Re: A Type Theory for Defining Logics and Proofs

[Lawrence Crowell]

I don't think I will get around to reading this. It is awfully dense. The 
subject looks interesting though. I am not sure, but this sort of type 
theory might be of importance with Laughlin wave functions. The expectation 
on a 2-dim surface, either a boundary or a graphene sheet <φ(z')φ(z)> ~ 
-log|z' – z| leads to the Laughlin wave for ψ ~ e^{φ(z')√q}e^{φ(z)√q} and 
then Z ~ |z' –-z|^q exp(¼|z'|^2), which in general is computed for a 
product of these in a path integral. These products <φ(z')φ(z)> emerge from 
the Baker-Campbell-Hausdorff theorem and this has some bearing with the 
idea of the 2-slit experiment as a logic gate, but where now instead of 
slits one has fields. However, this paper looks a bit dense to read.

LC

On Tuesday, August 6, 2019 at 5:38:54 AM UTC-5, Philip Thrift wrote:
>
>
> https://arxiv.org/abs/1905.02617 :
>
> *A Type Theory for Defining Logics and Proofs*
> Brigitte Pientka, David Thibodeau, Andreas Abel, Francisco Ferreira, 
> Rebecca Zucchini
> (Submitted on 7 May 2019)
>
> We describe a Martin-Löf-style dependent type theory, called Cocon, that 
> allows us to mix the intensional function space that is used to represent 
> higher-order abstract syntax (HOAS) trees with the extensional function 
> space that describes (recursive) computations. We mediate between HOAS 
> representations and computations using contextual modal types. Our type 
> theory also supports an infinite hierarchy of universes and hence supports 
> type-level computation thereby providing metaprogramming and (small-scale) 
> reflection. Our main contribution is the development of a Kripke-style 
> model for Cocon that allows us to prove normalization. From the 
> normalization proof, we derive subject reduction and consistency. Our work 
> lays the foundation to incorporate the methodology of logical frameworks 
> into systems such as Agda and bridges the longstanding gap between these 
> two worlds.
>
> @philipthrift
>

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A Type Theory for Defining Logics and Proofs

[Philip Thrift]

https://arxiv.org/abs/1905.02617 :

*A Type Theory for Defining Logics and Proofs*
Brigitte Pientka, David Thibodeau, Andreas Abel, Francisco Ferreira, 
Rebecca Zucchini
(Submitted on 7 May 2019)

We describe a Martin-Löf-style dependent type theory, called Cocon, that 
allows us to mix the intensional function space that is used to represent 
higher-order abstract syntax (HOAS) trees with the extensional function 
space that describes (recursive) computations. We mediate between HOAS 
representations and computations using contextual modal types. Our type 
theory also supports an infinite hierarchy of universes and hence supports 
type-level computation thereby providing metaprogramming and (small-scale) 
reflection. Our main contribution is the development of a Kripke-style 
model for Cocon that allows us to prove normalization. From the 
normalization proof, we derive subject reduction and consistency. Our work 
lays the foundation to incorporate the methodology of logical frameworks 
into systems such as Agda and bridges the longstanding gap between these 
two worlds.

@philipthrift

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Re: Some modal logics for conscious agents research

[Bruno Marchal]

> On 29 Sep 2018, at 10:18, Philip Thrift  wrote:
> 
> 
> 
> On Saturday, September 29, 2018 at 2:23:05 AM UTC-5, Bruno Marchal wrote:
> 
>> On 28 Sep 2018, at 20:33, Philip Thrift > 
>> wrote:
>> 
>> 
>> The Other-Condemning Moral Emotions - A Modal Logic Approach
>>  https://dspace.library.uu.nl/handle/1874/319982 
>> 
>> 
>> A logic for reasoning about counterfactual emotions
>> https://www.sciencedirect.com/science/article/pii/S0004370210002110 
>> 
>> 
>> A logic for intention
>> https://www.ijcai.org/Proceedings/99-1/Papers/026.pdf 
>> 
>> 
>> 
>> Intensionality and Intentionality: Phenomenology, Logic, and Mind
>> https://escholarship.org/content/qt25m22937/qt25m22937.pdf 
>> 
>> 
>> Emotional Belief-Desire-Intention Agent Model: Previous Work and Proposed 
>> Architecture
>> https://pdfs.semanticscholar.org/521c/d68e96579db8cf5dadbbc51ca3a78f790c71.pdf
>>  
>> 
>> 
>> Modality, The Synthetic Apriori, and Phenomenology
>> http://von-wachter.de/lv/08-2-modality/ 
>> 
>> 
>> Logic, Neuroscience and Phenomenology: In Cahoots?
>> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1.4683=rep1=pdf
>>  
>> 
>> 
>> A Modal Logic for Gödelian Intuition 
>> https://philarchive.org/archive/KHUAML 
>> 
>> 
>> An Introduction to Löb’s Theorem in MIRI Research
>> http://intelligence.org/files/lob-notes-IAFF.pdf 
>> 
>> 
> 
> 
> The last entry is not too bad for Gödel and Löb, but all entries should be 
> largely updated, and Here is not the place. They are most aristotelians 
> having still no idea of the mind)body problem.
> 
> Read my papers, and you will be able to make the update by yourself, and the 
> last entry on Lôb can be helpful, but his application in philosophy makes no 
> sense, as you will understand (use the mechanist assumption).
> 
> Bruno
> 
> 
> 
> Feel free to add modal logic references that could be applicable to 
> consciousness science.

See my papers and the reference therein. I work on this since very long. 



> 
> Have you ever attended one of the TSC conferences? Next one: 
> https://www.tsc2019-interlaken.ch/ 

I should but no, I don’t. Might think about it. Or not. 

