Define Truth

Internal vs. External Justification
Traditionally, it is assumed that when a belief is justified, there is something that makes it justified (or unjustified).

• Internalism (usually associated with evidentialism) holds (essentially) that our beliefs are justified by our mental states.
• Externalism (usually associated with reliabilism) holds that this is not the case. It is not one's evidence, but the reliability of that evidence that justifies a belief. Thus, while the source of our justification may qualify as being mental, the reliability does not. Because of this, it will not always be possible to recognize what it is that justifies a belief. And thus justification is considered to be external to our mental workings.

Radically Deceived Subject
Let us distinguish between two imaginary people: Tim and Tim*. The first is a normal person, just like me or you. The second experiences everything that Tim does, but he is a Brain In a Vat (BIV).

Now, according to the traditional view, when Tim believes he has hands, he is correct; but when Tim* believes he has hands, he is mistaken. Yet when Tim* is asked, he will (just as Tim would) indicate that he believes he has hands. Evidentialism would say that Tim* is justified in this belief because the evidence he is privy to supports the belief. But Reliabilism says Tim* is not justified, because the evidence is not reliable.

The DS Perspective
In the DS theory, the difference between Tim and Tim* is merely one of degrees. If you and I were BIVs, we would be no more capable of knowing this fact than Tim* is. Thus, we cannot say with absolute certainty that we are NOT BIVs. Thus, the real difference between us and Tim* is that we are certain that Tim* is a BIV, and we are NOT certain that we are BIVs. From our own perspective, (based on what we know of the science--in our own framework), the probability that we are BIVs would seem to be quite low. But, if that were actually the case, from the perspective of those maintaining our vats the probability would be 100%. Just as (from our perspective) it is for Tim*.

The illusion is that (for some reason) Tim* is not justified in his belief that he has hands, but (somehow) we are justified in the same belief.

The way I see it, it is quite obvious that Tim* has hands. What IS in question is whether those hands are physical in the same way that Tim's hands are physical. The traditional view is that Tim and Tim* are alike mentally, but dissimilar physically. But the truth is that we don't actually know if they are alike physically or not. If we are all BIVs, then we are all the same as Tim*. From our perspective, Tim* is a BIV, which means that if we are BIVs then Tim* is a BIV in a VAT. But Tim* doesn't become even less physical (than the rest of us simulations) just because he is a simulation inside another simulation.

Even if we assume that we are not BIVs... the problem is still only the
smoke and mirrors of perspective. For if we take another perspective it becomes obvious that Tim is also mostly non-physical. The atoms that constitute Tim's hands are composed primarily of "empty space". And this is true, even using the quantum mechanics model of the atom--where the electron is spread out in a probability smear over the whole of its orbit. The orbit is still a very small portion of the total area occupied by the atom, and the probability of the electron appearing outside of that orbit is virtually zero. Thus, from this perspective, we can say that Tim's hands are closer to being non-physical (composed of empty space) than they are to being physical (compose of a solid, stationary physical object).

In the DS theory, we are not justifying our belief: that would be simultaneously futile and redundant. Instead, we are justifying our expression of that belief. If John (my close friend) says he has a lot of work to do, but he'll try to make it to my party... and I then tell Tommy, "I believe John will come to my party." I justify this expression by demonstrating how closely it reflects my actual belief.

I know it is possible John won't have time to come. But in the past, when he's said the same thing, he has usually found the time to come, but he was just a little late. Thus, based on past experience, this would seem to be the most probable outcome--which means that it is the most [reliable (i.e. likely) possibility. Since "coming late" is a type of "coming", I am relatively justified in the expression of my belief that John will come. I am not as fully justified in this belief as I am in the belief that I have hands, however, because I know that it is possible John will not come.

As I've said, from the perspective of Tom*, Tim* clearly has hands. They are not physical in the same sense that my hands are physical, but they are hands, and they are physical in the way that 'physical things' in the world of Tim* are physical. Thus, from the perspective of Tom*, Tom* is fully justified in his belief that he has hands--and even that those hands are physical. From my perspective, the belief that Tom* has expressed (that he has physical hands) is not fully justified--because from my perspective I know that his hands are not physical in the same way that my hands are physical. However, I also intuitively understand the nature of perspective and so even from my perspective Tom* is still partially justified in his belief.

We can isolate three key distinctions between the DS theory and traditional Theories.

1. We justify the expression--not the belief.
2. Justification is a relative spectrum, allowing some expressions to be more fully justified than others--instead of understanding justification to be an absolute where a belief is either fully justified or fully unjustified, with no middle ground possible.
3. We cannot judge a justification without also knowing the details of the perspective. This allows for the speaker and the hearer of an expression to hold different (but equally valid) views about whether an expression is justified.

I often get the impression that either [English is your second language] or [you're trying to amuse yourself rather than trying to illuminate something for others]. And this seems to be one of those times, since I can't really figure out what you might be trying to say.

However, I've been wanting to discuss the if/then truth take, and this seems like a good lead in to that topic... whatever it is supposed to mean.

Truth Tables In Light of DS Principles
We can radically change the way we think about something based solely on whether we think of it as (1) a whole, or (2) as a sum of individual parts. One way of identifying these reciprocal aspects is by referring to the set or element aspects of the thing in question. So, if we let [α] and [β] be propositions in a logical statement, [P], then [α] and [β] can be understood as the elements in [set P].

Understanding [α], [β] and [P] in this way, can lead us to a new understanding of the truth-tables.

When we focus our attention on the [set aspect] and look specifically for [truth], we observe that (α & β) is [false] if [either α or β are false], because in order for the [whole set P] to be [absolutely true], [all of its parts must be true]. On the other hand, if we focus our attention on the [element aspect] while again looking specifically for [truth], we observe that (α & β) is [true] if [either α or β are true], because if any [part of the whole] is true then we have found [truth]. And since we aren’t worried about [P] as a [whole set] the fact that there may also be [false propositions] is simply incidental to our search for [truth].

In a sense then, it might be said that when we focus on the elements (rather than the set) the values of the [& connector] get inverted, becoming (in effect) the [v connector] with respect to the set.

As with any pair of reciprocals, we can see that the tables for [& and v] are at once the same and different. In both cases, we are looking for truth, but in one we are focusing on the [set aspect] and in the other we are focusing on the [element aspect].

We could also have inverted our perspective by focusing on finding [falsehood] instead of [Truth]. We might call the result of such an effort a falsehood table.

This time, when we focus our attention on the [set aspect] and look specifically for [falsehood], we observe that (α & β) is [true] if [either α or β are true], because in order for the whole set [P] to be [entirely false], [all of its parts must also be false]. On the other hand, if we focus our attention on the [element aspect] while again looking specifically for [falsehood], we observe that (α v β) is [false] if [either α or β are false], because if any [part of the whole is false] then we have found [something false]. And since we aren’t worried about [P] as a [whole set] the fact that there may also be [true propositions] is simply incidental to our search for [falsehood].

Obviously, there are two reciprocal properties involved here--which is a common characteristic of any aspect of the DS theory.

The first property is defined by the [set and element] aspects. For this property, we move from one aspect to the other by moving from the [& connector] to the [v connector]. The other property is defined by the [truth and falsehood] aspects. When we move from the [truth-table] to the [falsehood-table] we essentially invert the [& connector] to the values of the [v connector], and vice versa. This is because the [v connector] is the mirror image opposite of the [& connector].

Redefining the Conditional Statement
Now that we have a new way of looking at the truth tables, we can expand our examination to the [if... then...] connector. Traditionally, the table for [if... then...] has been given truth-values for four possible combinations.

This truth table for conditionals, however, leads to logical difficulties like Curry’s paradox.

I believe that one of the reasons it makes sense to call the [if... then...] connector a conditional is because the value in the table is conditional upon the nature of the relationship that [α] and [β] have to one another. For instance, consider these two [true statements]:

It is necessarily true that: An apple is a fruit
It is possibly true that : A fruit is an apple

The first statement is true because, by definition, [all apples are fruit]. This means we can use the term [fruit] to help define what it means to be an [apple], because without exception all apples are fruit. The second statement is true because we know that [some apples exist] and since [all apples are fruit], we can infer that [some of the fruit that exist will necessarily be apples]. In other words, given a truly arbitrary fruit, it is possible that it could be an apple. Obviously, we can not say this about a specific instance of a fruit—but if someone tells us they have a bag of fruit, and they intend to give us one, then (assuming they’re not lying) what they give us (from our perspective at least) could be an apple. By introducing two symbols, we can capture the subtlety of this new idea in very clear notation.

[□] = [necessarily];
[◊] = [possibly].

In the DS system, a Necessary set is a set that [currently does] or [has in the past] existed physically. (Yes, this is a different definition than the traditional one.) There are two types of Possible sets, although we typically do not bother to distinguish between them:

1. Those sets that happen to be necessary--but are unknown. So, for instance, if John brings me an apple in a bag and says, "I brought you a fruit." He knows that the bag necessarily contains an apple. But I only know that it is possible that the bag contains an apple.
2. Those sets that are not currently necessary, but may become necessary at some point in the future.

