floyd
merrell
Purdue
University
West
Lafayette, IN 47907
OVERDETERMINACY, UNDERDETERMINACY, INDETERMINACY
INORDINATELY UNRULY SIGNS?
Charles
S. Peirce occasionally alluded to what he termed a ‘logic of vagueness’ (i.e. of
‘possibility’ or ‘continuity’) as a ‘logic’ in ‘the broadest possible sense’, a
‘logic’ fit for all seasons and all reasons.
Obviously, such a logic would go against the grain of classical logic
insofar as it had been developed in Peirce’s time by Boole, de Morgan, Whatley,
Schröder, and others. In fact, one
would expect that it would follow the lines of some ‘triadic logic’ of some
sort or other. But it must be more than
that. In order that it form a ‘logic’
in ‘the broadest possible sense,’ it should be, as Peirce himself occasionally
put it, a ‘logic of vagueness,’ offering foreshadowings of today’s ‘fuzzy
logic.’[1] Peirce never quite made good on his promise
to construct this general ‘logic’.
However, in 1908 he did envision and outline the makings of a ‘triadic
logic’ of sorts based on ‘real possibility’, ‘actuality’, and ‘real necessity’.
Peirce
points out that a proposition asserting actual existents (Seconds) lies
at the half-way house between the poles of assertion of possibility
(Firstness) and those of necessity (Thirdness). [2]
While assertions regarding actuals follow the tenets of classical logic,
assertions of possibility and necessity do not, not necessarily, that is. In Peirce’s words:
that which characterizes and defines an assertion of
Possibility is its emancipation from the Principle of Contradiction, while it
remains subject to the Principle of Excluded Third; while that which
characterizes and defines an assertion of Necessity is that it remains subject
to the Principle of Contradiction, but throws off the yoke of the Principle of
Excluded Third; and what characterizes and defines an assertion of Actuality,
or simple Existence, is that it acknowledges allegiance to both formulae, and
is thus just midway between the two rational ‘Modals’, as the modified forms
are called by all the old logicians (MS 678:34-35).
What
lies within the sphere of possibility (Firstness) by and large violates the Principle
of Noncontradiction, which reigns in the ‘semiotically real’ world of
Secondness and classical logical principles.
Within the sphere of pure Firstness, contradictories can quite
comfortably exist side by side. For,
given the nature of unactualized Firstness as a superposed set of
possibilities, everything is there. It
composes an unimaginably massive, continuous collage of compatible and
incompatible, consistent and inconsistent, and complementary and contradictory,
nonessences. In this sphere of
pure chance, spontaneity, and infinitely diluted vagueness, nothing is
(yet) specified and everything is at one with everything else: there are as yet no distinctions, no
borders, no taxonomies. There is no
static plenum, per se, but rather, effervescent, fluctuating,
flickering, superposed possibilia in expectancy of their actualization
into some ‘semiotically real’ domain or other.
Thus the sphere of vagueness is thoroughly overdetermined. There is no knowing whether what would
otherwise be considered contradictory terms might not be considered equally
‘true’ at different times and places (e.g. the ‘Earth’ as center of the
universe before Copernicus, the ‘Sun’ as center of the universe after
Copernicus) (see Goodman 1978).
The
realm of necessity (Thirdness) includes mediary terms, with no end in
sight. Since any and all corpora of
signs remain invariably incomplete, something more can always be
added. Hence, unlike the eithers
and the ors of Secondness, within Thirdness the Excluded-Middle
Principle threatens to fall by the wayside. Between any two signs, given sufficient time and change of
context and complexity, the potential always exists for other signs and their
meanings, or the same signs and other meanings, to emerge. It is not a matter of the ‘center’ of the
universe either as the Earth (Ptolemy) or the Sun (Copernicus),
but neither the one nor the other. In other words, the ‘center’ for Ptolemy and the ‘center’ for
Copernicus is not simply a matter of either-or alternatives: with the demise of classical physics, the
‘center’ is now conceived to be something else altogether (i.e. something
entered the gap between the erstwhile either/or categories to render
them neither-nor). Yet since at
any given point in time the ‘center’ cannot be construed as both the
Earth and not the Earth, the Principle of Noncontradiction
remains in force—albeit tenuously at best.
Consequently, at a given point in time, any and all conceptual schemes
are destined to incompleteness, since no matter how replete the
previously considered gap between the either and the or is
filled, there will always be room for something else. Due to this persistence of incompleteness, underdetermination
necessarily prevails.
Overdetermination
includes the sphere within which a sign is not yet definitely or
authoritatively decided, settled, or fixed--though according to the
circumstances it presumably can be--and as such it is unbounded by definite
limits or restrictions. Might I venture
to suggest that overdetermination is related to the Peircean category of
Firstness, as well as to the concepts of vagueness and inconsistency. However, overdetermination in the
purest sense is actually tantamount to what we might label pre-Firstness,
before there is or can be consciousness of a sign (Baer 1988). Consciousness of a sign, during the
very moment it is emerging into the light of day, remains vague, to be
sure. As consciousness of the
sign becomes more pronounced and vagueness gives way to increasing
precision, a small number of the indeterminate range of possible specifications
of the sign can become actualized as Seconds to take their place in what is
perceived and conceived to be the ‘semiotically real’ world. But whatever specification might have been
actualized, others remain as possibilities, some of them contradictory with
respect to that which was actualized.
In other words, regarding the Secondness and Thirdness of signs of
which there is consciousness and regarding which specification of meaning can
be made more precise, underdetermination (related, I would suggest, to generality
and incompleteness) makes its presence known here and there. In another way of putting it, within the
sphere of overdetermination, mutually incompatible possibilities of
meaning can cohabit without undue conflict (and as a result, the Principle
of Noncontradiction loses some of its sting). In contrast, within the sphere of underdetermination, an
actualized meaning within one space-time slice can become something slightly to
radically different within another space-time slice (hence the Excluded
-Middle Principle is abrogated).[3]
It
becomes apparent, then, that the sphere of vagueness, of possibilia
(Firstness), is timeless, while that of generality (actuals developing
toward the fullness of Thirdness) is time-bound. By the very nature of this interrelationship, signs of generality
are destined, in the long run of things, to suffer a fate complementary with
that of signs of vagueness. In
this spirit, Peirce wrote that ‘[n]otwithstanding their contrariety, generality
and vagueness are, from a formal point of view, seen to be on a par’ (CP:5.447). Vague signs cannot be construed as vague
unless endowed with at least a tinge of generality, and general
signs, given their inevitable degree of incompleteness, are invariably
somewhat vague. Peirce readily
conceded that no sign can be vague and general from the same
perspective and from within the same space-time slice, since insofar as the
determination of a sign is extended to the interpreter--i.e. the case of generality--it
is by and large denied to the utterer, and insofar as it is extended to the
utterer--i.e. the case of vagueness--it lies largely beyond the grasp of
the interpreter (CP:1.463-69, 5.447-57). By no means, however, do I wish to imply that Firstness has a
monopoly on vagueness, but rather, vagueness to a greater or
lesser degree pervades any and all signs.
This is in keeping with Peirce’s abolition of clear and distinct, and
precisely demarcated, boundaries. I
must also add that the interrelationships herein implied between vagueness
and generality--and overdetermination and underdetermination--is
not usually forthcoming in twentieth-century philosophical discourse. Bertrand Russell (1923), for instance,
relates the law of excluded-middles exclusively to vagueness. Williard V. O. Quine (1953, 1960) has
focused almost obsessively on underdetermination with respect to
scientific theories, and by extension, natural language (Føllesdal 1975). More recently, Donald Davidson (1984) has
thrown vagueness into the same bag with generality and incompleteness
without showing how they are agonistically set apart and at the same time
intricately intertwined (Evnine 1991:105-14).
Every
sign is in the Peircean sense at least partially determined, and its partial
determination is contingent upon its varying degrees of context-dependent vagueness
and generality:
A sign (under which designation I place every kind
of thought, and not alone external signs), that is in any respect objectively
indeterminate (i.e. whose object is undetermined by the sign itself) is
objectively general in so far as it extends to the interpreter the
privilege of carrying its determination further. Example: ‘Man is
mortal’. To the question, What man? the
reply is that the proposition explicitly leaves it to you to apply its
assertion to what man or men you will.
