ebola?
Bluelight Crew
Here is another pertinent essay, also perhaps a rough draft... 
On Royce, Russel, Contradiction, and Experience
According to Royce, the system of logic emergent in an agent’s process of inquiry is based on a contradiction, a contradiction introduced immediately by exclusive disjunction. Why does this contradiction arise and what does it mean? Royce’s contradiction appears when we attempt to place the actions of the reasoning agent within our system of logic in an attempt to make our system adequate. This contradiction is not a problem, for example, if we adopt a nominalist ontology where the classifying actions of the rational agent are not included within the system, but such a system is inadequate as it does not capture all of the present activity. In this way, Royce’s contradiction is a problem of self-reference. Russell found a similar contradiction when investigating self-reference and set-theory, but chose to set aside the contradiction, incorporating additional axioms into his system in order to avoid such contradictions. For Royce and his pragmatist ontology, however, this fundamental contradiction of logic has meaning. The contradiction shows us that the agent’s logical system is not adequate; it cannot capture the totality of the of the agent-environment interaction at hand. The contradiction also shows us that the discrete delineations emergent from logical action do not capture the continuum of reality in its fullness. Rather, the continuum is constrained by our choices, but always presents an implicit context for future conceptual distinctions and an indefinite field of future possibilities.
In order to understand the implication’s of Royce’s contradiction, we must first understand what the contradiction is and where it appears. Imagine, for a second, an agent examining an object, object “p”. By designating this object object “p”, the agent has in effect split his universe in two. Half of her world is classified as “p”. The other half of her world is classified as “not-p”. Thus, when classifying a newly encountered object, the agent may classify it as either p or not-p, but not both, symbolized, “p v not-p”. Alternately, exclusive disjunction can be viewed in terms of Royce’s philosophy of logic where modes of action are the fundamental relation among particulars, not static classifications, and we begin from the existence of a choosing agent. Here, we could think of the agent as making a choice. Either she could do p, or she could do something else, something that is not p, but she could not do both. Regardless of whether we think in terms of classes or actions, though, we can see immediately that exclusive disjunction, p v not-p, emerges the second we distinguish p from the rest of the universe of discourse.
Given the deliniations we have made, we could symbolize the universe of discourse here with a circle split down the middle, one side labeled p, the other labeled not-p. The question arises, however, as to what, exactly, this line on our diagram is. We could argue that this line represents the action of our classifying agent. She has deliniated between the objects before her in terms of p and not-p, but this deliniation itself, the split the agent has chosen to make, is part of the agent, her activities, and is not part of the universe she is attempting to classify. The line on the diagram symbolizes a choice the agent made and not part of our miniature logical universe. To argue along these lines is to argue for a nominalist ontology where there is strict dualism between the agent and the external universe, the objective world, and where distinctions are made merely according to the agent’s whims. This sort of ontology is inadequate because it does not include the agent or her actions and is inaccurate because it does not account for the universe’s role in shaping the agent’s actions; we cannot maintain this sort of dualism.
What we need to do, then, is include the agent and her actions in our ontology. In turn, we need to include the division between p and not-p in our logical universe as an item in that universe. What, then, is the nature of the border between p and not-p? It is not p. Nor is it p. In fact, it is not p or not-p. We can then symbolize the division between p and not-p as not-(p v not-p). We have a problem though. This statement is contradictory. If we apply DeMorgan’s equivalence to our statement, we have p and not p, p . not-p, an obviously contradictory statement. We can see, then, that we introduced this contradiction into our logical universe by introducing the actions of the reasoning agent into that universe. The problem we created is a problem of self-reference. The self-referential structure of the reasoning agent’s description of her own actions is what creates this problem.
Russell, investigating set theory, found a similar logical contradiction involving self-reference in logic. He details this problem in the Introduction to Mathematical Philosophy among other places, speaking in terms of classes which contain sub-classes. He asks the reader to consider the class of all classes that are not members of themselves (Russell, 136). Then he asks the reader to decide, is the previously mentioned class a member of itself or not? Let us assume, for a second, that this class is a member of itself. If this class is not a member of itself, then, since it is the class of all classes that are not members of themselves, it should then be a member of itself. This is obviously contradictory, so by reductio ad absurdum, the opposite of our assumption should be true, so this class should be a member of itself. So let us assume, then, that this class is a member of itself. If this class is a member of itself, then, since this class is the class of all classes that are not members of themselves, this class should not be a member of itself. This, too, leads to an obvious contradiction. We can conclude, then, that the class of all classes that are not members of themselves presents an immediate contradiction. In much the same way as Royce’s exclusive disjunctive contradiction stems from the self-reference in the action of disjunction, the contradiction in Russell’s paradox also stems from its self-reference.
