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===6.14. Issue 3. The Status of Variables===
===6.14. Issue 3. The Status of Variables===
<pre>
Another issue on which the three styles of usage diverge most severely is with respect to a crucial problem about the status of variables. Often this is posed as a question about the "ontological status" of variables, what kinds of objects they are, but it is better treated as a question about the "pragmatic status" of variables, what kinds of signs they are used as. In this section, I try to accommodate common practices in the use of variables in the process of building a bridge to the pragmatic perspective. The goal is to reconstruct customary ways of regarding variables within a overarching framework of sign relations, while disentangling the many confusions about the status of variables that obstruct their clear and consistent formalization.
Variables are the most problematic entities that have to be dealt with in the process of formalization, and this makes it useful to explore several different ways of approaching their treatment, either of accounting for them or explaining them away. The various tactics available for dealing with variables can be organized according to how they respond to two questions: Are variables good or bad, and what kinds of things are variables anyway? That is: (1) Are variables a good thing to have in a purified system of interpretation or a target formal system, or should variables be eliminated by the work of formalization? (2) What sorts of things should variables be construed as?
The answers given to these questions determine several consequences. If variables are good things, things that ought to be retained in a purified formal system, then it must be possible to account for their valid uses in a sensible fashion. If variables are bad things, things that ought to be eliminated from a purified formal system, then it must be possible to "explain away" their properties and utilities in terms of more basic concepts and operations.
One approach is to eliminate variables altogether from the primitive conceptual basis of one's formalism, replacing every form of substitution with a form of application. In the abstract, this makes applications of constant operators to one another the only type of combination that needs to be considered. This is the strategy of the so called "combinator calculus".
If it is desired to retain a notion of variables in the formalism, and to maintain variables as objects of reference, then there are a couple of partial explanations of variables that still afford them with various measures of objective existence.
In the "elemental construal" of variables, a variable x is just an existing object x that is an element of a set X, the catch being "which element?". In spite of this lack of information, one is still permitted to write "x C X" as a syntactically well formed expression and otherwise to treat the variable name "x" as a pronoun on a grammatical par with a noun. Given enough information about the contexts of usage and interpretation, this explanation of the variable x as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by "x".
In the "functional construal" of variables, a variable is a function of unknown circumstances that results in a known range of definite values. This tactic pushes the ostensible location of the uncertainty back a bit, into the domain of a named function, but it cannot eliminate it entirely. Thus, a variable is a function x : X >Y that maps a domain of unknown circumstances, or a "sample space" X, into a range Y of outcome values. Typically, variables of this sort come in sets of the form {xi : X >Y}, collectively called "coordinate projections" and together constituting a basis for a whole class of functions f : X >Y sharing a similar type. This construal succeeds in giving each variable name "xi" an objective referent, namely, the coordinate projection xi, but the explanation is partial to the extent that the domain of unknown circumstances remains to be explained. Completing this explanation of variables, to the extent that it can be accomplished, requires an account of how these unknown circumstances can be known exactly to the extent that they are in fact described, that is, in terms of their effects under the given projections.
As suggested by the whole direction of the present work, the ultimate explanation of variables is to be given by the pragmatic theory of signs, where variables are treated as a special class of signs called "indices".
Because it was necessary to begin informally, I started out speaking of things called "variables" as if there really were such things, taking it for granted that a consistent concept of their existence could be formed that would substantiate the ordinary usages carried out in their name, and contemplating judgments of their worth as if it were a matter of judging existing objects rather than the very ideas of their existence, whereas it is precisely the whole question at issue whether any of these presumptions are justified. As concessions to common usage, encounters with these assumptions are probably unavoidable, but a formal approach requires one to backtrack a bit, to treat the descriptive term "variable" as nothing more substantial than a general name in common use, and to examine whether its uses can be maintained in a purely formal system. Further, each of the "variables" that is taken to fall under this term has to allow its various indications to be reconsidered in the guise of mere signs and to permit the question of their objective reference to be examined anew.