Now, we can indicate the relationship between [Apples] and [Fruit] with the following notation:

A□F …means [apples are necessarily fruit]
F◊A …means [ fruit are possibly apples]

Note: the part in blue is usually superscript, but this forum doesn't have that capability.

we can create the following conditional statements, one of which is [necessarily true] and the other which is only [possibly true].

Necessarily true: If I’m holding an [apple] then I’m holding a [fruit]
Possibly true: If I’m holding a [fruit] then I’m holding an [apple]

If we let [H] stand for [I am holding], then we can use the following notation to reflect the meaning of these two statements:

HA□F -> □HFA
.....….....…....If I am holding an apple (which is necessarily a fruit) then I am necessarily holding a fruit (which is an apple).
HF◊A -> ◊HAF
.....….........…If I am holding a fruit, (which is possibly an apple), then it’s possible I’m holding an apple (which is a fruit).

Now that we have developed a system of notation, we can reconstruct the truth table for conditionals by focusing on one of two possible aspects

1. what is actually the case, and
2. what is potentially the case.

If we make a [specific statement] that is [actually the case], then it is [necessarily true]. But an [unspecific statement] that is [actually the case] is only [potentially true] because there are other cases where it could be false. And, of course, a [specific statement] that is [potentially the case] is also [potentially true]. We will start with the [actual aspect].

Consider the following sentence:

If I am holding an apple then I am holding a fruit.
If.............[HA□F is true] then...........[□HF will be true].

In other words, if it is true that [I’m holding an apple (which is necessarily a fruit) in my hand] then it will also be true that [I’m necessarily holding a fruit in my hand]. Because of the definitions of [Apple] and [Fruit], no other possibilities exist. However, just as the first sentence was [necessarily true], the following statement is [necessarily false]:

If I am holding an apple then I am not holding a fruit.
If.............[HA□F is true] then...............[□HF will be false].

Clearly, it will never be the case that [I’m holding an apple (which is necessarily a fruit) in my hand] but [I am not necessarily holding a fruit in my hand]. Thus, the possibility for such a situation simply does not exist—and the statement is [necessarily false].

The situation changes drastically when dealing with a conditional that begins with a falsehood. For example, let’s assume that the only thing I’m holding in my hand is an [orange]. In this case, it is [not true] that I’m [holding an apple]—but it is true that I’m holding a fruit. We might reflect this situation with the statement:

If [HA□F is false] then [HF◊A will be true].

The problem with this statement, however, is that it isn’t a necessary truth. In this particular case, it happens to be true. But what if I was holding a [hammer] instead? [HA□F] is false because I’m [not holding an apple], but [HF◊A] is also false because I’m [not holding a fruit]. Thus, the most we can say from, [HA□F -> HF◊A] is that it is possibly true—which means that it is also possibly false.

Or in other words, the truth of the statement only becomes clear when we have a specific example of the general statement. This is a serious problem because the whole purpose of a truth statement is to predict the truth of a statement—despite the unspecified, variable parameters. Starting with a falsehood in an [if... then...] statement prevents us from being able to make a [necessarily true statement].

What this implies is that when we are focused on [being necessary], we can not make a specific conclusion about an [if... then...] statement that begins with a falsehood. Therefore, when we are focusing on truth, the if/then table should look like the following table.

The Falsehood perspective
This, of course, does not imply that we can’t make specific conclusions about statements that are false. All we have to do is invert our way of thinking with respect to the [necessary aspect], so that we are dealing with something that is functionally equivalent. Since we were focusing on [necessary truths] before, we can consider the reciprocal perspective by focusing on [possible falsehoods].

Whether we’re focusing on [necessity] or [possibility] depends on whether we put [A□F] or [F◊A] first in the conditional. So when we invert the [if... then...] statement we get:

F◊A -> A□F

Now, we can make the following statement, which will necessarily be true:

If I am not holding a fruit then I am not holding an apple.
If...............[HF◊A is false]......then.........[□HA will be false].

In other words, if [I’m not holding a fruit in my hand] then it is [necessarily the case that I’m not holding an apple in my hand]. Clearly, it is never the case that: If [I’m not holding a fruit] then [I am holding an apple]. The possibility for such a situation simply does not exist—because an apple is a fruit by definition. Conversely, if we change the consequent to a true statement, we falsify the whole.

If I am not holding a fruit then I am holding an apple.
If..............[HF◊A is false]...then..........[□HA will be true].

When we bring the [necessary] and the [possible] perspectives together we end up with two different (and reciprocal) truth tables. And, just as the [& table] and the [v table] are mirror images of each other, so too the [if/then tables] that are derived from [alternate perspectives] are inverse images of one another.

It should be fairly obvious that if we put the single value parts of this table together we would end up with the if-and-only-if connector, or the [<-> table]. What may not be quite so obvious is that we can do the exact same thing while only using the -> connector.

α□β -> β□α

is the same thing as

α <-> β

This means that it is possible (in the DS theory) to identify an equivalency relationship using a number of different (but functionally equivalent) notations:

α□β....->....β□α
α......<->....β
α=β...->.....β=α
α.......=......β

Curry’s Paradox
One of the advantages of understanding the truth tables in the light of the previous post is that it allows us to understand certain paradoxes in a new way. For instance, Curry’s paradox is a negation free paradox that involves a list of sentences, collectively called The List. One version of this list is the following:

1. Tasmainian devils have strong jaws.
2. The second sentence on The List is circular.
3. If the third sentence on The List is true, then every sentence is true.
4. The List comprises exactly four sentences.

The chain of logic runs as follows: Let’s suppose that the antecedent in line (3) is true. If this is so, then it is true that:

The third sentence on The List is true.

This in turn entails that what the third sentence states is true, so it is true that:

If the third sentence on The List is true, then every sentence is true.

Which entails the truth of:

Every sentence is true.

Which entails the truth of our first statement:

The third sentence on The List is true.

Which obviously starts the round of entailment all over again.

Although not a contradiction, the chain of logic is circular in a situation where it would (according to the argument) seem reasonable to expect that we should be able to avoid an infinite regression.

Solving Curry's Paradox
Given that statements (1, 2 and 4) are necessarily true, we can break statement (3) down as follows:

Let,
T = the third sentence on The List is true
E = every sentence is true

In this particular scenario, where all of the other sentences are true, it is clearly the case that [if either T or E is true then so is the other], and [if either one is false then the other is too]. No other possibilities exist when the other three statements are true. Thus, we can use the following notion:

T□E......…means [T necessarily implies E]
E□T......…means [E necessarily implies T]

We can now evaluate statement (3) as follows:

If [the third sentence on The List is true] then [every sentence is true].
...............................................[T□E]T → [□E]T
If [the third sentence on The List is false] then [every sentence is false].
................................................[T□E]F → [□E]F

Notice that (according to the functional equivalencies that I gave at the end of the last post) the structure of the sentence is simply an equivalency statement.

The problem is just that we have no meaningful way to determine whether [T□E] is true or false, without knowing whether [□E] is true or false; and vice versa. Thus, it would seem logical to assume that we should assign the value [□T v □F] to [T], which implies that [E] must also be [□T v □F]. And, of course, [□T v □F] is the same thing as saying [◊T & ◊F].

Taking a Closer Look at the Real Problem
In the case of Curry’s paradox, it happens that every statement on [The List] is necessarily true, with the possible exception of (3)—which is only possibly true. Thus, (3) might be said to have a deep structure of [T <-> E].

On the other hand, if we allow (1), (2) and (4) to be arbitrary statements (and thus allow that one of them might be false) then the [T <-> E] deep structure disappears and (3) necessarily becomes one-directional. Using our new notation, we can represent this unidirectional relationship as follows:

T◊E.....…means [T possibly implies E]
E□T.....…means [E necessarily implies T]

Now, we can represent (3) as follows:

T◊E → E□T

Now (according to the DS theory) it should be easier to see what the most serious problem with the Curry paradox is--and that is that the expression, [T◊E → E□T], represents a relationship in the wrong direction. If (1), (2) and (4) are arbitrary, it makes sense to say, “If every sentence is true, then the third sentence must be.” But in the other direction, it doesn’t make any sense, because we know the other sentences are arbitrary, and so they might not be true. The paradox gives the [T◊E → E□T] structure the appearance of making sense, by ensuring that (1), (2) and (4) are true. That, however, doesn't change the fact that the structure of (3) is backwards--at least if we want to obtain a single Truth value. What we obtain instead is the vague expression:

(◊T & ◊F) v (□T v □F)

According to the DS theory, this is exactly what we should expect. It is the value that appears on the truth table for this sort of construction.

The author of Curry's paradox manages to make his LIST seem paradoxical by making the rest of the sentences true... instead of being arbitrary. But this is similar to the case where there was an orange in the bag instead of a hammer. It doesn't change the fact that the orange is only a [possible fruit]--it is not a [necessary fruit].

