Context is everything. It makes no sense to speak of an oposite until you deine the context under which you will be evaluating the relationship.
I have a hard time thinking of an “absolute” opposite. If we are speaking from a materialist standpoint, I might suggest matter/anitmatter. Mutual annihilation seems a reasonable standard for opposite when speaking of matter.
Spiritus, I’m interested in that whole “context” issue. Are you implying that contextual references can only be made by conscious entities? That is, there can be no contexts without people? We must first postulate people before contexts?
I’m not sure I follow your question exaclty. I am saying that “opposite” is a relative evaluation and that relative evaluations make sense only within a defined frame of reference: i.e. context.
Since we are discussing language/labels then you can certainly extend this point to imply that consciousness is a necessary predicate for the recognition/identification/labeling of a pair of “opposites”.
I think Spiritus said yes, arl. [heh heh heh]
I sorta see what you mean about a relative context. But I also, then, sorta disagree.
Take your antimatter example. Even that wouldn’t be an “absolute” opposite because—well, take anti-hydrogen. It bonds with anti-oxygen to make anti-water. Not much oppositeness there absolutely.
Now, not going to get back into our epistemological argument here, but lets assume for the sake of argument that what we know in physics is an accurate assessment of things. That being the case, context becomes somewhat tangential to the issue of opposite-ness.
In fact, scratch that. At any level of context, it seems, we may make a case for oppositeness. That being the case, would you feel that the actual context is irrelevant? I’m looking at this from a limit-style argument. That is, no matter what the context is (no matter how small, how large, etc) we find oppositeness, then it doesn’t really matter what the context is. Opposites exist.
If we were to remove context, would you remove the oppositeness? I don’t think one can remove context. In fact, it would seem to me that context has no opposite either, much in the same way existence doesn’t. I cannot see that one can start off without context…
Ugh. You’re such a formalist. 
This is simply another example of using context to frame the concept. I have certainly not argued that an absolute opposite, independent of context, can be found for anything. Your focus upon isolated structure (molecular “shape”) as a denial of “oppositeness” is like claiming that 3 and -3 are not opposites because they have the same magnitude.
I don’t see that. Context is integral in forming any relative evaluation. Now, if you are arguing that the concept of oppositeness can be applied in any context, you might be correct. Demonstrating this will be tricky, though, since the concept itself must be defined within a context.
So far, it seems to me that you are saying that within any relative framework we can define a quality that we call opposite; therefore opposites always exist. In fact, this is actually an argument for the universality of definition. For it to say something particular about “opposite”, you will need to find a specific definition which applies across all contexts.
As you say, removing context is a difficult operation if we wish to be abe to say anything. I would say that shifting context has the potential to cancel the oppositeness of any particular pair of opposites. This is not a particularly startling result, of course. It is just the type of exercise which inspired this thread (and which you demonstrated with your antiwater argument).
Can you prove that?
This sums it up for me. That is, we may remove any one opposite by “shifting” the context but, like a bad wallpaper job, another one pops up somewhere else. I almost feel the the concept of context itself embodies oppositeness, in a way. That is, a context is gained at the exclusion of non-alike things. You see what I’m saying here about it being a little tangential? That’s like arguing that the sky isn’t blue, it just has a predominance of electromagnetic radiation with a wavelength of (whatever the wavelength around “blue” is). It doesn’t really answer the question. It is, in fact, the question itself. Do opposites exist? Does context exist? Sure.
Prove you’re a formalist? Ah, the levels of humor embodied in the SDMB. 
Does it? This is what you would have to show. Certainly within any context we can define a relationship as “opposite”, but you will need to show that the new “opposite” shares one or more properties that warrant its inclusion in the same category as our previous (and all other) “opposite” relationships.
Why are we justified in calling additive inverses, magnetic poles, and matter/antimatter pairs all “opposite”?
I don’t see alikeness as having any direct relationship to context. Context is a frame of reference from which to observe one or more things, whether they be alike, different, or identical.
I’m afraid I don’t. Until we establish a characteristic of “oppositeness” which applies across contexts I see the a primary relationship between any particular “opposite” and the context in which it holds.