At this point, it is worth trying to apply the insights of nominalism to these questions, if only to see where they lead.
It is the general advice of nominalism not to confuse a general name with the name of a general. To this, pragmatism adds the distinct recommendation not to confuse an individual name with the name of an individual, because a particular that seems perfectly determinate for some purposes may not be determinate enough for other purposes.
In the perspective that results from combining these two points of view, general properties and individual instances, alike, can take on from the start an equally provisional status as objects of discussion and thought, in the meantime treated as interpretive fictions, as mere potentials for meaning, awaiting the settlement of their reality at the end of inquiry. Meanwhile, the individual can be exactly as tentative as the general, and ultimately, the general can be precisely as real as the individual. Still, their provisional treatment as hypothetical objects of reasoning does not affect their yet to be determined status as realities. This is so because it is possible that a hypothesis hits the mark, and it remains so as long as a provenient fiction, something called a likely story on account of its origin, can still succeed in guessing the truth aright.
Unlike generals, individuals, and numerous other forms of logical and mathematical objects, whose treatment as fictions does not affect their status as realities, one way or the other, there does not seem to be any consistent way of treating variables as objects. Although each one of the elemental and the functional construals appears to work well enough when taken by itself in the appropriate context, trying to combine these two notions into a single concept of the variable can lead to the mistake of confusing a function with one of its values.
Whether one tries to account for variables or chooses to explain them away, it is still necessary to say what kinds of entities are really involved when one is using this form of speech and trying to reason with or about its terms, whether one is speaking about things described as "variables" or merely about their terms of description, whether there are really objects to be dealt with or merely signs to be dispensed with.
According to one way of understanding the term, there is no object called a "variable" unless that object is a sign, and so the name "variable name" is redundant. Variables, if they are anything at all, are analogous to numerals, not numbers, and thus they fall within the broad class of signs called "identifiers", more specifically, as "indices". In the case of variables, the advice of nominalism, not to confuse a variable name with the name of a variable, seems to be well taken.
If the world of elements appropriate to this discussion is organized into objective and syntactic domains, then there are fundamentally just two different ways of regarding variables, as objects or as signs. One can say that a variable is a fictional object that is contrived to provide a variable name with a form of objective referent, or one can say that a variable is a sign itself, the same thing as a variable name. In the present setting, it is convenient to arrange these broad approaches to variables under the NOS's where one finds them most often pursued.
1. The IL approach to the question takes the "objective construal" of variables as its most commonly chosen default. The IL style that is used in ordinary mathematical discussion associates a variable with a determinate set, one that the variable is regarded as "ranging over". As a result, this NOS is forced to invoke a version of set theory, usually naive, to account for its use of variables.
2. The FL styles are manifestly varied in their explanations of variables, since there are many ways to formalize their ordinary uses. Two of the main alternatives are: (a) formalizing the set theory that is invoked with the use of variables, and (b) formalizing the sign relations in which variables operate as indices. Since an index is a kind of sign that denotes its object by virtue of an actual connection with it, and since the nature and direction of these actual connections can vary immensely from moment to moment, a variable is an extremely flexible and adaptable kind of sign, hence its character as a "reusable sign".
3. The CL styles are also legion in their approaches to variables, but they can be divided into those eliminate variables as a primitive concept and those that retain a notion of variables in their conceptual basis.
a. An instructive case is presented by what is the most complete working out of the computational programme, the "combinator calculus". Here, the goal is to eliminate the notion of a variable altogether from the conceptual basis of a formal system. In other words, it is projected to reduce its status as a primitive concept, one that applies to symbols in the object language, and to reformulate it as a derived concept, one that is more appropriate to describing constructions in a metalanguage.
b. In CL contexts where variables are retained as a primitive notion, there is a form of distinction between variables and variable names, but here it takes on a different sense, being the distinction between a sign and its HO sign. This is because a variable is conceived as a "store", a "component of state" (COS) of the interpreting machine, that contains different values from time to time, while the variable name is a symbolic version of that store's address. The store when full, or the state when determinate, constitutes a form of numeral, not a number, and so it is still a sign, not the object itself. This makes the variable name in this setting a type of HO sign.
