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===6.12. Issue 1. The Status of Signs===
===6.12. Issue 1. The Status of Signs===
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This section considers an issue that affects the status of signs and their mode of significance, as it appears under each of the three NOS's. The concerns that arise with respect to this issue can be divided into two sets of questions. The first type of question has to do with the default assumptions that are made about the meanings of signs and the strategies that are used to deal with signs that fail to have meanings. The second type of question has to do with higher order signs, or signs that involve signs among their objects.
Only certain types of signs are able to make their appearance in a given medium or a particular style of text, while many others are not. But a sign is a sign by virtue of the fact that it is interpreted as a sign, and thus plays the role of a sign in a sign relation, and not of necessity because it has any special construction other than that of being construed as a sign.
The theory of formal languages, as pursued under the FL perspective, is closely related to the theory of semigroups, as pursued under the IL perspective, in the sense that arbitrary formal languages can be studied as subsets of the semigroups that embody the primitive concatenation of linguistic symbols within their algebraic laws of composition. Thus, in staging any discussion of formal languages, the theory of semigroups is often taken for a neutral, indifferent, or undifferentiated background, but the wisdom of using this setting is contingent on understanding the distinct outlooks of the casual and formal NOS's. What divides the two styles and their favorite subjects in practice is a certain difference in attitude toward the status and role of their subject materials. Namely, it turns on the question of whether their primitive and derived elements are valued as terminal objects in and of themselves or whether these syntactic objects and constructions are interpreted as mere signs and sundry expressions whose true value lies elsewhere.
In taking up the IL attitude toward any mathematical system, semigroups in particular, one assumes that signs are available for denoting a class of formal objects, but the issue of how these notational matters come to be constellated is considered to be peripheral, lacking in a substantive weight of concern and enjoying a purely marginal interest.
In the discussion of formal languages the presumption of significance is shifted in the opposite direction. Signs are presumed to be innocent of meaning until it can be demonstrated otherwise. One begins with a set of primitive objects, formally called "signs", but treated as meaningless tokens or as objects that are bare of all extraneous semantic trappings. From these simplest signs, a law of composition allows the construction of complex expressions in regular ways, but other than that anything goes, at least, at first.
A first cut taken in the space of expressions divides them into two classes: (a) the "grammatical", "well formed" or "meaningful, maybe", versus (b) the "ungrammatical", "ill formed" or "meaningless, for sure". This first bit of semantic information is usually regarded as marking a purely syntactic distinction. Typically one seeks a recursive function that computes this bit of meaningfulness as a property of its argument and thereby decides (or semi decides) whether an arbitrary expression ("string", "strand", "sequence") constitutes an expressive expression ("word", "sentence", "message"), or not. The means of computation is often presented in the form of various "grammars" or "automata" that can serve as "acceptors or "generators" for the language.
Depending on one's school of thought, the syntactic bit of computation for interesting cases of natural languages is thought to be either (1) formally independent of all the more properly semantic features, or (2) heavily reliant on the construal of further bits of meaning to make its decision. Accordingly, the semantics proper for such a language ought to begin either (1) serially after or (2) concurrently while the syntactic bit is done. The first standpoint is usually described as a "declaration of syntactic independence", while the second opinion is often called a "semantic bootstrapping hypothesis".
Over and above both of these positions the pragmatic theory of signs poses a stronger thesis of irreducibility or non independence that one might call a "pragmatic bootstrapping hypothesis". Even though it is a more complex task initially to work with triadic relations themselves instead of their dyadic projections, this hypothesis suggests that the structural integrity of interesting natural languages, when taken over the long haul, may well depend on them. One part of this thesis is not a hypothesis but a fact. There do indeed exist triadic relations that cannot be reconstructed uniquely from their dyadic projections, and thus are called "irreducibly triadic" (IT). The parts of the thesis that are hypothetical, and that need to be cleared up by empirical inquiry, suggest that many of the most important sign relations are IT, and that interesting cases of natural languages depend heavily on these kinds of sign relations for their salient properties, for example, their relevance and adaptability to the objective world, their structural integrity and internal coherence, and their learnability by human agents and other species of finitely informed creatures.