Bruno



> 
> 
> Another reference:
> 
> Reasoning about another agent through empathy
> https://pdfs.semanticscholar.org/f5c9/0902584db239f5341c387f894251185a1e49.pdf
> 
> 
> (As Philip Goff notes: Our picture of matter needs to be updated to include 
> consciousness. Modal language is one approach to doing that.)
> 
> - pt
> 
> 
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Re: Some modal logics for conscious agents research

[Philip Thrift]

On Saturday, September 29, 2018 at 2:23:05 AM UTC-5, Bruno Marchal wrote:
>
>
> On 28 Sep 2018, at 20:33, Philip Thrift > 
> wrote:
>
>
> The Other-Condemning Moral Emotions - A Modal Logic Approach
>  https://dspace.library.uu.nl/handle/1874/319982
>
> A logic for reasoning about counterfactual emotions
> https://www.sciencedirect.com/science/article/pii/S0004370210002110
>
> A logic for intention
> https://www.ijcai.org/Proceedings/99-1/Papers/026.pdf
>
>
> Intensionality and Intentionality: Phenomenology, Logic, and Mind
> https://escholarship.org/content/qt25m22937/qt25m22937.pdf
>
> Emotional Belief-Desire-Intention Agent Model: Previous Work and Proposed 
> Architecture
>
> https://pdfs.semanticscholar.org/521c/d68e96579db8cf5dadbbc51ca3a78f790c71.pdf
>
> Modality, The Synthetic Apriori, and Phenomenology
> http://von-wachter.de/lv/08-2-modality/
>
> Logic, Neuroscience and Phenomenology: In Cahoots?
>
> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1.4683=rep1=pdf
>
> A Modal Logic for Gödelian Intuition 
> https://philarchive.org/archive/KHUAML
>
> An Introduction to Löb’s Theorem in MIRI Research
> http://intelligence.org/files/lob-notes-IAFF.pdf
>
>
>
> The last entry is not too bad for Gödel and Löb, but all entries should be 
> largely updated, and Here is not the place. They are most aristotelians 
> having still no idea of the mind)body problem.
>
> Read my papers, and you will be able to make the update by yourself, and 
> the last entry on Lôb can be helpful, but his application in philosophy 
> makes no sense, as you will understand (use the mechanist assumption).
>
> Bruno
>
>
>
Feel free to add modal logic references that could be applicable to 
consciousness science.

Have you ever attended one of the TSC conferences? Next 
one: https://www.tsc2019-interlaken.ch/ 


Another reference:

Reasoning about another agent through empathy
https://pdfs.semanticscholar.org/f5c9/0902584db239f5341c387f894251185a1e49.pdf


(As Philip Goff notes: *Our picture of matter needs to be updated to 
include consciousness.* Modal language is one approach to doing that.)

- pt

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Re: Some modal logics for conscious agents research

[Bruno Marchal]

> On 28 Sep 2018, at 20:33, Philip Thrift  wrote:
> 
> 
> The Other-Condemning Moral Emotions - A Modal Logic Approach
>  https://dspace.library.uu.nl/handle/1874/319982
> 
> A logic for reasoning about counterfactual emotions
> https://www.sciencedirect.com/science/article/pii/S0004370210002110
> 
> A logic for intention
> https://www.ijcai.org/Proceedings/99-1/Papers/026.pdf
> 
> 
> Intensionality and Intentionality: Phenomenology, Logic, and Mind
> https://escholarship.org/content/qt25m22937/qt25m22937.pdf
> 
> Emotional Belief-Desire-Intention Agent Model: Previous Work and Proposed 
> Architecture
> https://pdfs.semanticscholar.org/521c/d68e96579db8cf5dadbbc51ca3a78f790c71.pdf
> 
> Modality, The Synthetic Apriori, and Phenomenology
> http://von-wachter.de/lv/08-2-modality/
> 
> Logic, Neuroscience and Phenomenology: In Cahoots?
> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1.4683=rep1=pdf
> 
> A Modal Logic for Gödelian Intuition 
> https://philarchive.org/archive/KHUAML
> 
> An Introduction to Löb’s Theorem in MIRI Research
> http://intelligence.org/files/lob-notes-IAFF.pdf
> 


The last entry is not too bad for Gödel and Löb, but all entries should be 
largely updated, and Here is not the place. They are most aristotelians having 
still no idea of the mind)body problem.

Read my papers, and you will be able to make the update by yourself, and the 
last entry on Lôb can be helpful, but his application in philosophy makes no 
sense, as you will understand (use the mechanist assumption).

Bruno



> 
> - pt
> 
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Some modal logics for conscious agents research

[Philip Thrift]

The Other-Condemning Moral Emotions - A Modal Logic Approach
 https://dspace.library.uu.nl/handle/1874/319982

A logic for reasoning about counterfactual emotions
https://www.sciencedirect.com/science/article/pii/S0004370210002110

A logic for intention
https://www.ijcai.org/Proceedings/99-1/Papers/026.pdf


Intensionality and Intentionality: Phenomenology, Logic, and Mind
https://escholarship.org/content/qt25m22937/qt25m22937.pdf

Emotional Belief-Desire-Intention Agent Model: Previous Work and Proposed 
Architecture
https://pdfs.semanticscholar.org/521c/d68e96579db8cf5dadbbc51ca3a78f790c71.pdf

Modality, The Synthetic Apriori, and Phenomenology
http://von-wachter.de/lv/08-2-modality/

Logic, Neuroscience and Phenomenology: In Cahoots?
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1.4683=rep1=pdf

A Modal Logic for Gödelian Intuition 
https://philarchive.org/archive/KHUAML

An Introduction to Löb’s Theorem in MIRI Research
http://intelligence.org/files/lob-notes-IAFF.pdf


- pt

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Re: Propositions, Properties and Logics

[Stephen P. King]

On 2/26/2012 4:43 PM, Bruno Marchal wrote:


On 26 Feb 2012, at 20:37, Stephen P. King wrote:


On 2/26/2012 12:27 PM, Bruno Marchal wrote:


On 25 Feb 2012, at 20:01, Stephen P. King wrote:


snip

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.





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.


The first person content are not encoded, they are just true belief, 
or correct inference with respect to plausible local universal numbers.


 Dear Bruno,

I mean encoded in the sense of the propositions; the p in Bpp, 
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?


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).


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:


[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.]