A sign that is objectively indeterminate in any respect is objectively vague
in so far as it reserves further determination to be made in some other
conceivable signs, or at least does not appoint the interpreter as its deputy
in this office. Example: ‘A man whom I could mention seems to be a
little conceited’. The suggestion
here is that the man in view is the person addressed, but the utterer does not
authorize such an interpretation or any other application of what she
says. She can still say if she likes,
that she does not mean the person addressed. Every utterance naturally leaves the right of further exposition
in the utterer, and therefore, in so far as a sign is indeterminate, it is
vague, unless it is expressly or by a well understood convention rendered
general. (CP:5.447; also 1.434)
Thus, ‘a sign can only escape from being either
vague or general by not being indeterminate’.
Yet no sign ‘can be absolutely and completely indeterminate’ (vague) (CP:5.506). For a sign, ‘however determinate, may be
made more determinate still, but not ... absolutely determinate’ (general) (CP:3.93). If a sign were totally determinate, it would
always be as it is, its attributes remaining intact and changeless.
In
everyday situations, when the plethora of potentially variant space-time slices
comes into the picture, the possibility of any absolutely determinate sign
dissolves. There was a George Bush of
‘Read my lips’, of ‘No new taxes’, of ‘Perhaps new taxes’, of ‘New taxes’, and
of ‘New taxes, but the democrats made me do it’. But there is no George Bush impervious to any and all
change. These days we have a Bill
Clinton of the Democratic Party as now neoliberal, now for social programs, now
wooing the conservatives, now catering to the business community, now also of
the working class and capable of eating hamburgers with the best of them, now
favorable to the educators, now sympathetic with women and minority groups and
gays, now friendly with the women folks but doing nothing improper, now intimate
with members of the opposite sex but still morally upstanding. Bill Clinton, like all signs, can be many
things to many people, or he can be virtually an empty set capable of taking in
almost any sign, according to the interpretation.[4] Like all signs, he simply cannot stand
still. Were a changeless sign actually
to exist, it would be absolutely autonomous, individual, and indivisible. However, such absolutes ‘can not only not be
realized in sense or thought, but cannot exist, properly speaking. For whatever lasts for any time, however
short, is capable of logical division, because in that time it will undergo
some change in its relations’ (CP:3.39 n1). So every sign must relate to some
not-quite-absolutely-general ‘semiotic object’. The ‘object’ cannot be the absolutely ‘real object’ as it is, for
all ‘objects’ are related to all other ‘objects’ of a given field of
signs. To be sure, all signs relate to
some singular ‘object’, at least potentially understood by all semiotic
agents. But since the ‘really real’
lies perpetually beyond our grasp, there must exist some lesser sphere
containing signs and their ‘semiotic objects’.
That sphere is partly shared by the semiotic agents involved in dialogic
exchange, and those signs and ‘semiotic objects’ are to a greater or lesser
degree general, though never absolutely so, and hence they are to a greater or
lesser degree vague. Vagueness and
generality are in this sense complementary forms of indeterminacy. A sentence can be determinately judged
either ‘true’ or ‘false’ in the ‘here-now’, though in the ‘there-then’ its
value will have suffered a change, however small--Peirce’s conception of
‘logic’ in the ‘broadest possible sense’ embraces temporality. And a sentence that has been determined
either ‘true’ or ‘false’ in one respect may be neither ‘true’ nor ‘false’ in
another. A sound can be neither blue
nor red in the literal sense, though it may conceivably be either the one or
the other in the synaesthetic sense.
Consequently, the predicates ‘shrill’ or ‘mellow’, ‘bitter’ or ‘sweet’,
or ‘blue’ or ‘red’ attached to the sign can be both ‘true’ and ‘false’ from
within the range of all possible conceptions.
Generality
includes the Peircean terms potentiality, convention, necessity, conditionality,
and regularity--all of the category of Thirdness--which implies process,
growth, intellect, and mind (CP:1.340).
Generality thus calls for ever greater account of particular signs and
their attributes as types. Yet to
expect absolute determinacy through generality is out of the question: there can be no more than an approximation
toward a sign in its most general sense.[5] Vagueness, given its nature as indefinite,
ambiguous, and indeterminate, takes the terms possibility, chance, spontaneity,
and novelty into its embrace. While
generality entails relations to ‘semiotic objects’, vagueness bears no form or
fashion of relatedness of signs to other signs established by
some semiotic agent. Pure vagueness
(Firstness) is the superposition of all possibilities without any of them being
actualized. However, vagueness of
actual signs (Secondness) requires their concrete contextualization and their
being related to other signs. Such
actualized signs, according to their interpretation, can now take on generality
(Thirdness). It is for this reason that
while the onus of further determination of a general sign is left to the
conceptual scheme, the criteria, and the style of reason and the wishes and
whims of its interpreter. In contrast,
determination of a vague sign depends upon further revelation and specification
of its meaning by its author and the context of its engenderment.
Regarding
the complementarity of vagueness and generality, Peirce writes that no
general description can serve indubitably to identify the object of a sign or
establish its meaning. A certain degree
of identification of the object is always left to ‘common sense’ (Firstness,
vagueness). For:
the common sense of the interpreter of the sign will
assure him that the object must be one of a limited collection of objects. Suppose for example, two Englishmen to meet
in a continental railway carriage. The
total number of subjects of which there is any appreciable probability that one
will speak to the other perhaps does not exceed a million, and each will have
perhaps half that million not far below the surface of consciousness, so that
each unit of it is ready to suggest itself.
If one mentions Charles the Second, the other need not consider what
possible Charles the Second is meant.
It is no doubt the English Charles the Second. Charles the Second of England was quite a different man on
different days; and it might be said that without further specification the
subject is not identified. But the two
Englishmen have no purpose of splitting hair in their talk; and the latitude of
interpretation which constitutes the indeterminacy of a sign must be understood
as a latitude which might affect the achievement of a purpose. (CP:5.448
n)
In
addition to common sense, purpose is a watchword here. If two somewhat different conceptions of the
same sign--one person’s estimation of Charles the Second and that of another
person--yielded meanings that were for all possible purposes equivalent, then
the signs could conceivably be considered equivalent. There apparently would be no latitude of purpose, the sign would
be general in the fullest possible sense.
Nor would there seem to be any room for vagueness, for the sign would
have taken on the fullness of its generality, in the minds of its interpreters
at least. However, in the context of
human communication by way of natural language--and all other sorts of
communication as far as that goes--there is no absolute identity of purpose. For, the motivating force behind purpose
itself involves common sense (intuition, inclination, belief, disposition, all
of which have a foothold in Firstness and are inevitably tinged with some
degree or other of vagueness).
Vagueness, then, is irreducible to the rank and file absolute
determinacy of the ‘semiotic object’, since there is always something
indeterminable and left indeterminate.
Yet vagueness is every bit as essential to thought as is
generality. For, a particular sign, its
‘semiotic object’, or its interpretant, cannot be properly cognized in the total
absence of the general nature of the semiotic entity in question. And unless there is some element of
vagueness, there can hardly be any account of the entity’s change over time: a changeless, timeless sign would be none
other than a Parmenidean eternally invariant domain of some form or other
jam-packed with a host of timeless essences into an artificial plenum.
To sum up, in a finite community of
fallible semiotic agents, there can be no unadulterated sign of generality
without at least a tinge of vagueness.
And there can be no purely vague sign, for once actualized in order that
it be made intelligible, a vague sign must take on at least some modicum of
generality according to its interpreters’ inevitable beliefs, habits,
presuppositions, prejudices, and preconceptions. If any form or fashion of a ‘logic in the broadest possible
sense’ there may be, it must include the spheres of both vagueness and
generality, and hence the Principles of Noncontradiction and the
Excluded-Middle will not always be able to wield their terrible swift
sword. The upshot is that insofar as we
finite, fallible semiotic agents are concerned, all generals are also possibly
false (i.e. the incompleteness of underdetermination), therefore they can be
taken only conditionally as necessary, those conditions always remaining
subject to their partial fulfillment, or in the event that they are false, to
their unfulfillment.
Now for
a further look at the complementary role of a sign’s author and its
interpreters--themselves also signs.
SIGNS ARE ALWAYS SOMEWHERE ELSE
Taking
into account the composite characteristics of possibility (Firstness),
actuality (Secondness), and potentiality (Thirdness), a certain ‘Principle of Indeterminacy’
is crucial to an understanding of Peirce’s notion of semiosis.
Quite
obviously, Peirce was keen on the idea that we dwell in a vague and
inconsistent, and general but perpetually incomplete, world of signs. The ubiquity of vagueness and inconsistency
breeds a tendency to embrace contradiction and paradox. And the inevitability of incompleteness in
all signs of general nature allows for the entrance of unexpected thirds
without conceivable end. Yet, Peirce
writes in so many ways that the collusion of possibility, actuality, and
potentiality makes up our ‘semiotically real world’ as we perceive and conceive
it, which, if we are fortunate, stands an outside chance of approximating some
portion of the ‘real’. Any and all
‘semiotic worlds’, in this light, must remain radically uncertain, for, ‘when
we busy ourselves to find the answer to a question, we are going upon the hope
that there is an answer, which can be called the answer, that is, the final
answer. It may be that there is none’.