How, then, should we treat these paradoxes? Russell chose to set his paradox aside in an attempt to avoid such contradictory paradoxes. Russell argued that “classes are a logical fiction”, so to speak of a class that cannot be reduced to a primitive logical type is, although not strictly false, to speak nonsense, is to fail to say anything of meaning (Russell, 137). On this view, classes are not logical objects. In order to avoid his paradox of self-reference and avoid speaking of classes as logical objets, Russell built a modified logical system with an inherent hierarchy of logical types which would prevent self-reference of the sort that would cause inherent contradictions in the system. In creating a logical system free of paradoxes, Russell rendered his logical system inadequate. That is, because Russell’s system cannot manipulate classes as logical objects, classes of the sort that are emergent in our every-day inquiries, Russell’s logical system is limited and cannot describe our logical actions as rational choosers.
Royce, too, noticed this limitation of Russell’s logical system. Royce writes, “Given the ‘logical constants,’ Mr. Russell regards the order-systems as creatures of definition; although, from his point of view, definition also appears to be a process by which one reports the existence, in the logician’s realm, of certain beings, namely, classes relation, series, orders. . .” (Royce, 371). In other words, Royce argues that the meaning of statements and their logical types and the relations among them in Russell’s system is given by the manner in which these statements and types are defined; Russell’s system is based on analytic truths. The question arises, however, as to what these truths, these logical relations in Russell’s system, mean to us in our experience. Do these statements actually relate to our experience or do all chains of analytic inferences arrive merely at arbitrary definitions? By creating a hierarchy of types and barring self-reference from his system, Russell has greatly limited the ways in which his system is applicable to our experience, rendering his system inadequate. Although this is not strictly a problem of analyticity, Russell has still fallen into the trap of anchoring his system on arbitrary definitional relations while failing to check whether he has given an adequate description of our experience as rational choosers.
The question then arises as to what paradoxes such as Russell’s paradox and the contradiction immediately introduced by disjunction in Royce’s system mean to us. We have seen that attempts to eliminate these paradoxes halt rather than further our logical inquiry, so we must come to grips with them. These paradoxes show but do not tell us in propositional form that logic, as a tool, can take slices of the continuum of experience and describe and manipulate those slices but cannot capture the whole of experience. If we attempt to capture the totality of experience within our logical descriptions thereof, including the rational agent within our logical description, our system falls into contradiction which belies its inability to do so.
This shortcoming of logic is also reflected in our introspective experience of ourselves as subjects and the external world as our objects. In the self-othering relationship between subject and object which constitutes our experience, we can also reflect on this self-othering process in an attempt to capture our experience. In doing so, however, there appears a part of the subject that is meta-aware, looking down upon the self-othering process but not captured by our description of it. In a further attempt to capture our experience, we can include this meta-aware part of ourselves, looking upon the self-othering process, in our descriptions of our experience of self-othering. The problem here, though, is that in doing so, a meta-meta-aware part of ourselves appears in the examination of our meta-awareness and the self-othering process. This part is not captured by our description of our experience. This process can continue ad infinitum, but we will never wholly capture our experience and ourselves.
Royce’s contradiction of disjunction also shows our experience as underdetermined by our choices and descriptions of it, always presenting an indeterminate number of future possibilities in spite of the constraints placed thereon by the choices we have already made. Recall, again, the disjunctive contradiction. Because our logical description of this disjunction is contradictory, this shows the failure of our logical distinctions to wholly capture the continuum of experience before us. Thus, even though certain distinctions we have made render parts of our experience determinate and certain choices we have made constrain future options open to us, the continuum of experience still serves as a context for future choices. Not having been captured by our description of it, the continuum of experience continues to be ripe with an indeterminate number of possibilities. For example, even if we have chosen join a rural commune, this choice having constrained a number of future possibilities, there is still an indeterminate number of future possibilities open to us. Although we could not get a job as an investment banker, we can still choose whether to farm, build houses, pick flowers, run wildly, our mouths agape and screaming, murder our house-mates, etc.