[If any of the symbols × ? ? ? ? ? ? ? ? is a box then your browser is 
lacking some of the HTML 4.0 mathematical symbols as listed in 
http://www.cs.tut.fi/~jkorpela/html/guide/entities.html. Consider 
upgrading your browser to Firefox. Safer too. If only the last symbol is 
box, it denotes not-?.]



   Basic Concept

/*Short Form*/ A Chu space is a transformable matrix whose rows 
transform forwards while its columns transform backwards.


/*Generality of Chu spaces*/ Chu spaces unify a wide range of 
mathematical structures, including the following.


# 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.


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.


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.


/*Definition*/

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.


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 

Re: Logics

[Bruno Marchal]

On 28 Sep 2011, at 16:44, Stephen P. King wrote:


On 9/27/2011 10:47 AM, Bruno Marchal wrote:



On 27 Sep 2011, at 13:49, Stephen P. King wrote:


On 9/26/2011 7:56 PM, Jason Resch wrote:


snip

For well-defined propositions regarding the numbers I think the  
values are confined to true or false.


Jason

--

[SPK]
Not in general, unless one is only going to allow only Boolean  
logics to exist. There have been proven to exist logics that have  
truth values that range over any set of numbers, not just {0,1}.  
Recall the requirement for a mathematical structure to exist: Self- 
consistency.


Consistency is a notion applied usually to theories, or (chatty)  
machines, not to mathematical structures.
A theory is consistent if it does not prove some proposition and  
its negation. A machine is consistent if it does not assert a  
proposition and its negation.


[SPK]
Is not a machine represented mathematically by some abstract  
(mathematical ) structure?  I am attempting to find clarity in the  
ideas surrounding the notion of machine and how you arrive at the  
idea that the abstract notion of implementation is sufficient to  
derive the physical notion of implementation.


This follows from the UD Argument, in the digital mechanist theory. No  
need of AUDA or complex math to understand the necessity of this, once  
we accept that we can survive with (physical, material) digital  
machines.








In first order logic we have Gödel-Henkin completeness theorem  
which shows that a theory is consistent if and only if there is a  
mathematical structure (called model) satisfying (in a sense which  
can be made precise) the proposition proved in the theory.


[SPK]
What constraints are defined on the models by the Gödel-Henkin  
completeness theorem? How do we separate out effective consistent  
first-order theories that do not have computable models?


What do you mean by computable models?






Also, it is true that classical (Boolean) logic are not the only  
logic. There are infinitely many logics, below and above classical  
propositional logic. But this cannot be used to criticize the use  
of classical logic in some domain.

[SPK]
OK. My thought here was to show that classical (Boolean) logic  
is not unique and should not be taken as absolute. To do so would be  
a mistake similar to Kant's claim that Euclidean logic was absolute.


OK, but then why to use that fact to criticize Jason's defense of  
arithmetical truth independent of humans.








All treatises on any non classical logic used classical (or much  
more rarely intuitionistic) logic at the meta-level. You will not  
find a book on fuzzy logic having fuzzy theorems, for example. Non  
classical logics have multiple use, which are not related with the  
kind of ontic truth we are looking for when searching a TOE.


[SPK]
Of course fuzzy logic does not have fuzzy theorem, that could be  
mistaking the meaning of the word fuzzy with the meaning of the  
word ambiguous. I have been trying to establish the validity of  
the idea that it is the rules (given as axioms, etc) that are used  
to define a given mathematical structure, be it a model, or an  
algebra, etc. But I think that one must be careful that the logical  
structure that one uses of a means to define ontic truths is not  
assumed to be absolute unless very strong reasons can be proven to  
exist for such assumptions.




Usually non classical logic have epistemic or pragmatic classical  
interpretations, or even classical formulation, like the classical  
modal logic S4 which can emulate intuitionistic logic, or the  
Brouwersche modal logic B, which can emulate weak quantum logic.  
This corresponds to the fact that intuitionist logic might modelize  
constructive provability, and quantum logic modelizes  
observability, and not the usual notion of classical truth (as used  
almost everywhere in mathematics).


[SPK]
I use the orthocomplete lattices as a representation of quantum  
logic. My ideas are influenced by the work of Svozil, Calude and  
von Benthem, and others on this. I am not sure of the definition of  
weak quantum logic as you use it here.


Svozil, Calude and van Benthem thought on the subject are very good.  
Weak quantum logic is the logic of sublattice of ortholattices, like  
in the paper of Goldblatt that I have often refer to you. Basically it  
is quantum logic without the orthomodularity axiom. It does not  
distinguish finite dimensional pre-Hilbert space from Hilbert spave,  
for example.






One question regarding the emulations. If one where considering  
only finite emulations of a quantum logic (such as how a classical  
approximation of a QM system could be considered), how might one  
apply the Tychonoff, Heine–Borel definition or Bolzano–Weierstrass  
criterion of compactness to be sure that compactness obtain for the  
models? If we use these compactness criteria, is it necessary that  
the collection

Re: Logics

[Bruno Marchal]

On 27 Sep 2011, at 20:02, meekerdb wrote:


On 9/27/2011 5:28 AM, Jason Resch wrote:




On Tue, Sep 27, 2011 at 6:49 AM, Stephen P. King stephe...@charter.net 
 wrote:

On 9/26/2011 7:56 PM, Jason Resch wrote:








Okay, there may be other subjects, besides number theory and  
arithmetical truth where other forms of logic are more  
appropriate.  For unambiguous propositions about numbers, do you  
agree with the law of the excluded middle?


Jason


I think this an assumption or another axiom.  Consider the  
conjecture that every even number can be written as the sum of  
twoprimes.  Suppose there is no proof of this from Peano's  
axioms, but we can't know that there is no proof; only that we can't  
find one.  Intuitively we think the conjecture must be true or  
false, but this is based on the idea that if we tested all the evens  
we'd find it either true or false of each one.  Yet infinite testing  
is impossible.  So if the conjecture is true but unprovable, then  
it's undecidable.


Undecidable does not entails the negation of the law of the excluded  
middle.


Undecidable (by PA, say) = ~Bp  ~B~p. Law of the excluded middle = (p  
V ~p)


(~Bp  ~B~p) - ~(p V ~p) is NOT a theorem of G*.