In the same way, it doesn't matter that (1), (2) and (4) are true... because the structure of (3) is still such that it cannot provide a specific truth value.

A Visual Proof of Curry's Paradox (without the visual)
Suppose that you have two overlapping circles [A] and such that they form three distinct areas:

(1) A[-B]
(2) A intersection B
(3) B[-A]

In this case, the [union of A and B] is clearly different than either [A] or alone. We can, however, make [A] and the visually equivalent with [A and B] by moving the two circles together until they exactly overlap. At this point, [union of A & B] is indistinguishable from [intersection of A & B]. We can imagine this [circle] as either [one circle with two names] or as [two circles with one name]. In other words, we can conceptualize it as a single circle with the name [A & B]; or as two overlapping circles with the names [A] and —which is functionally equivalent to saying [A v B].

The result is that (at least visually) [A & B] has become the same thing as [A v B].

In essence, I would suggest that this is essentially what the author of Curry’s Paradox has done (to confuse philosophers) by ensuring that (1, 2 and 4) are true. Because now, if [T] is true it necessarily implies that [E] is true—and vice versa. Curry’s paradox places [T] first in the sentence to emphasize its paradoxical nature. But the fact that we can produce a circular chain of logic from sentence (3) is no more surprising or paradoxical than the fact that when we bring [circle A] and [circle B] together, then [circle A & B] necessarily implies [cirlce A] v [circle B], which necessarily implies [circle A & B], and so on.

By making all the other statements true, we have made [T◊E] functionally equivalent to
[E□T]... just as with the circles, [A v B] was made equivalent to [A & B].

Impredication per wiki please discuss?

History

The vicious circle principle was suggested by Henri Poincaré (1905-6, 1908)[2] and Bertrand Russell in the wake of the paradoxes as a requirement on legitimate set specifications. Sets which do not meet the requirement are called impredicative.

The first modern paradox appeared with Cesare Burali-Forti's 1897 A question on transfinite numbers[3] and would become known as the Burali-Forti paradox. Cantor had apparently discovered the same paradox in his (Cantor's) "naive" set theory and this become known as Cantor's paradox. Russell's awareness of the problem originated in June 1901[4] with his reading of Frege's treatise of mathematical logic, his 1879 Begriffsschrift; the offending sentence in Frege is the following:
"On the other hand, it may be also be that the argument is determinate and the function indeterminate".[5]
In other words, given f(a) the function f is the variable and a is the invariant part. So why not substitute the value f(a) for f itself? Russell promptly wrote Frege a letter pointing out that:
"You state ... that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. There we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection does not form a totality".[6]
Frege promptly wrote back to Russell acknowledging the problem:
"Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic".[7]

While the problem had adverse personal consequences for both men (both had works at the printers that had to be emended), van Heijenoort observes that "The paradox shook the logicians' world, and the rumbles are still felt today. ... Russell's paradox, which makes use of the bare notions of set and element, falls squarely in the field of logic. The paradox was first published by Russell in The principles of mathematics (1903) and is discussed there in great detail...".[8] Russell, after 6 years of false starts, would eventually answer the matter with his 1908 theory of types by "propounding his axiom of reducibility. It says that any function is coextensive with what he calls a predicative function: a function in which the types of apparent variables run no higher than the types of the arguments".[9] But this "axiom" was met with resistance from all quarters.

The rejection of impredicatively defined mathematical objects (while accepting the natural numbers as classically understood) leads to the position in the philosophy of mathematics known as predicativism, advocated by Henri Poincaré and Hermann Weyl in his Das Kontinuum. Poincaré and Weyl argued that impredicative definitions are problematic only when one or more underlying sets are infinite.
Ernst Zermelo in his 1908 A new proof of the possibility of a well-ordering presents an entire section "b. Objection concerning nonpredicative definition" where he argued against "Poincaré (1906, p. 307) [who states that] a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it".[10] He gives two examples of impredicative definitions -- (i) the notion of Dedekind chains and (ii) "in analysis wherever the maximum or minimum of a previously defined "completed" set of numbers Z is used for further inferences. This happens, for example, in the well-known Cauchy proof of the fundamental theorem of algebra, and up to now it has not occurred to anyone to regard this as something illogical".[11] He ends his section with the following observation: "A definition may very well rely upon notions that are equivalent to the one being defined; indeed, in every definition definiens and definiendum are equivalent notions, and the strict observance of Poincaré's demand would make every definition, hence all of science, impossible".[12]

Zermelo's example of minimum and maximum of a previously defined "completed" set of numbers reappears in Kleene 1952:42-42 where Kleene uses the example of Least upper bound in his discussion of impredicative definitions; Kleene does not resolve this problem. In the next paragraphs he discusses Weyl's attempt in his 1918 Das Kontinuum (The continuum) to eliminate impredicative definitions and his failure to retain the "theorem that an arbitrary non-empty set M of real numbers having an upper bound has a least upper bound (Cf. also Weyl 1919.)"[13]

Ramsey argued that "impredicative" definitions can be harmless: for instance, the definition of "Tallest person in the room" is impredicative, since it depends on a set of things of which it is an element, namely the set of all persons in the room. Concerning mathematics, an example of an impredicative definition is the smallest number in a set, which is formally defined as: y = min(X) if and only if for all elements x of X, y is less than or equal to x, and y is in X.

Burgess (2005) discusses predicative and impredicative theories at some length, in the context of Frege's logic, Peano arithmetic, second order arithmetic, and axiomatic set theory.

I'm not sure what prompted this post, but I found it interesting.
I think the resolution of the vicious circle problem can be found in a more accurate definitions of the terms [set] and [element]. Traditionally it is held that (since a set can also be an element) there is essentially no difference between the two--except that an element is contained by a set.

I don't agree with this assessment. And there are two analogies that I like to use to illustrate the way my DS theory understands these terms.

The Glass of Water Analogy
When you have a [glass of water] you have two distinct and reciprocal aspects. You have the container (or the glass) and you have that which is contained (or the water). Water cannot contain the glass. Nor can the [glass of water] truly be said to contain itself. But the [glass of water] can be contained by something else--such as another, [larger glass of water]. If the [first glass] is small enough it can sit entirely inside the [second glass]; and it still contains water, while also being fully contained by the [larger glass] (with its water).

In this analogy, the [glass] is the set and the [water] is the elements contained by the set. And, in DS terminology, the [glass of water] is a class--being the union of both the [set aspect and the element aspect].

The Box Analogy
Similarly, a box is analogous to a [set], while the contents of the box is analogous to the [elements in the set]. Now, obviously a box cannot contain itself. But a [smaller box] can be contained by a [larger box]. When the [small box] is inside the [larger box], however, the [small box] is--with respect to the [larger box]--that which is contained. It is not a container within this particular relationship.

Applying to Math and Logic
It is semantically meaningless to say that you can substitute f(a) for f in the function f(a). This is analogous to saying that you can make the container [that which is contained]. In other words, it contains itself. On the other hand, you can substitute f(a) for the g in the function g(a), that's just like saying that you can place [box A] into [box B].

Gödel's Incompleteness Theorem
A variation of this problem can be found in the Liar's Paradox, which can be said to form the foundation of Gödel's Incompleteness Theorem. Over simplifying greatly, the dilemma can be expressed by saying that Gödel supposedly defines a theorem (in his system) which basically says:

(G)... G is not provable

But as we've seen, you can't put a box (G) inside of itself. So this theorem is semantically meaningless. To see why, consider that if it were true that you could nestle a theorem (or set) in this way, then you could also substitute (G) for the the G in [G is not provable]. This produces:

(G)... (G is not provable) is not provable

But G still remains, so we can substitute again:

(G)... ((G is not provable) is not provable) is not provable

But G still remains. And it will always remain, no matter how many times you substitute.

I've done a much more formal proof to show where Gödel's proof goes awry, but that's a little too involved for this forum, I think.

Cantor's Paradox

Cantor's Theorem: given any finite or infinite set S, the power set of S has larger cardinality (greater size) than S.

Cantor's paradox considers the set of all sets. Let us call this set the universal set and denote it by U. The power set of U is denoted ℘(U). Since U contains all sets it will in particular contain all elements of ℘(U). Thus ℘(U) must be a subset of U and must thus have a cardinality which is less than or equal to the cardinality of U. However, this immediately contradicts Cantor's theorem.

I believe I've already presented a counter-proof (on this thread) showing why Cantor's Theorem is wrong. In fact, the cardinality of U and ℘(U) are different only because they involve different [units of measure]--which invalidates any magnitude comparison between them. Thus, it is meaningless to say that the cardinality of ℘(U) is larger than U. But we can look at this problem in a less formal light as well.

