# Re: Propositions, Properties and Logics

```On 2/26/2012 4:43 PM, Bruno Marchal wrote:
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On 26 Feb 2012, at 20:37, Stephen P. King wrote:

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```On 2/26/2012 12:27 PM, Bruno Marchal wrote:
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On 25 Feb 2012, at 20:01, Stephen P. King wrote:
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<snip>

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Likewize Bp & Dt, and Bp & Dt & p, are other important variants. I will say more when I get more time, but by searching 'S4Grz' or 'hypostase' in the archive you might find the many explanations I already give. See my papers and the reference therein. Ask precise question when you don't understand, so I can help.
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Thank you for this brief set of remarks. I would like to see an elaboration of the Löbian entity such that we can see the means by which the 1p content is encoded.
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The first person content are not encoded, they are just true belief, or correct inference with respect to plausible local universal numbers.
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Dear Bruno,

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I mean "encoded" in the sense of the propositions; the "p" in Bp&p, for example. All of the examples that I have seen of Löbian entities are too simple, or am I thinking at the wrong level? Are the Löbian entities that which the universal numbers are computing? I need a diagram of some sort to understand the relations and levels better. Is there one in your French papers?
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A brain does not create a person, it helps a person to manifest herself with respect to other universal numbers (some being person themselves, and others might be less clear).
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Yes, I agree with this in the sense that or bodies (brains) are the "interfaces" between universal numbers. But if we follow Pratt's Chu space idea then the categories of these "interfaces" is dual to the category of the predicates. Pratt discusses this here <http://chu.stanford.edu/>. Here is a except:
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"[Note: In the following, whereas in the past we've written A and X for the respective sets of points and states of a Chu space, Steve Vickers has them the other way round for the points and opens of the topological systems defined in his book ``Topology via Logic.'' We had originally justified this by matching up Chu spaces with frames rather than locales. However we felt it might be less confusing to orient Chu spaces to agree with the ``natural direction'' of topological systems, taking X to consist of points and A of states. The downside of this switch is the possibility of confusion and inconsistency during the transition. We considered the alternative notation X_* and X^* used originally by Y. Lafont in 1988 and more recently by P.-L. Curien, but we prefer Vicker's notation as having the benefits of a less cluttered look while leaving the superscript and subscript positions free for other uses.]
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Basic Concept

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/*Short Form*/ A Chu space is a transformable matrix whose rows transform forwards while its columns transform backwards.
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/*Generality of Chu spaces*/ Chu spaces unify a wide range of mathematical structures, including the following.
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# Relational structures such as sets, directed graphs, posets, and small categories. # Algebraic structures such as groups, rings, fields, modules, vector spaces, lattices, and Boolean algebras. # Topologized versions of the above, such as topological spaces, compact Hausdorff spaces, locally compact abelian groups, and topological Boolean algebras.
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Algebraic structures can be reduced to relational structures by a technique described below <http://chu.stanford.edu/#algrel>. Relational structures constitute a large class in their own right. However when adding topology to relational structures, the topology cannot be incorporated into the relational structure but must continue to use open sets.
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Chu spaces offer a uniform way of representing relational and topological structure simultaneously. This is because Chu spaces can represent relational structures <http://chu.stanford.edu/#reprel> via a generalization of topological spaces which allows them to represent topological structure <http://chu.stanford.edu/#gen> at the same time using the same machinery.
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/*Definition*/

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Surprisingly this degree of generality can be achieved with a remarkably simple form of structure. A Chu space (X,r,A) consists of just three things: a set X of points, individuals, or subjects, a set A of states or predicates, and a lookup table or matrix r: X×A ? K which specifies for every subject x and predicate a the value a(x) of that predicate for that subject. The value of a(x) is given by the matrix r as the entry r(x,a). These matrix entries are drawn from a set K.
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K can be as simple as {0,1}, as when representing topological spaces or Boolean algebras. Or it can be as complex as the set of all complex numbers, as when representing Hilbert spaces. Or it can be something in between, such as the set of 16 subsets of {0,1,2,3} when representing topological groups. The full generality of Chu spaces derives from the ability of predicates to take other values than simply /true/ or /false/ (the case K={0,1}).