Bruno

http://iridia.ulb.ac.be/~marchal/



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Re: Logics

[Stephen P. King]

On 9/27/2011 10:47 AM, Bruno Marchal wrote:


On 27 Sep 2011, at 13:49, Stephen P. King wrote:


On 9/26/2011 7:56 PM, Jason Resch wrote:


snip

For well-defined propositions regarding the numbers I think the 
values are confined to true or false.


Jason

--

[SPK]
Not in general, unless one is only going to allow only Boolean 
logics to exist. There have been proven to exist logics that have 
truth values that range over any set of numbers, not just {0,1}. 
Recall the requirement for a mathematical structure to exist: 
Self-consistency.


Consistency is a notion applied usually to theories, or (chatty) 
machines, not to mathematical structures.
A theory is consistent if it does not prove some proposition and its 
negation. A machine is consistent if it does not assert a proposition 
and its negation.


[SPK]
Is not a machine represented mathematically by some abstract 
(mathematical ) structure?  I am attempting to find clarity in the ideas 
surrounding the notion of machine and how you arrive at the idea that 
the abstract notion of implementation is sufficient to derive the 
physical notion of implementation.




In first order logic we have Gödel-Henkin completeness theorem which 
shows that a theory is consistent if and only if there is a 
mathematical structure (called model) satisfying (in a sense which can 
be made precise) the proposition proved in the theory.


[SPK]
What constraints are defined on the models by the Gödel-Henkin 
completeness theorem? How do we separate out effective consistent 
first-order theories that do not have computable models?




Also, it is true that classical (Boolean) logic are not the only 
logic. There are infinitely many logics, below and above classical 
propositional logic. But this cannot be used to criticize the use of 
classical logic in some domain.

[SPK]
OK. My thought here was to show that classical (Boolean) logic is 
not unique and should not be taken as absolute. To do so would be a 
mistake similar to Kant's claim that Euclidean logic was absolute.




All treatises on any non classical logic used classical (or much more 
rarely intuitionistic) logic at the meta-level. You will not find a 
book on fuzzy logic having fuzzy theorems, for example. Non classical 
logics have multiple use, which are not related with the kind of ontic 
truth we are looking for when searching a TOE.


[SPK]
Of course fuzzy logic does not have fuzzy theorem, that could be 
mistaking the meaning of the word fuzzy with the meaning of the word 
ambiguous. I have been trying to establish the validity of the idea 
that it is the rules (given as axioms, etc) that are used to define a 
given mathematical structure, be it a model, or an algebra, etc. But I 
think that one must be careful that the logical structure that one uses 
of a means to define ontic truths is not assumed to be absolute unless 
very strong reasons can be proven to exist for such assumptions.




Usually non classical logic have epistemic or pragmatic classical 
interpretations, or even classical formulation, like the classical 
modal logic S4 which can emulate intuitionistic logic, or the 
Brouwersche modal logic B, which can emulate weak quantum logic. This 
corresponds to the fact that intuitionist logic might modelize 
constructive provability, and quantum logic modelizes observability, 
and not the usual notion of classical truth (as used almost everywhere 
in mathematics).


[SPK]
I use the orthocomplete lattices as a representation of quantum 
logic. My ideas are influenced by the work of Svozil 
http://tph.tuwien.ac.at/%7Esvozil/publ/publ.html, Calude 
http://www.cs.auckland.ac.nz/%7Ecristian/10773_2006_9296_OnlinePDF.pdf  and 
von Benthem http://staff.science.uva.nl/%7Ejohan/publications.html, 
and others on this. I am not sure of the definition of weak quantum 
logic as you use it here.


One question regarding the emulations. If one where considering 
only finite emulations of a quantum logic (such as how a classical 
approximation of a QM system could be considered), how might one apply 
the Tychonoff, Heine--Borel definition or Bolzano--Weierstrass criterion 
of compactness to be sure that compactness obtain for the models? If we 
use these compactness criteria, is it necessary that the collection of 
open sets that is used in complete in an absolute sense? COuld it be 
that we have a way to recover the appearence of the axiom of choice or 
the ultrafilter lemma?


Could it be possible to have a notion of accessibility to 
parametrize or weaking the word every as in the sentence:  A point 
/x/ in /X/ is a *limit point* of /S/ if every open set 
http://en.wikipedia.org/wiki/Open_set containing /x/ contains at least 
one point of /S/ different from /x/ itself. to A point /x/ in /X/ is a 
*limit point* of /S/ if every open set 
http://en.wikipedia.org/wiki/Open_set , that is assessible from some 
S, containing /x/ contains at least one point of /S/ different from /x

Re: Logics

[Stephen P. King]

On 9/26/2011 7:56 PM, Jason Resch wrote:



On Mon, Sep 26, 2011 at 12:14 PM, Stephen P. King 
stephe...@charter.net mailto:stephe...@charter.net wrote:


On 9/26/2011 11:52 AM, Jason Resch wrote:



On Mon, Sep 26, 2011 at 9:44 AM, Stephen P. King
stephe...@charter.net mailto:stephe...@charter.net wrote:

snip
Jason,

I really would like to understand how it is that the
truth valuation of a proposition is not dependent on our
knowledge of it can be used to affirm the meaning of the
referent of that proposition independent of us?


That sentence was hard to parse!  If I understand it correctly,
you are asking how a truth, independent of our knowledge, can
confer meaning to something without us?

[SPK]
Essentially, yes.




Things unknown to anyone can have consequences which are
eventually do make a difference to beings which are aware of the
difference.  A comet colliding with the Earth and hitting a pond
of unicellular organisms may have drastically altered the course
of evolution on our planet.  That such a comet impact ocurred is
a fact which is either true or false, despite it being
independent of anyone's knowledge of it.  Yet it has perceptable
results.


[SPK]
The web of causes and effects is an aspect of the material
universe. I am taking that concept into consideration.



Correspondingly, the existence of some mathematical truth (even
if not comprehended by anyone) can have effects for observers, in
fact, it might explain both the observers themselves and their
experiences.