According to the DS theory, we might say that the [class of all sets] contains itself as an [element], but does not contain itself as a [set]. Remember, a class is the union of the set aspect and the element aspect. When we identify the elements that are the [class of all sets], the result is the [enumeration of all sets]--which is exactly what the [class of all sets] is. Thus, when we refer to the element aspect of the [class of all sets] we are simply being self-referential. But it is meaningless to say that you can place the set aspect of the [class of all sets] into the [class of all sets]. The [class of all sets] is already in itself... as the elements that are [all the sets that exist]. And as the [set aspect], it cannot possibly be an element in this relationship--it can't be [that which is contained] if it is [that which contains].

We can use another analogy to illustrate this relationship. My body 'contains' numerous parts. Thus, we can create the [set of all the parts of my body]. (PMB). Is my [body] an element in (PMB)? Of course not! My body is [all the parts collectively.

Now, my body can also be seen as a part--as in a [part of my family]. But the sense in which my body is a part (i.e. a element) is different from the sense in which it is a whole (i.e. a set). Thus, [my body] can be an element in another set, but not in (PMB). Now, if (PMB) were all that existed, then there would be no meaningful part (or element) aspect to consider. Similarly, the [Class of all sets]--since it contains all the sets which exist--has no meaningful [part aspect]. And thus, the [set aspect of this class] cannot be placed in any other set. All that remains is the self-referential [part aspect] that enumerates all the [elements] of the [set aspect].

Russell's Paradox
The traditional understanding of this paradox goes like this: Some sets, such as the [set of all teacups], are not members of themselves. Other sets, such as the [set of all non-teacups], are members of themselves. Call the set of all sets that are not members of themselves "R". If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.

According to the DS theory, a set cannot contain (or belong) to itself. Thus, Russell's R would be the non-set. (Or in traditional terminology the empty set.) The non-set is [that which isn't a set].

Grelling's Paradox
Involves a predicate defined as follows. Say a predicate is heterological if it is not true of itself, that is, if it does not itself have the property it expresses. Thus the predicate "German" is heterological, since it is not itself a German word, but the predicate "deutsch" is not heterological. The question that leads to the paradox is now:

Is "heterological" heterological?

First, using the same DS logic already presented, we can distinguish between the element or word "English" and the set or language [English]. The word "English" is certainly an element of the [English] language. But the language [English] is not an element of the [English] language; nor is the word "English" the same as the set aspect [English].

Applying this to heterological, we get the word/element "heterological" and the [class of sets that are heterological in nature]. And the paradox asks, "Is the word "heterological" a member in the [class of heterological sets]?

The problem is that the answer is vague. We cannot decide if the word "heterological" is a member in the [class of heterological sets] until we define whether "heterological" is a member in the [class of heterological sets].

In a way, this scenario isn't all that unique. For instance, suppose a man is on the verge of death. Maybe his heart has stopped and he's stopped breathing. Is this man dead? We can only answer that if we know what it means for a man to be dead. If a dead man is someone who's heart has stopped, then he is obviously a dead man. But if we define a dead man as someone whose brain activity has totally ceased, then he is not a dead man. Until we know what the definition of a dead man is, we can not decide whether or not the man is actually dead.

Now, the heterological paradox is structured in such a way that we cannot meaningfully answer the question. If we arbitrarily decide to define "heterological" in either of the possible ways, it necessarily leads to a contradiction.

I would suggest, however, that this is not so much a paradox as it is an abuse of defining privileges. The real definition is necessarily vague, and cannot be arbitrarily answered in any other way.

This is not unlike the situation that exists when we try to measure the side of a box with rounded corners. Given this task, we have two options. 1) We can measure only the part of the edge that is actually straight--in which case the measurement is too short, because it doesn't include the rounded portion of the "edge". 2) We can imagine an imaginary line extending from the perpendicular edges and measure from one of those imaginary lines to the other--but in this case, the measure will be too long, because it includes area that isn't actually on the edge of the box, but is more accurately the corner of the box.

We can arbitrarily accept one answer or answer and ignore the other--but that doesn't change the fact that reality is necessarily and so it includes both aspects.

The only difference with the heterological paradox is that the structure which has been set up for it is essentially:

H is (~H)

By distinguishing between the [set aspect] and [element aspect] we can define the paradox in such a way that we can choose between two meanings.

"H" is ~[H]
[H] is ~"H"

Because we are saying that the [set aspect] is not the [element aspect] and vice versa, there is nothing truly paradoxical about this arrangement. It only seems paradoxical because we are not used to consciously thinking about the difference between these aspects.

Another analogy that serves to illustrate the relationship between an [element] and a [set] is what I call the Father and Son Analogy.

The Father/Son Analogy
The words father and son are clearly opposites when seen from within the framework of a particular relationship between two people. For example, if [Charles1] is [my father] then it is clearly not possible for [Charles1] to be [my son] or for [me] to be the [father of Charles1]. The relationship between father and son necessitates that the later designations are incompatible with the first. However, it is just as obvious that my father, [Charles1] must also be [someone else’s son], and that I, the [son of Charles1] can myself be a [father to someone other than Charles1].

Despite the fact that [father] and [son] are logical opposites--I can be both father and son at the same time. What keeps this situation from being a contradiction of Aristotle’s (PNC) is Aristotle’s own stipulation that the things we are comparing must be compared:

In the same respect and at the same time.

This is where the importance of being very precise comes into play, for clearly, when [I can be both father and son] is true, it is because I am referring to the [multiple relationships with different people]. And when [I can be both father and son] is false, it is because I am referring to a perspective that involves a [single, specific relationship between two specific people].

When we think precisely, it is obvious that these are two very different things. But when we define things in a vague way, we leave the door open for a vague interpretation, and this can appear to turn [two things that are different-in-an-absolute-sense] into [two things that are similar-in-a-relative-sense]—invalidating our conclusions, and making them seem paradoxical.

Sets and Elements: In the same way, a given [thing] can be both an [element] and a [set], but it cannot be both at the same time and with respect to the same perspective. This is the basic rule of logic which is being broken when we try to place [set A] inside itself, as [Element A]. [Set A] can easily be an element in another set--and, in fact, almost all sets necessarily are elements in other sets. But the [element aspect] and the [set aspect] are logical opposites, so when we think of the "element aspect" of [set A] we are necessarily thinking of something that is [not set A]. Given this fact, it is literally impossible to put [set A] inside itself, without changing what [set A] is.

It's a bit like imagining a [square/circle]. We can imagine the concept, and speculate about what it might mean to place a set inside itself... but (in both cases) the result is bound to lead to error and paradox, because it is necessarily the case that the concept does not have a corresponding possibility in any real world scenario.

This is only true with respect to the realm of set theory. Outside of this realm, I suspect that the [class of all sets] can be seen as a part of some larger group.

I haven't spent any time considering how this should be done. But one obvious possibility that presents itself can be shown by acknowledging that:

(G)... {G, x, y}

is not a valid set. But it is a valid idea... and an idea is a type of thing. Thus, it would seem, there are ideas that are not sets--and if we consider all of these things, then the [class of all sets] is a part of that larger category of "things".

To understand how this prevents "dead ends" in our reasoning... see my previous post on TROPES... and my discussion of vicious circles.

I'm pretty sure those were made on this thread.

One final note on this topic.

There are a few self-referential, real world instances that may seem to support the idea that you can substitute f(a) for the f in f(a). For instance, if a camera is pointed at a monitor that is broadcasting its feed, the result will be a picture of a monitor ((with a picture of a monitor) with a picture of a monitor)... etc. Or a Mirror that reflects the image of another mirror will create another real-world, infinite regression.

But this is not a valid argument against my position--in fact, it is further proof that my position is correct. Think about it like this: a monitor normally shows us what is actually present in the world. But when we create such a self-referential feedback loop, the monitor shows us something that does not exist in the world. Thus, by creating the feedback loop, we have changed the nature of the monitor--and what it shows us.

This is the same situation we find when we insert f(a) for the f in f(a). Doing so changes the essential nature of what f(a) is--or perhaps, more accurately, what it does. And (just like in the real-world, with the mirror or the monitor) the result is a paradoxical, non-reality.

Self Reference
I think I've already looked at this subject briefly, but I think it would be interesting to take a more indepth consideration, and look at the subtle ways that it applies to the DS theory.

It seems to me that the traditionally definition of a self-reference implies that [A] refers to [A]. But according to the DS theory, the irony of a "self reference" is that it doesn't actually refer to itself. It may be said to refer to part of itself, but it can never refer wholly to it's total self.

Sometimes when we modify one word with another, doing so simply defines a smaller set than the original word. For instance if we modify [pony] to create [Shetland pony], the term [Shetland pony] is simply a [more specific type of pony]. In other words, [Shetland ponies] are a subset of [all ponies].

By contrast, a [starfish] is not a [fish] (nor is it a star) and a [peanut] is not a [nut] (nor is it a pea). The irony of self-reference is that it follows the pattern of the second group, for (at least according to the DS theory) it is not truly a [reference] (nor is it truly a self). Like a starfish, it is an entirely new and unique category of "thing".