(CP:4.61)
To be
more specific, Peirce does not use the pair of Gödelian terms, inconsistency
and incompleteness, now commonplace in mathematics, logic, and physics. However, his vagueness-generality dyad is
brought in line with something reminiscent of a Gödelian framework by Rescher
and Brandom (1979:124-26), though admittedly for a different purpose (see
Merrell 1991, 1995a, Nadin 1982, 1983).
The relationship between vagueness-generality and
inconsistency-incompleteness and their relevance to indeterminacy becomes
apparent if one sufficiently contemplates Peirce’s suggestion that ‘[e]very
utterance naturally leaves the right of further exposition in the utterer; and
therefore, in so far as a sign is indeterminate, it is vague, unless it is
expressly or by a well-understood convention rendered general’ (CP:5.447). In other words, the indeterminately vague
sign calls out to its maker for further clarification, since that which can
render it less vague is more accessible to the possibilities that lie before her
that before the sign interpreter.
An
example of this principle is illustrated by an anecdote about Bertrand
Russell. In a social setting while
discussing conditional statements, he remarked that a false statement can imply
virtually anything and everything. A
skeptical colleague thereupon challenged him to prove that if 2 = 1, then he is
the pope. ‘Why’, Russell responded,
‘the pope and I are two, but two equals one, therefore the pope and I are
one’--the point being that it is useless to deal with inconsistent systems (in
Bronowski 1978:79). During this
exchange, confusion initially ensued upon Russell’s uttering the vague signs,
‘false statement’, ‘anything’, and ‘everything’. It was up to him, the signs’ author, to provide their further
determination, which he was most willing to do. One cannot deny that with Russell’s utterance of additional
specifying signs, the originally vague signs took on additional interpretative
baggage. But the process was ongoing,
for now, further determination of those signs as generalities rested on the
shoulders of their interpreters. In
this sense, the vague signs, ‘false statement’, ‘anything’, and ‘everything’,
remained alongside other specifying, yet to a degree general, signs, which
combined to spell a potential for future semiotic activity on the part of both
sign makers and sign takers.
If a
sign of vagueness includes contradictions, then the sign’s meaning for one
community might be incompatible with its meaning for another community at
another time. And if a sign of
generality is never determined to the extent that it cannot be determined
further, then an unordered set of potential interpretations exists with the
characteristic that between any given pair of interpretations there can always
be a third one. In other words, as we
have noted, the Excluded-Middle Principle loses part of its sting. A small group of mathematicians, the
intuitionists, deny the Excluded-Middle Principle altogether. They would discard statements the likes of
‘Either there is a string of 18 consecutive 5s somewhere in the decimal
expansion of p or there is not’, since
they can most likely enjoy no proof in our finite world. That is to say, ‘truth’ is intimately linked
to provability. For quite different
reasons, a handful of quantum theorists also reject the excluded-middle, in
roughly the sense of Jan Lukasiewicz, the Polish logician of the 1920s, whose
‘3-valued logic’ includes ‘true’, ‘false’, and ‘undetermined’ (indeterminate,
intermediate). In fact, John von
Neumann pioneered an alternate ‘logic’, ‘quantum logic’, especially tailored to
the needs of quantum phenomena.
Following the general implications of quantum theory and quantum logic,
a sign’s becoming a genuine sign depends upon the interpreter’s interaction
with it. Just as no ‘wave packet’ is an
actualized ‘particle-event’ until it enters into relationship with some aspect
of its surroundings, so also no sign is a full-blown sign until it has been
actualized (and interpreted) by some interpreter in some respect or capacity.[6]
An
additional example may serve to illustrate the idea that (1) a sign is not a
genuine sign until it has interacted with some semiotic agent, (2) within the
(vague) realm of all possible signs, inconsistency or contradiction inevitably
prevails, and (3) given the range of all actualized (general) signs, past,
present, and future, there is no guarantee that the excluded-middle applies,
hence the meaning of any and all signs will be incomplete. Assuming I have little knowledge regarding a
particular event reported in the newspaper, I can read each individual sentence
with rather wide-eyed, innocent--and exceedingly vague--belief. Yet at a more general level I may also
believe that this article, like all others, is in all probability the victim of
at least some degree of biased reporting.
I tend to believe each individual sentence as it stands, but at the same
time I am willing to concede to the possibility that my belief in a given
sentence can embrace contradiction, since I also believe that, lurking
somewhere in the report, there is undoubtedly some distortion of the
‘truth’. So I take the article as a
whole with a grain of disbelief, though I have not yet encountered any sign of
deceit: it remains as a sign of
possibility. Even though I might not
have been able to catch the reporter at her devious game, I may still retain my
faith that a closer reading will in all likelihood reveal some sort of
inconsistency (i.e. that the sign of possibility will be actualized). In other words, I believe the article is
neither wholly ‘true’ nor wholly ‘false’, but somewhere in between (we once
again realize that banishing any and all contradictions and paradoxes is an
interminable and hence futile enterprise).
Extrapolating from Peirce, it seems to follow that: (1) an assertion of possibility (Firstness),
having found newborn freedom from the Principle of Noncontradiction, rests
chiefly within the domain of vagueness, (2) an assertion of necessity
(Thirdness), liberated from the fetters of the Excluded-Middle Principle,
pertains primarily to generality, and (3) an assertion of actuality
(Secondness) by and large, and for practical purposes, remains quite obedient
to the demands of classical logic.
This
collusion of vagueness and generality constitutes a fundamental principle,
noted above, of what Peirce envisioned for his ‘logic in the broadest possible
sense’. According to the tenets of
classical logic, once the identity of a proposition has been determined, it is
either ‘true’ or ‘false’. But for
Peirce’s more general ‘logic’, as long as a proposition remains
indeterminate--which must always be the case to a greater or lesser degree--it
is not necessarily ‘true’ that it is either ‘true’ or ‘false’. In fact, it may also be neither ‘true’ nor
‘false’, for some newly born ‘truth’ may exist somewhere between the erstwhile
horns of the presumed extremes of ‘truth’ and ‘falsity’. And until the proposition is an absolutely
determinate actuality--which will never be the case in a finite setting of
fallible semiotic agents--it may be ‘true’, given its vast range of all
possible determinations at diverse space-time slices, that it is both ‘true’
and ‘false’. Peirce’s ‘logic’, it
tentatively appears, reflects a tension and potential mediation between
vagueness and generality, the individual and the universal, and discontinuity
and continuity, as well as between self and other and self
and sign, in such a manner as to defy precise description. This accounts for the elusiveness of his
hopeful ‘logic’, and his obvious difficulty in bringing it to fruition. It also endows the terms in question with a
flavor somewhat reminiscent of Bohr’s complementarity regarding the
wave/particle duality, of Heisenberg’s uncertainty, which, he argued
repeatedly, is more a methodological and epistemological than an ontological
necessity, and of Gödel’s incompleteness-inconsistency.
Now,
since (1) complementarity and the uncertainty principle entail one’s knowing
now one character of an entity, now another character, without the possibility
of knowing both characters in simultaneity, and since (2) Peirce’s ‘logic in
the broadest possible sense’ is time-bound, (3) a brief incursion--albeit
tangentially by way of Kurt Gödel, if I may--into the nature of time behooves
us.
IT’S ABOUT TIME.
According
to Gödel’s theorem, there are certain questions neither a machine nor we
sapient human semiotic agents can answer with a firm ‘yes’ or a firm ‘no’, for
a degree of inconsistency (vagueness) inexorably inheres. In our nitty-gritty world of human praxis,
on the other hand, a number of questions exist that apparently cannot be
completely (in the most general sense) answered at any particular point in
time. But, given sufficient time and
experience, and the numbing range of variable possible contexts, eventually a
satisfactory answer may be forthcoming.
Moreover,
if a question is posed we can--though with some vacillation--choose to answer
neither with a definite ‘yes’ nor a definite ‘no’, which is nonetheless also a
decision. This pro tempore
license to vacillate between this and that and yes and no
creates the possibility, at each new moment, of a slightly to radically
different context. And context and time
are all-important, for they hold some of the keys to the significance (meaning)
of signs and of the semiotic agent’s very existence. It is not that time heals all change. Rather, through time, change ushers in new possibilities (Firsts)
a minute portion of which are at particular space-time bifurcations and within
particular contexts actualized (as Seconds) due to happy, and at times
unexpected, collisions and collusions of memories, of present habits,
dispositions, and conventions, and of anticipations of the future by the
semiotic agent (via Thirdness).