[SPK]
Slow down! existence of some mathematical truth??? Do you
mean the truth value of some existing mathematical statement? That
is what I mean in my question by the phrase truth valuation of a
proposition. Is a truth value something that exists or does not
exist?


I am not sure what you mean by exists in this case so let me say 
this, the state of being true, or the state of being false, for the 
proposition in question, was settled before a human made a 
determination regarding that proposition.


[SPK]
Is the state of being true a physical state, like the state of 
having 10 units of momentum? Is there a truth detector? Are you sure 
that state and true are words that go together? AFAIK, true (or 
false) are values, like numbers. In fact logics can have truth values 
that range over any set of numbers. This puts truth valuations in the 
same category as numbers. No?





How does the sentence 17 is prime is a true statement
confer implicit meaning to its referent?


What is the referent in this case?  17?  And what do you mean by
meaning?  17's primality is a fact of nature.  The statement's
existence or non-existence, comprehension or non-comprehension
makes no difference to 17, only what you could say we humans have
discovered about 17.


[SPK]
Is the symbol 17 the same extant as the abstract number it
refers to?


No, as I mentioned to Brent in a post the other day, we ought not 
confuse the label for the thing.  Nor should we confuse our idea of a 
thing for the thing itself.

[SPK]
OK, does not this imply that there are (at least) two separate 
categories: Labels and Things? What relation might exist between these 
categories?



Do you believe that symbols and what they represent are one and
the same thing???


No, we can apply some simple rules to the symbols in certain way to 
learn things about the object in question.

[SPK]
What relation might exist between the rules of symbols and the 
rules of things?




How does not the fact that many symbols can represent one and the
same extant disprove this hypothesis? Is the word tree have a
brownish trunk and greenish foliage?  What about the case where
sets of symbols that have more than one possible referent?
Consider the word FORD. Does it have wheels and a motor? What is
the height of the water that one displaces when we might walk
across it? There is a categorical difference between an object and
its representations and the fact that one subobject of those
categories exists is not proof that a subobject in another
category has a given truth value. BTW, truth values are not
confined to {True, False}.


For well-defined propositions regarding the numbers I think the values 
are confined to true or false.


Jason

--

[SPK]
Not in general, unless one is only going to allow only Boolean 
logics to exist. There have been proven to exist logics that have truth 
values that range over any set of numbers, not just {0,1}. Recall the 
requirement for a mathematical structure to exist: Self-consistency.


Onward!

Stephen

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Re: Logics

[Jason Resch]

On Tue, Sep 27, 2011 at 6:49 AM, Stephen P. King stephe...@charter.netwrote:

  On 9/26/2011 7:56 PM, Jason Resch wrote:



 On Mon, Sep 26, 2011 at 12:14 PM, Stephen P. King 
 stephe...@charter.netwrote:

   On 9/26/2011 11:52 AM, Jason Resch wrote:



 On Mon, Sep 26, 2011 at 9:44 AM, Stephen P. King 
 stephe...@charter.netwrote:

   snip
  Jason,

 I really would like to understand how it is that the truth valuation
 of a proposition is not dependent on our knowledge of it can be used to
 affirm the meaning of the referent of that proposition independent of us?


 That sentence was hard to parse!  If I understand it correctly, you are
 asking how a truth, independent of our knowledge, can confer meaning to
 something without us?

  [SPK]
 Essentially, yes.



 Things unknown to anyone can have consequences which are eventually do
 make a difference to beings which are aware of the difference.  A comet
 colliding with the Earth and hitting a pond of unicellular organisms may
 have drastically altered the course of evolution on our planet.  That such a
 comet impact ocurred is a fact which is either true or false, despite it
 being independent of anyone's knowledge of it.  Yet it has perceptable
 results.

   [SPK]
 The web of causes and effects is an aspect of the material universe. I
 am taking that concept into consideration.


  Correspondingly, the existence of some mathematical truth (even if not
 comprehended by anyone) can have effects for observers, in fact, it might
 explain both the observers themselves and their experiences.


  [SPK]
 Slow down! existence of some mathematical truth??? Do you mean the
 truth value of some existing mathematical statement? That is what I mean in
 my question by the phrase truth valuation of a proposition. Is a truth
 value something that exists or does not exist?


 I am not sure what you mean by exists in this case so let me say this,
 the state of being true, or the state of being false, for the proposition in
 question, was settled before a human made a determination regarding that
 proposition.


 [SPK]
 Is the state of being true a physical state, like the state of
 having 10 units of momentum?


If the object under consideration is a physical object, you might be able to
say that.  If the object under consideration is 17, I would say no.


 Is there a truth detector?


There can be truth detectors, in some sense we may be truth detectors, but
us discovery of a truth is not what makes it true.


 Are you sure that state and true are words that go together?


I am at a loss for an english word that conveys the status of true or
false.  We have the word parity for the status of even or odd, for example,
but I could not think of such a word that conveys the same for true or
false, which is why I used the state of being true or false.


 AFAIK, true (or false) are values, like numbers. In fact logics can have
 truth values that range over any set of numbers. This puts truth valuations
 in the same category as numbers. No?


True and false can be represented by two different numbers, but I am not
sure that makes them values in the same sense of numbers.






   How does the sentence 17 is prime is a true statement confer implicit
 meaning to its referent?


 What is the referent in this case?  17?  And what do you mean by
 meaning?  17's primality is a fact of nature.  The statement's existence
 or non-existence, comprehension or non-comprehension makes no difference to
 17, only what you could say we humans have discovered about 17.

   [SPK]
 Is the symbol 17 the same extant as the abstract number it refers to?


 No, as I mentioned to Brent in a post the other day, we ought not confuse
 the label for the thing.  Nor should we confuse our idea of a thing for the
 thing itself.

 [SPK]
 OK, does not this imply that there are (at least) two separate
 categories: Labels and Things? What relation might exist between these
 categories?


Labels are a human invention to support communication of ideas, which you
might say is yet another category of things.

The relation ship might be as follows: if I tell you to multiply 1200 x
1800, you could arrange 1800 rows of 1200 beans and count them all, or you
could follow some simple rules of transformation applied to the labels
'1200' and '1800' and have a shortcut to the answer, without having to do
all that counting.





