A collection of infinitely many distinct elements

[doronshadmi]

Since the logical connectives depends on Non-locality \ Locality, logical connectives are used here to discover what principle sands as their basis.

Then you have a circular definition.

Not at all.

There is nothing circular by going after the smell (logical connectives) in order to find the food (non-local\local).

Referring to that post means reiterating it, so that's a third lie today. It really gets pathological, doesn't it?

Yes, but it is only in the mind of the words' counter.

 

[doronshadmi]

Maybe you should then write what you mean and not something else. I asked you five times what had been changed and you didn't reply. The sixth time you give a misleading reply. You're really afraid people might actually have time to read that load of crap, aren't you?

ETA: that reference is the fourth lie.

I did not answer to you because there is no use to answer to you, because you are unable to get the organic paradigm.

The only reason that I continue to reply to you, is that you help a lot to expose the context dependent crap.

 

[jsfisher]

Referring to that post means reiterating it, so that's a third lie today. It really gets pathological, doesn't it?

While I completely agree that doron has trouble making truthful statements, I'm not sure I'd make the leap to willful liar.

Unfortunately for doron, he has demonstrated a cognitive difficulty carrying conclusions developed in one step to anything beyond only the very next step. By step three, it can all be forgotten.

This is why he cannot comprehend the meaning of "union of a set's members", nor can he see the circular reasoning inherent in his complementation follows from logic and logic then follows from his complementation, nor can he cope with change-of-variable substitutions, nor can he recognize the absurdity in his interpretations of Hilbert's speech, Cantor's proof, and Gödel's premise.

 

[doronshadmi]

From time to time when jsfisher or ddt do not get what I say, then I am a liar.

I realy do not care.

Let us go to visit Aristotle’s law of noncontradiction ( http://en.wikipedia.org/wiki/Contradiction ):

Aristotle’s law of noncontradiction states that “One cannot say of something that it is and that it is not in the same respect and at the same time.”.

Wall, Aristotle’s law of noncontradiction is not universal because non-local thing can be a member and not a member of a given domain.

Actually without non-locality, logical connectives like Nor, And, Nxor, etc... do not hold.

People like ddt or jsfisher will tell you that logical connectives simply are.

They are afraid to realize that logical connective are not fundamental as they think they are, because in this case their reasoning will be changed by a paradigm shift from the context dependent paradigm to the organic paradigm.

The struggle for new paradigms is at least 5000 years old, and was learned by people like Thomas Kuhn ( http://en.wikipedia.org/wiki/Thomas_Kuhn ).

 

[ddt]

I did not answer to you because there is no use to answer to you, because you are unable to get the organic paradigm.

And you don't think it good form anyhow to tell what you changed in your paper?

The only reason that I continue to reply to you, is that you help a lot to expose the context dependent crap.

I see the crowd of converts flocking to you like flies to the honey.

While I completely agree that doron has trouble making truthful statements, I'm not sure I'd make the leap to willful liar.

While I agree that doron's initial remark in post #781 may be construed as a misunderstanding, this is not the case with his reaction in post #788. My post made clear I was looking for the differences in the documents using the word "set". Doron chose on purpose to perpetrate the "misunderstanding", and thus definitively turn it into a lie.

From time to time when jsfisher or ddt do not get what I say, then I am a liar.

I realy do not care.

It helps with the martyrdom aspect of crackpottery, doesn't it?

 

[jsfisher]

Wall, Aristotle’s law of noncontradiction is not universal because non-local thing can be a member and not a member of a given domain.

Doron, since you are rejecting the Law of Non-contradiction, you are also rejecting all of the standard frameworks for Mathematical Logic.

The axiomatic systems for set theory have First-Order Logic as their underpinning, and FOL includes the Law of Non-contradiction. So, you also completely reject set theory.

Even Intuitionistic Logic includes it, so you cannot build your set theory based on Intuitionistic Logic, either.

That puts you in a world of logic unfamiliar to the rest of us. Care to explain the foundations for your logic framework? From what I've seen of how you apply it, it looks very inconsistent. Perhaps you can provide some insight as to how it works.