Most importantly, choices of one sort or another are exercised at each
space-time juncture.
Now, if
we replace choice by decision we are on the road toward
approximating Gödel’s turf. We decide
and then choose, or we mindlessly choose, and then create the illusion we have
judiciously arrived at a decision. In
whichever case, a decision is made. In
mathematical language, to have a proof entails the ability to make a decision
regarding the ‘truth’ of an axiom. That
is all quite rigorous, however. For the
moment best we stick to our everyday language use. From within natural languages, just as much as from within formal
languages, inconsistency and incompleteness play havoc with the power of decidability,
which depends upon manageable degrees of complexity. The problem is that, given a relatively rich and sophisticated
field of natural language signs, the degree of complexity is such that it
simply defies our finite, fallible human capacity for specifiability and
decidability. What has been called the
‘Berry Paradox’ may give us a handle on the issue. This paradox comes in the form of an injunction: ‘Find the smallest whole number that cannot
be specified by a string of words with less than twenty-nine syllables’. Attempting to solve the problem by entering
through back door, we can declare that the number of syllables in the Berry
sentence itself, twenty-eight, is capable of describing that smallest
number. And that smallest number is
equal to the smallest number which cannot be specified by a string of words
with less than twenty-nine syllables.
We must conclude, then, that the least whole number not namable by a
string of words with fewer than twenty-nine syllables can in fact be named in
twenty-eight syllables. The problem is
that the Berry sentence specifies a whole number which by its own definition it
contains too few words to specify.
Logically speaking, it should not be able to make a decision regarding
such a number, for it cannot ‘jump outside’ itself to specify the number from
some ‘transcendental’ vantage. If in
this vein we take human finitude into due consideration, ultimately, the
smallest number not namable by the Berry sentence is for practical purposes
virtually equivalent to the total number of our possible brain states: we cannot possibly hold each and every one
of that mind-bogglingly monstrous collection of brain states in our purview for
the purpose of deciding on and specifying its magnitude, for logically
speaking, we cannot do so without stepping outside our own brains, which we
cannot do.
This impossibility of our grasping and
specifying the whole of a given corpus has a temporal-existential counterpart,
which was quite forcibly made evident in Wittgenstein’s (1956) remarks on
mathematics (see also Bloor 1976, 1983; Shanker 1987). A natural language rendition of this
temporal-existential counterpart is revealed by another quandary known as the
Prisoner Paradox. The paradox goes like
this. It is Sunday. The warden tells the prisoners that the
judge has decreed their execution on one day of that week. But they will not be informed which day it
will be until the arrival of that very day, hence it will be a surprise. The prisoners, however, happen to have found
a quite astute lawyer. She reasons
that, assuming the warden has told them the truth, they cannot be executed, for
if the fatal day is to be Saturday, then it cannot be a surprise, since it will
be the only day remaining. By this mode
of reasoning neither can it be Friday, for Saturday now having been eliminated,
Friday is no longer a viable candidate.
The same can be said of Thursday, and so on down to Monday. Therefore they cannot legitimately be
executed.
Now
there is a flaw here. The lawyer’s
reasoning is strictly by atemporal logical means; she can certainly
afford to be logical, for her life is not at stake. Her field of signs, conveniently conforming to logical
principles, is quite manageable and for her apparently decidable. In contrast, the prisoners’ very existence
is in jeopardy. They are rightly
concerned over how much time remains of their life, and time is precisely the
issue here. The lawyer’s logic is timeless,
and within this framework, entailing a God’s-Eye grasp of things, the paradox
springs forth in full force. In other
words, as far as the lawyer is concerned, all events exist timelessly in the before
or the after (i.e. J. M. E. McTaggart’s [1927] B-series). There can’t be a ‘day after’,
regarding the prisoners’ demise, for if there were, there could be no surprise,
hence neither can there be a ‘day before’. So the event of the prisoners’ death at the hands of the firing
squad can’t occur, according to the lawyer’s logic that is. But the prisoners, their emotions having
understandably taken precedence over their reasoning faculties, are condemned
to time. They live in another world
entirely, with a past, a future and a knife-edged present
racing from the former toward the latter (i.e. McTaggart’s [1927] temporal
A-series). At any given present
the warden can make his decision, the firing squad will be called up, and as
far as the prisoners are concerned they will die. Hence try as their lawyer may to convince them otherwise, she
will not be able to reason away their expectations of an unexpected moment
announcing their doom. Condemned to a
time-bound set of possibly, actually, and potentially
unexpected signs the complexity of which is beyond their grasp, they can
conceive of no solution. There is for
them no timeless God’s-Eye perspective of the sort apparently enjoyed by their
lawyer.
The
Berry Paradox traps the sentence ‘within’ itself and the interpreter within the
sentence. The Prisoner Paradox traps
the real flesh and blood objects of predication, the prisoners, ‘within’ the
sentence, though a neutral interpreter can presumably remain ‘outside’,
maintaining a timeless logical slant on the whole. It is ultimately a matter of the capacity or incapacity to survey
and give account of, and of the knowability or unknowability of, the whole of
things. The lawyer thinks she can view
the whole from a timeless perspective, as if she were gazing upon the undivided
sphere of Firstness or of Thirdness completed once and for all. She sees an inconsistency, and, applying it to
the prisoners’ ‘semiotically real’ world of Secondness, declares that the
judge’s decreed event, the fulfillment of Thirdness, cannot logically come to
pass. Caught within their temporal
existence and unable to survey the whole, the prisoners believe that an event,
so decreed by the judge, is surely inevitable, but they cannot know the point
of its occurrence along the race of time.
The judge claims he knows what the prisoners and their lawyer don’t
know; the lawyer claims she knows the judge cannot (logically) know what he
thinks he knows; the prisoners know they cannot know what the judge knows, in
spite of their lawyer’s refutation of the judge’s knowledge.
Is there
no happy meeting ground uniting such apparently incommensurable mind sets?
THE WAY OF LEARNED IGNORANCE
Yes,
there is a meeting ground of sorts. It
plays on the limitations of knowability, that is, on the incompleteness
and inconsistency of knowledge.
The
judge, of the Prisoner Paradox, thinks he can justifiably set the day of the
prisoners’ execution, but the lawyer has discovered an inconsistency in his
reasoning. The prisoners think they
know not the day of the execution, and even though the lawyer points out the
error of the judge’s ways, they are not deterred from their learned sort of
ignorance. They know their knowledge is
destined to remain radically incomplete, for between a given future time frame
and a past time frame, an instantiation of the present can always pop up within
which their doom becomes manifest. In
other words, at the very instant knowledge of the time of their execution is at
hand, they will be executed: their
knowledge will now be complete, but at the expense of their very
existence. Whichever day the judge
decides upon, an inconsistency will inhere.
Whatever the prisoners think, their knowledge will be incomplete. The lawyer thinks she has dissolved the
inconsistency by mentally strait-jacketing the judge and bringing the system to
completion by discarding the possibility of a decision: things will remain as they are,
timelessly. But the prisoners’
‘semiotically real’ world dictates otherwise, for the entire scheme is, from
whichever vantage, either inconsistent or incomplete--or perhaps both--up to
the instant their very existence is terminated. Each party, it would appear, is either right for the wrong
reasons or wrong for the right reasons.
The
reason behind all this madness is the following. The lawyer’s timeless realm of logic, when placed in the living
and breathing world of time-bound Seconds and Thirds, is not existentially
valid, for it allows of no temporality, the very stuff life is made of. So from the subjective world of the
prisoners, the lawyer’s form of logic is overdetermined: inconsistent signs are superposed as quite
unruly bed partners. The lawyer, in
contrast, wishes objectively to interject the timeless orb of her classical
logic into the actualized sphere of Seconds, which allows for neither
contradictory signs nor a proliferation of middles. But the lawyer’s logic, from within the prisoners’ own
existential world, is a time bomb ticking out their destiny. It remains for them in their concrete
living and breathing incomplete:
underdetermined. They cannot
know at what point in time the expected unexpected event of their death will
occur, though they think they know it will occur. When it does occur, their knowledge will have reached completion
and the uncertainty of proliferating temporal middles between the judge’s
decree and their execution will no longer exist. But all will have been to no avail, for they will be no more.