 Do you believe that symbols and what they represent are one and the same
 thing???


 No, we can apply some simple rules to the symbols in certain way to learn
 things about the object in question.

 [SPK]
 What relation might exist between the rules of symbols and the
 rules of things?


I think I covered this above.





  How does not the fact that many symbols can represent one and the same
 extant disprove this hypothesis? Is the word tree have a brownish trunk
 and greenish foliage?  What about the case where sets of symbols that have
 more than one possible

Re: Logics

[Stephen P. King]

On 9/27/2011 8:28 AM, Jason Resch wrote:



On Tue, Sep 27, 2011 at 6:49 AM, Stephen P. King 
stephe...@charter.net mailto:stephe...@charter.net wrote:


On 9/26/2011 7:56 PM, Jason Resch wrote:



On Mon, Sep 26, 2011 at 12:14 PM, Stephen P. King
stephe...@charter.net mailto:stephe...@charter.net wrote:

On 9/26/2011 11:52 AM, Jason Resch wrote:



On Mon, Sep 26, 2011 at 9:44 AM, Stephen P. King
stephe...@charter.net mailto:stephe...@charter.net wrote:

snip
Jason,

I really would like to understand how it is that the
truth valuation of a proposition is not dependent on our
knowledge of it can be used to affirm the meaning of the
referent of that proposition independent of us?


That sentence was hard to parse!  If I understand it
correctly, you are asking how a truth, independent of our
knowledge, can confer meaning to something without us?

[SPK]
Essentially, yes.




Things unknown to anyone can have consequences which are
eventually do make a difference to beings which are aware of
the difference.  A comet colliding with the Earth and
hitting a pond of unicellular organisms may have drastically
altered the course of evolution on our planet.  That such a
comet impact ocurred is a fact which is either true or
false, despite it being independent of anyone's knowledge of
it.  Yet it has perceptable results.


[SPK]
The web of causes and effects is an aspect of the
material universe. I am taking that concept into consideration.



Correspondingly, the existence of some mathematical truth
(even if not comprehended by anyone) can have effects for
observers, in fact, it might explain both the observers
themselves and their experiences.

[SPK]
Slow down! existence of some mathematical truth??? Do
you mean the truth value of some existing mathematical
statement? That is what I mean in my question by the phrase
truth valuation of a proposition. Is a truth value
something that exists or does not exist?


I am not sure what you mean by exists in this case so let me
say this, the state of being true, or the state of being false,
for the proposition in question, was settled before a human made
a determination regarding that proposition.


[SPK]
Is the state of being true a physical state, like the state
of having 10 units of momentum?


If the object under consideration is a physical object, you might be 
able to say that.  If the object under consideration is 17, I would 
say no.

[SPK]
OK. So it is your belief that , in general, objects (of any 
categorical type) have specific and definite properties absent the 
specification of the means of observation? How do you explain the 
existence of conjugate observables in QM?




Is there a truth detector?


There can be truth detectors, in some sense we may be truth detectors, 
but us discovery of a truth is not what makes it true.


Are you sure that state and true are words that go together?


I am at a loss for an english word that conveys the status of true or 
false.  We have the word parity for the status of even or odd, for 
example, but I could not think of such a word that conveys the same 
for true or false, which is why I used the state of being true or false.


AFAIK, true (or false) are values, like numbers. In fact logics
can have truth values that range over any set of numbers. This
puts truth valuations in the same category as numbers. No?


True and false can be represented by two different numbers, but I am 
not sure that makes them values in the same sense of numbers.

[SPK]
I was mentioning the fact that logics with truth values that range 
over different sets of values have been proven to exist. Logic is not 
limited to truth values over {0,1}, only Boolean logics are so 
restricted by their defining rules.







How does the sentence 17 is prime is a true statement
confer implicit meaning to its referent?


What is the referent in this case?  17?  And what do you
mean by meaning?  17's primality is a fact of nature.  The
statement's existence or non-existence, comprehension or
non-comprehension makes no difference to 17, only what you
could say we humans have discovered about 17.


[SPK]
Is the symbol 17 the same extant as the abstract number
it refers to?


No, as I mentioned to Brent in a post the other day, we ought not
confuse the label for the thing.  Nor should we confuse our idea
of a thing for the thing itself.

[SPK]
OK, does not this imply that there are (at least) two separate
categories: Labels and Things? What relation might

Re: Logics

[Bruno Marchal]

On 27 Sep 2011, at 13:49, Stephen P. King wrote:


On 9/26/2011 7:56 PM, Jason Resch wrote:


snip

For well-defined propositions regarding the numbers I think the  
values are confined to true or false.


Jason

--

[SPK]
Not in general, unless one is only going to allow only Boolean  
logics to exist. There have been proven to exist logics that have  
truth values that range over any set of numbers, not just {0,1}.  
Recall the requirement for a mathematical structure to exist: Self- 
consistency.


Consistency is a notion applied usually to theories, or (chatty)  
machines, not to mathematical structures.
A theory is consistent if it does not prove some proposition and its  
negation. A machine is consistent if it does not assert a proposition  
and its negation.


In first order logic we have Gödel-Henkin completeness theorem which  
shows that a theory is consistent if and only if there is a  
mathematical structure (called model) satisfying (in a sense which can  
be made precise) the proposition proved in the theory.


Also, it is true that classical (Boolean) logic are not the only  
logic. There are infinitely many logics, below and above classical  
propositional logic. But this cannot be used to criticize the use of  
classical logic in some domain.


All treatises on any non classical logic used classical (or much more  
rarely intuitionistic) logic at the meta-level. You will not find a  
book on fuzzy logic having fuzzy theorems, for example. Non classical  
logics have multiple use, which are not related with the kind of ontic  
truth we are looking for when searching a TOE.


Usually non classical logic have epistemic or pragmatic classical  
interpretations, or even classical formulation, like the classical  
modal logic S4 which can emulate intuitionistic logic, or the  
Brouwersche modal logic B, which can emulate weak quantum logic. This  
corresponds to the fact that intuitionist logic might modelize  
constructive provability, and quantum logic modelizes observability,  
and not the usual notion of classical truth (as used almost everywhere  
in mathematics).


