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20th WCP: Conflict without Contradiction: Noncontradiction as a Scientific Modus
Operandi
Logic and Philosophy of
Logic
Conflict without Contradiction:
Noncontradiction as a Scientific Modus Operandi
Don Faust
Northern Michigan University
dfaust@nmu.edu
ABSTRACT: We explicate the view that our ignorance of
the nature of the real world R, more so than a lack of ingenuity or sufficient time to
have deduced the truth from what is so far known, accounts for the inadequacies of our
theories of truth and systems of logic. Because of these inadequacies, advocacy of
substantial correctness of such theories and systems is certainly not right and should be
replaced with a perspective of Explorationism which is the broadest possible investigation
of potential theories and systems along with the realization that all such theories and
systems are partial and tentative. For example, the position of classical logic is clearly
untenable from the perspective of explorationism. Due to ignorance regarding R and,
consequently, the partial and evidential nature of our knowledge about R, an
explorationist foundational logical framework should contain machinery which goes beyond
that of classical logic in the direction of allowing for the handling of confirmatory and
refutatory evidential knowledge. Such a foundational framework (which I call Evidence
Logic) is described and analysed in terms of its ability to tolerate substantial
evidential conflict while not allowing contraditions.
0. Overview
The variegated landscape of theories of truth and systems of logic, wherein each is
cogently argued while yet inconclusive, is substantially accounted for by the fact that we
just don’t know enough yet about the nature of our universe, let us call it R, to be
able to settle on one or the other of these theories and systems as adequate for the
representation and processing of our knowledge about R. In this paper firstly we discuss
this thesis, that it is primarily our ignorance of R, and not any failure to rigorously
construct our theories and systems, that is a fundamental cause of the inadequacies of
these theories and systems. Secondly we will delineate a scientific perspective, Explorationism,
which, if the thesis first considered is correct, is deserving of advocacy. Finally, we
exemplify this perspective by exhibiting a logic, Evidence Logic (EL), which incorporates
a broadened concept of negation which (1) provides for the representation and processing
of both confirmatory and refutatory evidential knowledge including the possibility of a
generous range of conflicting evidence while yet (2) enforces noncontradiction.
1. The inadequacy of our theories of truth and systems of logic
Any survey of the gamut of theories of truth so far constructed makes clear that, while
each may be presented cogently, each manages to tell only part of the story. That story
may be thought to begin naïvely with Aristotle’s well-known observation, "to
say of what is that it is or of what is not that it is not, is true; to say of what is
that it is not or of what is not that it is, is false". But quickly things become
complicated, and theories in the main either largely retreat from R into elaborate
linguistic accounts which in the end tell us little about how the truth of a sentence
(which is in some sense supposed to be speaking about R) is to be ascertained or
plunge unwarrantedly into accounts which assume far more than is indeed known about R and
so in the end provide a construction whose utility is brought seriously into question by
the tenuous character of its hypotheses about the nature of R.
In contrast, a clear virtue of Tarski’s explication of the matter is his avoidance
of both extremes, his concentration on the center of the issue: he sets out the T-scheme
clearly while succeeding in avoiding a labyrinth of linguistic complications, and he
explicates Aristotle’s naïve correspondence theory clearly while succeeding in
avoiding the morass of our ignorance of R by deftly defining satisfaction in terms of a
crisp mathematical notion of structure (a nonempty set together with appropriate n-ary
relations defined on it).
So each theory is helpful in one direction or another, while none can be said to give a
complete explication of the concept of truth: each is only at most a partial explication
and all have essential inadequacies traceable to the fact that our ignorance of R remains
great. As Russell remarked in concluding his insightful paper "Vagueness"
[Australasian J. of Philosophy and Psychology 1923, pp. 91-92]: "My own belief is
that most of the problems of epistemology, in so far as they are genuine, are really
problems of physics and physiology …". The partial nature of each theory
involves areas unaddressed by the theory. For example, both coherence and correspondence,
even if taken together, are clearly only part of the story. Further, our theories are in
many ways tentative. For example, even the object-predicate approach itself, ubiquitous in
our modern constructions, builds tentativeness into our theories of truth to the extent
that, although they seem to work well often, we don’t really know enough about R to
know to what degree they correctly model R. In fact, to the extent that a theory of truth
is timid in not reaching out substantially toward R, it is partial although possibly less
tentative, while to the extent that the theory boldly reaches toward R, it is tentative
although possibly less partial.
In summary, then, inadequacies there clearly are, in our theories of truth. Our
theories are partial only, and they are tentative; and, to a large extent, this is because
of our substantial ignorance of R. Facing this fact openly is healthy. For it adds to a
theory to be clearer about its limitations, and it helps us to move on to further
improvements if we are clearer about the ways in which the theory is only partial and
tentative.