“Context is a frame of reference from which to observe one or more things, whether they be alike, different, or identical.”
Indeed. Context groups things by a certain quality…say, color.
For example, lets use the happy axiom of choice, which states that given any number of sets, each containing a finite number of elements, a new set can be created by taking one member of each of the previous sets.
However, I may point out, once any set is constructed we know what its contents are…or rather, what they are not.
In this way we may construct any new set from any existing sets, or if you prefer, make any new context with regards to existing contexts. Is it a stretch to say that from any context we may create a new context that contains oppositeness? Oppositeness need not even be defined in the “normal” way.
Continuing on the axiom of choice idea, we can create a whole array of new sets from one single set. In fact, the new set just made can then be used, with the original, to create a third, and so on until the original set itself has been reconstructed. This would happen in a finite number of steps, of course, which would be [# of elements in original set +1].
What we’ve done here is create a whole array of sets all containing elements of the original set. In this way, too, we may take any two objects and list all their properties as elements of a set.
Using these two latter sets, any new set we create compares the two. Indeed, we may come across some meaningless pairings, such as [length[sub]1[/sub]], color[sub]2[/sub]], but if we go through every permutation of sets using the axiom of choice on the original two sets alone we find that there are, also, a finite number of sets comparing what must be similar qualities. If these sets then contain multiples of the same element then, indeed, they are similar.
What I think is crucial here is that “all” the properties of any two particular objects will not have the same number of elements in them unless those two objects are the same object. that is, a blue ball and a red ball are not opposite since they would have corresponding elements, even though those elements may not be identical in magnitude.
Am I making one bit of sense here before I continue?
The Axiom of choice states that given any set (of sets) finite or not, we can construct a set containing one element from each element of the given set. If you throw finite in, you have the therom of choice.
You completely lost me in the rest of your post.
It seems to me that opposites exist only in context. If fact, existance only makes sense in context. If I say negative one is the opposite of one, it must be in some context that allows negative numbers to exist.
No. Context is simply the frame of reference. We can form groupings of things if we wish, and said grouping will depend upon our contextual base, but context itself does not require any such action.
Yes. Oppositeness is simply a relationship between elements. It is a relative measure, like “greater than” or “preceding” or “symmetrical”. I think, really, that a concrete example will help here. You say:
The point is: there is no normal way. Definitions of opposite are entirely dependent upon context. Think about the derivation you are trying to make. You wish to derive a context that contains what? think explicitely. How might you define this concept of oppositeness which you can derive from the set of all contexts? What will its relationship be to the “opposite” defined within each of the contexts from which you have exercised your choice?
It appears you are trying something close to this with your last few paragraphs, but I think you are taking a couple of wrong turns.
This presumes that the objects in question share similar qualities. You have made no such stipulation to our selection of sets/objects, though.
Not true. We might, for instance, have the integers +2 and -2 which differ in one property yet have the same “number of elements” in their “set of properties” since they differ only in sign.
Are you saying that objects cannot be opposite if they have corresponding elements? Or are you saying that objects with no corresponding elements are opposite? I see problems with both.
1) If you hold to your presumption above, that all comparisons must eventuall find a set of similar properties, then you have by definition eliminated the property of “oppositeness”. No two integers, for example, could be opposites of each other since they would share the common properties of all integers.
2) If you define opposite to mean “share no corresponding elements” you have the same problem. Additionally, if you escape the presumption that any two objects must share similar properties you find yourself defining a relationship of “opposite” between any pair of objects that have absolutely no other relationship to each other. “Rock”, for instance, might be the opposite of “tempo”, “anger” the opposite of “inclined plane”.
Well, the path is marked, but I don’t think it leads where you are trying to go.
Hmm. Think of this.
Set X = {a;b;c}
Set Y = {d;e;f}
Now define a relationship R such that a R b and d R e.
Left like this there isn’t a whole lot we can say about R. We can’t even say that b R a (imagine that R is >). Even suggesting that if a R b and b R c then a R c needs to be proved.