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Symmetry of subject and predicate

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This definition can be reconciled with the short form definition above by taking A to be a subset of K^X , namely by representing the predicate a as the function ?x.r(x,a): X?K mapping each individual to the value of the predicate on that individual, and taking A to be the set all such functions from X to K. In this case there is no separate matrix, just the two sets consisting of respectively individuals and predicates. However this breaks the symmetry of subject and predicate. The more symmetric matrix encoding of this information makes it equally reasonable to view X as a subset of K^A when this is helpful, bearing in mind however that the matrix may contain repeated rows and/or columns, i.e. in neither case need these be extensional subsets of K^X or K^A .
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A more satisfactory reconciliation with the short form definition is to view the subject-predicate relationship as a symmetrically expressed relation r. We do not /have/ to interpret the notion of predicate on a set X of subjects as a function a:X?K, any more than we have to interpret a subject as a function x:A?K. Instead we have three options: either of those two, or just leaving r as the symmetric expression of the relationship. Unlike both algebras and topological spaces, Chu spaces do not take subjects to be primitive and predicates to be derived, but rather take both to be primitive.
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In this symmetric view, ``subject'' is more natural than ``individual.'' One envisages an individual as having an independent existence. A subject on the other hand is a subject of something: it forms one half of an elementary proposition r(x,a) that combines a subject x with a predicate a.
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This reconciliation has a historical link with the discovery and resolution of the paradoxes of Cartesian Dualism. If we identify the sets X and A with Descartes' 1647 division of the universe into physical and mental components respectively, then r is the mediator of these components sought by many philosophers during the following century. The respective proposals of Hume and Berkeley to make one side or the other primitive correspond to taking respectively X or A to be primitive and deriving the other in terms of functions from the former to K. That Hume won out would seem to be correlated with mathematics' preference for basing mathematical objects on their constituent individuals rather than their constituent predicates.
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If we view the open sets of a topological space as its permitted predicates defining its structure, a topological space is an example of an object structured by the interaction of its subjects and predicates. It is by no means the only example however. Although both abelian groups and vector spaces are traditionally represented as algebras, they also have well-known alternative representations as what amounts to Chu spaces. Finite (and more generally locally compact) abelian groups are representable as the sets consisting of their elements and their homomorphisms (continuous in the infinite case, where locally compact becomes necessary) into the multiplicative group of nonzero complex numbers. And vector spaces V over a field F are representable as the sets of points and dual points of V, with the latter defined as the set of functionals on V, or linear transformations V?F where F in this context denotes the one-dimensional vector space over F."
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Can, for example, we include a free or atomic boolean algebra in a Löbian entity?
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Algebraically Löbian machines can be handled by diagonalizable algebra (that is boolean algebra endowed with a transformation operator verifying the Löbian axioms, the fixed point property.
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Is this fixed point property just the Kleen fixed point <http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem>? What I am asking about is what makes up the content of the "complete partial order" that is required (in addition to the continuous and monotone function) to obtain the fixed point. I see the continuous and monotone function on that is local in the sense of 1p and not a global or 3p identification or preorder. This is why I am making such a fuss about how you are not looking at the time problem carefully enough. We cannot simply use the natural order of the Integers as it is a global order and does not allow for consideration of concurrency issues.
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Maybe the concurrency problem is strictly in 1p and thus is not so problematic... Maybe.... But how do you continue to use 1p type language to discuss 3p concepts. This causes at least confusions of the level at which the concept is defined.
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But what the machine can observe is non boolean, and cannot, I presume be extended in a Boolean reality. It is an open problem if all coherent dreams could define a unique physical reality. I doubt it.
