A collection of infinitely many distinct elements

[doronshadmi]

Nonsense. It is not at all formal.

Just for starters, € and its negation are operators. They do not have values (0 or 1, in or out, false or true, whatever). They have meaning when expressing relationships between operands.

Try as you may, € cannot be offered up by itself as something meaningful as you have tried to do. It has to be in the form (A € B) -- it is a binary operator after all.

Let us write it in this way:

[] = domain

in = €

out = ~€


€ ~€
0 0 --> [ ] (in,out are the same) (non-locality) is the same as:

(0 in []) nor (0 out []) --> [ ]


€ ~€
0 1 --> [ ]. (in,out are not the same (locality) is the same as:

(0 in []) xor (1 out []) --> [ ].


€ ~€
1 0 --> [.] (in,out are not the same (locality) is the same as:

(1 in []) xor (0 out []) --> [.]


€ ~€
1 1 --> [_]_ (in,out are the same) (non-locality) is the same as:

(1 in []) and (1 out []) --> [_]_

 

[ddt]

Let us write it in this way:

It only gets less comprehensible.

Now, how difficult can it be to express the relation €? It is a binary relation between two elements in your domain of discourse. You just conceded your whole domain contains only two elements: 0 and 1. Then there are only 4 cases to consider.

0 € 0 is true
0 € 1 is false
1 € 0 is false
1 € 1 is true

for instance.

 

[doronshadmi]

It only gets less comprehensible.

Now, how difficult can it be to express the relation €? It is a binary relation between two elements in your domain of discourse. You just conceded your whole domain contains only two elements: 0 and 1. Then there are only 4 cases to consider.

0 € 0 is true
0 € 1 is false
1 € 0 is false
1 € 1 is true

for instance.

No,

[] = domain

in = "in the domain"

out = "out of the domain"


0 = nothing

1 = somthing


(0 in []) nor (0 out []) is true --> [ ] (in,out are the same) (non-locality)

(0 in []) xor (1 out []) is true --> [ ]. (in,out are not the same (locality)

(1 in []) xor (0 out []) is true --> [.] (in,out are not the same (locality)

(1 in []) and (1 out []) is true --> [_]_ (in,out are the same) (non-locality)

 

[ddt]

No,

[] = domain

in = "in the domain"

out = "out of the domain"


0 = nothing

1 = somthing


(0 in []) nor (0 out []) is true --> [ ] (in,out are the same) (non-locality)

(0 in []) xor (1 out []) is true --> [ ]. (in,out are not the same (locality)

(1 in []) xor (0 out []) is true --> [.] (in,out are not the same (locality)

(1 in []) and (1 out []) is true --> [_]_ (in,out are the same) (non-locality)

We were talking about the membership relation €. Your reply says nothing about it - the symbol doesn't occur in your reply. So your reply is meaningless w.r.t. the current discussion.

I further note that you didn't contradict me saying that you have only two elements in your domain of discourse - 0 and 1. So it's true.

What's this whole business then about? You have a trivial universe, with only two elements!

 

[Apathia]

It's not clear to me what could come before logic. Logic is the meta-level of mathematics; it's the fabric that mathematical proofs are made from, the language they are stated in. Mathematics is about deriving theorems from initial assumptions: the axioms we first lay down in a (new) field of mathematics. Logic deals with the ways how we may derive theorems, lemmata, etc. from those axioms.

I'm just getting to this because I'm having issues with my ISP, modem, both, or whatever.
I can't participate as I'd like.

Point accepted. Doron is offering a logic. But it's not one based on the law of noncontradiction but the relations of polar opposites. One aspect of of the relations is paired off as Symmetry/Asymmetry, and that is where Doron's law of non-contradiction rests as a special case. Meanwhile the over all realtion of Locality/Non-Locality includes realtions where the law of non-contradiction does not hold. Remember, in Doron's book, a set isn't a fixed complete. There are virtual elements that from the view of Non-Locality are both members and non-members.

I know this hardly helps. I'm not trying to shore up Doron's inconsistances. I just try to understand what he's trying to say and his intentions with that.

so I don't see anything modal

Yes, Doron's logic is not a modal or multivalue logic. It's a logic of polarities. It's as I said in a previous post, "digital."