By the way, I am specifically not interested in the philosophy of your logic framework, just how it works mathematically. Behavior of the logic operators, construction of well-formed formulae, and the principles for reasoning.

 

[ddt]

Since the logical connectives depends on Non-locality \ Locality, logical connectives are used here to discover what principle sands as their basis.

Then you have a circular definition.

Not at all.

There is nothing circular by going after the smell (logical connectives) in order to find the food (non-local\local).

Anyone with half a brain now smells a rat. You nowhere indicate that the logical connectives depend on (non-)locality. In fact, at the bottom of page 2 of your NXOR\XOR paper, you present a truth table for the connectives where they do not depend on that. So you're contradicting yourself.

Referring to that post means reiterating it, so that's a third lie today. It really gets pathological, doesn't it?

Yes, but it is only in the mind of the words' counter.

You're the one who was unwilling to indicate what you changed about your paper, and on purpose deceitful and lying when indicating it.

 

[ddt]

People like ddt or jsfisher will tell you that logical connectives simply are.

They are afraid to realize that logical connective are not fundamental as they think they are, because in this case their reasoning will be changed by a paradigm shift from the context dependent paradigm to the organic paradigm.

The struggle for new paradigms is at least 5000 years old, and learned by people like Thomas Khon ( http://en.wikipedia.org/wiki/Thomas_Kuhn ).

The meaning of the logical connectives has not changed from Aristotle to now.

Kuhn's paradigm shifts apply to science - like the shift from phlogiston to oxygen, or the shift from Newtonian to Einsteinian mechanics - and not to mathematics. He doesn't give a single example of such.

 

[doronshadmi]

The meaning of the logical connectives has not changed from Aristotle to now.

Kuhn's paradigm shifts apply to science - like the shift from phlogiston to oxygen, or the shift from Newtonian to Einsteinian mechanics - and not to mathematics. He doesn't give a single example of such.

You are right.

But a paradigm shift is a paradigm shift, and the organic paradigm is a paradigm shift of the context dependent paradigm.

The organic paradigm is not limited to any particular science (including the mathematical science).

 

[ddt]

But a paradigm shift is a paradigm shift, and the organic paradigm is a paradigm shift of the context dependent paradigm.

Except that the emperor has no clothes. There is no organic paradigm. In your 892 posts to date on this forum, you've not been able to show it to anyone with a math background. And I'm not even counting the posts you made on other boards.

There has been a new paradigm in logic, about a century ago: intuitionism. While most people didn't subscribe to it, they at least recognized it as math. What you're doing is no math.

 

[doronshadmi]

Doron, since you are rejecting the Law of Non-contradiction, you are also rejecting all of the standard frameworks for Mathematical Logic.

The axiomatic systems for set theory have First-Order Logic as their underpinning, and FOL includes the Law of Non-contradiction. So, you also completely reject set theory.

Even Intuitionistic Logic includes it, so you cannot build your set theory based on Intuitionistic Logic, either.

That puts you in a world of logic unfamiliar to the rest of us. Care to explain the foundations for your logic framework? From what I've seen of how you apply it, it looks very inconsistent. Perhaps you can provide some insight as to how it works.

By the way, I am specifically not interested in the philosophy of your logic framework, just how it works mathematically. Behavior of the logic operators, construction of well-formed formulae, and the principles for reasoning.

Since you cannot use the familiar context dependent paradigm in order to get the organic paradigm, you are like a child that learns how to walk (what is called baby steps).

The first step is to get non-locality.

We start by some concrete (even physical) example, and then try to find out what is needed in order to translate it into an abstract thing.

It must be done by you as an active participator of the translation.

The first problem is:

There is a drawn circle on the floor, and there is a piece of stick, which is naturally both in and out of the circle.

Please try to define a general abstract rule of the described case.

 

[doronshadmi]

Except that the emperor has no clothes. There is no organic paradigm. In your 892 posts to date on this forum, you've not been able to show it to anyone with a math background. And I'm not even counting the posts you made on other boards.

There has been a new paradigm in logic, about a century ago: intuitionism. While most people didn't subscribe to it, they at least recognized it as math. What you're doing is no math.

Wrong.

You simply do not get the organic paradigm, because you are afraid of the paradigm shift of your context depentent point of view.

