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Old 09-14-2006, 03:26 AM
Bobbo Bobbo is offline
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The a b c proposition

Consider this standard proposition:
Given three objects a, b, and c.
If a = b and a = c, then b = c.

For real objects (the objects exist in what we call the physical universe):

Each of the three objects is declared by the proposition to be unique. How? Each is identifiable (somehow) as separate and distinct from the other two. Note that if the objects are declared to be the same object, the proposition is trivial, since there will be only one object.

An object’s existence (in the physical universe), or presence, is expressed by the object’s interactions with its surroundings. An object interacts with its surroundings by a change in state. The state of an object includes the object’s position relative to its surroundings. Thus, even if all other aspects of the objects’ states are the same, the three objects will not have exactly the same state (including position) at exactly the same time. Note that for the (imaginary) case where there are only the three objects in their universe, and their relative positions do not uniquely identify them, there can be no observer to initiate or confirm the proposition (i.e., the proposition is not presented).

This example is for discrete objects (objects that cannot occupy the same place at the same time). For non-discrete objects (currently only theorized) to retain their identity when collocated, there must be some aspect of their state (not currently defined) that will provide that identity, and that unique identity of each will make them not equal.

So, for real objects, we see that objects will not be equal (have the same state at the same time) and not only does the proposition fail, but the basic postulate (of equality) is incorrect.

For objects in logic:
The proposition declares that one object is equal to two others. In the fantasy of logic one may attempt to define objects, their possible states and surroundings in such a way as to allow this to be true. In order to have even fantasy objects as identifiable, they must have some property that will make each unique, and therefore not equal to any other object.

Ultimately a rigorous examination (proof) should show that the construction of the fantasy is flawed. Then either the logic will have to be abandoned or the proposition declared “self-evident”, “a priori”, or something else that needs no proof. In other words, the logic may be accepted even though it is demonstrated to be false.

For objects in mathematics:
Any sort of mathematics that involves something other than real objects is logic. The manipulation of the quantities of fantasy objects does not change the objects. The logic remains flawed.

For mathematics involving real objects, the proposition reduces to each of the three objects representing a quantity of a real object. Just as for fantasy objects, manipulation of the quantities of the real objects does not change the objects. The objects are not equal.

The case in which the objects may be seen as only numbers is trivial. Numbers have no meaning when not associated with (e.g., indicating the quantity of) an object. Manipulating numbers alone is meaningless aside from demonstrating the rules of the particular mathematics.

For ourselves:
The proposition fails for real objects, logic and mathematics. Still, we teach it and use it for solving real problems and even for developing theories of how our universe works. This sort of not thinking is probably at least a part of why so many are so confused by so many others about so much.
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Old 09-21-2006, 01:24 AM
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Antone Antone is offline
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Quote:
Originally Posted by Bobbo
Each of the three objects is declared by the proposition to be unique. How? Each is identifiable (somehow) as separate and distinct from the other two. Note that if the objects are declared to be the same object, the proposition is trivial, since there will be only one object.
NOTE: my comments are not mainstream philosophical thinking. But then neither are yours, so I think maybe you'll give my comments a fair listening to.

I've long maintained that much of the problem with this sort of thinking is that we misconstrue the meaning of the [=] sign.
If each of the objects is indeed unique, then they obviously aren't exactly the same thing as one another. One apple is obviously not the same thing as another apple or else it would not be [other].

Thus, it seems fairly clear that what we mean in this case by [equal to] is really [conceputally equivalent to], written [=c=]. Thus:
a =c= b;
might be applied when we have two apples. One apple is conceputally equivalent to the other apple, simply means that both objects belong to the [type of universal entity] called [apple].