Of
course we would like to assume that such paradoxes are not ordinarily
pernicious and that we can always ‘jump out’ of the signs within which they are
dressed to specify whatever we wish: we
persist in our desire to think we are master of our signs. However, though we can occasionally exercise
a move from one system to another of greater complexity, we are often able to
manhandle that ‘lower’ system from what we imperiously believe to be our
‘metaperspective’. But we can usually
do so only insofar as our own system--ultimately the brainmind--is of
greater complexity than that ‘lower’ system, and above all, only insofar as by
some inconceivable stretch of the imagination it stands outside time. If not, like the Berry sentence or the
lawyer of the Prisoner Paradox, we run the risk of futilely attempting to
survey the unsurveyable, decide the undecidable, specify the unspecifiable,
know the unknowable.
That is
to say, given the sign fabricator and its interpreter--both hopeless
meaningmongers in the event that they are high-handed humans--what is taken out
of the sign is actually what was put there in the first place. What was put there is always subject, in
time, to some change of minor to radical sorts, and what is taken out, since
invariably incomplete, is always subject, also in time, to further possible
additions and deletions. In short, no corpus
of knowledge in the time-bound world of our severely restricted capacities can
be both entirely consistent and complete, though our thinking would like to
make it so. As an afterthought, the
assertions of the last few paragraphs will likely open me to the charge that I
am mixing time and timelessness. This
has been, of course, a perennial bugbear of Western metaphysics, and I harbor
no illusions of being able to resolve the issue in one fell swoop. What I am very modestly attempting to
illustrate by the prisoner and Berry parables is the impossibility of divorcing
what is presumed to be timeless, objective thought from the concrete
life-world. In other words, I am
attempting to unite mind and body, a topic the further development of which I
offer elsewhere (Merrell 1997a, 1997b, 1998a, 1998b).
And yet,
our thinking can to a greater or lesser extent be made to give the appearance
that what we think is the case as a mind-act divorced from the concrete world
is indeed the case. This making of our
thought and of our thought’s making what appears to be the case the case, at
least for us at a given space-time juncture, would seem germane to the
implications of what Peirce terms the ‘pragmatic maxim’. Proper development of the ‘maxim’ is
impossible here, given the time and space limitations of this paper. Nevertheless, it deserves at least a minimal
share of the spotlight at this juncture.
Peirce’s 1878 incarnation of the maxim says:
Consider what effects, that might conceivably have
practical bearings, we conceive the object of our conception to have. Then our conception of these effects is the
whole of our conception of the object. (CP:5.402; also 5.2, 5.9, 5.l8,
5.427, MS 327)
It
specifies that the meaning of a sentence regarding what appears to be the case
is the product of all conceivable consequences presented by other
sentences--and their own consequences--engendered from the original
sentence. This product of all
conceivable consequences entails the translation of the initial sign or
sentence into a series of conditional sentences the antecedents of each of
which prescribe certain interactions between the interpreter and the signs in
question. The consequences, ideally,
consist of observable sign phenomena that should or would make themselves
manifest in the event that the original signs or sentences are indeed ‘true’. But ‘truth’ is not really the goal. Rather, the task at hand is to draw meaning
from the signs being processed by way of interpreter-sign interaction. The interpreter takes the initial signs and
creates a hypothetical situation by imagining what would most likely ensue. Then she puts her hypothetical signs to the
test in terms of a thought experiment ‘in here’ or by interacting with the
signs’ objects ‘out there’ in order to see if she was right. If her hypothesis turns out to appear
correct for the time being, the possibility nonetheless remains that other hypotheticals
may at future moments present themselves, compelling her to repeat the
operation. If her initial hypothesis is
found deficient, then back to the drawing board for an alternative
hypothetical, in which case she also repeats the operation. And so on (Skagestad 1981).
By its
very nature this process leads to an increasingly complex series of
sentences. For example, the statement
‘This is salt’ in relation to some small crystals calls for the imaginative
construction of all the conceivable consequences in all conceivable
contexts. In fact, the inclusion of all
conceivable contexts knows no ultimate boundary, nor does it prioritize any
particular mode of knowing. It entails
an entire range of possible sentences drawn from the sphere of vagueness. These sentences stipulate what would most
likely occur if the salt were dissolved in water, dumped in the water softener,
sprinkled on a hamburger, used to treat a slab of pork, analytically subjected
to spectroscopy, qualitatively tested in the high-school laboratory for
chlorine and sodium, tossed on the city streets during a snowstorm, or
whatever. Thoroughly to interpret the
sentence in question--to arrive at its ultimate or final interpretant within
the sphere of generality--demands knowledge of the behavior of salt in all
possible situations and all possible contexts.
The ramifications are virtually limitless. Consequently, the plenitude of meaning regarding the initial
sentence, ‘This is salt’, given the plethora of subsidiary sentences derived from
it, is in its fullness of numbing complexity.
It exhausts our individual cognitive capacities and our capacity to know
our world. I write ‘our capacity to
know our world’ with concrete living in mind.
This includes signs of feeling as well as intellect, of intuition as
well as reason, of experience as well as of logic. That is, it includes signs of the body as well as of the
mind. But since I refuse to divorce
body and mind, it includes signs of bodymind, bodymindsigns
(Merrell 1998a, 1998b).[7]
In formal
systems (mathematical, geometry, logic, ‘computer thought’) the search is in
essence not for new ‘truths’ either, but for reasons unlike those of the
‘maxim’. The object of the quest,
rather than meaning, is for axioms with which to enrich the system by erasing
inconsistencies and moving toward completeness. Along complementary lines, the task of a human semiotic agent
engaged in the everyday affairs of natural language use is that of interpreting
signs--interacting with them in such a way that meaning emerges--in order to
enrich her conception of the semiosic whole. It is not simply a matter of adding new ‘facts’--new meanings--to
an already bloated corpus of theories and their putative evidence, but of
engendering new sentences capable of clarifying--and at times of
overthrowing--that corpus (Kline 1980).
Quite justifiably, in this vein, it has been observed that Gödel’s
theorem can be viewed as ‘a consequence of the limited complexity of any formal
arithmetic system, a limitation affecting human minds as well as machine
programs’ (Paulos 1985:99).
‘So it
might appear’, one wishes to retort, ‘but this hardly separates the chaff of
machines from the wheat of thinking and feeling humans; it merely illustrates
that a machine endowed with time-binding human attributes is remotely like a
human’. Fortunately, however, human praxis
is not limited to the rigorous arguments of strictly formal languages--as the
above observations on vagueness and generality suggest. In everyday human affairs we make decisions,
the consequences of which at the time and within that particular context seemed
plausible. But further experience often
serves to determine that that particular decision was not prudent after all. For the set of implications presented by the
whole system remained beyond our grasp, and we could therefore not be aware of
the degree of vagueness and generality, and inconsistency and incompleteness of
our knowledge of that system in terms of its future consequences. In retrospect we might be able to become
aware of the error of our ways, or we can congratulate ourselves on our
sagacity, but at the time our decision was made, we exercised intuitive
capacities the total ramifications of which we could not possibly have been
aware. In other words, we are tenderly
and rather helplessly fallible semiotic agents.
The
conclusion? Of course any and all
conclusions are problematic, when taking processual semiotics into
account. Yet at this juncture it seems
safe to suggest that actuality, vagueness, and generality,
coupled with inconsistency-incompleteness, affords at least a
premonition of the infinite series of conceivable consequences and their
products arising out of the implications inherent in a given sentence
engendered by the ‘pragmatic maxim’.
Vagueness marks the presence of an absent set of possible signs. Actuality is the presence of that which
constitutes signs relating to the ‘furniture’ of our ‘semiotically real’
world. And generality, a mediator and
moderator between the other two members of the triad, tones them, keeps them
within reasonable bounds, and synthesizes them. In concert, vagueness, actuality, and generality
not only pattern Peirce’s categories, they are also the motivating force behind
Peirce’s unfinished ‘logic’ of semiosic praxis, which is quite
compatible with much contemporary discourse, as we shall observe.
In conjunction with any disquisition on
vagueness and inconsistency and generality and incompleteness, Peirce’s
categories should be properly foregrounded before we move on.
Firstness is the possibility
of a sign’s becoming in the realm of Secondness, such becoming governed by the
mediating force of the mind by way of convention, habit, and all other
propensities lying in wait in the realm of Thirdness. Regarding this role of mind, given our human habits of thought,
it seems that acts of Firstness are invariably pervaded with ‘subjectivism’ and
‘idealism’, Secondness with ‘realism’, and Thirdness with ‘objectivism’ and
‘realism’. But these categories do not
correspond to disjunctive ‘realms’ at all.
They are mutually interdependent, a constantly folding in and over one
another. Their interdependence is
essential to their very nature as categories.