With regards to our discussion of epsitemology, a reference is when some thing "points to" or "defines" something else. More specifically, a reference is when a sign defines either a [conceputal entity] or a [physical object].

A sign is anything that stands for something else.

So, a reference is the correlation between the [sign] and the [thing the sign stands for]. For instance, if the phrase, ¡°this man¡± is being used to refer to a [specific physical man], then the letters in ¡°this man¡± are the sign which stand for (or represent) the [specific physical man] who is being referred to.

Notice that if the letters ¡°this man¡± occur alone on a piece of paper, without some context as to what they might refer to, then they literally do NOT refer to anything. They are still a sign, because they stand for the [idea of a specific man] when said in the appropriate context. Thus the letters still stand for some meaning, but they do not refer to anything.

Now, it should be quite obvious that the sign, ¡°this man¡± is very different from (say) [specific, living and physical man]. The sign is simply a collection of lifeless letters; while the [specific, living man] is a breathing organism. Similarly, if "this man" refers to a character in a book, the conceptual nature of the character is again very different from the conceputal meaning of the letters, "this man".

A painting is another type of sign that can reference the [physical aspect of an object or entity. So, for instance, if the painting is of a unicorn, the painting doesn¡¯t depict the unicorn as a formless idea¡ªit depicts the unicorn as a physical object. The fact that no corresponding physical object can be found in physical reality does not change this fact. Again, the [sign] and [what it stands for] are very different. The painting is canvas and pigmentation in oil¡ªwhile the thing being referred to is a living breathing animal, albeit one that doesn¡¯t actually exist.

In every possible case, a reference uses [sign A] to refer to [something that is not sign A]. But a self-reference breaks this fundamental rule of semantic logic and uses [sign A] to "refer" to [sign A].

Semantic logic is perfectly okay with [Sign A] referring to [Sign B]. Afterall, signs are [physical objects] in their own right--thus, a [sign] can refer to [another sign]. But when it tries to refer to itself, it ceases to be a reference, because there isn't anything for the [sign] to "point to" or "define". Thus, it is not truely a reference.

Nor does a [self-reference] actually refer to [itself]. This may seem like a strange assertion, but think about it. A sign is always an expression--and as such, it will always be somewhat less than an absolutely perfect representation of the thing it is referring to. The thing being referred to is always (necessarily) a state-of-being, which is absolutely what it is. This means that the expression is always something less than (or at least different from) the thing that the expression refers to. So how could [Sign A] possibly be referring to itself? It can't. The best it can do is to refer to something relatively similar to itself. More over, we can look at real-life situation to experimentally confirm this conclusion.

Partial Self-Reference
If you point a video camera at a monitor that is displaying what the camera is recording, and the camera is tightly focused so that it is recording the whole screen, and nothing but the screen (we can call this a 1:1 self-reference) then the monitor will look normal--indistinguishable from any other screen of static. But if you pull the camera back a bit, so that it records the whole monitor (not just the screen) and perhaps a good bit of the table the monitor is sitting on, then the monitor will show a table with a monitor sitting on it, showing another table with a monitor sitting on it and so forth, into infinity.

Notice that when the camera is in a true 1:1 self-reference, the screen looks pretty much like it would if there were no camera. It isn't exactly the same because the reproduction process isn't perfect, but it's close enough to be very difficult to tell the differnce.

This close-but-not-quite scenario is analogous to expressions of identity, such as X=X. For (as I've said before) there is a sense in which all such statements are false. In this case, the [x on the left] is not the same as the [x on the right]. Like the self-recording screen, it is easy to overlook the ways in which the [right x] is not a self-reference with respect to the [left x]. But that doesn't mean that the distinction doesn't exist.

More on Self-Reference
The irony of self-reference is that it always turns the [thing being referred to] into something that other's of its kind are not.

Self-References can be carried on a number of different mediums, for example:

1. Images
2. Works of Fiction
3. Scientific References
4. Sentences
5. Mathematical Expressions

I would argue that, in each case, the act of self-reference changes the essential nature of the thing being referred to--in essence, inverting it along at least one property-spectrum into it's logical opposite.

Images: We've already seen how self-reference turns a finite scene into one that has infinitely many reiterations.
Works of Fiction: Typically, a book of fiction is about 1) things that are not real; and 2) things that are not the book. But a book of fiction that refers to itself is (to some extent) about something that is real--and it is (at least partially) about itself.
Scientific References: Scientific references generally cite outside sources to give added authority to the work. But sometimes such a work will include a self-reference. Such a self-reference is not about an external work, and it does not imbue the current work with any additional scientific authority.
Sentences: In the DS theory, a sentence can only be true if there is a correspondence between the [signs of the sentence] and one or more [conceputal enity] or [physical object] which is external to the sentence. If this correspondence is lacking, then it is false. A sentence that refers to itself ceases to have the necessary structure. And therefore we can say that it is neither true nor false. Like two mirrors that reflect each other in a perfect 1:1 self-reference, their light reflects back and forth infinitely many times--so too a self-referential sentence alternates back and forth between the [sign] aspect and the [referent] aspect. When both are true, (or both are false) the sentence appears to have a singular truth value--but the truth that is that it has two infinitely alternating values that just happen to be the same. When those values are different, then the paradoxical nature of the sentence becomes easily apparent.
Mathematical Expressions: Mathematical expressions are typically about something specific. But a self-reference in mathematics isn't about anything specific. In the sense that X=X is true, it doens't tell us anything. It doesn't do anything to explain what X is. Like the monitor that looks exactly like itself, a true 1:1 mathematical self-reference is just the expression itself, so the true self-reference of X is just X. Like all other self-references, a mathematical self-reference like function (G), which is defined as: f(G) is only a partial self-reference. The very act of including (G) in the function changes the nature of the (G)... just as the monitor that wasn't in a 1:1 self-reference created an infinite reiteration of new and categorically differnt images than when the monitor is in a 1:1 self-reference. But as I see it, the basic problem is that the "so-called" self reference is constantly changing. This is categorically quite different from an expression that captures some idea of infinity. The idea may be infinite--but it is a specific idea that does not change over time. But the self-reference expression is not like that. It is an idea that does change over time--progressing through the infinite reiterations in real-time, so to speak. As such, it is not a valid mathematical expression--and I believe that any valid mathematical system should exclude such expressions via the system's rules of formation. I'll have more to say about this as I look closely at some of the paradoxes associated with self reference.

Paradoxes of Self-Reference generally come in three varieties: Semantic paradoxes, Set-theoretic paradoxes, and Epistemic Paradoxes.

Semantic paradoxes

1. Liar's Paradox:
   
   This sentence is false.
2. Grelling's Paradox: If a predicate is heterological if it is not true of itself, then we can ask,
   
   Is "heterological" heterological?

Set-theoretic paradoxes

1. Russell's Paradox: Consider the Russell set R, which is the set of all sets that are not members of themselves.
   
   Is R an element of R?
2. Cantor's Paradox: According to Cantor's theorem, any set (except the empty set) will always have a power set that is larger. But now, consider U, the set of all sets. Presumably, it must also have a power set, which must be larger than it is--but if so, it can't be the set of all sets.

Epistemic Paradoxes

1. Paradox of the Knower: I'll describe this one in detail when I get to it.

The structure of both the paradoxes of Russell and Grelling can be understood by defining the extention of a predicate to be the set of all objects about which the predicate is true. For predicate P we can denote its extension as ext(P).

Russell's paradox creates the set of all sets that are not elements of themselves, which can be expressed as:

{x:x is not an element of x}

As I've previously noted, one of the most fundamental principles of the DS theory of sets is that a set never contains a set--it only contains elements. Some of those elements can be sets in their own right--but within their specific relationship to this particular set, those sets are entirely elements, and nothing but elements.

Since it is impossible for any set to contain a set, it is most certainly not possible for a set to contain itself. Analogously, this is exactly like trying to define a word using that same word. Or, you might think of it like saying that you're going to paint [picture A] inside of [picture A]--when what picture A is is a blank canvas. If the blank canvas is painted as a 1:1 self-reference then the result is simply a blank canvas. But if the painting includes background, then the only way to include [picture A] is to make it something other than a blank canvas.

In the same way, when we insist on separating the notions of element and set, then [element R] is literally a 1:1 self reference of [set R], meaning that it doesn't change at all. It remains exactly what it is. But when we fail to keep these notions separate, then the self reference is only partial in nature, and we get the infinite cascading nature that is universal to all such structures.

Grelling's paradox is essentially identical, but instead of elements and sets we are dealing with words and their definitions. Or a [name] and the [thing named].

Grelling's paradox supposedly creates an extension for the predicate, heterological, which can be expressed as:

{P is not an element of P}

The first thing that should be observed is that this is true in every case for every possible predicate. Because, again, a set can not be an element of itself.

Given this fact, what then is really going on with a predicate like heterological? According to the DS theory, a [name] is not the same thing as the [thing named]--which is just the non-set oriented way to say that a set can't be an element of itself.