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So we can consider Heyting logic type extensions. OK. Kitada's paper shows that the dreams cannot define a single physical reality as it would be inconsistent, but this is a solution to the time problem, so is a good result. ;-)
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I would also appreciate your comments on this paper by Barry Cooper: http://www1.maths.leeds.ac.uk/~pmt6sbc/preprints/rome.paper.pdf
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Here is its Abstract:

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"Amongst the huge literature concerning emergence, reductionism and mech- anism, there is a role for analysis of the underlying mathematical constraints. Much of the speculation, confusion, controversy and descriptive verbiage might be clari fied via suitable modelling and theory. The key ingredients we bring to this project are the mathematical notions of defi nability and invariance, a computability theoretic framework in a real-world context, and within that, the modelling of basic causal environments via Turing's 1939 notion of interac- tive computation over a structure described in terms of reals. Useful outcomes are: a re finement of what one understands to be a causal relationship, includ- ing non-mechanistic, irreversible causal relationships; an appreciation of how the mathematically simple origins of incomputability in defi nable hierarchies are materialized in the real world; and an understanding of the powerful ex-
```planatory role of current computability theoretic developments."
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Interesting, but still not taking into account the comp mind-body problem, or the comp first person indeterminacy. Might say more on this later. It would have been nice I (re)discovered that paper soon, but many thanks :)
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Please also see http://homepages.inf.ed.ac.uk/jrl/Research/laplace1.pdf which contains many of the same questions that I have been asking but expressed in a more formal and erudite manner.
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You cannot ask to read 50 pages long technical pages at each paragraph, and guess what are your non understing of the UDA is from that. It looks like not too bad material though, but does not really address the question we are discussing here.
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OK, I will post excerpts from the paper to show you want I mean by my remark. For example we find the following:
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"Laplace's imagery of a hypothetical predictive 'intelligence' (nowadays widely referred to as Laplace's demon) provides a valuable prop for the imagination, and a convenient metaphor for expressing many of our ideas. Broadly speaking,
```we will be interested in questions such as the following:
. How exactly is the instantaneous state of the universe (or more modestly,
that of some 'closed' physical system) presented to the demon? That is,
. What kinds of 'analysis' is the demon able to perform on this information?
For example, what kinds of infinitary deductions or computations are
permitted?
To elaborate a little further on what lies behind each of these questions:
. Exactly what 'information' is deemed to be present in a physical system
at a given instant in time? In particular, which entities or quantities are
considered to be 'physically real'? As we shall see, the attempt to make
Laplace's claim precise forces us to be very explicit about the ontological
assumptions that underlie a physical theory.
. What exactly is meant by saying that certain information can be 'deduced'
or 'computed' from certain other information (under the assumption that
certain physical laws hold)? To answer this question, one is naturally led
to draw on ideas from mathematical logic, and here one finds that many
different conceptions of truth, provability and computability have been
elaborated. Moreover, some of the choices here turn out to be related to
deep metaphysical issues. Again, trying to clarify the idea of determinism
forces us to be very explicit about what we are presupposing."

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I am still not seeing how you define the philosophical terms that you are using, as the way that you are using words, such as "dualism" and "monism" are inconsistent with their usage by others in philosophy.
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I use them in the sense of the wiki you did provide to me.

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Neutral monism, in the "philosophy of mind" consists in explaining mind and matter, and the relation between, in term of something else.
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Yes, but I see numbers as belonging to the category of mental content and thus not capable of forming a neutral "something else".
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But this is basically, with all my respect, a mistake. You confuse the theory of numbers, with the meta and psychological theory (which assumes much more things) of how humans mentally handle the numbers. Unless you make clear your ontology, and what is your theory, or initial theory, you might just beg the question.
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You are assuming that the differentiation of the properties of numbers is ontologically primitive. I do not know what ontological theory you are using to think that that is not problematic. I am following the reasoning in Ayn Rand's Objectivist Epistemology <http://en.wikipedia.org/wiki/Introduction_to_Objectivist_Epistemology>. Which epistemology are you using?
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It is not a question of true or false, but understanding a reasoning. You have to go through the thought experiment until you have the aha!