Thus Firstness without Secondness and Thirdness is nothing. Secondness without Firstness and Thirdness
is surely dead. And Thirdness without
Firstness and Secondness is well-nigh unthinkable. Together, when on their best of behavior, then stand tall;
divided, and they will surely fall.
Signs of
Firstness cannot but remain vague, and quite often inconsistent. Signs of Secondness, after emerging into the
light of day, can--albeit partly arbitrarily—take on what at the outset appear
to be crystal clear lines of demarcation.
But as particulars, their moment of glory cannot but be ephemeral. For they are destined to pass on into
something other than what they are/were, even though the differences between
each of their momentary flashes of existence are well-nigh infinitesimal--hence
the classical identity principle also runs the risk of falling by the
wayside. Signs of Thirdness, it is
assumed, must possess some form of continuity of existence. They are hopefully identical with themselves
from one moment to the next, and they can be distinguished from other signs in
terms of their character as generalities--though they cannot help being tinged
with some degree of vagueness, for they are never free of Firstness via
Secondness. But as generalities they
are destined to remain incomplete, since there will always exist the
possibility of other signs filling in the gaps between what had hitherto been
construed as a set of precise categories.
The upshot is that by and large there is a definite move toward some
sort of idealism in terms of sign generalities, yet, incompleteness there will
always be. Underdetermination is the
order of the day in this domain of generalities, since whatever sign happens to
be underdetermined at a given time and place, it could always have been
something other than what it is. As a rule
of thumb, overdetermination ultimately entails a superposition of all
possibilities without any of them having been actualized into Secondness;
underdetermination is the juxtaposition of what at a give slice in space-time
is considered ‘real’ and what is relegated to the status of ‘unreality’.
The
underdeterminationist assumption has it that intuitively we believe something
but not everything is ‘real’. Since we
cannot by empirical means discover what is ‘real’ without a shadow of a doubt,
the matter is left to our judgment, according to our persuasions and
propensities and wishes and whims.
Underdetermination implies incompleteness, for, what is ‘real’ could
always have been construed otherwise, and what is ‘unreal’ may yet stand some
outside chance of becoming ‘real’ at another time and another place. Underdetermination regarding scientific
theories stipulates that competing and equally legitimate theories--equally
legitimate from within their particular conceptual schemes, that is--can be
generated on the basis of the same set of observations. In this vein, at the turn of the century,
Pierre Duhem (1954) and Henri Poincaré (1952), and more recently, Nancy
Cartwright (1983) and Hilary Putnam (1983), argue that there will always be
equally satisfactory alternatives to a given theory or general theoretical
framework (paradigm). Consequently, no
single story can account for all the furniture of the world in one
fell-swoop. This is, in essence, the
Duhem-Quine scenario--in which Peirce is a principle actor, though his role in
this respect is often overlooked--predicated on the radical underdetermination
of theories (i.e. they are empirically equivalent but logically incompatible)
(see also Gähde and Stegmuller 1986, Roth 1987, Sacks 1989).
Quine (1969) is one of the more ardent
propagators of the underdeterminationist thesis--by way of Duhem’s
methodological ‘holism’ and the essentials of Peirce’s ‘pragmatic maxim’. He argues that a theoretical sentence in
physics can have the same underdetermined relation to experiments and
observation sentences that a sentence of natural language has to the observed
objects, acts, and events that it is about (Vuillemin 1986). He writes that since experience is never an
infallible adjudicator for rejecting or embracing individual theoretical
sentences, theoretical physics cannot be other than an interconnected web of
sentences, procedures, and formalisms in contact with the world only at its
edges, if at all. Any impact observation
sentences may have on the web becomes distributed throughout the web such that
no part of it is immune to change and no part stands alone in bearing the brunt
of that impact. Additions, deletions,
and adjustments of diverse sorts can often be made in the whole to accommodate
the experience, but there is no infallible or unique method for making these
adjustments. Four naturally occurring
elements or many of them, phlogiston or oxygen, Euclidean geometry or
Reimannian or Lobachevskyan geometry, Darwinian or Lamarckian evolution, all
during certain periods have been aided and abetted by proper ‘empirical
evidence’ from one perspective or another.
According to the dictates of a community’s desires, what now appear to
us as the most bizarre of theories could be, and at times have been, granted ‘truth
value’. And when fads, fashions, and
tastes have suffered from the introduction of alternatives, theories have
either followed suit, or they have served as stimuli for the most likely
candidates from among those alternatives.
Given
the nature of underdetermination, then, it is quite often possible to embrace
logically incompatible but empirically equivalent theories—albeit at different
times and in different places. As a
consequence, competing and mutually exclusive theories may always be available
to account for the observational data at hand.
Arguments for determining absolute ‘Truth’ are thus rejected: we can at best only know what we (think we)
know, for we can’t know whether what we know is infallibly ‘true’. That is, by Peircean refutation or Popperian
falsification, we can’t know that what we know is not ‘false’. So the dominoes are set up only to be
knocked down. Yet the hope persists in us that to all questions an answer can
eventually be found, otherwise there would hardly be any motivation for
continuing to play the game of inquiry.
In other words, thought can potentially cure all ills, though when put
into signs for communication with other semiotic agents, it often threatens to
become undecidable.
It would
appear, then, that our ideals are perpetually out of line with our real
capacities. Such is the general nature
of Peirce’s ‘objective idealism’, his ‘pragmatic maxim’, and his doctrine of
fallibilism. Thus we see with greater
force that overdetermination and underdetermination applies to the very idea of
fictionality, and especially to the inexorable fuzziness between fictions and
the ‘semiotically real’. The exact
quantity of gold in Pike’s peak, the cause of Hamlet’s dementia, the reason for
Napoleon’s decision at the Battle of Waterloo, Don Quixote’s height, the use of
Ö-1 in quantum theoretical
equations, the absolutely precise nature of the sun with respect to all other
entities in the firmament, are all underdetermined in that they are never so
complete as to be immune to further determination. In fact, all signs are to a greater or lesser degree
underdetermined, their ‘reality’ status or their fictionality status
notwithstanding. Consequently, a
community’s fabric of signs is read into experience, and in the process it becomes
the world that is, the ‘semiotically real’. ‘Semiotically real’ signs from diverse time periods and from a
variety of belief that are pregnant with meaning (‘mass’, ‘energy’,
‘Eucharist’, ‘Big Foot’, ‘Zeus’, ‘UFOs’, ‘mana’, ‘witches’, ‘AIDS’, ‘cholesterol’,
and the ‘Cross’ and ‘Swastika’) have become so impregnated because of the role
they play and the place they occupy in their respective interwoven semiosic
fabric. They do not describe
experience; they are ‘intersubjective idealizations’ of experience. Whether dressed in relatively concise and
complete abstract language or in everyday language and enshrouded in vagueness,
much of their meaning remains implicit.
Yet experience and the signs tentatively
painting it are often, and in spite of their agents’ wishes to the contrary,
caught up in a quandary comparable to that of the logical positivist’s
‘observation sentences’ and ‘theoretical sentences’, following their
objectivist ‘correspondence theory of truth’.
It was assumed that one could link up theory with experience by
specifying rules with which to bridge the gap between the two domains of
sentences, as in the case of identifying a statement about observable water
droplets with a statement about electrons--i.e. the Millikan experiment. The problem was that such identification was
always to a degree dependent upon an arbitrary operation. Lower-level theoretical statements regarding
the capillary properties of water that are easily observable can perhaps be
deduced and derived from higher-level theoretical statements regarding the
attraction of particles of opposite charge and molecular polarity. But statements concerning direct experience
cannot always be in the same fashion deduced and derived directly from
higher-level theoretical statements.
There will always remain a certain unbridgeable gap between everyday
lived experience and abstract theory, which testifies to the latter’s perpetual
incompleteness (companion to underdetermination in the sphere of semiotic
generality) and the former’s inevitable dovetailing with vagueness (companion
to overdetermination in the sphere of semiotic inconsistency). In other words, experience (of presumed
‘facts’, Seconds, entailing indexicality) is in a formal sense and at certain
points incompatible with overdetermination-underdetermination, since it adheres
to the tenets of classical logic.
THE EMPIRICISM-RATIONALISM DILEMMA AGAIN
After
all has been said and done, the overdetermination
(vagueness)-underdetermination (incompleteness) pair of terms is itself
perhaps most economically viewed as two complementary approaches toward
knowing what is (see especially CP:2.322-23).[8]
The two
approaches pattern the Heraclitus-Parmenides and Aristotle-Plato
antagonisms. In their purest form, one
is messy and unkempt; the other is orderly.