The name "food" is not the same thing as the physical objects that I cook and eat, which we might distinguish with the sign [food]. Food, however, is a predicate (or property) which is either true or false of various other things. [Dirt] is not a type of [food], but [Carrots] and [beef] are. But here's the kicker, "food" is not a type of [food]--it simply is food.

We might logically say that "food" is the set that contains various elements, such as {[carrots], [beef], [peas], [liver]...}. The element aspects are the actual physical objects that provide substanence. I can eat [carrots], but I cannot eat the name "carrots"; I can eat [beef], but I cannot eat the name "beef". Collectively, all of the elements in "food" are singularly [food]. I cannot eat and receive substanence from the name "food", but I can receive substanence from [food].

Similarly, the name "heterological" refers to the property of being [heterological]. The name "German" is [heterological] because it is NOT a [German] word; while the name "Deutsch" is [NOT heterological] because it IS a [German] word. But just as "food" is not a type of [food], so too "heterological" is not a type of [heterological] thing.

Why not? Well, notice that each pairing requires two things which must be compared. One is a [name] and the other is [the thing being named]. We compare the name "German" to the set of words which are [German]... and through this comparison we can determine if the relation between these two entities qualifies as the [heterological property]. But when we compare "heterological" to [heterological] the comparison process becomes somewhat redundant--and in this case, contradictory.

The redundant problem occurs if we define "homological" as the property of being true of itself, and ask, "Is "homological" [homological]." It's still analogous to a mirror reflecting another mirror--but because the structure produces a consistent answer, we tend to ignore the infinitely recursive nature of its structure. We only focus on that when the structure is apparently:

x is not x.

or

x is not what it is.

but neither of these suggestions are acceptable--for the first and most fundamental principle of the DS theory is that every thing is exactly what it is. Thus, x has to be x.

Traditionally, there is apparently no good way for this fact to weed out constructions like,

heterological is heterological

or

x is not x

when a schema, theorem or so forth provides a method for constructing it. But in the DS theory, we can still use such construction methods, and then cull the unsatisfactory specific structures from the resulting set, based on the fact that it violates the more fundamental rules of formation--which do not allow for structures like [x is not x].

Cantor's Paradox
This paradox goes astray because it treats two different names for the SAME thing, as if they were names for Different things.

Again, I believe I've previously posted the proof for this on this thread. But briefly, the set,

(A)... {1, 2, 3}

and it's power set

P(A)... {[1], [2], [3], [1,2], [1,3], [2,3], [1,2,3]}

(A) and P(A) are simply different names for the same set, since both sets contain only [1], [2] and [3]... and nothing else.
But P(A) contains all the various groupings of these elements. It can only be said to contain more elements because the elements themselves are of a distinctly different type.

For instance, suppose you are going to define the set (food) to list all the types of food that you're going to have at your thanksgiving dinner.

(F)... {[turkey], [potatoes], [corn], [peas], [yams],...}

Now, many of the people who come to eat this food do not like one or more of the items being served.
(Allen) ... likes only {turkey}
(Bob) ... likes only {potatoes}
(Cal) ... likes only {corn}
and so forth.

In fact there is exactly one person for each possible combination of all the different foods. Which means there are a lot of people... Far more than the number of foods that were being served. But does this mean that the amound of food they ate was more than the amount of food that was served? Does this mean that there were more types of food that they ate than the types of food that were served. No! Of course not.

What we are dealing with is two different ways of looking at the exact same scenario. And the same is true with a set and it's power set. Nothing has changed with respect to the state-of-being that we are dealing with. But we are using two different "names" or ways of looking at that state-of-being.

According to the DS theory, the important thing to understand is that a given state-of-being is always exactly what it is; and just because we may choose to call it by two different names doesn't change the fact that it is what it is.

The last paradox (other than the liar) is the paradox of knowablilty. But before I get to that, let me sidetrack a bit.

The differences that I've given between traditional theories and the DS theory lead to some very significant and far reaching consequences. And I want to take a few moments to address a few of them.

T-Schema: Q <-> T<Q>

Here T stands for truth, Q is any arbitrary sentenence and <Q> is a name for that sentence. Thus, T-schema basically says that for every sentence Q, Q holds only if "Q is true" holds.

Tarski 'proved' that the liar's paradox can be formalized into any formal theory that contains his T-schema, and thus any such system is necessarily inconsistent. And yet, most theoreticians seem to think that T-schema is intuitively obvious.

The DS theory has two points worth making on this:

First, as I've already said, a [name] is not the same thing as the [thing named]. Most of the time we can ignore this fact, because the differences are of such a subtle nature that usually they don't change or interfere with logical manipulations. But it is possible to create certain logical structures that are capable of exploiting the differences to create paradoxes.

Second, the way the DS theory defines truth renders the very notion of the T-schema meaningless. For the DS theory doesn't define something as true (at least not in the practical relative sense) because it is what it is... That's something that we can't know or express.

Instead, the DS theory defines truth as a correlation between one definition (a sign) and another (the thing being expressed). Thus, to formalize this idea we would have to create two separate and distinct ways of referring to these two very differnt notions. For instance, a DS version of the truth-schema might look something like:

DST-Schema: T<Q> <-> (Q ~ <Q>)

Here T stands for truth, <Q> is a name for Q, and ~ stands for "corresponds adequately to".

Thus, the DST-schema can be stated as: "Q" is true iff [Q] corresponds adequately to "Q".

This makes creating a formal system from the DS theory a rather challenging task, since it would be very difficult (if not impossible) to formalize what is true and what is not--since the paradoxical nature of reality must be built into the formal system itself. Thus, the best we can do is formalize what we currently believe is true.

The last sentence is true of any formal system, but most define truth as if it were an absolute--and then they must live with the necessary paradoxical results. The DS strategy would be to formalize the paradoxical nature of reality, so that results of the system are never paradoxical. But I am not altogether certain that this can actually be done.

Diagonal Lemma

Let S be a theory extending first-order arithmetic. For every formula Q(x) there is a sentence Y such that S :- Y <-> Q<Y>

Here S :-Y means that Y is provable in theory S. Thus, the lemma basically says that every sentence that is provable in S must express the property represented by Q(x). For example, Q(x) might be the truth property. In which case, the diagonal lemma gives rise to sentence Y which satisfies the biimplication Y <-> Q<Y>. Or, in other words, Y becomes a sentence expressing of itself that it has property Q--and since Q is the truth property, Y says of itself that it is true.

This isn't true self-reference, because it is the [name of Y], <Y>, that is being said to have the property--not Y itself, but Y behaves like a self-reference. Which means that we can now construct paradoxical self-references that resemble the liar's paradox.

The DS Perspective
In the DS theory a 1:1 self-reference is impossible. And the [name of something] is not the same thing as the [thing being named]. Thus, the name of a sentence (i.e. <Y>) would never be a permissible part of a DS formal theory. And, even if it was, a construction like

Y <-> Q<Y>

would never be permitted.

However, we can still create a partial self-reference by using a two step process. The first step is to define two reciprocal aspects. So for example with respect to a truth value, T is the logical opposite of F. Thus, we might say that

1/F f= T ...OR
The inverse of False is True

This allows us to talk about property T using the property 1/F. In essence, what we are doing us using the [definition of T] to talk about [T]. In this way, all properties are defined, for we can also use 1/T to talk about F. Using this strategy would give a formal system the expressiveness of self-reference, without the complications that result when we treat the [name of something] as if it were identical to the [thing named]. If done correctly, I think it would also make the distinction between first-order and second-order entirely meaningless.

Consequences of Set-Theoretic Paradoxes

Unrestricted Comprehension:
@u (u E {x: Q(x)} <-> Q(u)) ... or
for any u, u is an element of the (abstraction) set {x such that x has property Q} if and only if u has property Q.

Here, @u stands for "for any u'. And E stands for "is an element of".
So, in other words, The Unrestricted Comprehension Principle (UCP) says that for any property (expressed by Q) there is a set of entities that satisfy the property. This sounds reasonable, and captures the intuitive concept of what a set should be. Unfortunately, it is believed to give rise to Russell's paradox. For all that is necessary is to consicer the property of non-self-membership, which we can express as:

x ~E x

Thus, if we let the property Q(x) be the formula, x ~E x then the set

{x: Q(x)}

becomes the Russell set R. which UCP renders as:

@u (u E R <-> u ~E u)

Or, in other words, u is an [element of itself] only if it is [not an element of itself].

The DS Perspective
Again, the problem here is the assumption that the formula x ~E x has any actual meaning--aside from the obvious observation that a set cannot contain a set, and so it canot contain itself. Remember our previous discussion where we distinguished between the singular word "German" and the set that contained all "German" words, and can be identified as [German]. [German] cannot be an element of [German] because [German] is literally all of the words that belong to that set. The only way that [German] can be considered an element of the set [German] is when we use the abstraction set {x: is "German"}. But the nature of this set is reciprocal to the enumeration set, for it contains only one element, x. And this single element is literally all of the words that belong in the enumerated aspect of the class that is [German]. (Remember, in the DS theory, a class is the union of the [enumeration set] and the [abstraction set] aspects.)