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You are assuming something that is fallacious, sorry Bruno, but the problem that I see in COMP is not its claims or statements, it is the tacit and unquestioned assumptions that you are bringing into it. You assume too much. You are assuming that the existence and truth value (which can be unassailably argued to be independent necessary possibilities) of universal numbers also define the propositional particulars of those numbers. This is equivalent to the idea in naive realism that "...objects are also able to retain properties of the types we perceive them as having, even when they are not being perceived. Their properties are perception-independent." This assumption is shown by Rand to be flawed beyond all possibility of rehabilitation. So again I ask, what is your epistemology theory?
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```OTOH, if we stick to your consideration that minds are only the 1p
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associated with true beliefs.
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Yes, but this requires the assumption that all the correlata of those true beliefs exist a priori in a definite sense. This is the realist assumption that I have a problem with that I mentioned above. It assumes a natural one-to-one relation between observables and concepts/representations that we can easily show is naive and wrong. The only solution to this that I see is to expand the universe of possible observers to include all the physical worlds of possible observers, as we see in Tegmark's paper, but this would require us to answer the open problem of a unique set of physical laws question in the negative as I mentioned above.
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then your argument that COMP is a neutral monism is consistent modulo finite considerations. I think that considerations of 3p spoils this neutrality (the Laplace draft paper above touches on this), but let us see what happens in our discussions.
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OK.

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If your theory is scientific, the something else must be clearly specifiable, that is itself described by a reasonable theory, so that the explanation of mind and body from it makes (sharable) sense.
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With comp, in short, a TOE is given by RA (ontological), and its epistemological laws is given by the variants of relative self-reference of all the (Löbian) numbers. Physics consists in some of those variants (hypostases).
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Some believe that the numbers belongs to the mind, but with comp it is more natural to define the mind, in a large sense, by the universal numbers imagination.
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This is my prejudice and I am working hard to overcome it. My resent comment that we have computations (as abstract logical machines) and physical processes as orthogonal intersections on 1p is an attempt in this direction. I am trying to remain consistent with Pratt's idea of state transitions via chained residuation.
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The mind is, notably, what computer can explore, quasi by definition with comp.
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Local computers, like the one you are using right now, are universal number written in physical universal sublanguage of physics. And normally UDA should help you to convince yourself that physics becomes necessarily a sort of projective limit of the mind, with comp.
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I see physics, in the sense of groups and other relations, as the mutually consistent sharable content of 1p, but this does not alone cover us to solve the problem of time. The only solution that I have seen that is semi-congruent with COMP is Hitoshi Kitada's proposal (http://arxiv.org/abs/physics/0212092 ).
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No problem. I don't try to solve a problem, just to formulate it, in a way such that humans, but also machine, can understand. AUDA is machine's answer, somehow.
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We have to explain the appearances, even if they are illusions or our explanations will fail.
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With comp, the only way to singularize you or your neighborhood consists in layering down the substitution level in the transfinite. Why not? The study of comp can help to build rigorous non comp theory. Sets and hypersets can be helpful for this, indeed. For comp too, probably.
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Ben Goertzel has a very nice paper discussing the use of hypersets and consciousness here <goertzel.org/consciousness/consciousness_paper.pdf>. Craig's discussion of it is here <http://s33light.org/post/17993511503>.
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Yes. It is not bad, but I use combinators or lambda terms to handle the non foundations, or the second recursion theorem, or the modal logic (based on the use of those diagonalizations), which is natural in the comp meta-theory. Ben was a participant of this list years ago. We had good discussions. It is also a not too bad material. But polishing too much tools for solving a problem can distract from solving the problem, or even from formulating it (or a subproblem of it). I already told I am skeptical on the notion of sets in general. I like very much ZF, which I have studied in deep, but I see it just as a sort of very imaginative Löbian machine. Jean-Louis Krivine, Jech, and recently Smullyan and Fitting wrote very nice books on set theory. They explain the Cohen forcing technic with a nice modal construction in S4.
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Do you see how the self-referencing of the Löbian entities demands hyperset and thus we have to be careful that our reasoning is consistent with this? Hypersets allow for a kind of ontological primitive set theory, but one that is different from ZF. You have to deal with comprehension, regularity and foundedness issues.
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Are there any papers discussing Cohen's forcing in S4 that you recommend?
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Onward!

Stephen

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