This notion of the field of experience as presenting an indeterminate number of possibilities for choice and distinctions is further exhibited by the boundless number of logical borders we can create from the fundamental disjunctive division. Recall our continuum of experience divided into p and not-p, its border symbolized as not-(p v not-p) or simply p . not-p. We can see that additional divisions can be made between p and p . not-p or between not-p and p . not-p. These divisions can each be symbolized as not-(p v (p . not-p)), i.e. p . not-(p . not-p), and not-(not-p v (p . not-p), i.e. not-p . (p . not-p), respectively. These secondary borders too are contradictory, again displaying our logical system’s inability to capture the whole of experience. Furthermore, we see that we could also make tertiary borders between our secondary borders and the first border or the secondary borders and p or not-p, quaternary borders, fifth-order borders, and so on. There is an infinite regress of borders we can make. What this shows us is that, even given the constraints previous distinctions and previous choices made place on our experience, there are still an indeterminate number of possibilities for choices and future distinctions open to us.
What, then, do all these contradictions show us about the socially situated organism-environment interaction that constitutes our world? Firstly, they provide evidence that logic is a tool emergent in this interaction, used to describe and manipulate the interaction, but not exhaustively or completely determinately so. Consequently, even with the emergence of logic, our organism-environment interaction will continue as an on-going process. The continuum of experience will never be exhausted by our use of logic; new distinctions will emerge and new possibilities will continue to present themselves. These above patterns characteristic of our experience close the door on the modern quest for complete certainty because experience cannot be captured completely and will continue to present surprises. We will not do well to sit and lament, however, because logic presents us with resources to shape the possibilities before us, allowing us to attain our ends in view in novel and adaptively appropriate ways, allowing us to succeed.
On Royce, Russel, Contradiction, and Experience
According to Royce, the system of logic emergent in an agent’s process of inquiry is based on a contradiction, a contradiction introduced immediately by exclusive disjunction. Why does this contradiction arise and what does it mean? Royce’s contradiction appears when we attempt to place the actions of the reasoning agent within our system of logic in an attempt to make our system adequate. This contradiction is not a problem, for example, if we adopt a nominalist ontology where the classifying actions of the rational agent are not included within the system, but such a system is inadequate as it does not capture all of the present activity. In this way, Royce’s contradiction is a problem of self-reference. Russell found a similar contradiction when investigating self-reference and set-theory, but chose to set aside the contradiction, incorporating additional axioms into his system in order to avoid such contradictions. For Royce and his pragmatist ontology, however, this fundamental contradiction of logic has meaning. The contradiction shows us that the agent’s logical system is not adequate; it cannot capture the totality of the of the agent-environment interaction at hand. The contradiction also shows us that the discrete delineations emergent from logical action do not capture the continuum of reality in its fullness. Rather, the continuum is constrained by our choices, but always presents an implicit context for future conceptual distinctions and an indefinite field of future possibilities.
In order to understand the implication’s of Royce’s contradiction, we must first understand what the contradiction is and where it appears. Imagine, for a second, an agent examining an object, object “p”. By designating this object object “p”, the agent has in effect split his universe in two. Half of her world is classified as “p”. The other half of her world is classified as “not-p”. Thus, when classifying a newly encountered object, the agent may classify it as either p or not-p, but not both, symbolized, “p v not-p”. Alternately, exclusive disjunction can be viewed in terms of Royce’s philosophy of logic where modes of action are the fundamental relation among particulars, not static classifications, and we begin from the existence of a choosing agent. Here, we could think of the agent as making a choice. Either she could do p, or she could do something else, something that is not p, but she could not do both. Regardless of whether we think in terms of classes or actions, though, we can see immediately that exclusive disjunction, p v not-p, emerges the second we distinguish p from the rest of the universe of discourse.