 

[doronshadmi]

I see the crowd of converts flocking to you like flies to the honey.

I am not the important thing here. Evolution is the name of the game. Give it any limit and it will find the ways to be developed beyond it.

 

[ddt]

I am not the important thing here. Evolution is the name of the game. Give it any limit and it will find the ways to be developed beyond it.

Evolution?



You know that most mutations are lethal?

I notice you've failed thus far to back up your claim in post #776:

Furthermore, your reasoning is circular because you define x by using x as a part of the definition that defines x, for example:

There exists x such that x … bla bla bla.

that I've used circular definitions.

 

[jsfisher]

The first step is to get non-locality.

Don't assume I don't "get" non-locality. I actually do. However, just because I understand what you seem to mean by it does not mean I agree with you.

We start by some concrete (even physical) example, and then try to find out what is needed in order to translate it into an abstract thing.

It must be done by you as an active participator of the translation.

The first problem is:

There is a drawn circle on the floor, and there is a piece of stick, which is naturally both in and out of the circle.

Please try to define a general abstract rule of the described case.

You seem to forget I developed the famous French bread paradox. Nothing new here. Let's move on.

How do your logical operators behave? How are well-formed formulae constructed? What are your rules for inference?

 

[jsfisher]

It should be obvious to everyone that Doron is quite the phrase scavenger. He quickly incorporates new phrases presented to him into his lexicon, typically without regard to meaning.

I hereby claim credit for dichotomy, context dependence, and Law of Non-contradiction.

 

[PixyMisa]

Since you cannot use the familiar context dependent paradigm in order to get the organic paradigm, you are like a child that learns how to walk (what is called baby steps).

The first step is to get non-locality.

We start by some concrete (even physical) example, and then try to find out what is needed in order to translate it into an abstract thing.

It must be done by you as an active participator of the translation.

The first problem is:

There is a drawn circle on the floor, and there is a piece of stick, which is naturally both in and out of the circle.

Please try to define a general abstract rule of the described case.

General abstract rule: It's a stick.

Wrong.

You simply do not get the organic paradigm, because you are afraid of the paradigm shift of your context depentent point of view.

You have not provided any coherent definition of this "organic paradigm", so little wonder that no-one "gets" it. And in those rare instances where you have been coherent, you have been infallibly wrong.

I am not the important thing here. Evolution is the name of the game. Give it any limit and it will find the ways to be developed beyond it.

How does mathematics "evolve"?

 

[ddt]

This is what comes of Doron using logic notations for what should be regarded as pre-logic relations.

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. Let's have a look at the most basic theory of logic.

Propositional logic

Propositional logic^WP doesn't deal at all with the mathematical content of propositions. It just studies how we may reason with propositions to form new ones. To abstract away from the mathematical content, we work with a list of atomic formulae A1, A2, A3, ... Think of those as truth values. From those, you can form more complex formulae by using the usual logical connectives.

Now there are two possible approaches to see which formulae are true. One is to draw up truth tables. Then you'll see that, e.g. A1 OR (NOT A1) is true for whatever truth value of A1; or that A1 AND (NOT A1) is false for whatever truth value of A1.

The other is to analyse what steps we take in actual proofs and draw up rules according to which we may infer true formulae. A couple of examples.

Modus ponens. Suppose we have derived a formula X -> Y from a list of assumptions Gamma, and suppose we have also derived the formula X from the same list of assumptions Gamma. Then we may conclude that the formula Y follows from that same list of assumptions. In a formula, we have as inference rule:

[latex]$$\frac{\Gamma \vdash X \Rightarrow Y ~~~~~ \Gamma \vdash X}{\Gamma \vdash Y}$$[/latex]

The list of assumptions Gamma is needed because we always derive in real life formulae in a context: the axioms which we've taken up as assumptions in the theory we're developing.

ETA: The format of a clause X1, X2, ..., Xn |- Y may look familiar to programmers; indeed, when you turn around the clause, you recognize the Horn clauses
Y :- X1, X2, ..., Xn
as they are used in logic programming languages such as Prolog.

