cosmology. Seen this way, category and topos theory are worth studying for their own sake. I don't think it is likely that every conceivable universe with consistent laws of mathematics has actual existence (to nutshell my understanding of Tegmark's theory) is actually true (whatever that means). Nor do I take Schmidhuber's all running programs notion very seriously. Interesting ideas to play with, and to use some tools on. Strangely, then, I view these notions as places to apply the math I'm learning to. And I'm thinking small, in terms of simple systems. A paper I have mentioned a couple of times is directly in line with this approach: Fotini
Re: modal logic and possible worlds
On Tue, Aug 13, 2002 at 10:08:50AM -0700, Tim May wrote: * Because toposes are essentially mathematical universes in which various bits and pieces of mathematics can be assumed. A topos in which Euclid's Fifth Postulate is true, and many in which it is not. A topos where all functions are differentiable. A topos in which the Axiom of Choice is assumed--and ones where it is not assumed. In other words, as all of the major thinkers have realized over the past 30 years, topos theory is the natural theory of possible worlds. How does this compare to the situation in classical logic, where you can have theories (and corresponding models) that assume Euclid's Fifth Postulate as an axiom and theories that don't?
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 06:16 PM, Wei Dai wrote: On Tue, Aug 13, 2002 at 10:08:50AM -0700, Tim May wrote: * Because toposes are essentially mathematical universes in which various bits and pieces of mathematics can be assumed. A topos in which Euclid's Fifth Postulate is true, and many in which it is not. A topos where all functions are differentiable. A topos in which the Axiom of Choice is assumed--and ones where it is not assumed. In other words, as all of the major thinkers have realized over the past 30 years, topos theory is the natural theory of possible worlds. How does this compare to the situation in classical logic, where you can have theories (and corresponding models) that assume Euclid's Fifth Postulate as an axiom and theories that don't? Because such a dichotomy (and theories that don't) means the logic is ipso facto modal. The very form tells us that a modal (and hence intuitionist) assumption is at work: If it were the case that the parallel postulate were valid, then... and Suppose the parallel postulate is not true, then... If the Fifth Postulate is independent of the others, then within the framework of the other postulates one may have one branch where the Fifth holds (Euclidean Geometry) and another branch where it doesn't hold (all of the various non-Euclidean geometries). Now this turns out to be a not very important example, as various geometries with various geodesics on curved surfaces are sort of mundane. And the details were mostly worked out a hundred years ago, starting with Gauss, Bolyai, Lobachevsky, Riemann, and continuing to Levi-Cevita, Ricci, and Cartan. The fact that by the mid-19th century we could _see_ clear examples of geometries which did not obey the parallel postulate, e.g., triangles drawn largely enough on a sphere, great circles, figures drawn on saddle surfaces and trumpet surfaces, etc., meant that most people didn't think much about the modal aspects. But they are certainly there. (I believe it's possible to cast differential geometry, including the parallel postulate or its negation, in topos terms. Anders Kock has done this with what he calls synthetic differential geometry, but I haven't read his papers (circa 1970-80), so i don't know if he discusses the parallel postulate explicitly.) Both category theory and topos theory have been used to prove some important theorems (e.g., the Weyl Conjecture about a certain form of the Riemann zeta function, and the Cohen forcing proof of the independence of the Continuum Hypothesis from the Zermelo-Frenkel logical system), but it is misleading to think that either will give different results from conventional mathematics. It is not as if Fermat's Last Theorem is true in conventional logic or in conventional set theory but false in intuitionist logic or category theory. I'm going to have to slow down in my writing. You ask a lot of short questions, but these short questions need long answers. Or, perhaps, I feel the need to make a lot of explanations of terminology and motivations. I'll have to tune the length of my responses to the length of your questions, I think! --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
On Tue, Aug 13, 2002 at 03:51:49PM -0700, Tim May wrote: I also don't know what your goals are, despite reading many of your posts. If, for example, you are looking for tools to understand a possible multiverse, or how multiverses in general might be constructed, I'm not at all sure any such tools have ever existed or _will_ ever exist, except insofar as tools for understanding toposes, lattices, etc. exist. I think the theory of everything is a multiverse theory. So I want to understand the implications that following from the idea that multiple universes exist. These include philosophical, practical, and scientific implications. Right now I really want to know the answers to these questions: 1. What do probabilities mean? 2. How should one reason and make decisions? 3. What is the structure of the multiverse? Which class of universes does it contain? For example does it contain non-computable universes? 1 and 2 are philosophical questions, but clearly very practically relevant. For 3, I'm only interested in the coarsest level of detail for now. It needs to be answered because the answer makes a difference for question 2. The MWI/Tegmark/Egan stuff is very far out on the fringes, as we know, and there is unlikely to be anything one can do calculations of. Still, it seems likely that a _lot_ of mathematics is needed...a lot more math than physics, almost certainly. I think there are a lot of philosophical and practical questions that can be answered without detailed investigation into the fine structure of the multiverse. Certainly understanding the fine structure, including the structure of all of the universes that it contains, requires a lot of math (in fact it requires ALL of math if Tegmark is correct), but I'll leave that to the future. Modal logic seems to me to be _exactly_ the right logic for talking about possible states of existence, for talking about possible worlds, for talking about branching universes. So the issue is not But can't I find a way to do everything in ordinary logic? but is, rather, to think in terms of modal logic offering a more efficient basis (in the conceptual vector space), a basis with a smaller semantic gap between the formalism and the hypothesized world. I don't know. When I hear a modal sentence, I have to interpret it in terms of possible worlds. It seems easier to just talk directly about possible worlds. I haven't seen where the efficiency comes from. I'm sure it is more efficient for some purposes, but I'm not convinced that it is for mine. Seen this way, category and topos theory are worth studying for their own sake. I don't think it is likely that every conceivable universe with consistent laws of mathematics has actual existence (to nutshell my understanding of Tegmark's theory) is actually true (whatever that means). Nor do I take Schmidhuber's all running programs notion very seriously. Interesting ideas to play with, and to use some tools on. Well why don't you take these ideas seriously?