Given the deliniations we have made, we could symbolize the universe of discourse here with a circle split down the middle, one side labeled p, the other labeled not-p. The question arises, however, as to what, exactly, this line on our diagram is. We could argue that this line represents the action of our classifying agent. She has deliniated between the objects before her in terms of p and not-p, but this deliniation itself, the split the agent has chosen to make, is part of the agent, her activities, and is not part of the universe she is attempting to classify. The line on the diagram symbolizes a choice the agent made and not part of our miniature logical universe. To argue along these lines is to argue for a nominalist ontology where there is strict dualism between the agent and the external universe, the objective world, and where distinctions are made merely according to the agent’s whims. This sort of ontology is inadequate because it does not include the agent or her actions and is inaccurate because it does not account for the universe’s role in shaping the agent’s actions; we cannot maintain this sort of dualism.
What we need to do, then, is include the agent and her actions in our ontology. In turn, we need to include the division between p and not-p in our logical universe as an item in that universe. What, then, is the nature of the border between p and not-p? It is not p. Nor is it p. In fact, it is not p or not-p. We can then symbolize the division between p and not-p as not-(p v not-p). We have a problem though. This statement is contradictory. If we apply DeMorgan’s equivalence to our statement, we have p and not p, p . not-p, an obviously contradictory statement. We can see, then, that we introduced this contradiction into our logical universe by introducing the actions of the reasoning agent into that universe. The problem we created is a problem of self-reference. The self-referential structure of the reasoning agent’s description of her own actions is what creates this problem.
Russell, investigating set theory, found a similar logical contradiction involving self-reference in logic. He details this problem in the Introduction to Mathematical Philosophy among other places, speaking in terms of classes which contain sub-classes. He asks the reader to consider the class of all classes that are not members of themselves (Russell, 136). Then he asks the reader to decide, is the previously mentioned class a member of itself or not? Let us assume, for a second, that this class is a member of itself. If this class is not a member of itself, then, since it is the class of all classes that are not members of themselves, it should then be a member of itself. This is obviously contradictory, so by reductio ad absurdum, the opposite of our assumption should be true, so this class should be a member of itself. So let us assume, then, that this class is a member of itself. If this class is a member of itself, then, since this class is the class of all classes that are not members of themselves, this class should not be a member of itself. This, too, leads to an obvious contradiction. We can conclude, then, that the class of all classes that are not members of themselves presents an immediate contradiction. In much the same way as Royce’s exclusive disjunctive contradiction stems from the self-reference in the action of disjunction, the contradiction in Russell’s paradox also stems from its self-reference.
How, then, should we treat these paradoxes? Russell chose to set his paradox aside in an attempt to avoid such contradictory paradoxes. Russell argued that “classes are a logical fiction”, so to speak of a class that cannot be reduced to a primitive logical type is, although not strictly false, to speak nonsense, is to fail to say anything of meaning (Russell, 137). On this view, classes are not logical objects. In order to avoid his paradox of self-reference and avoid speaking of classes as logical objets, Russell built a modified logical system with an inherent hierarchy of logical types which would prevent self-reference of the sort that would cause inherent contradictions in the system. In creating a logical system free of paradoxes, Russell rendered his logical system inadequate. That is, because Russell’s system cannot manipulate classes as logical objects, classes of the sort that are emergent in our every-day inquiries, Russell’s logical system is limited and cannot describe our logical actions as rational choosers.
Royce, too, noticed this limitation of Russell’s logical system. Royce writes, “Given the ‘logical constants,’ Mr. Russell regards the order-systems as creatures of definition; although, from his point of view, definition also appears to be a process by which one reports the existence, in the logician’s realm, of certain beings, namely, classes relation, series, orders. . .” (Royce, 371). In other words, Royce argues that the meaning of statements and their logical types and the relations among them in Russell’s system is given by the manner in which these statements and types are defined; Russell’s system is based on analytic truths. The question arises, however, as to what these truths, these logical relations in Russell’s system, mean to us in our experience. Do these statements actually relate to our experience or do all chains of analytic inferences arrive merely at arbitrary definitions? By creating a hierarchy of types and barring self-reference from his system, Russell has greatly limited the ways in which his system is applicable to our experience, rendering his system inadequate. Although this is not strictly a problem of analyticity, Russell has still fallen into the trap of anchoring his system on arbitrary definitional relations while failing to check whether he has given an adequate description of our experience as rational choosers.