Case distinction. Suppose we have derived a formula X OR Y from a list of assumptions Gamma. What do you do in a mathematical proof? Well, you make a case distinction. First you go on reasoning from Gamma plus X and then you derive a new formula Z. Then you reason from Gamma plus Y and try to reach the same formula Z. Again, as an inference rule:

[latex]$$\frac{\Gamma \vdash X \vee Y ~~~~~ \Gamma, X \vdash Z ~~~~~ \Gamma, Y \vdash Z}{\Gamma \vdash Z}$$[/latex]

This way, you can draw up some eleven inference rules: basically, for each connective an introduction rule which introduces the connective in the conclusion, and an elimination rule which has the connective in the antecedent but not in the conclusion. All those rules are intuitively obviously correct. A last rule I'd like to mention is the introduction rule for the implication:

[latex]$$\frac{\Gamma, X \vdash Y}{\Gamma \vdash X \Rightarrow Y}$$[/latex]

The notation I used above is from the Sequent calculus^WP, but then restricted to having only consequent; the rules are those from Natural deduction^WP.

When you can derive a formula X from a list of assumptions Gamma, it is said to be provable.

The wonderful thing about propositional logic is that it is both consistent (sound) and complete. Consistent means that every formula X that is provable is also true (whenever the assumptions are true) according to the truth tables. Complete, on the other hand, means that every formula X that is true according to the truth tables is also provable by means of the inference rules.

Thus far, we've essentially only analyzed the structure of mathematical proofs and the way such a proof may be built.

I see no room here, whatsoever, for any other notion which may influence our logic.

He's trying to define his Non-Locality.

It's like this:

If you draw a boundary of enclosure (say a circle) on an empty plane and give the interior a different content than what is exterior, the distinction creates a Local configuration or value. X is the contrasting value within. Without is not X. This is Local.

Okay. Let's say X is orange (sorry for the chauvinism; it'll be over in two weeks ). So the interior of the circle is orange - for the sake of argument, I restrict your "content" to the colour aspect of the content. But what colour does the exterior have? It can be blue, yellow, green, black, etc. So "not X" is not "not X" in a logical sense. On a logical level, there are only two values: true and false. In a real mathematical setting, we have of course many values, but that's on another level. It is wrong to speak of the content of the exterior as "not X"; you should label it Y, and then you may indeed claim that those are different: NOT (x=y). That brings us to the next chapter in mathematical logic.

First-order Logic
In First-order Logic^WP (FOL), we combine the logic language of propositional logic with the language of some mathematical theory. A FOL consists of the kind of logic formulae we built in propositional logic, but now we don't build them up from some atomic formulae A1, A2, ..., but from terms from a mathematical theory. That theory has a structure, i.e., it is built from some atomic terms t1, t2, ..., some function symbols f1, f2, ... and some relation symbols R1, R2, ... Each mathematical theory has prescribed in its structure how many terms there are, how many functions and how many relations (and how many arguments each function/relation has). For instance, the (simplest) theory of natural numbers has

• one atomic term, 0
• one unary function succ
• one binary relation =

whereas set theory has one term (the empty set) and one binary relation (membership).

Furthermore, we introduce variables x1, x2, ... which range over values of the domain of discourse. With this, we can build further terms of our domain of discourse; say, with the natural numbers, succ(x) is again a natural number; and we can build logical formulae, e.g.
succ(x) = succ
Note that we thus have two kind of values: terms denote values in the domain of discourse, whereas formulae denote truth values. To sum it up, a term is built by the following rules:

• an atomic term is a term
• a variable is a term
• a function symbol with the correct number of terms as arguments, makes a term

and logical formulae are built by the following rules:

• a relation symbol with the correct number of terms as arguments makes a formula
• one or two formulae connected with a logical connective make a formula
• a quantifier (see below) with a variable and a formula makes a formula

Finally, FOL introduces the quantifiers "forall" and "exists" which range over values in the domain of discourse, so we can make formulae as:

[latex]$$\forall x: \forall y: \mbox{\textbb{succ}}(x) \mathbb{=} \mbox{\textbb{succ}} \Rightarrow x \mathbb{=} y$$[/latex]

and some extra inference rules for those quantifiers.