To go any further, I think that we need a good notation system and set theory seems to provide a pretty good foundation. Oppositeness is a relationship between two members of a set. Relationships can have any or all of three properties, which are (to remind those who need reminding)
a R b implies b R a means that R is symmetric.
a R b, b R c implies a R c means that R is transitive
And to complete the trinity, if x R x for all members of the set, R is reflexive.
A relationship that has all three properties, such as “=”, is an equivalence.
Now consider “oppositeness”. To me it seems intuitive that oppositeness is symmetric. If a is opposite to ~ a (written henceforth as a ^ ~a) then ~a ^ a. I can’t think of a quick way to prove this though.
It is also intuitive that ^ is not reflexive. An object cannot be its own opposite.
Transitiveness though is probably the most interesting of the three. Is ^ transitive? Does x having more than one opposite in a given context even make sense? And what is context in this notation?
I’d consider context to be the set itself. That is, in the example used at the beginning of this post, X is a context that makes sense of relationships between a, b and c whilst Y is a context that makes sense of relationships between d, e and f.
However in this sense we can combine sets and get, for example, Z = {a;b;d;e}, a context that makes sense of a, b, d and e.
Do relationships survive the crossover - if a^b in X then does a^b in Z?
aynrandlover - I think that you are assuming that the answer to the above question is yes when you invoke the theorem of choice. But I’m pretty sure that this assumption is not sound.
I’d say that the empty set represents a contextless situation. Note that we cannot define any relationship to be opposite within this. The set must have members before we can define the ^ relation.
Anybody want to go any further with this?
Yes, but what I’m trying to show is that one can always create a context, this seems to be the debatable point. As well, the context isn’t that negative numbers exist, its is in what we are calling opposite.
I was basically trying to get to: of all the sets created by taking elements of two other sets (which are the properties of the two objects) in every permutation one will create, as well as all the permutations of those sets, etc, until one has built the largest set without repeated elements from the original two sets (whew!) that everything intuitively that can be compared would be compared. This is the arbitrary context: it always exists (which was what to be shown, halfway). Now I need to find out if in between any two objects, in this general sense, there exists opposties. That is, can we always create a context wherein a given relationship holds.
:shrug:
spiritus
The definition of context I read seems to imply that both you and I are right as to what it means… you may use something like a “frame of reference” or “grouped by quality.” It is the interrelations of things.
You then go on to say that there is neither a “normal” way to define opposite nor can we create a context that contains an opposite. I disagree completely, since those two ideas together seem to make no sense (in any context ;)). If there is no normal way to define it, since it is, indeed, embedded in the idea of context itself (as a corallary, almost), then it seems to me that we can indeed always create a context that contains an opposite.
Putting it into a rigorously logical context seems pretty tough, but the result is damn intuitive to me.
Also, you might note that 2 and negative 2 do not mathematically have the same number of qualities. Negative two is not a natural number.
kabbes
the ^ operator, as defined here, is not provable. You may consider the additive inverse property in algebra, which guarantees (axiomatically) the existence of an additive inverse, such that
For all a there exists the additive inverse of a, written -a, such that a + -a = 0.
AFAIK this is never proved.
In rereading this response so far, I seem to get the feeling for something here, especially now that you, kabbes, have made things a little more visual for me. That is, we are merely defining a new operation.
It is intuitively, and provably, true that one can make a context such that all things are equal (mathematically, if one allows division by zero every number equals every other number).
It seems very intuitive that out of any context one can create a new context which allows for oppositeness. It would be akin to showing that, under some form of valuation, a != b. So long as a!=b, there is some context for oppositeness of the form A/~A, or that [quality of]a + [quality of]b = 0, where [quality of] is the new context.
That is, we are not just comparing color, but the brightness of the color, or the saturation of the color, etc…we keep the original context and enhance it.
Let us take a small example, comparing 2 to 3. In what way are they “opposite”? Let’s use A/~A oppositeness.
We compare 2 to 3 by creating the set (context) of some mathematical qualities (only some for brevity of typing).
2 : {even, prime, element of N,…}
3 : {odd, prime, element of N,…}
Let us create all combinations of sets. Somewhere in there we find {even, odd}.