The question then arises as to what paradoxes such as Russell’s paradox and the contradiction immediately introduced by disjunction in Royce’s system mean to us. We have seen that attempts to eliminate these paradoxes halt rather than further our logical inquiry, so we must come to grips with them. These paradoxes show but do not tell us in propositional form that logic, as a tool, can take slices of the continuum of experience and describe and manipulate those slices but cannot capture the whole of experience. If we attempt to capture the totality of experience within our logical descriptions thereof, including the rational agent within our logical description, our system falls into contradiction which belies its inability to do so.
This shortcoming of logic is also reflected in our introspective experience of ourselves as subjects and the external world as our objects. In the self-othering relationship between subject and object which constitutes our experience, we can also reflect on this self-othering process in an attempt to capture our experience. In doing so, however, there appears a part of the subject that is meta-aware, looking down upon the self-othering process but not captured by our description of it. In a further attempt to capture our experience, we can include this meta-aware part of ourselves, looking upon the self-othering process, in our descriptions of our experience of self-othering. The problem here, though, is that in doing so, a meta-meta-aware part of ourselves appears in the examination of our meta-awareness and the self-othering process. This part is not captured by our description of our experience. This process can continue ad infinitum, but we will never wholly capture our experience and ourselves.
Royce’s contradiction of disjunction also shows our experience as underdetermined by our choices and descriptions of it, always presenting an indeterminate number of future possibilities in spite of the constraints placed thereon by the choices we have already made. Recall, again, the disjunctive contradiction. Because our logical description of this disjunction is contradictory, this shows the failure of our logical distinctions to wholly capture the continuum of experience before us. Thus, even though certain distinctions we have made render parts of our experience determinate and certain choices we have made constrain future options open to us, the continuum of experience still serves as a context for future choices. Not having been captured by our description of it, the continuum of experience continues to be ripe with an indeterminate number of possibilities. For example, even if we have chosen join a rural commune, this choice having constrained a number of future possibilities, there is still an indeterminate number of future possibilities open to us. Although we could not get a job as an investment banker, we can still choose whether to farm, build houses, pick flowers, run wildly, our mouths agape and screaming, murder our house-mates, etc.
This notion of the field of experience as presenting an indeterminate number of possibilities for choice and distinctions is further exhibited by the boundless number of logical borders we can create from the fundamental disjunctive division. Recall our continuum of experience divided into p and not-p, its border symbolized as not-(p v not-p) or simply p . not-p. We can see that additional divisions can be made between p and p . not-p or between not-p and p . not-p. These divisions can each be symbolized as not-(p v (p . not-p)), i.e. p . not-(p . not-p), and not-(not-p v (p . not-p), i.e. not-p . (p . not-p), respectively. These secondary borders too are contradictory, again displaying our logical system’s inability to capture the whole of experience. Furthermore, we see that we could also make tertiary borders between our secondary borders and the first border or the secondary borders and p or not-p, quaternary borders, fifth-order borders, and so on. There is an infinite regress of borders we can make. What this shows us is that, even given the constraints previous distinctions and previous choices made place on our experience, there are still an indeterminate number of possibilities for choices and future distinctions open to us.
What, then, do all these contradictions show us about the socially situated organism-environment interaction that constitutes our world? Firstly, they provide evidence that logic is a tool emergent in this interaction, used to describe and manipulate the interaction, but not exhaustively or completely determinately so. Consequently, even with the emergence of logic, our organism-environment interaction will continue as an on-going process. The continuum of experience will never be exhausted by our use of logic; new distinctions will emerge and new possibilities will continue to present themselves. These above patterns characteristic of our experience close the door on the modern quest for complete certainty because experience cannot be captured completely and will continue to present surprises. We will not do well to sit and lament, however, because logic presents us with resources to shape the possibilities before us, allowing us to attain our ends in view in novel and adaptively appropriate ways, allowing us to succeed.
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![:\](https://bluelight.org/xf/BL_Images/Orig_Emoji/wrygrin.gif "wry grin :\")
As I've said before, either produce some evidence or shutup, because until you do, you're what is commonly known as "a crank". (Thats the second time in 3 days I've called someone that, the other guy beleives he's disproved the Fermats Last Theorem proof because "You can't find the largest cube". You and he should get together, your logical reasoning is about on par!). Those "white papers" you mention in your journal I look forward to seeing, I could do with a chuckle.
That's it though. 