 

[jsfisher]

Let us write it in this way:

[] = domain

in = €

out = ~€

And this means what? Are not in and out operators of arity-two (or maybe arity-one if the domain is implied).

In the past, you used these operators in the sense of "A in B" (as a binary operator). Do you mean something different now? In the past, you defined ~€ as the complement of €. Do you mean something different now? Is out the complement of in?

€ ~€
0 0 --> [ ] (in,out are the same) (non-locality) is the same as:

(0 in []) nor (0 out []) --> [ ]

Ah, there's nothing like standard notation to convey an idea, and this is nothing like standard notation.

Moreover, you are conflating an operator's "value" with the operand's "value".

Finally, the way you are using NOR (and later XOR and AND) here is not as any sort of logic operator. You put it here simply because F NOR F is T (or 0 NOR 0 is 1) as if that somehow makes it meaningful for here, but not in later lines of your table.

The expression ((0 in []) nor (0 out [])) is an expression in logic. It will be either true or false depending on the in and out operations. Never will it result in [ ].

 

[doronshadmi]

We were talking about the membership relation €. Your reply says nothing about it - the symbol doesn't occur in your reply. So your reply is meaningless w.r.t. the current discussion.

I further note that you didn't contradict me saying that you have only two elements in your domain of discourse - 0 and 1. So it's true.

What's this whole business then about? You have a trivial universe, with only two elements!

Come on man,

in = "in the domain" = €

out = "out of the domain" = ~€

0 = nothing

1 = somthing


(0 in []) nor (0 out []) is true --> [ ] (in,out are the same) (non-locality)

(0 in []) xor (1 out []) is true --> [ ]. (in,out are not the same (locality)

(1 in []) xor (0 out []) is true --> [.] (in,out are not the same (locality)

(1 in []) and (1 out []) is true --> [_]_ (in,out are the same) (non-locality)

As you can see nothing or somthing are representations of locality and non-locality.

Contradiction in my framework is this:

There is non-local such that it is local, notated as:

[latex]$$$ \exists$$$[/latex] __ : (.)

or

There is local such that it is non-local, notated as:

[latex]$$$ \exists$$$[/latex] . : (_)_

 

[ddt]

Come on man,

That's quite condescending. I find it insulting. You're the one confusing the hell with your notation and then loathe other people for not getting it?

in = "in the domain" = €

out = "out of the domain" = ~€

So you present two different notations for the same thing? Are you saying that the two formulae "x in A" and "x € A" mean precisely the same thing?

And you think we should understand that?

Moreover, on page 2&3 you call A a "system", whereas on page 4, you call A an "element". And now you say it is a "domain". It's quite obvious that you're not sure yourself what you're talking about.

0 = nothing

1 = somthing

Those were the only two elements in your domain. You said so before, I repeated that at least three times and you didn't contradict.

Now, before we go on sanitizing your "truth tables", let's first clean up the notation.

1) choose between "in/out" and "€/~€" and refrain from now on from using the other.

2) what kind of arguments does this € operator take? The left one is an element of your domain and the right one is a domain?

 

[ArmillarySphere]

in = €
out = ~€

(my bolding)
That's where you go wrong.

"out" is a completely different relation between x and A than "in" - it might be the negation of "in", but not necessarily.

~€ doesn't mean "out" as such - it means not in, which is something quite different.

Consider your favourite non-local example, the stick x lying across the border of a circle A. Is the stick inside the circle? Is it outside?

If we decide "in" to mean "any part is inside", and "out" to mean "any part is outside", then:

x in A is true
x not in A is false
x out A is true
x not out A is false

"in" and "out" are not each other's negations - they're two independent relations between x and A.

 

[doronshadmi]

That's quite condescending. I find it insulting. You're the one confusing the hell with your notation and then loathe other people for not getting it?


So you present two different notations for the same thing? Are you saying that the two formulae "x in A" and "x € A" mean precisely the same thing?

And you think we should understand that?

Moreover, on page 2&3 you call A a "system", whereas on page 4, you call A an "element". And now you say it is a "domain". It's quite obvious that you're not sure yourself what you're talking about.


Those were the only two elements in your domain. You said so before, I repeated that at least three times and you didn't contradict.

Now, before we go on sanitizing your "truth tables", let's first clean up the notation.

1) choose between "in/out" and "€/~€" and refrain from now on from using the other.