Notice how the reciprocal definitions prevent the construction of Russell's set R, while still allowing us to refer to the property of not belonging to a given set. The DS version of property Q, traditionally written as

x ~E x

would be something like:

(Q)... "x" ~E [x]

. Now, suppose we have the set

[x]... {"u", "v", "y", "z"}

. Clearly "x" is not an element of [x]. So it would have property Q. But the set

[x]... {"x", "y", "z"}

would not have property Q.

Notice how the reciprocal nature of the definitions exponentially increases the expressive nature of the theory, while at the same time making it much more precise and intuitively accurate.

To me, it seems the Biggest problem with constructing a DS formal theory is simply finding a way to indicate all the different reciprocal aspects of a given state-of-being--for example, at the very least we have to find a way to indicate the three distinct aspects of x. The [enumerated set] aspect, the [abstraction set] aspect, and the [class] aspect, which is the union of both the others. And similarly, we have to find a way to indicate when x is being used as an element and when it's being used as a set. And there are many more reciprocal definitions which would need to be distinguished adequately in notation in order for a DS formal theory to be correctly expressed.

As I think I've said before, one of the few weaknesses of the DS theory is that it requires a significant increase in the complexity of a theory at the fundamental level. But the advantage is that it reduces the complexity (while increasing the accuracy) of the theory at the higher levels. Thus, once the aspects are clearly delineated, writing an Unrestricted Comprehension Principle that doesn't lead to a paradox is quite simple. Ridiculously simple, in fact, when compared to the bizzare abstractions and permutations of thought that traditional theories put themselves thru in order to accomplish essentially the same thing.





Similarly, "English

Consequences of the Epistemic Paradoxes
At this point, I'd like to go back and address the knower's paradox. First, traditional theories introduce a special predicate of "knowability" which is written as K; thus K<Q> expresses the idea that Q is knowable.

Traditionally, it is believed that there are 4 basic logical principles that K must satisfy in order for a theory to adequately formalize knowability.

1. All knowable sentences must be true:
   
   K<Q> -> Q
2. The above listed (1) must itself be knowable:
   
   K(K<Q> -> Q)
3. All theorems of first-order arithmetic ought to be knowable:
   
   @Q: K<Q>
4. Knowability must be closed under logical consequence:
   
   K<Q -> Y> -> (K<Q> -> K<Y>)
   If we (know Q & Y) then we (know Q) and we (know Y)

The DS Perspective
According to the DS theory, (1) is only true with respect to absolute knowledge... which cannot be accurately expressed. Which means that we cannot know that we know (1), so (2) is also false.

We can replace these with the following DS principles:

1. @P: *P -> P
2. @P: ~(KP) ...or... @P: ~(K(*p))

Where @P: stands for "any P such that" and
*P stands for "P is absolutely knowable" and
P stands for "absolute P".

Now we have to define relative truth, which I haven't completely done in a formalized notion format: but I think it might look something like:

@P: KP -> ((b[P]) <-> (b"P")) & "P" := [P]

Here, := stands for "is a relatively adequate expression of". Thus, we can read the above principle as: "For any P: we know P if we believe [P] if and only if we believe "P" and "P" is a relatively adequate expression of [P].

We can combine the above with a axiom that says something like:

@P: KP -> ([P] f= 1/"P") & ("P" f= 1/[P])

Or, in other words, For any P, if we know P then we know that [P] is the reciprocal opposite of "P" and "P" is the reciprocal opposite of [P]. Thus we might say that P := "P"/[P] and P := [P]/"P".

In other words, we have defined P as the class that is the union of [P] and "P' . And we have defined [P] as "1/P" and we have defined "P" as [1/P], creating two reciprocal terms that are functionally equivalent and can thus be used to talk about one another.

According to the Self-Reference article on the Stanford Encyclopedia of Philosophy website, there are three basic strategies which have traditionally been used to resolve most paradoxes of self-reference.

1. Explicit Hierarchies (EH)
2. Implicit Hierarchies (IH)
3. Fixed Point Approaches (FP)

Explicit Hierarchies
Russell's Theory of Types is a good example. The strategy here is to stratify the universe of discourse (sets, sentences, etc) into levels, called types. And any set of a given type may only contain sets of a lower type.

Tarski's Hierarcy of Language is another good example, but here the strategy is to build a layering of languages where only languages on a "higher" level can apply predicates (such as a truth predicate) to lower languages. This creates a higher-level meta-language that is used to talk about a lower-level object language; and a meta-prime-language which is used to talk about the meta-language, which now becomes the object-language with respect to this meta-prime-language... and so forth.

While these strategies use different methodologies, boty rely on the same basic principle of separating ideas through stratification. Self-reference paradoxes like Russell's paradox and the Liar depend on circular self-reference notions, such stratification effectively circumvents the semantic and set theoretic paradoxes. A similar strategy could easily be applied to block the epistemic paradoxes, by making a distinction between first-order knowledge (which is about the external world) and second-order knowledge (which is about first-order knowledge) and so forth.

The DS Perspective
It is worth noting that the DS theory creates the same stratification by defining [elements] and [sets] in the stratifying way that it does. (Or, with respect to knowledge, by distinguishing between a [name] and the [thing named].)

The advantage for the DS strategy is that this separation is created as part of the theory's most fundamental definitions--rather than being an after-the-fact, tacked on, fix-it strategy. Most theoreticians (including Russell and probably Tarski) tend to agree that the fix-it strategy of explicit hierarchies not only seem to be overly drastic and heavy handed, but rather ad hoc as well, since such a notion of hierarchies is not part of ordinary discourse.

In addition, Explicit Hierarchies introduce a number of complicating technicalities no present in a so-called 'flat-universe' (without such hierarchies). And although the paradoxes do disappear, so do all the non-paradoxical occurrences of self-reference. For instance if we let (N) be the following statement, made by Nixon,

(N)... All of Jones' utterances about Watergate are true.

and let (J) be the following statement, made by Jones,

(J)... Most of Nixon's utterances about Watergate are false.

Tarski's hierarcy of language cannot deal with this scenario because both (N) and (J) would have to be on a higher language level in order to talk about the other. And similarly, being different types would necessarily lead to a similar problem.

And finally, there are some liar-like paradoxes that do not rely on self-reference. For example, Yablo's paradox consists of an infinite chain of sentences, each one professing the falsehood of all the previous sentences in the chain. I believe a DS formalization would prevent such constructions, but stratification approaches do not.

Implicit Hierarchies
Here the strategy is to adopt a three-value logic, where the third truth-value is undefined. Essentially, a partially defined predicate only receives one of the classic truth-values when it is applied to one of the terms for which the predicate has been defined.

One strategy is to create a partial model for a first-order language, wherein each n-place predicate symbol P is interpreted by an extension of P (which includes all the objects about which P is true) and an anti-extension (which includes all the objects about which P is false); all other objects are undefined. The partial model is defined by the Pair, (U,V) where U is the extension of P and V is the anti-extension of P.

Armed with this partial model, we can construct partially interpreted languages where L is a first-order language adn M is a partial model of L. In this manner, we can create a recursive sequence of languages, L0, L1, L2... where the only difference is ther interpretation of the truth predicate T. The first language, L0, is taken to be an arbitrary language in which both the extension and antiextension of T are the empty set. One of the characteristics of this system is that both the extension and anti-extension of T increase as we move to the higher-level languages. So U and V are larger sets in L1 than they are in L0; larger still in L2, and so forth.

Further manipulations eventually produce a language that stabilizes, and contains its own truth predicate--implying that for any sentence Q, it is true (or false) if and only if the sentence expressing it is true (or false).

It should be reasonably clear that this whole (rather complicated) processs
is very similar to Tarski's hierarcy of languages. In fact, the SEP article suggests that they are virtually the same, except that in this later approach there is no syntactic distinction between the different languages and their truth predicates.

This strategy deals with the liar's paradox by saying that the final "stabilized" language has a truth-value gap, which remains undefined, even when presented in the structure of the highest level languages.

Most theoreticians consider the implicit hierary strategy to be less than completely satisfactory, since after all that, the truth-value gap is still present for the liar, and other paradoxes of self-reference. In addition, the resulting language is expressively weak. For example, it is not possible to have a predicate that expresses non-truth, because non-truth includes both the set of things which are false and the set of things which are undefined. Thus, if a partially interpreted language were to contain such a predicate, the following version of the liar could be constructed:

This sentence is non-true

The Fixed Point Approach
This strategy is closely related to the previous one--but it can be said to be even more complex and convoluted. I don't plan to get into its details.

I will simply make the point that, even if such an approach is capable of circumventing the paradoxes of self-reference without creating any of the various problems which have been discussed (or new ones) it is still quite apparent that the system itself has become so over-whelmingly complex, convoluted, and counter-intuitive that you have to be a virtual genius just to follow what it's trying to say. And at this level of complexity, it is very easy to create theorems that seem to be logically sound, but are in fact based on faulty reasoning, or a error in the naming conventions, etc.