It is not just the influence of different conventions about language use that forms the source of so much confusion. Different conventions that prevail in different contexts would generate conceptual turbulence only at their boundaries with each other, and not distribute the disturbance throughout the interiors of these contexts, as is currently the case. But there are higher order differential conventions, in other words, conventions about changing conventions, that apply without warning all throughout what is pretended to be a uniform context.
For example, suppose I make a casual reference to the following set of pronouns:
{I, you, he, she, we, they}.
Then chances are that the reader will automatically shift to what I have called the "sign convention" to interpret this reference. Even without the instruction to expect a set of pronouns, it makes no sense in this setting to think I am referring to a set of people, and so a charitable assumption about my intentions to make sense will lead to the intended interpretation.
However, suppose I make a similar reference to the following set of variables:
{x1, ... , xn}.
Then it is more likely that the reader will take the suggested set of variable names as though they were the names of some fictional objects called "variables".
The rest of this section deals with the case of boolean variables, that are soon to be invoked in providing a functional interpretation of propositional calculus.
This discussion draws on concepts from two previous papers (Awbrey, 1989 & 1994), changing notations as needed to fit the current context. Except for special sets (B, N, R, Z) and sign relational domains (O, S, I), I use plain capital letters for ordinary sets, singly underlined capitals for coordinate spaces and vector spaces, and doubly underlined capitals for the "alphabets" and "lexicons" that generate formal languages and logical universes of discourse.
If X = {x1, ... , xn} is a set of n elements, it is possible to construct a "formal alphabet" of n "letters" or a "formal lexicon" of n "words" that exists in a one to one correspondence with the elements of X and can be notated as follows:
X = Lit (X) = {x1, ... , xn}.
The set X is known in formal settings as the "literal alphabet" or the "literal lexicon" associated with X, but on more familiar grounds it can be called the "double" of X. Under conditions of careful interpretation, any finite set X can be construed as its own double, but for now it is safest to preserve the apparent distinction in roles until the sense of this double usage has become second nature.
This construction is often useful in situations where has to deal with a set of signs {"s1", ... , "sn"} with a fixed or a faulty interpretation. Here, one needs a fresh set of signs {x1, ... , xn} that can be used in analogous ways to the original, but free enough to be controlled and flexible enough to be repaired. In other words, the interpretation of the new list is subject to experimental variation, freely controllable in such a way that it can follow or assimilate the original interpretation whenever it makes sense to do so, but critically reflected and flexible enough to have its interpretation amended whenever necessary.
Interpreted on a casual basis, the set X can be treated as a list of "boolean variables", or, according to another reading, as a list of "boolean variable names", but both of these choices are subject to the eventual requirement of saying exactly what a "variable" is.
The overall problem about the "ontological status" of variables will also be the subject of an extended study at a later point in this project, but for now I am forced to side step the whole issue, merely giving notice of a signal distinction that promises to yield a measure of effective advantage in finally disposing of the problem.
If a sign, as accepted and interpreted in a particular setting, has an "existentially unique" (EU) denotation, that is, if there exists a unique object that the sign denotes under the operative sign relation, then the sign is said to possess a "EU denotation", or to have a "EU object". When this is so, the sign is said to be "eudenotational", otherwise it is said to be "dysdenotational".
Using the distinction accorded to eudenotational signs, the issue about the ontological status of variables can be illustrated as turning on two different "acceptations" of the list X = {x1, ..., xn}.
1. The natural (or naive) acceptation is for a reader to interpret the list as referring to a set of objects, in effect, to pass without hesitation from impressions of the characters "x1", ..., "xn" to thoughts of their respective EU objects x1, ..., xn, all taken for granted to exist uniquely. The whole set of interpretive assumptions that go into this acceptation will be referred to as the "object convention".