One is rich in the variety of its concrete particulars; the other is a
formal and parsominious. The one is a
maze of tropical flora; the other is a barren desert converted into a grid of
meticulously cultivated plots. Quite
significantly, along these lines, Pierre Duhem (1954), himself a sort of
hopeful Platonist, distinguished between these complementary approaches. They evince characteristics essentially the
equivalent of (1) experienced signs (empiricism, drawn from the overdetermined
sphere of vagueness, collected as Secondness, and accommodated to the mind as
Thirds) and (2) mental-signs (drawn from the sphere of Thirdness, but in
conjunction with experienced Seconds, they invariably remain underdetermined).[9]
The first
risks producing myriad details but relatively little conceptual depth, while
the second runs the opposite risk. The
first, in line with classical empiricism Duhem would like us to believe, is
generally the modus operandi of the English mind; the second,
hypothetico-deductive in nature, is quite typical of the French mind. The English scientist loves ‘facts’
(Seconds) and is relatively unconcerned with putting things into a tidy,
compact, formalized package: her
empiricism always manifests a tendency toward nominalism, though she
occasionally avoids that trap. The
French scientist strives for a unified, elegant, abstract grasp of the world
(conceptualization, Thirdness): she
tends toward, but does not necessarily fall victim to, realism (i.e. of the
rationalist-Platonic sort). The English
investigator attempts to hold a perplexing concoction of particulars
together. With these particulars she
hopes to coordinate the gears, pulleys, nuts, and bolts of her world into a
harmonious whole while imposing a minimum of abstract forms of organization on
them, and by all means, of resisting the temptation actively to intervene in
their affairs. The French scientist
considers that no laws is a legitimate law unless it is expressed in elegant,
abstract mathematics. The world of the
English scientist is much like a nonlinearly developing aperiodic fractal; that
of her French counterpart ideally bears the image of a beautifully cut crystal. Or, in the terms of the present inquiry, in
spite of her wishes to the contrary the first scientist has a penchant for
trying to jam-pack the world into the purest of generalities which are the
makings of her own mind. In contrast,
the second scientist begins with vague assumptions--or models--with the idea
that, subject to a few deletions here and some appendages there, her mental
construct will become a map of the world.
In
short, the French garner the dream of a deep holist account of
scientific theories, conceptual schemes, and even, we might surmise, modes of
discourse and artistic styles--though Duhem does not go quite this far. At its most extreme, holism stipulates that
theories, thoughts, and beliefs face the tribunal of experience and judgment
not on a sentence by sentence basis but as corporate bodies. All sentences, past, present, and
potentially to come, are interlocked into an intricate web without ready-made
structures, though they exact constraints on their speakers-writers. As a potential, these interconnections make
up a fabric without fixed joints or junctures.
The continuum’s ruptures are partly conventional, normative, and perhaps
necessary, but also at all times and all places there exists the possibility of
partly arbitrary cuts. From a
holist--and, I might add, antipositivist--point of view, theories, conceptual
schemes, and languages are so laden with prejudices and presuppositions that it
is virtually impossible to determine precisely which sentences divulge ‘facts’,
which fictions, and which merely fantasy or nonsense. Indeed, all sentences should perhaps be taken as provisionary
‘fictions’ eliminable, if at all, by some sort of move toward the ‘truth’.
From the
holistic view, all theories, conceptual schemes, and languages are radically
underdetermined: conflicting discourses
generated from conflicting views can account, with equal cogency and validity
from their respective views within their conceptual scheme, for the same items
of experience. This is both the beauty
and the bane of holism. There is no
foreseeable limit to the number of explanations with which to give account of a
particular corpus of data.
Consequently, any ardent effort to eliminate inconsistencies insofar as
possible--since they are a liability rather than an asset--is guided by the
hope that an alternative theory, conceptual scheme, or language, can be
found. For there must exist, following
this hope, a viable alternative somewhere ‘out there’, whether compatible,
incommensurable, or contradictory with the present theory, conceptual scheme or
language, that could originally have been found but was not. Consequently, for the ideal Duhemian holist
scientist of cognitive depth, nature is
never so completely known that it cannot be subject to further
inquiry. However, since, pace
Peirce, nature cannot be so known by any finite community of knowers, knowledge
is destined to remain incomplete, at best serving as a hopeful approximation to
‘truth’.
Take,
for instance, Boyle’s laws of gases.
They stipulate what ideally would be the case were certain conditions to
be in effect. All actual gases come
close to this ideal, some more than others.
But as of this writing no gas has been discovered that is absolutely
identical to the ideal. Moreover, no
matter how internally consistent they are, Boyle’s laws evince a degree of incompleteness. For
there is no way we can know with absolute certainty that at some
juncture or other new data will not pop up to demonstrate one or another of
their its inadequacies. The laws, to be
sure, offer themselves up as a general account of a particular aspect of
nature: they quite effectively
describe, and in a manner of speaking even explain, the domain for which they
were intended. But they offer little in
the way of understanding. In light of
the above on vagueness and generality, it might be said that for the holist,
Boyle’s laws are general insofar as ‘the principle of excluded middle does not
enjoy iron-clad application’, since the possibility always exists that some
other theory might appear that is equally or more adept at describing and
explaining the phenomena in question.
In contrast, the English scientist’s tentative assertions--as
actualizations from the realm of possibilia, of vagueness--are valid
only insofar as ‘the principle of contradiction does not necessarily
hold’. For, in the empirical world of
particulars, one must be mindful that things do not have the talent for
remaining absolutely identical to themselves for all time.
At
bottom line, the French scientist, with ultimate visions of cognitive depth,
can optimistically embrace ontological realism--coupled with a vague sort of
epistemological idealism--but only if she realizes final complete and
consistent knowledge will not be at hand.
In other words, since absolute depth of knowledge is a receding horizon,
indeterminacy cannot help but exercise its force. This must be the case, since there is no knowing whether a given
sentence relates faithfully to the ‘real’ or whether it merely appears to do
so, since it is conveniently coherent with all other sentences in the whole fabric. In contrast, the English scientist in search
of empirical breadth can adopt methodological realism--coupled with a
cloudy hint of ontological idealism--as long as she is aware that
inconsistencies will always be lurking at some unsuspected bend in the
road. That is, to repeat William
James’s (1950 I:287-88) words, since the mind ‘is at every stage a theatre of
simultaneous possibilities.... [i]t works on the data it receives very much as
a sculpture works on his block of stone’.
Since there are countless statues that could have been actualized in
place of the one that now exists, ‘the sculpture alone is to thank for having
extricated this one from the rest’. In
this manner, ‘the world for each of us, however different our several views of
it may be, all lay embedded in the primordial chaos [vagueness] of sensations,
which give the mere matter to the thought of all of us differently
[which at times bears on inconsistency]’ (brackets added).
In this
sense signs can be as indeterminate for Duhem’s Englishman as they are for his
Frenchwoman. On the one hand, at the
level of Firstness, of pure possibility, there is no determining absolutely a
sign’s meaning due to the elements of chance and spontaneity. And on the other hand, at the level of
Thirdness, of necessity via habit and convention, the sign’s meaning is
also indeterminate, given the impossibility of knowing absolutely and without a
shadow of a doubt whether a given meaning is ‘true’ and when that meaning will
not be exchanged for some other ‘truth’.
Just as a potential infinity of statues can be sculpted from the stone,
with no knowing precisely which one will have been sculpted at a further moment
down the line, so also the gap between experience and preconceived theory
cannot be absolutely bridged. For, as
Duhem (1954:l68-72) puts it, ‘we can make an infinity of different formulas or
distinct physical laws correspond to the same group of facts.... [A]ny other law representing the same
experiments with the same approximations may lay as just a claim at the first
to the title of a true law or, so to speak more precisely, of an acceptable
law’. Between theories (sentences)
along an unordered series, another account can always be found. On the other side of the coin, given the
infinite number of possible theories (sentences), contradictory accounts may
stand a democratic chance of being selected at different times and places. The indefinite and unlimited range of
possibilities cannot be crammed into the one and only ‘true’ finite theory
(sentence). Furthermore, over time,
given the same infinite possibilities of experience, inconsistent and
contradictory theories (sentences) can with little difficulty be used--albeit
at times erroneously--as catch-alls for the whole of experience (more on this
later, when the terrain has been properly tilled).
PEIRCE HAS HIS SAY, FINALLY
If we
combine Duhem’s two classes of scientists, we will have something akin to
Peirce’s indeterminacy of meaning and knowledge via his ‘objective
idealism’. This is a rarefied
combination of a practical, hands-on sort of methodological ‘realism’ and
ontological ‘idealism’ in conjunction with a vague, visionary nod toward
methodological ‘idealism’ and ontological ‘realism’--to say nothing of
‘subjectivism’ and ‘objectivism’. It is
a collusion of Peirce’s evolutionary cosmology coupled with his no-nonsense
‘realism’ tinged with ‘idealist’ metaphysics (Rescher and Brandom 1979). It is not merely a matter of the indeterminacy
of underdetermination according to Duhem’s French hedgehog in
contradistinction to the English fox.