To invoke the existence of non classical logic to throw a doubt about  
the universal truth of elementary statements in well defined domain,  
like arithmetic, would lead to complete relativism, given that you can  
always build some ad hoc logic/theory proving the negation of any  
statement, and this would make the notion of  truth problematic. The  
contrary is true. A non classical logic is eventually accepted when we  
can find an interpretation of it in the classical framework.


A non standard truth set, like the collection of open subsets of a  
topological space, provided a classical sense for intuitionist logic,  
like a lattice of linear subspaces can provide a classical  
interpretation of quantum logic (indeed quantum logic is born from  
such structures). It might be that nature observables obeys quantum  
logic, but quantum physicists talk and reason in classical logic, and  
use classical mathematical tools to describe the non classical  
behavior of matter.


Bruno


http://iridia.ulb.ac.be/~marchal/



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Re: Logics

[meekerdb]

On 9/27/2011 4:49 AM, Stephen P. King wrote:

[SPK]
Not in general, unless one is only going to allow only Boolean logics to exist. 
There have been proven to exist logics that have truth values that range over any set of 
numbers, not just {0,1}. Recall the requirement for a mathematical structure to exist: 
Self-consistency.


Onward!

Stephen


How do you define consistency for fuzzy or probabilistic logics?  If you prove P(x)=0.1 
and P(x)=0.2 is that inconsistency?


Brent

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Re: Logics

[meekerdb]

On 9/27/2011 5:28 AM, Jason Resch wrote:



On Tue, Sep 27, 2011 at 6:49 AM, Stephen P. King stephe...@charter.net 
mailto:stephe...@charter.net wrote:


On 9/26/2011 7:56 PM, Jason Resch wrote:



On Mon, Sep 26, 2011 at 12:14 PM, Stephen P. King stephe...@charter.net
mailto:stephe...@charter.net wrote:

On 9/26/2011 11:52 AM, Jason Resch wrote:



On Mon, Sep 26, 2011 at 9:44 AM, Stephen P. King stephe...@charter.net
mailto:stephe...@charter.net wrote:

snip
Jason,

I really would like to understand how it is that the truth 
valuation
of a proposition is not dependent on our knowledge of it can be 
used to
affirm the meaning of the referent of that proposition independent 
of us?


That sentence was hard to parse!  If I understand it correctly, you are 
asking
how a truth, independent of our knowledge, can confer meaning to 
something
without us?

[SPK]
Essentially, yes.




Things unknown to anyone can have consequences which are eventually do 
make a
difference to beings which are aware of the difference.  A comet 
colliding
with the Earth and hitting a pond of unicellular organisms may have
drastically altered the course of evolution on our planet.  That such a 
comet
impact ocurred is a fact which is either true or false, despite it being
independent of anyone's knowledge of it.  Yet it has perceptable 
results.


[SPK]
The web of causes and effects is an aspect of the material 
universe. I am
taking that concept into consideration.



Correspondingly, the existence of some mathematical truth (even if not
comprehended by anyone) can have effects for observers, in fact, it 
might
explain both the observers themselves and their experiences.

[SPK]
Slow down! existence of some mathematical truth??? Do you mean 
the truth
value of some existing mathematical statement? That is what I mean in my
question by the phrase truth valuation of a proposition. Is a truth 
value
something that exists or does not exist?


I am not sure what you mean by exists in this case so let me say this, 
the state
of being true, or the state of being false, for the proposition in 
question, was
settled before a human made a determination regarding that proposition.


[SPK]
Is the state of being true a physical state, like the state of 
having 10
units of momentum?


If the object under consideration is a physical object, you might be able to say that.  
If the object under consideration is 17, I would say no.


Is there a truth detector?


There can be truth detectors, in some sense we may be truth detectors, but us discovery 
of a truth is not what makes it true.


Are you sure that state and true are words that go together?


I am at a loss for an english word that conveys the status of true or false.  We have 
the word parity for the status of even or odd, for example, but I could not think of 
such a word that conveys the same for true or false, which is why I used the state of 
being true or false.


AFAIK, true (or false) are values, like numbers. In fact logics can have 
truth
values that range over any set of numbers. This puts truth valuations in 
the same
category as numbers. No?


True and false can be represented by two different numbers, but I am not sure that makes 
them values in the same sense of numbers.






How does the sentence 17 is prime is a true statement confer 
implicit
meaning to its referent?


What is the referent in this case?  17?  And what do you mean by meaning? 
17's primality is a fact of nature.  The statement's existence or

non-existence, comprehension or non-comprehension makes no difference 
to 17,
only what you could say we humans have discovered about 17.


[SPK]
Is the symbol 17 the same extant as the abstract number it refers 
to?


No, as I mentioned to Brent in a post the other day, we ought not confuse 
the label
for the thing.  Nor should we confuse our idea of a thing for the thing 
itself.

[SPK]
OK, does not this imply that there are (at least) two separate 
categories:
Labels and Things? What relation might exist between these categories?


Labels are a human invention to support communication of ideas, which you might say is 
yet another category of things.


The relation ship might be as follows: if I tell you to multiply 1200 x 1800, you could 
arrange 1800 rows of 1200 beans and count them all, or you could follow some simple 
rules of transformation applied to the labels '1200' and '1800' and have a shortcut to 
the answer, without having to do all that counting.




Do you believe that symbols and what they represent are one

Re: Logics

[Jason Resch]

On Tue, Sep 27, 2011 at 1:02 PM, meekerdb meeke...@verizon.net wrote:

  Not in general, unless one is only going to allow only Boolean logics
 to exist. There have been proven to exist logics that have truth values that
 range over any set of numbers, not just {0,1}. Recall the requirement for a
 mathematical structure to exist: Self-consistency.


 Okay, there may be other subjects, besides number theory and arithmetical
 truth where other forms of logic are more appropriate.  For unambiguous
 propositions about numbers, do you agree with the law of the excluded
 middle?