2. The reflective (or critical) acceptation is to see the list before all else as a list of signs, each of which may or may not have a EU object. This is the attitude that must be taken in formal language theory and in any setting where computational constraints on interpretation are being contemplated. In these contexts it cannot be assumed without question that every sign, whose participation in a denotation relation would have to be indicated by a recursive function and implemented by an effective program, does in fact have an existential denotation, much less a unique object. The entire body of implicit assumptions that go to make up this acceptation, although they operate more like interpretive suspicions than automatic dispositions, will be referred to as the "sign convention".
In the present context, I can answer questions about the ontology of a "variable" by saying that each variable xi is a kind of a sign, in the boolean case capable of denoting an element in B = {0, 1} as its object, with the actual value depending on the interpretation of the moment. Note that xi is a sign, and that "xi" is another sign that denotes it. This acceptation of the list X = {xi} corresponds to what was just called the "sign convention".
In a context where all the signs that ought to have EU objects are in fact safely assured to do so, then it is usually less bothersome to assume the object convention. Otherwise, discussion must resort to the less natural but more careful sign convention. This convention is only "artificial" in the sense that it recalls the artifactual nature and the instrumental purpose of signs, and does nothing more out of the way than to call an implement "an implement".
I make one more remark to emphasize the importance of this issue, and then return to the main discussion. Even though there is no great difficulty in conceiving the sign "xi" to be interpreted as denoting different types of objects in different contexts, it is more of a problem to imagine that the same object xi can literally be both a value (in B) and a function (from Bn to B).
In the customary fashion, the name "xi" of the variable xi is flexibly interpreted to serve two additional roles. In algebraic and geometric contexts "xi" is taken to name the ith "coordinate function" xi : Bn >B. In logical contexts "xi" serves to name the ith "basic property" or "simple proposition", also called "xi", that goes into the construction of a propositional universe of discourse, in effect, becoming one of the "sentence letters" of a truth table and being used to label one of the "simple enclosures" of a venn diagram.
Rationalizing the usage of boolean variables to represent propositional features and functions in this manner, I can now discuss these concepts in greater detail, introducing additional notation along the way.
1. The sign "xi", appearing in the contextual frame "_ : Bn >B", whether explicitly or implicitly, can be interpreted as denoting the ith coordinate function xi : Bn >B. The entire collection of coordinate maps in X = {xi} contributes to the definition of the "coordinate space" or "vector space" X : Bn, notated as follows:
X = <X> = <x1, ..., xn> = {<x1, ..., xn>} : Bn.
Associated with the coordinate space X are various families of boolean valued functions f : X >B.
a. The set of all functions f : X >B has a cardinality of 22^n and is denoted as follows:
X-> = (X -> B) = {f : X -> B}.
b. The set of linear functions f : X >B has a cardinality of 2n and is known as the "dual space" X* in vector space contexts. In formal language contexts, in order to avoid conflicts with the use of the "Kleene star" operator, it needs to be given an alternate notation:
X+> = (X +> B) = {f : X +> B}.
c. The set of singular functions f : X >B has a cardinality of 2n and is notated as follows:
X!> = (X !> B) = {f : X !> B}.
d. The set of positive functions f : X >B has a cardinality of 2n and is notated as follows:
X@> = (X @> B) = {f : X @> B}
e. The set of coordinate functions, also referred to as "basic" or "simple" functions", has a cardinality of n and is denoted in the following ways:
X = (X :> B) = {f : X :> B} = {xi : X->B}.
2. The sign "xi", read or understood in a propositional context, can be interpreted as denoting one of the n "features", "qualities", "basic properties", or "simple propositions" that go to define the n dimensional "universe of discourse" X[], also notated as follows:
X[] = [X] = [x1, ..., xn] = <X, X->> : Bn&->B.
</pre>
===6.15. Propositional Calculus===
===6.15. Propositional Calculus===