2) what kind of arguments does this € operator take? The left one is an element of your domain and the right one is a domain?

ddt,

I must say that you are right about the mass that I made in TOUM.pdf by using different notations and words, in order to represent the same value.

After I understand much better your standard point of view, because of jsfisher's argument about the law of contradiction, I think that I can find
better ways to explain my non-standard point of view about fundamental mathematical\logical stuff.


I wish to thank you for the patience that you had until now, to continue the dialog no matter how hard it is, and I really appreciate that.

The law of contradiction is a very accurate term to start with, and I'll do my best in order to be understood from now on.

Have a good night, and we shell continue (if you wish) tomorrow.

 

[doronshadmi]

(my bolding)
That's where you go wrong.

"out" is a completely different relation between x and A than "in" - it might be the negation of "in", but not necessarily.

~€ doesn't mean "out" as such - it means not in, which is something quite different.

Consider your favourite non-local example, the stick x lying across the border of a circle A. Is the stick inside the circle? Is it outside?

If we decide "in" to mean "any part is inside", and "out" to mean "any part is outside", then:

x in A is true
x not in A is false
x out A is true
x not out A is false

"in" and "out" are not each other's negations - they're two independent relations between x and A.

Non-locality: the stick can be both in and out of the circle.

Locality: the stick can be completely in or completely out of the circle.

Non-locality: the stick can be both in and (not in and not out) of the circle (where not in and not out is exactly on the circle's border).

Non-locality: the stick can be both out and (not in and not out) of the circle (where not in and not out is exactly on the circle's border).

Locality: the stick (if it is a flexible one) can be not in and not out of the circle (where not in and not out is exactly on the circle's border).



Locality: a point can be completely in the circle, completely out of the circle, not in and not out of the circle (where not in and not out is exactly on the circle's border).



Locality means: to be exactly in a one relation.

Non-locality means: to be in more than a one relation.

 

[ddt]

It's not clear to me what could come before logic. Logic is the meta-level of mathematics; it's the fabric that mathematical proofs are made from, the language they are stated in. Mathematics is about deriving theorems from initial assumptions: the axioms we first lay down in a (new) field of mathematics. Logic deals with the ways how we may derive theorems, lemmata, etc. from those axioms.

I'm just getting to this because I'm having issues with my ISP, modem, both, or whatever.
I can't participate as I'd like.

Point accepted.

OK, so you agree with the point that logic comes before anything else?

BTW, sorry about your connection issues, hope they'll be over soon.

Doron is offering a logic. But it's not one based on the law of noncontradiction but the relations of polar opposites.

Thus far, however, what we've seen of Doron's logic is consistent with classical logic.

One aspect of of the relations is paired off as Symmetry/Asymmetry, and that is where Doron's law of non-contradiction rests as a special case. Meanwhile the over all realtion of Locality/Non-Locality includes realtions where the law of non-contradiction does not hold.

But if you say that the locality/non-locality influences whether non-contradiction holds, you put again the notion of locality on a level before logic - contradicting your earlier statement.

Remember, in Doron's book, a set isn't a fixed complete. There are virtual elements that from the view of Non-Locality are both members and non-members.

So what? That's on the level of mathematical theory, not on the level of logic.

Yes, Doron's logic is not a modal or multivalue logic. It's a logic of polarities. It's as I said in a previous post, "digital."

It's not quite clear to me what you mean with the polarities. I see no difference between "polarities" as "antitheses" then in the classical true/false dichotomy.

 

[ArmillarySphere]

You original version of this post had defined "in" and "out" as "completely in" and "completely out". Since you've removed those definitions, the conclusions below depend highly on how the relations are defined.

Non-locality: the stick can be both in and out of the circle.

This depends on how "in" and "out" are defined. For some relations, this is possible, for others, it's not.

Locality: the stick can be completely in or completely out of the circle.

Trivially true if either can ever be true, since these are independent relations, no matter how in and out are defined or where the stick is.

Locality: the stick (if it is a flexible one) can be not in and not out of the circle (where not in and not out is exactly on the circle's border)..

This has nothing to do with the circle's border or flexible sticks and more to do with how "in" and "out" are defined. For some relations, it's possible, for others, it's not.

By the way, whether something is "exactly on the circle's border" is essentially meaningless - all that matters is how the "in" and "out" relations are defined.