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 08:47 PM, Wei Dai wrote: Seen this way, category and topos theory are worth studying for their own sake. I don't think it is likely that every conceivable universe with consistent laws of mathematics has actual existence (to nutshell my understanding of Tegmark's theory) is actually true (whatever that means). Nor do I take Schmidhuber's all running programs notion very seriously. Interesting ideas to play with, and to use some tools on. Well why don't you take these ideas seriously? Lack of even the slightest piece of evidence for all possible mathematical universes actually exist and/or the all runnable computer programs.' I also don't believe there are gods or other supernatural beings, for the same reason. If and when I see an experiment that points to there being other universes which have tangible existence, then I'll start to believe. --Tim May That the said Constitution shall never be construed to authorize Congress to infringe the just liberty of the press or the rights of conscience; or to prevent the people of the United States who are peaceable citizens from keeping their own arms. --Samuel Adams
Re: modal logic and possible worlds
On Monday, August 12, 2002, at 12:07 PM, Wei Dai wrote: According to possible world semantics, it's necessary that P means that P is true in all worlds accessible from this one. Different modal logics correspond to different restrictions on the accessibility relation. Before the invention of possible world semantics, people argued about which modal logic is the correct one, but now philosophers realize that different notions of accessibility (and the corresponding notions of modality) are useful at different times, so there is no single correct modal logic. That's my one paragraph summary of possible world semantics. Please correct me if I'm wrong, or read these articles if you're not familiar with this topic: http://www.xrefer.com/entry.jsp?xrefid=552831 http://www.xrefer.com/entry.jsp?xrefid=553229 My questions is, why not just quantify over the possible worlds and refer to the accessibility relation directly? This way you can talk about multiple accessibility relations simultaneously, and you don't have to introduce new logical symbols (i.e. the box and the diamond). Is modality just a syntactic shorthand now? Modal logic is a lot more than syntactic shorthand. Consider this example, phrased in MWI terms. It is possible that WWIII will happen before the end of this year. In one possible world, A, many things are one way...burned, melted, destroyed, etc. In another possible world, B, things are dramatically different. There can be no implication from one world to the other. That is, we can't say A implies B or B implies A. This branching future is exactly what I was talking about a week or so ago in terms of partially ordered sets. If the order relationship is occurs before or at the same time as, which is equivalent to less than or equal to, A and B cannot be linearly ordered. In fact, since both A and B are completely different states, neither can be said to be a predecessor or parent of the other. In fact, A and B are not comparable. We cannot say A or not-A. We have thus left the world of classical logic and are in the world of non-classical, or intuitionistic, or Heyting logic. Posets are not just a different syntactic shorthand from linearly-ordered sets. Branching worlds, aka possible worlds, aka MWI (when QM is involved) is a more accurate way of talking about time and successions of events than is attempting to force time into a strait-jacket of linearly-ordered sets (chains). Besides the topos work of Saul Kripke, Vaughan Pratt at Stanford has written a lot on concurrency, lattices, and posets. Lee Smolin's book Three Roads to Quantum Gravity is very good at explaining how this relates to cosmology. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
Tim, I'm afraid I still don't understand you. On Mon, Aug 12, 2002 at 06:00:26PM -0700, Tim May wrote: It is possible that WWIII will happen before the end of this year. In one possible world, A, many things are one way...burned, melted, destroyed, etc. In another possible world, B, things are dramatically different. Ok, but what about my point that you can state this by explicit quantification over possible worlds rather than using modal operators? I.e., There exist a world accessible from this one where WWIII happens before the end of this year. instead of It is possible that WWIII will happen before the end of this year.? There can be no implication from one world to the other. That is, we can't say A implies B or B implies A. What does that have to do with my question? Anyway A and B are supposed to be worlds here, not propositions, so of course you can't say A implies B. I don't know what point you're trying to make here. This branching future is exactly what I was talking about a week or so ago in terms of partially ordered sets. If the order relationship is occurs before or at the same time as, which is equivalent to less than or equal to, A and B cannot be linearly ordered. In fact, since both A and B are completely different states, neither can be said to be a predecessor or parent of the other. In fact, A and B are not comparable. I'm with you so far in this paragraph. We cannot say A or not-A. Now I'm lost again. Again A is a world not a proposition so what would A or not-A mean even if A and B are comparable? If anyone else understand the point Tim is making please help me out...