Our original context was properties of individual numbers. We’ve now created a new context, parity in this case. We now want to find if ~even = odd for A/~A oppositeness.
Note we have not lost any original context, merely added more context on, or made the context more strict. The original context is still there. That is, the property of the property of the property of…these two numbers is opposite.
So, for ~even=odd to be true, every number must be either even or odd. An even number is defined by 2n, where n is a natural number. An odd number is defined as 2n+1, where n is a natural number.
Assume 2n=2n+1, that is, assume that there is some number n where the final number is both even and odd.
2n = 2n+1
-2n -2n
0 = 1
So, obviously there is no number n which results in a number which is both even and odd. I am not about to use the induction to show that every number is either even or odd, however, over N. I think we all know that.
So, we’ve created a context where 2 is opposite to 3 in a A/~A way.
It seems almost trivial that one may always do this by word length, mapping the alphabet onto the number system and adding, etc, to get to a situation similar to this. I lack the notation to write this out. Does anyone disagree, or is anyone disatisfied?
Am I talking out of my ass?
Granted. The distinction I have been trying to keep clear is that between reference frame and evaluation. Once you start grouping things, you have already begun evaluation.
I am afraid you have misread my post. I did say there is no “normal” way to define opposite. I did not say it was impossible for us to create a context that contains an opposite.
If you refuse to limit what relationships we label “ooposite”, it is trivial to create a context with an opposite relationship. It also serves no purpose. You must first define your relationship before you decide whether the relationship can be applied universaly to any context (or constructed universally from any set of contexts). Otherwise, as I did say, you are actually constructing an argument for the universality of definition.
You do seem, however, to be attempting something of the sort later in your post, albeit in an implied rather than explicit manner.
I have to ask, though, is it really so hard to say: I define the relationship opposite, regardless of context, to have the following properties . . . It might clear up any number of issues if you would/could do so.
Can we? Does it not depend on what you mean by opposite? Is it not a little bit important to know whether when you make that statement you are saying something about the existence of any relationship between elements (which we can label opposite no matter its properties) or a specific relationship betweem elements which will share those properties we have defined to be intrinsic to “oppositeness” no matter the context?
If you eliminate the null context and the solipsistic context (empty set and single element) I would agree to the first, though I don’t find it particularly interesting. The second might be an interesting idea, but it depends entirely on the qualities defined as “intrinsic”.
I have not seen a clear enough definition from you to decide whether I find the result intuitive. Regardless, while intuition is often a valuable tool for directing study, relying upon ignorance in the absence of supporting reason is just another mask for ignorance.
Well, I could raise some arguments about whether membership in a group hwose definition lies outside the context specified (the context of integers does not contain the descrimintaion “natural number?”). But . . .
Instead I will just note that “not a natural number” and “natural number” are each a quality, so the number of qualities is indeed idential.
some specifics
If so, and if we accept that opposite is not an equivalence relationship, then this invalidates your idea that we can find an opposite under any context.
Brightness and saturation seem poor metaphors for the relationship you end up arguing, which is simply an unambiguous bifurcation.
Actually, this would represent not an additional context (enhancement) but a reduction in context (minimalization). You have jetisoned all but one property of your original context.
Your test of ~even=odd tells you whether your binary test is unambiguous. Is it really your intention to argue that any unambiguous bifurcation represents an opposite relationship? The opposite of 2 is {±1,±3,±5,±7 . . .}? does oppositenes depend upon how we classify our options? A not-red ball is opposite a red ball, but a blue ball isn’t?
BTW – your odd even test fails your stated relationship [quality of]a + [quality of]b = 0 unless we define the “+” operation under odd/even to specifically grant that result. A more intuitive (for me) way of framing “+” under odd/even would look like a truth table for the XOR operation.
Well, this quite strongly implies that the definition for opposite you are working under is nothing more or less than an unambiguous dichotomy. This cerainly focuses the debate. As I see it, there are three issues left (assuming this definition was indeed your intent. You seem to have abandoned your earlier view that objects could not be opposite if they have corresponding elements. Perhaps this is simply a function of reducing our context to a single property.):
1) Is this relationship useful?