Non-locality: the stick can be both in and (not in and not out) of the circle (where not in and not out is exactly on the circle's border).

Non-locality: the stick can be both out and (not in and not out) of the circle (where not in and not out is exactly on the circle's border).

These two cases contain a contradiction no matter how in and out are defined.

Locality: a point can be completely in the circle, completely out of the circle, not in and not out of the circle (where not in and not out is exactly on the circle's border).

All at once? I don't think so...
Sarcasm aside, see the above comments for the line cases; there's no difference.

Locality means: to be exactly in a one relation.

Non-locality means: to be in more than a one relation.

Whoah! Now you're talking - I think this is the first time I've seen an actual definition.

Too bad that it's pointless - set theory has exactly one relation to speak of, that being membership. "not membership" is always the logical negation, so everything is local to any set. And yes, I did mean everything.

 

[nathan]

Good, and how ~ is X operator?

Well, aside from this not being a sentence. Who cares? You asked what relates X and ~X. Whether X is an operator is immaterial.

It cannot be done if there is no common ground that enables this, because ~ is not X.

Of course ~ is not X in general. In the specific case of X being bound to ~, then ~ is X. But that's irrelevent.

You don't know any maths, do you?

 

[ddt]

I must say that you are right about the mass that I made in TOUM.pdf by using different notations and words, in order to represent the same value.

After I understand much better your standard point of view, because of jsfisher's argument about the law of contradiction, I think that I can find
better ways to explain my non-standard point of view about fundamental mathematical\logical stuff.

I wish to thank you for the patience that you had until now, to continue the dialog no matter how hard it is, and I really appreciate that.

Thanks for those words. It is apparent that you have little in the way of formal mathematical training. So I'd advise you, when pursuing further this, to acquire some good learning books on mathematical foundations, and/or to enroll in some undergrad courses on these subjects. AFAIK, both Tel Aviv University and the Weizmann institute have very competent math/CS departments. Other posters might suggest good (English) books.

I'm not saying that so that you'd relinquish your ideas - on the contrary, that you acquire familiarity with the mechanics of mathematical theories; how you define things, the place of axioms, how you set up a proof, etc.; those are things that are sorely lacking in your writings. In particular, you might focus on the place of logic in the building of mathematics - though I get the impression from your last post that there is no reason why you'd want to depart from classical logic.

 

[ddt]

Non-locality: the stick can be both in and out of the circle.

Locality: the stick can be completely in or completely out of the circle.

Non-locality: the stick can be both in and (not in and not out) of the circle (where not in and not out is exactly on the circle's border).

Non-locality: the stick can be both out and (not in and not out) of the circle (where not in and not out is exactly on the circle's border).

Locality: the stick (if it is a flexible one) can be not in and not out of the circle (where not in and not out is exactly on the circle's border).

Locality: a point can be completely in the circle, completely out of the circle, not in and not out of the circle (where not in and not out is exactly on the circle's border).

OK. These examples much better illustrate what you want to do than the various formulae you've given. I'll try to rephrase them in slightly more mathematical terms, but I'll not yet make them completely formal.

Standard set theory is built on one relation, membership, and we can say "x in A" to express that x is a member of set A.

Your theory, however, is built on two relations, "in" and "out". Informally, we may think of "x in A" as "x is (partly) inside A", and of "x out A" as "x is (partly) outside A". The two relations are not mutually exclusive, i.e., they may be both true at the same time.

Another example of the theory might be the interpretation: "person x lives in country A" resp. "person x lives outside country A". For a person with two homes in different countries, both statements are true.

Right so far?

Now, of course, to make this theory interesting, the next step is to draw up axioms for the two relations.

Locality means: to be exactly in a one relation.

Non-locality means: to be in more than a one relation.

With the relations in these two phrases, you only mean the "in" and "out" relations - not arbitrary relations you may define later on?

And a second question: you here say that "locality" means that one of them is true. However, the last two examples of the list above are cases where none of them is true. That's contradictory. So you have to decide whether you allow such a case or not.

I'll assume for the moment it's not allowed. Then you have a first axiom:

[latex]$$\forall x, A: x \mbox{\textbb{in}} A \vee x \mbox{\textbb{out}} A$$[/latex]

and you have definitions for "local" and "non-local":

Definition. x is local w.r.t. A iff (x in A) xor (x out A).
Definition. x is non-local w.r.t. A iff (x in A) and (x out A).

and you have a first theorem:

Theorem. x is non-local w.r.t. A iff not x is local w.r.t. A.
Proof. Straightforward and left to the reader.

