Changes

To express the nature of this integration task in logical terms, it combines elements of both proof theory and model theory, interweaving:  (1) A phase that develops theories about the symbolic competence or ''knowledge'' of intelligent agents, using abstract formal systems to represent the theories and phenomenological data to constrain them;  (2) A phase that seeks concrete models of these theories, looking to the kinds of mathematical structure that have a dynamic or system-theoretic interpretation, and compiling the constraints that a recursive conceptual analysis imposes on the ultimate elements of their construction.

To express the nature of this integration task in logical terms, it combines elements of both proof theory and model theory, interweaving:  (1) A phase that develops theories about the symbolic competence or ''knowledge'' of intelligent agents, using abstract formal systems to represent the theories and phenomenological data to constrain them;  (2) A phase that seeks concrete models of these theories, looking to the kinds of mathematical structure that have a dynamic or system-theoretic interpretation, and compiling the constraints that a recursive conceptual analysis imposes on the ultimate elements of their construction.

The set of sign relations {''A'', ''B''} is an example of an extremely simple formal system, encapsulating aspects of the symbolic competence and the pragmatic performance that might be exhibited by potentially intelligent interpretive agents, however abstractly and partially given at this stage of description.  The symbols of a formal system like {''A'', ''B''} can be held subject to abstract constraints, having their meanings in relation to each other determined by definitions and axioms (for example, the laws defining an equivalence relation), making it possible to manipulate the resulting information by means of the inference rules in a proof system.  This illustrates the ''proof-theoretic'' aspect of a symbol system.

The set of sign relations <math>\{ L_\text{A}, L_\text{B} \}</math> is an example of an extremely simple formal system, encapsulating aspects of the symbolic competence and the pragmatic performance that might be exhibited by potentially intelligent interpretive agents, however abstractly and partially given at this stage of description.  The symbols of a formal system like <math>\{ L_\text{A}, L_\text{B} \}</math> can be held subject to abstract constraints, having their meanings in relation to each other determined by definitions and axioms (for example, the laws defining an equivalence relation), making it possible to manipulate the resulting information by means of the inference rules in a proof system.  This illustrates the ''proof-theoretic'' aspect of a symbol system.

Suppose that a formal system like {''A'',&nbsp;''B''} is initially approached from a theoretical direction, in other words, by listing the abstract properties one thinks it ought to have.  Then the existence of an extensional model that satisfies these constraints, as exhibited by the sign relation tables, demonstrates that one's theoretical description is logically consistent, even if the models that first come to mind are still a bit too abstractly symbolic and do not have all the dynamic concreteness that is demanded of system-theoretic interpretations.  This amounts to the other side of the ledger, the ''model-theoretic'' aspect of a symbol system, at least insofar as the present account has dealt with it.

Suppose that a formal system like <math>\{ L_\text{A}, L_\text{B} \}</math> is initially approached from a theoretical direction, in other words, by listing the abstract properties one thinks it ought to have.  Then the existence of an extensional model that satisfies these constraints, as exhibited by the sign relation tables, demonstrates that one's theoretical description is logically consistent, even if the models that first come to mind are still a bit too abstractly symbolic and do not have all the dynamic concreteness that is demanded of system-theoretic interpretations.  This amounts to the other side of the ledger, the ''model-theoretic'' aspect of a symbol system, at least insofar as the present account has dealt with it.

More is required of the modeler, however, in order to find the desired kinds of system-theoretic models (for example, state transition systems), and this brings the search for realizations of formal systems down to the toughest part of the exercise.  Some of the problems that emerge were highlighted in the example of ''A'' and ''B''.  Although it is ordinarily possible to construct state transition systems in which the states of interpreters correspond relatively directly to the acceptations of the primitive signs given, the conflict of interpretations that develops between different interpreters from these prima facie implementations is a sign that there is something superficial about this approach.

More is required of the modeler, however, in order to find the desired kinds of system-theoretic models (for example, state transition systems), and this brings the search for realizations of formal systems down to the toughest part of the exercise.  Some of the problems that emerge were highlighted in the example of <math>\text{A}</math> and <math>\text{B}</math>.  Although it is ordinarily possible to construct state transition systems in which the states of interpreters correspond relatively directly to the acceptations of the primitive signs given, the conflict of interpretations that develops between different interpreters from these prima facie implementations is a sign that there is something superficial about this approach.

The integration of model-theoretic and proof-theoretic aspects of ''physical symbol systems'', besides being closely analogous to the integration of denotative and connotative aspects of sign relations, is also relevant to the job of integrating dynamic and symbolic frameworks for intelligent systems.  This is so because the search for dynamic realizations of symbol systems is only a more pointed exercise in model theory, where the mathematical materials made available for modeling are further constrained by system-theoretic principles, like being able to say what the states are and how the transitions are determined.

The integration of model-theoretic and proof-theoretic aspects of ''physical symbol systems'', besides being closely analogous to the integration of denotative and connotative aspects of sign relations, is also relevant to the job of integrating dynamic and symbolic frameworks for intelligent systems.  This is so because the search for dynamic realizations of symbol systems is only a more pointed exercise in model theory, where the mathematical materials made available for modeling are further constrained by system-theoretic principles, like being able to say what the states are and how the transitions are determined.