In FOL, we now again can use inference rules to prove all kinds of theorems from assumptions - the assumptions typically being the axioms of the theory in question. An example of such an axiom is the reflexivity of equality:

[latex]$$\forall x: x \mathbb{=} x$$[/latex]

or the formula two paragraphs above which states that succ is injective.

In propositional logic, we had "provable" on the one hand, and "truth tables" on the other hand. In FOL, we still have "provable", but on the other hand, there can be all kinds of models of the theory: an actual set, with an actual element 0 and an actual function succ and an actual relation =. In theory, you can check for each such model whether a certain formula is true. From that, we can extend to saying if a formula is true in all models of the theory. That is the counterpart of "truth tables" in FOL.

However, it is obvious that it would be very tedious to check if a formula is valid in one model of the theory, let along in all models. The great thing about FOL is that we still have both consistency and completeness!

Consistency now thus means that every provable formula is valid in all models of the theory. Completeness means that every formula that is valid in all models of the theory is also provable. The completeness theorem is due to Gödel.

Gödel, completeness - I hear you mutter - didn't that guy prove incompleteness? Yes, he did too. There is one proviso to the completeness theorem I mention. It says that valid formulae are only provable if you have a finite list of assumptions (axioms). The theory of natural numbers, however, needs an infinite list of axioms.

If the there is a contrast in value where X is the value for the exterior while the interior is not X, this distinction too amounts to a Locality.

But if the interior and exterior both have the value of X, there is no distinction and no boundary. There is just same old space. This is the Non-Local. There's no contrast to be found, so no place to be recognized, so no Locality.

The same is the case if the interior and exterior have the same value of not X. There is no contrast, no interior vrs, exterior, no Locality.

If there is no contrast between the interior and the exterior, there is a symmetry of value. So another disignation he gives for Non-Locality is Symmetry.

If the interiior and exterior are of contrasting value, there is an asymmetry of Value. So another designation he gives the Locality is Asymmetry.

I see in this exposé (and the following parts I culled) no reason to put "locality" before logic. It seems to me to be just part of the mathematical language in the FOL framework. Just introduce a relation "local" on the values of your domain of discourse.

Just like a sphere is a unity with two polar opposites, all complementary poles are the the extremes of a single Unity he calls "The VALUE."

The VALUE has no characteristics of its own. Characteristics are the result of the relations of polar contrasts. The VALUE is the Unity of all complements, so it has no identifying or locating features, as these are all matters of contrast, and it contains all contrasts.

He goes on to frame a Logic where A is one pole and ~A another. In The VALUE, A = ~A is a given as one combination, but only as one, not the whole ballgame.

I can only take words like "THE VALUE" at face value - if there's only one value in your domain of discourse, it is utterly uninteresting. As to in Doron's logic being A and ~A being poles: wherever he does say things about his logic, he firmly sticks to classical two-valued truth tables, so I don't see anything modal or otherwise revolutionary about them. A formula and its negation simply cannot be true at the same time.

 

[ddt]

The first step is to get non-locality.

Your definition is abundantly clear: it cannot occur. We already established that.

There is a drawn circle on the floor, and there is a piece of stick, which is naturally both in and out of the circle.

That means you're a louse stick thrower.

Please try to define a general abstract rule of the described case.

You present this as if there is some paradox, but there really isn't.

The relational approach
Define
O(t) := "part of the stick is out of the circle at time t"
and
I(t) := "part of the stick is in the circle at time t"

Then we can express "all of the stick is in the circle at time t" as
I(t) AND ~ O(t)
This even accounts for a stick that moves with time.

The stochastic approach
Define the stochasts
X_t := "the fraction of the stick that is in the circle at time t"
and kick the stick at every clock tick. Now you have a Markov chain^WP of stochasts, provided you don't memorize how you kicked last time.

The calculus approach
Define the indicator function
i(x,y) := "at point (x, y) lies a piece of stick"
and integrate this function over the area of the circle (the function is Riemann integrable).

Do you want more approaches?

 

[doronshadmi]

Don't assume I don't "get" non-locality. I actually do. However, just because I understand what you seem to mean by it does not mean I agree with you.

I am sure that you don't get non-locality.

Show me that I am wrong about that.