 

[PixyMisa]

Point accepted. Doron is offering a logic. But it's not one based on the law of noncontradiction but the relations of polar opposites. One aspect of of the relations is paired off as Symmetry/Asymmetry, and that is where Doron's law of non-contradiction rests as a special case. Meanwhile the over all realtion of Locality/Non-Locality includes realtions where the law of non-contradiction does not hold.

Which means that Doron automatically loses.

Them's the rules. If you're arguing by logic (and I use the term advisedly) that allows for contradictions, then by your own rules, all your statements are provably false (even when they are factually true).

 

[jsfisher]

Doron,

You seem to be struggling trying to reconcile an intuitive notion of non-local with the rigors of formal logic. Have you explored 3-value logic as an alternate? E.g. True, False, and cannot be determined.

You will still be constrained by the Law of Non-contradiction (~(X and ~X)), but the Law of the Excluded Middle doesn't have to be a theorem (X or ~X) in a three value system.

The path you have been on trying to define non-local seems doomed to contradiction. On the other hand, (non-local <==> cannot be determined) may fit your intuitive model.

 

[Apathia]

Which means that Doron automatically loses.

Them's the rules. If you're arguing by logic (and I use the term advisedly) that allows for contradictions, then by your own rules, all your statements are provably false (even when they are factually true).

I believe that Doron would agree that "thems the rules" for "the researchable," as he calls it, meaning anything that is an object of thought or "Local."

 

[doronshadmi]

Doron,

You seem to be struggling trying to reconcile an intuitive notion of non-local with the rigors of formal logic. Have you explored 3-value logic as an alternate? E.g. True, False, and cannot be determined.

You will still be constrained by the Law of Non-contradiction (~(X and ~X)), but the Law of the Excluded Middle doesn't have to be a theorem (X or ~X) in a three value system.

The path you have been on trying to define non-local seems doomed to contradiction. On the other hand, (non-local <==> cannot be determined) may fit your intuitive model.

I'll do it informal.

We have, let's say, at least two (there can be infinitely many) poles (as Apathia suggested) .

Now I say to myself:"How it is possible to define in the first place, that there is a thing that is a pole of another thing?

My informal research:

1) No pole is the cause of the other, otherwise there is a hierarchy of cause\result and no polarity can be found.

2) These poles cannot be considered as poles if not both of them are simultaneously considered.

3) There must be a point of view that enables this simultaneous consideration, and it is not any one of the poles, otherwise we do not have a simultaneous consideration of the poles.

4) If one of the poles is the point of view of the other, then a non-simultaneous consideration of the poles is used.

5) A non-simultaneous consideration is a serial consideration of the poles.

6) Let Serial be non-simultaneous consideration.

7) Let Parallel be simultaneous consideration.

8) The parallel point of view is not polar. In that case it is considered as the middle between the poles.

9) The middle is not any of the poles, but without it there cannot be a parallel point of view, and no pole can be found.

10) The middle is not "a one of many" thing (I call it non-local, symmetry, invariant).

11) Each pole is "a one of many" thing (I call it local, asymmetry, variant).

13) I call the middle Relator, function, mapping, connective.

14) I call a pole Related, object.

15) Complementation is the result of Relator\Related relation.

16) The organic paradigm is Relator\Related relation.

17) I try to address asymmetry in terms of XOR.

18) I try to address symmetry in terms of NXOR.

19) I try to define Logic in terms of Non-locality\Locality relations.

20) Organic Natural Numbers (ONN) are Non-locality\Locality relations.

21) By ONN both parallel\serial point of views are used.

22) From a parallel point of view, each ONN's object is in superposition of identities of the rest of the objects.

23) From a serial point of view, each ONN's object has a unique identity.

24) Given any cardinal, an ONN is any possible Non-local\Local relations of this cardinal.

25) No collection of identities (whether there are in a superposition or not) is logically the Relator (which is not "a one of many" thing).

26) As a result no collection is logically complete, where complete is: not "a one of many" thing.

Let us stop here, check it and try to formalize it, before we continue.