In practice, this question has little consequence for the present study, on account of the extremely simple and artificial kinds of languages that are needed to carry out its aims. If some reason develops to emulate the properties of interesting natural languages in this microcosm, then a decision about which strategy to use can be made at that time. For now it seems worthwhile to keep exploring all of the above options.
In a FL context one begins with the imposition of general inhibition against the notion that a specific class of signs has any meaning at all, or at least, that its elements have the meanings one is accustomed to think they do. It is significant that one does not proscribe all signs from having meaning, or else there is no point in having a discussion, and no point from which to carry on a discussion of anything at all. Therefore, the arena of formal discussion is a limited one and, except for the occasional resonance that its action induces in the surrounding discursive universe, most of the signs outside its bounds continue to be used in the habitual ways.
What can be done with the signs in question? Apparently, signs viewed as objects in the formal arena, temporarily cut off from their usual associations, treated as terminal values in themselves, and put under review to suggest explanations for themselves, can still be discussed. Doing this involves the use of other signs for denoting the signs in question. These extra signs, whose sense and use are not in question at the moment in question, are called into play as "higher order" (HO) signs, and it is their very meaningfulness and effectiveness that one must rely on to carry out the investigation of the "lower order" (LO) signs that are in question.
An apt and proper discussion of a set of signs in question requires the ability to classify the tokens of these signs according to their types. Doing this calls on the use of other HO signs to denote these "tokens", the transient instances of signs, and their "types", the propertied classes of tokens that correspond to what is typically valued as a sign. The invocation of HO signs can be iterated in a succession of HO's that extends as far as one pleases, but no matter how much of this order is progressively formalized one eventually must resort to signs of such a high order that they are taken for granted as resting, for the moment, in an informal context of interpretation.
What is the sense and use of such a proceeding? Evidently, the signs in question, as a class, must present the inquirer with phenomena that are somehow simpler than, and yet convey instructive information about, the phenomenon known as the "whole objective world" (WOW). If their orders of complexity and perplexity are just as great as the world at large, then their investigation affords no advantage over the general empirical problem of trying to account for the WOW. If they enjoy no informative connection with the greater wonders of why the world is the way it is, and therefore fail to present a significant representation of the original question, then their isolated inquiry can serve no larger purpose in the world.
In situations like the one just described, where functions and relations on one order of arguments are clarified, defined, or explained in terms of functions and relations on another order of arguments, it is natural to understand the effort at clarification, definition, or explanation as a recursive process. What raises the potential for confusion in the given arrangement of formal and casual contexts is the circumstance that what seems natural to call the LO arguments are being discussed in terms of what seems natural to call the HO arguments. What is going on here? As it happens, the ordering of signs from LO to HO that seems obvious from the standpoint of their typical construction and their order of appearance on the stage of discussion does not reflect the measure of complexity that is relevant to the effort at recursive exposition.
The measure of complexity that is relevant to the formal exposition is the measure of doubt, uncertainty, or perplexity that one entertains about the sense and use of a sign beset by questions, whether this occurs by force of a voluntary effort to bracket its habitual senses or by dint of a puzzling event that brings its automatic uses to a halt.
It is the language being discussed that is the formal one, to be treated initially as an object, while the language that is used to carry out the discussion tries to maintain its informal viability, expecting in effect to be taken on faith as not undermining or vitiating the effort at inquiry due to unexamined flaws of its own. Nevertheless, if inquiry in general is expected to be self correcting, then a continuing series of failures to conclude inquiries by means of a given arrangement, that is, an inability to resolve uncertainties through a particular division of labor between FL and IL contexts, must lead to the grounds of attack being shifted.
In working out compromises between the FL and IL styles of usage one faces all the problems usually associated with integrating different "frameworks of interpretation" (FOI's), but compounded by the additional factors (1) that this conflict of attitudes, or its practical importance, is seldom openly acknowledged, and (2) that the frameworks in and of the negotiation to be transacted are rarely capable of being formalized, or even of being made conscious, to the same degree at the same time. These circumstances make the consequences of the underlying conflict difficult to address, and thus they continue to obstruct the desired implementation of a common CL environment that could serve as a resource for work on both sides of the frame.
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===6.13. Issue 2. The Status of Sets===
===6.13. Issue 2. The Status of Sets===