By contrast, a DS system might be a little more complex at the beginning... but once the basic definitions have been created the rest of the formal system would be comparitively simple and intuitive, making it much easier to use and its theorems more reliable. In addition, it would be much more assessible to virtually anyone who wanted to learn about it.

In a previous post I gave a very informal visual 'proof' using a partial self-reference created by a monitor showing a camera feed of its own self.

I now have some additional thoughts on the subject.

Recap (and New TErminology)
AS the partial self-reference approaches a 1:1 self-reference it was also coming closer and closer to infinty--as illustrated by fact that it's Rate of Progress is getting higher and higher. Finally, when a true 1:1 self-reference is reached, infinity can be said to reach unity .
It passes through a limit point and instantaneously manifests its self in a functionally equilent form--a singular image in the mirror. This is very similar to the process that happens when the [positive numbers] pass through the [0] limit point, to become the [negative numbers]. In a like manner, when the self-reference crosses over the limit point as it becomes a true 1:1 self-reference, the infinite aspect instantaneously collapses to become a holistic oneness.

Another way to understand the Unitary Principle of Infinity is to realize that—like [1]—every single physical object (or conceptual entity) is sitting there waiting to be dissected into infinitely many parts.

Whether we choose to think of this singular thing as a [oneness], or as a [collection of infinitely many parts] is an entirely arbitrary choice. Both aspects are necessary to define every possible state-of-being.

Medium or Message
We should keep in mind that what is actually being self-referenced in the case of a [monitor that is showing its own image], is the medium—not the [message that the medium carries]. That’s why a 1:1 self-reference of this sort shows us nothing but the medium—a blank monitor screen, for instance.

When a self-reference is purely the medium, it is impossible to create a non-fractional self-reference, or in other words, a n:n self-reference where the first n is larger than the second, instead of smaller. What the monitor shows is the room with a monitor in it. That monitor shows a room with a monitor and so on. There is never, at any level, a non-self-referential image on the monitor. Same thing with mirrors, or a painting. In each case, the medium acts as a carrier for a message--the monitor carries electrical signals; the mirrors carry rays of light; the canvas carries an array of paints. In each case, part of the message is a self-reference that refers back to itself. And if the message doesn't do this, then how can it really be a self-reference?

Suppose that we started with a normal painting instead of a [self-referencing monitor]. Lets call this (painting A), then the 1:1 self-reference of that painting is simply the painting itself—nothing more and nothing less. And if we try to create a 1:2 self reference—there is nowhere on the painting for the image of the painting—because the whole painting is already about something other than itself. Any effort to include a smaller version of the painting inside itself would cover up part of the original painting.

However, we can go in the other direction and make the reproduced self-reference larger than the original painting. This would look like a small painting sitting on larger reproduction of its self.

Again, this is only a partial self-reference, since the center of the larger picture is covered up by the presence of the original picture. But, once again, we can imagine an infinite array of larger and larger images, which we might call a multiplying self-reference, or (MSR)

This sort of [multiplying self-references] (MSR) is reciprocal to the [fractional self-reference] (FSR) we've previously been considering--and so, according to the DS theory, we should expect them to be opposites in certain ways. Obviously, with the FSR the medium gets smaller and smaller; while with the MSR the message grows larger and larger. In addition, however, there is another critically important distinction.

The FSR (fractional self-reference) occurs spontaneously, and extends into infinity. There is a practical limit to how many reiterations I can see in the monitor, for instance, but if the screen had a high enough pixel count, and we magnified the image, we would continue to see more and more images imbedded in the one before. The limits therefore are in our physical senses and the technology that we're using to capture the phenomena.

On the other hand, the MSR (multiplying self-reference) does not occur spontaneously. If we want to create a second-order reference we have to produce a larger reproduction of the original painting and place it beneath the original. To create a third-order reference, we have to produce a still larger reproduction, and so on. Thus, while the FSR is spontaneous and infinite, the MSR is deliberate and finite.

Fractals and Self-Reference
I believe that in the real world, the vast majority of self-reference structures are actually of the Multiplying variety--and we typically refer to them as being fractal in nature.

Many real world objects appear self-similar over a wide range of scales. A cloud, a coastline, and even a head of lettuce, are all examples of multiplying self-references.
No matter how large the head of lettuce (in the picture above) is, however, it will still have leaves that define its outer dimensions. And, going in the other direction, there is obviously a limit to how small the leaves can get before they cease to look all that much like the outer leaves—otherwise, the atoms (and all smaller sup-particles) would have to look like lettuce leaves.

Because the outer leaves look fairly self-similar all the way to the last leaf—while the inner leaves seem to look more and more dissimilar from those highly visible leaves as they get smaller, it's easy to think of the head of lettuce as a fractional self-reference, where the reproductions are getting smaller and smaller. But this makes no sense when you stop to think about the way a living organism is formed. It starts with a very small bud, that looks a lot more like the small cluster in the middle of the head than it does like the outer leaves.

Of course, the outer leaves are the first to be produced, but the individual leaf is just the [medium] upon which the original shape or [message] is being conveyed. The message is the center of the lettuce head, which gets reproduced in a larger and larger medium as the head grows.

Similarly, when we look at a cloud, the fundamental message is the smallest grouping of water droplets that take on the characteristic aspects of a cloud. This is where all clouds form--and larger clouds are just larger reproductions of this original 'image'.

Assuming this is so, where does the message come from? And what is the distinction between message and medium?

I think Rupert Sheldrake's theory of Morphic Resonance goes a long way towards explaining the MSR phenomenon that I've described. But his work is very detailed and complex, so I will put it in much simpler terms.

We can basically say that the medium is the [physical atoms that make up reality]--and the messages are the [Rules of Nature that define how those atoms can interact with one another].

Stephen Wolfram, in his seminal work, A New Kind of Science (NKS) shows how very simple programs (which he calls cellular automaton) can give rise to extremely complex patterns and shape, which appear to be fractal in nature.

In DS terms, here's how a basic cellular automaton works: the medium is a graphed page, where each individual cell can be either left blank (white) or filled in (black). The message is a simple set of rules, and a starting point--which is a single black cell in the center of the first line.

Here is an example of a rule:

The top row is a pattern, which, when encountered on the graph, forces the medium to reproduce the appropriate response, which is given in the "response line" of the rule. In other words, each cell of the rule can be interpreted as an, "If you see this then do this," statement.
The rule given above would produce the following graph:

The pyramid can get as large as we want it to get, but we must add a new line each time we want it to grow. And whatever level of iteration we are on, the pyramid will be fininite in nature. Conceputally, however, the pyramid can be understood as being infinitely large.

Now, this set up is obviously severely limited, but even with this very basic structure there are 256 possible rules. In fact, there is a very simple way to name these rules. Since there are two possible responses that can be produced in the response line, either (white) or (black), we can replace each cell of the rule with a number, either a [1] or a [0]. Thus, the first rule is [00000000], where every response line is white. This is rule 0, because it is the binomial number for [0]. [00000001] is Rule 1, because it is the binomail number for [1], and so forth. Thus, we can create [rule 0] through [rule 155].

You can see very small versions of these rules here and here.
There are a number of interesting observations that can be made about the graphs that are created by these rules. For one thing, some of the rules produce the exact same result. For example, the following rules:

{0, 8, 32, 40, 64, 72, 96, 104, 128, 136, 160, 168, 192, 200, 224, 232}

all produce a graph that is unchanged--that is a single black cell in the center of the first line.

Some graphs (such as rule 90) produce images that appear to be typical fractal-like structures, as we see on page 26 of NKS. Others (rule 30) result in graphs that seem to be truely chaotic, as we see on
Page 29 of NKS.

What is intersting about (rule 30) is that we can see different sizes of the same basic shape repeatd over in over. On the left of the image, we see that the sizes cluster together to form larger structures of similarity. But as we move to the right, those self-similar super-structures break up and turn into apparent chaos. In addition, as we move deeper and deeper into the graph, new super-structures appear--both instances of order and instances of chaos. Pages 30-38 of NKS show this progression in detail, but page 33 is perhaps the most illustrative of all the examples.

Not only is it possible to create 'chaos' from a simple set of rules, but if we begin with a random starting condition, some rules can produce very orderly structures, as we see on page 225 of NKS.

By making the rules (or the starting point) just a little more complicated, we can exponentially increase the number of possible graphs that can be created. And some of them are incredible complex and beautiful.

More importantly, many of them mimic patterns and shapes that we can find in nature. For example, on page 371 of NKS, we see a rule that mimics a crystaline structure. All sorts of structures can be replicated, and they don't have to be two dimensional, so we can create models for virtually anything, from a mountain, to a tree, to a virus. In fact, if we make our rules include time as an element, we can even model the behavior of such things as fluid flows, or the flight of a flock of birds.

Jesus