 Jason


 I think this an assumption or another axiom.  Consider the conjecture that
 every even number can be written as the sum of two primes.  Suppose there is
 no proof of this from Peano's axioms, but we can't know that there is no
 proof; only that we can't find one.  Intuitively we think the conjecture
 must be true or false, but this is based on the idea that if we tested all
 the evens we'd find it either true or false of each one.  Yet infinite
 testing is impossible.  So if the conjecture is true but unprovable, then
 it's undecidable.


Propositions can be undecidable in the context of a given set of axioms, but
there are stronger systems in which the proposition is decidable.

In any case, whether or not some proposition is decidable (can be
demonstrated as true or demonstrated as false in a series of logical steps
leading to the axioms in question) does not suggest that a mathematical
proposition is true or false dependently of us.  Conversely, I think it is
one of the strongest arguments against the idea that math is man-made.  Any
system of axioms we develop is imperfect in the sense that it cannot answer
all questions concerning the numbers.

Those who think that the objects of study in mathematics are human
inventions are living in the early 20th century.

Jason

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Re: Logics

[Jason Resch]

On Tue, Sep 27, 2011 at 8:02 AM, Stephen P. King stephe...@charter.netwrote:

  On 9/27/2011 8:28 AM, Jason Resch wrote:



 On Tue, Sep 27, 2011 at 6:49 AM, Stephen P. King stephe...@charter.netwrote:

  On 9/26/2011 7:56 PM, Jason Resch wrote:



 On Mon, Sep 26, 2011 at 12:14 PM, Stephen P. King 
 stephe...@charter.netwrote:

   On 9/26/2011 11:52 AM, Jason Resch wrote:



 On Mon, Sep 26, 2011 at 9:44 AM, Stephen P. King 
 stephe...@charter.netwrote:

   snip
  Jason,

 I really would like to understand how it is that the truth valuation
 of a proposition is not dependent on our knowledge of it can be used to
 affirm the meaning of the referent of that proposition independent of us?


 That sentence was hard to parse!  If I understand it correctly, you are
 asking how a truth, independent of our knowledge, can confer meaning to
 something without us?

  [SPK]
 Essentially, yes.



 Things unknown to anyone can have consequences which are eventually do
 make a difference to beings which are aware of the difference.  A comet
 colliding with the Earth and hitting a pond of unicellular organisms may
 have drastically altered the course of evolution on our planet.  That such a
 comet impact ocurred is a fact which is either true or false, despite it
 being independent of anyone's knowledge of it.  Yet it has perceptable
 results.

   [SPK]
 The web of causes and effects is an aspect of the material universe.
 I am taking that concept into consideration.


  Correspondingly, the existence of some mathematical truth (even if not
 comprehended by anyone) can have effects for observers, in fact, it might
 explain both the observers themselves and their experiences.


  [SPK]
 Slow down! existence of some mathematical truth??? Do you mean the
 truth value of some existing mathematical statement? That is what I mean in
 my question by the phrase truth valuation of a proposition. Is a truth
 value something that exists or does not exist?


 I am not sure what you mean by exists in this case so let me say this,
 the state of being true, or the state of being false, for the proposition in
 question, was settled before a human made a determination regarding that
 proposition.


 [SPK]
 Is the state of being true a physical state, like the state of
 having 10 units of momentum?


 If the object under consideration is a physical object, you might be able
 to say that.  If the object under consideration is 17, I would say no.

 [SPK]
 OK. So it is your belief that , in general, objects (of any categorical
 type) have specific and definite properties absent the specification of the
 means of observation?


Yes I believe objects have properties even if unobserved.  Do you really
believe the cat is both alive and dead (in the same universe) until it is
observed?


 How do you explain the existence of conjugate observables in QM?


If you are a CI proponent, you could say being in many places at once, or
having many simultaneous states simultaneously is a property of objects in
superposition.  The Everettian might say a more simply that a property of a
particle (or the universe) is that it obeys the Shrodinger equation.

I assume your point is that an particle cannot have a definite momentum and
position, but this is really a statement about observation (the observer
cannot know both simultaneously), not the object (or many objects) under
consideration.






  Is there a truth detector?


 There can be truth detectors, in some sense we may be truth detectors, but
 us discovery of a truth is not what makes it true.


  Are you sure that state and true are words that go together?


 I am at a loss for an english word that conveys the status of true or
 false.  We have the word parity for the status of even or odd, for example,
 but I could not think of such a word that conveys the same for true or
 false, which is why I used the state of being true or false.


  AFAIK, true (or false) are values, like numbers. In fact logics can have
 truth values that range over any set of numbers. This puts truth valuations
 in the same category as numbers. No?


 True and false can be represented by two different numbers, but I am not
 sure that makes them values in the same sense of numbers.

 [SPK]
 I was mentioning the fact that logics with truth values that range over
 different sets of values have been proven to exist. Logic is not limited to
 truth values over {0,1}, only Boolean logics are so restricted by their
 defining rules.


I think Bruno addressed this very well.








   How does the sentence 17 is prime is a true statement confer
 implicit meaning to its referent?


 What is the referent in this case?  17?  And what do you mean by
 meaning?  17's primality is a fact of nature.  The statement's existence
 or non-existence, comprehension or non-comprehension makes no difference to
 17, only what you could say we humans have discovered about 17.

   [SPK]
 Is the symbol 17 the same extant

Re: Logics

[Stephen P. King]

On 9/27/2011 1:40 PM, meekerdb wrote:

On 9/27/2011 4:49 AM, Stephen P. King wrote:

[SPK]
Not in general, unless one is only going to allow only Boolean 
logics to exist. There have been proven to exist logics that have 
truth values that range over any set of numbers, not just {0,1}. 
Recall the requirement for a mathematical structure to exist: 
Self-consistency.


Onward!

Stephen


How do you define consistency for fuzzy or probabilistic logics?  If 
you prove P(x)=0.1 and P(x)=0.2 is that inconsistency?


Brent

I am not a in a position to write out such definitions. You might find a 
thorough explanation of fuzzy logic in any of Bart Kosko's books on the 
subject. I am sure that there are papers and or books that explain the 
same for probabilistic logic.


Onward!

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