Quote:
Originally Posted by Bobbo
For objects in logic:
The proposition declares that one object is equal to two others. In the fantasy of logic one may attempt to define objects, their possible states and surroundings in such a way as to allow this to be true.
My family calls me [antone]. In grade school some of my friends called me [Tony], later I told one of my old bosses this and he started calling me [T-man].
Now, it is clear that each of these names is unique. They have a different number of letters (as well as different letters). Perhaps more importantly, the names themselves are known to different groups. For instance, my siblings wouldn't know who you were taking about if you called and asked for [Tony] or [T-man]. These names call up certain characteristics about me. [Antone] calls up images of the person who I am when I'm around my family. [Tony] calls up images of the child who used to play basketball with his neighbors. [T-man] calls up images of a pizza delivery driver.

In a sense, each of these "people" who the names call to mind are different. The child who played basketball with his friends is nothing like who I am as a middle aged adult, or who I am around my family, where I'm referred to as [Antone]. Thus, we might say that these names literally refer to different particular instances of the [collective whole which is the person who I am and have been].

Even if we deny that there is such a collective thing... and assume that I am always exactly the same physical object (from infancy through old age) then it must still be recognized that the names themselves are not the same.

Each refers to a uniquely different conceptual entity. If we allow, however, that I am the same physical object (from birth to death). Then we must admit that these different names refer to the same physical object. This is the reverse of the last case, where we used [=c=] to stand for [conceptually equivalent]. This time, we might use [=R=] to stand for [physically eqivalent].
And, if we let
A = Antone;
B = Tony;
C = T-man
then we might write:
A =R= B; B =R= C; A=R= C
In the first case, we had different particulars (physical objects) that were identified as being conceputal equivalent via both being the same universal (conceptual entity).

In this second case, we have different names (conceptual entities) that are being identified as being physically equivalent via the fact that they both refer to the same universal (physical object), namely me.

In both cases, what we have is an instance of equivalence--not a case of being equal to. At least not when [equal to] is defined as being exactly the same.

There is another case... the one where one side of the equation is a conceputal entity and the other is a physical object.

For instance when we point to a particular apple and say:
That apple is an apple
This may seem like a stupid and foolish example, because at first glance it seems to be totally circular and redundant, etc. But in a valid sense, it is not.

The first apple term refers to the physical object, and the second apple term refers to the [apple concept]. We can demonstrate this fact by noting that we could replace the [first apple term] with the phrase, [That object]. Thus the statement becomes:
That object is an apple.
[That object] is nothing more than a variable (a place holder) that stands for the [physical object that we call an apple].

The [second apple term] cannot be meaningfully replaced by [that object] (at least not without changing the meaning of the statement) because it does not refer to a [physical object]. It is a [physical instance of the conceptual notion "apple"], however, as can be seen by the fact that we could replace the second apple term with that phrase and the sentence would still make sense.
That object is a physical instance of the conceptual notion "apple".
Thus, we can create a third find of equivalence, which we might call mixed equivalence, written [=M=]. If
A = the physical apple, and
B = the conceptual apple
Then we can write the following equation:
A =M= B
Note that this eqivalency is one-directional, for while a [physical apple] is an instance of a [conceptual apple], the reverse is not true. Thus it is not the case that [B =M= A].

One more form of equivalency is what I call functional equivalency , written [f=]. The most important thing to keep in mind with this type of equivalency is that you are inverting a logical opposite.

A logical opposite has two distinctive characteristics
1. it is the diametric opposite of its opposite
2. in the same way that it is the same.

[Black] is the logical opposite of [white], because both deal with how much light (or color) is present... but whereas one is [all light or color] the other is [no light or color].

Now, [black] is the opposite of [white], thus if we invert one of these opposites, we create a functional equivalency.
B f= ~W
Again, [black] is not the same thing (in all respects) as [not white], but they are undoubtedly equivalent.

We can also see functional equivalency at work in the following equation:
[1/2] x [2/1] = [1/1/2] x [1/2/1]
All we have to do is understand that the inverse [1/n] is the logical opposite of [n]. Then we have
[Y] x [Z] = [1/Y] x [1/Z]
In this case, we have to invert both [1/2] and [2/1] becasue we don't have the [not function] to do the inverting for us. But the basic principle is the same. We are simply inverting two opposites instead of one.
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Last edited by Antone; 09-21-2006 at 01:33 AM.
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