Rather, the very existence of underdetermined theories (sentences)
depends upon the infinite possibilities of theories (sentences) that were there
for the taking from the overdetermined sphere of pure vagueness--raw experience
before it is mediated by the selecting mind.
The superposition
of possibilities--the sphere of Firstness--contains the full range of all
concrete items of the furniture of all possible worlds, past, present, and
future, a minuscule portion of which can be actualized (into Secondness) at a
particular time and place. This appears
prima facie to be an unruly concoction held together only tentatively at
best, and more often than not without stable rhyme or reason. Granted, the merging of contradictory possibilia
into one monstrous package apparently flies in the face of the best of logic
and of reason. But given the
underdetermination of theories, conceptual schemes, and languages, if they were
all piled one upon the other without limit, even the most outlandish idea could
stand a chance of becoming ‘real’ at some other time and place, and the most
logically cogent idea could become the most bizarre. This indeterminacy of overdetermination, complementary
with that of underdetermination, affords the notion of inconsistency
(vagueness) precariously coupled with incompleteness (generality).
Such,
ideally, would be the way of Duhem’s ‘French mind’ plus his ‘English mind’
striving respectively for cognitive depth and empirical breadth of
thought. For Duhem, thought maintains
the upper hand. The creator of his
universe is a Great Cogitator, the ultimate French mathematician. This omniscient seer begins with what is
hopefully the Absolute Posit, an unshakable cornerstone emerging from the
depths of her mind (Firstness, abduction), and she goes on to create a
beautifully elegant hypothetical construct (Third, deduction), capable of
accounting for everything that is (Secondness, induction). It is the scientist’s task to create a simulacrum
of that parsimonious construct our Grand Cogitator has left us, such simulacrum
being the ultimate sign of generality.
On the other hand, Duhem’s Anglophile mind inclines toward concrete
items of the world’s furniture. She
patterns her own creator’s untidy construction of the universe by quite random
hits and misses, finally accumulating a makeshift jumble (Seconds,
particulars), which, after being subjected to literally countless perceptual
grasps, can then somehow be collated and collected into certain tenuous
generalities (Thirds) hopefully forthcoming by way of instinctive leaps of the
imagination (Firsts). However, the sum
total of all descriptions and explanations of scientists from both traditions cannot
but be as overdetermined as they are underdetermined. For, from an ahistorical (overdetermined) view, conflicting
theories from different times and places can and have been embraced, and, given
a body of (underdetermined) data, there will always be equally satisfactory but
conflicting alternatives at other times and places. Overdetermination and underdetermination are apparently in open
competition with each other, yet they are not exactly an antagonistic, but
rather, a complementary, duo.
Peirce would perhaps be sympathetic
toward such a collusion of the Duhem holist-empiricist agonistics in his
obstinate effort to devise a method for determining ‘truth’, if only in the
‘theoretical long run’ of things. For
Peirce--and Karl Popper after him--the semiotic agent cannot directly know the
‘truth’, but only that which bears a degree of ‘untruth’. So she gets along quite well, now going with
the flow, now caught in an ephemeral whirlpool, now bucking a strip of unruly
whitewater, and all the while finding fault with successive bits and pieces of
conventional knowledge. With perhaps
more luck than management, she may exercise a few moves closer to the ideal,
though if she ever had it in its totality she would never know it. For, unable to see any ‘untruth’ in it, she
would be incapable of distinguishing it from some unknown standard to set
herself apart from it as a reactionary other in her dialogue with nature. In other words, the ideal could not be her
object of knowledge, for she would be inside it. It would be as if nature were saying to itself that which is
‘true’ of itself and of its very saying what it is: its saying would be part of the object of its saying.
This
paradox is implicit in Peirce’s infinite regressus and infinite progressus
regarding what Umberto Eco (1976) dubs his infinite semiosis. Given such indeterminacy, how can we know
where we began and where we are and where we are going? How can we realize any advance instead of
simply treading water? How can we know,
really know, instead of becoming aware at every step that what we thought we
knew we knew not. Peirce offers a few
tentative answers by way of his pragmatism, as a fallible and tentative
separator of the sheep of the properly ‘semiotically real’ from the goats of
the erroneous or merely fictive. To
embrace Peirce’s pragmatist, triadic concept of the sign is to embrace his
‘pragmatic maxim’. And to embrace the
maxim is by and large to forget about ‘truth’, capital ‘T’ Truth, take signs
and their meanings as they come, and get on with the game as best we can.
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[1] ‘Fuzzy logic’ has at least two chief sources over the past century. The first of these sources was initiated by Peirce in the form what he called a "logic of vagueness," the full development of which he held as a project for some future time that never arrived before his death. The concept of "vagueness" was later picked up by Max Black (1937), and has more recently become the focus of studies by Brock (1979), Engel-Tiercelin (1992), Merrell (1995a, 1996, 1997a, 1998a), and Nadin (1982, 1983), among others. The second source is an outgrowth of work with "fuzzy sets" in the 1960s and 1970s by Lofti Zadeh (1965, 1987). In a word, "fuzzy logic" reveals the sludge inherent in linguistic practices. As such, this new logic refuses to prioritize language over para-extra-linguistic modes: all communication is to a greater or lesser degree vague. It was, of all philosophers, the analytical Bertrand Russell (1923), who, in a paper on vagueness, suggested that language is invariably vague and that vagueness is a matter of degree.
[2] Firstness, Secondness, and Thirdness refer to Peirce’s three categories of thought. According to Peirce, any conceptual body of knowledge, no matter how complex, can be reduces to triadicity, but that triadicity cannot be further reduced without its suffering a loss. Although limited time and space do not permit my expounding on the categories, I trust their nature can be inferred within the context of my exposition (for further, see Savan 1987-88, Almeder 1980).
[3] For development of the notions of overdetermination and underdetermination and their relationship to the logical principles of noncontradiction and excluded-middle with respect to signs within broad cultural contexts, see Merrell (1998a, 1998b).
[4] I would like to believe that in Merrell (1998b) I have presented an effective case of signs and their various and sundry ‘logics’ regarding what is perhaps the most complex cultural milieu in our contemporary world, Latin American. In this study I suggest throughout that ‘cultural semio-logics’ are fabricated rather than discovered or coming from on high, they are invented rather than ready-made, and their interpretation depends upon a virtually incomprehensible array of possible perspectives within an indefinite number of possible contexts.
[5] The allusion here is to Peirce’s often maligned idea that science—and knowledge in general—is in a process asymptotically of approximating the truth (for a critique of Peirce’s convergence theory, see Rorty 1991; for a discussion of the pros and cons, Skagestad 1981; for a defense, Hausman 1993).
[6] Of course there exists a veritable spate of alternate ‘logics’, for example, three- and many-value logic, modal logic, dialectical logic, Buddhist logic, fuzzy logic, free logic, and, more in line with the premises underlying the present inquiry, Lupasco’s ‘logic of contradiction’ (1947), Melhuish’s ‘complementary contradictory logic’ (1967), Rescher and Brandom’s ‘logic of inconsistency’ (1979), and the ‘paraconsistent logic’ developed in Brazil (da Costa 1974), none of which I intend to pre-empt here. I wish merely to open the door to a smattering of the many possibilities revealed by Peirce.
[7] Admittedly, I am here offering an informal ‘ordinary language’ rendition of the ‘pragmatic maxim’ within the realm of ‘concrete living’, which goes against the grain of many considerations of Peirce’s philosophy, the focus of which customarily lies on the scientific, mathematical, and logical thrust of his thought (for a discussion of these formal aspects of the maxim, see especially Nesher 1983, 1990).
[8] I must acknowledge a debt to Nancy Cartwright (1983) and Mary Hesse (1966) for much of this section.
[9] Recent trends in the hard sciences seem to corroborate this view. The Platonic approach to understanding the universe, founded on the dream of invariant, symmetrical, laws has led to the search for a ‘grand unified theory’ [GUT], a ‘theory of everything’ [TOE]. Since the 1970s an Aristotelian alternative of sorts has been on the rise in the form of ‘chaos theory’, Prigogine’s ‘physics of complexity’, and Mandelbrot’s ‘fractals’. This alternative lays emphasis on observable happenings of everyday life rather than unobservable invariants behind them. Time and change have as a consequence become paramount [see Toulmin 1990:192].)