2) Does this relationship match intuitive expectaions of the word “opposite” closely enough to warrant giving it that label.
3) Can you demonstrate the claimed universality for this relationship across all contexts.
What is the sound of one poster ignoring a straight line?
Whew! Getting ugly now. Damn your girlfriend, kabbes!!!
I cannot see what you mean by reference frame that isn’t a quality in itself.
Blasphemy! Might as well of asked 2000 years’ worth of mathematicians to quit trying to prove Euclid’s parallel line postulate! 
The question was, can I prove that oppositeness exists in all contexts? I’m trying…haha.
I if you would like to make “not a natural number” a quality then my “proof” would be much simpler, wouldn’t you say? In that case oppositeness is clearly very easy; A/~A fashion as a car is a non bus, and so they (a car and a bus) must be opposites. In my case, two and three were not explicit opposites, one must construct the proper context to understand what exactly is opposite. But, my case was that from any one context one could create a context (now worded better) within that context to “create” an A/~A opposite. This was, I was hoping, to show that opposites exist regardless of context, not that context was disposable. In other words, I wanted to create the “not” as a result, not build it in to the set itself.
As for my every-number-equals-every-other-number world, that is a tricky one to place a context on. I need to think about that…
The idea I’ve got in my head is this:
0) Contexts exist, and are defined in many good dictionaries.
The first two statements are my axioms-of-context. The third statement is what I hoped to show. The result is trivial.
Hardly(but not completely). Objects are opposite in some way unless they are the same object. I have indeed abandoned the blue red distinction I made earlier, however, because given enough time I could reduce wavelengths of light to some dichotomy.
I guess it all comes back to the Ayn Rand debate we got to know each other in. I really believe that one can always reduce something to a dichotomy. I believe you had a problem with that, though I honestly don’t remember and I’ll burst into tears if I read that thread again, haha.
As well, I just referred to my original writings on it when dealing with them non-mathematically and I’ve said essentially the same thing: that all things can be reduced to a dichotomy. That dichotomy is an opposite-ness thingy.
I find myself having a hard time constructing arguemnts against you mainly because of my trepidation over the RToT thread, and its associated ideas. Every time I’m about to type something I always think, “Now, is he gonna say that I am not working within the system?”
http://dec59.ruk.cuni.cz/~peregrin/HTMLTxt/FRLogic.htm
I was looking for a paper called “Contingency of Relations” but it doesn’t appear to exist.
[sub]or: How many straight lines can one doper resist[/sub]
Fair enough. I can’t see where I said such a thing. A reference frame is not, however, a quality intrinsic to any object within the reference frame.
Nonsense. Closer to asking them to agree what Euclid’s postulate was before trying to prove it.
It depends upon which proof. You have changed, as I noted above, your requirements for “oppositeness”. the “number of qualities” argument is absolutely irrelevant to your later reductionist tack. As to defining our qualities, I cannot imagine an enumeration which would justify making inclusion in a set a quality but exclusion from a set not a quality.
Well, seting aside problems with terminology (which I tried to show you with the “blue” vs “non-red” illustration) this brings us back to your definition of opposite as dichotomous over any single quality. I asked a few questions about such a definition. I would appreciate it if you answered them.
How? By showing that a dichotomy can be found between any two objects which are not identical? “Fred is opposite all other things because none of them are Fred.” Why bother to label such a relationship “opposite”. We already have the term “different”.
On a strictly formal level, you forgot the step where you prove that the opposite relationship you define for your “more strict” context has application in the general context. Otherwise, you have demonstrated nothing about any contexts which contain more than a single quality.
On a more concrete level, I think your definition of opposite is seriously deficient.
Again, in what way is this relationship “opposite” of “different”.
[sub]long tangent about dichotomies delted. That is/was a different thread, and I don’t think we want/need to go there again.[/sub]
Sorry if I scared you on the RToT thread (You sure it wasn’t the Ayn Rand thread? I thought I was being nice on the RToT thread.) I just think that when we start spinning conjectures in the world of pure abstract we quickly lose our way if we aren’t careful in mapping our route. Clarity of expression aids clarity of thought.
Not even that strictly formal, Spiritus. I think that is a key point - one cannot just extrapolate a relationship (note - NOT an operation arl. ^ isn’t analogous to +, rather ^ is analogous to > or =).
Interestingly having just read my dictionary definition of “opposite”, I don’t think that they really know what it means either
How about this:
For property a and property b to be opposite (ie for a^b) in context X
Property a present only if property b not present
Property b present only if property a not present
er… thoughts? Clearly we need some way of indicating that a and b are “diametrically opposite” as my dictionary so quaintly and non-rigorously puts it. Some kind of annhilation property perhaps?
If we can define ^ satisfactorily maybe we can actually derive some properties of it.
pan
[QUOTE]
*Originally posted by Spiritus Mundi *
No no, 2 is not opposite -1, opposite doesn’t apply. There is some quality of two that is opposite of the corresponding quality of negative one. You brought context into it, now you want to remove it?
Yeah…or rather, the opposite test. That is what XOR is for. XNOR would be just as good when properly interpreted.
I feel it is. Contrasting elements is a useful property. I don’t know what you expect from me here. It isn’t always worthwhile, I guess I could say, because of the levels of context that must be piled on to obtain a clear opposite.
I would say so. That’s what I mean by opposite in a strict sense. Even in a weak sense, where I say “that’s the opposite of what you said” or some such…there clearly isn’t a mutually annihilating sentence, but there are elements/properties of each sentence that are diametrically opposite. Honestly, I think this is what everyone means when they say “opposite.”
Can I guarantee the existence of an opposite between any two objects in any context? I think I addressed that one: no, I don’t think I can, but I’m still trying anyway.
What worries me is what sort of “contexts” you are considering, like in the RToT thread where you said, “I can imagine an epistemology without logic” or the other epistemic monsters. On one hand, it is good that you challenge the meaning behind what I say; on the other, you fade out of your quality “2” above: the intuitiveness of the whole concept gets lost.
Well, I can see that this might bother people. :shrug: I was actually thinking harder about this one, and there were problems I came to in this, that is, assuming “not a member of N” is not a quality.
I would wonder how useful this is, as well, but I feel like there were some proofs I used to know where “not a {blank} and thus…” were used, though I cannot think of any off the top of my head.
As well, “is not a…” seems more useful once opposites have been accepted. That is, we want something specific to occur. We know a specific part(the ram) of A(swat team vehicle) causes this. We come across B. Does B have a corresponding specific part? Using this specific part as the context, are they different? What makes them different? Then we {can} form more strict contexts around this specific part until we find that a crucial quality (say, tensile strength) is opposite, or we may even have to go further still, finding only more “different” things.
Indeed, buried in “different” is “opposite.” I don’t doubt that there are times when “differentness” is all that is necessary to form a conclusion.
The problem with XOR is that the inputs can only have two values; the dichotomy is built into it. “different” and “opposite” have the same connotation (unless we consider what in electronics is called a high-impedance state, but I don’t think so).
If you use “not a member of” or “not possessing the quality of” this becomes transparent that in any two objects there exists a contextual dichotomy in comparing the two. This is how “different” becomes “opposite” and thus, as you hint, becomes useless (I don’t think it becomes useless, just too broad).
Opposite may also become useless, again, due to the level of context we must stoop to to put our finger on a dichotomy. That is, comparing the red ball to the blue ball. They are different in color, and far enough down the line one can construct (though you don’t seem to think so) a dichotomy, though–again–that might not be necessary.
kabbes
I would feel that “+” is a relationship of sorts, too, but perhaps that is stretching it.
Your 1 and 2 are basically directions for constructing an XOR gate: if the inputs are of different truth value, the result is true. Interpreted for us, if the two qualities are mutually exclusive/dichotomous, they are opposite.
So what we need to construct is an operator, like XOR. The relationship is the “= true” or “is opposite”, the operator is the “opposite test”.
No? What we would have is A ^ ~A = 1(true, whatever).
To make it a relation, such as A^~A, we need to figure out how to operate around it in a way that an “=” or a “<” may be worked around/across, etc.
Again, I feel like we are stating the logical eqivalent of the additive inverse axiom.