Turning to the consideration of systems of logic, since theories of truth are involved
in one way or another in the foundational aspects of any system of logic, to the extent
that a logical system is thereby considered part of a realization of some conception(s) of
truth, that system is partial and tentative as well. When we construct systems of logic
having, say, objects and (intensional or extensional) predicates, and we find that with
some degree of success the system models some aspects of our meagre understanding of R, we
are indeed building models which embrace some sense of the nature of R, but we should be
careful not to delude ourselves with the fallacious notion that we have in any absolute
sense actually described R itself. (Oh, maybe we have, a tiny bit; but surely we have a
long way to go.)
Classical logic is a good case in point. Its two-valued semantics declare a sentence
‘true’ if what it says is the case and ‘false’ if what it says is not
the case, and this is-the-caseness is supposed to obtain or not obtain in the structures
where the semantics is realized. Well, clearly those structures are not absolutely known
entities in R since we don’t know that much about R. Taking, for example, Tarskian
semantics, embracing as well a Kripke semantics to handle modalities if one wishes, these
structures are just neat mathematical constructs, and the partial efficacy we meet with in
applying such a logic results from the fact that these mathematical constructs fairly well
model R to the superficial extent to which the constructs of the logic penetrate the
nature of R.
Some insight into what is going on here may be gotten by recalling the
Ehrenfeucht-Fraisse games for elementary equivalence. Two structures are n-elementarily
equivalent if no n-sentence (a sentence with at most n alternations of quantifiers)
distinguishes them, that is if no n-sentence holds in one but not the other. If m>n, an
m-sentence potentially penetrates a structure somewhat further than does an n-sentence,
uncovering more of the structure’s nature / uniqueness. However, it is well to keep
in mind that the m-sentence’s penetration may yet be quite superficial, for example
in cases where the structure is of sufficient depth that there are in fact k-sentences for
arbitrarily large k which continue to strictly more fully elucidate the structure.
Analogously, a richer logic may be better suited to further uncover the nature of R and
hence the richer logic will be of some help, yet the further penetration the richer logic
provides may be in fact paltry in comparison with the actual depth of R. Of course, it may
also be the case that the logic’s added richness is in directions which are in fact
divergent from the nature of R, in which case use of the logic will lead to flags of
incongruity being raised, which information may also be helpful, although in admittedly
limited ways, in disclosing further the structure of R.
So the current plethora of nonclassical logics is generally a healthy phenomenon,
attempting to broadly reach out toward R with a wide variety of systems of logic, giving
us an ever-growing mass of evidence about that beast we wish to understand. Temporal,
multi-valued, fuzzy, Dempster-Shafer, and paraconsistent logics are all examples of worthy
contributors to the cause. But, as with theories of truth, it is well to remember that
each has its inadequacies, that each is both partial and tentative so long as our
ignorance of R remains great.
2. Explorationism
Let us refer to the maintenance of the thesis briefly discussed in Section 1 as Explorationism.
Explorationism embraces broad investigation of a wide variety of theories of truth and
systems of logic, each attempting to penetrate certain clearly delineated aspects of the
target beast R while being vigilant to openly declare known ways in which the theory or
system is partial and tentative. Although we could look to any number of theories or
systems to exemplify this point of view, let us focus on the foundational problem area of
what base logic might be a fruitful choice from an explorationist point of view. Of
course, tradition and Occam’s razor have contributed much to entrench classical logic
as the base logic of choice. But, as we saw in Section 1, given our great ignorance of R,
this choice is clearly untenable given the absolutist view of our knowledge which is
manifested in the semantics of classical logic. In contrast, an Explorationist Base Logic
(EBL) should clearly incorporate machinery for the representation of evidential knowledge,
knowledge which is gradational and which also allows for both the confirmatory and the
refutatory.
Recalling that it is our considerable ignorance of R that molds the fundamental
character of the explorationist position, let us consider the concept of an EBL and its
relation to the long-term evolutionary development of our knowledge of R. For a long time
now, most would certainly grant that at least since the Upper Paleolithic, man has been
focusing fairly consistently on trying to understand R better. Further, as we look into
the future, certainly we will continue to improve our grasp of R. If we never come to a
complete understanding of R, then an EBL will remain useful forever. On the other hand, if
we eventually do come to a complete understanding of R, then the question arises as to
whether that complete understanding of R is workably representable. If it isn’t, an
EBL will remain useful forever. On the other hand, if it is, then the question arises as
to whether that workably representable complete understanding of R is processible. If it
isn’t, again an EBL will remain forever useful. On the other hand, if it is, then the
question arises as to whether that processible workably representable complete
understanding of R is in fact two-valued. If it isn’t, yet again an EBL will remain
forever useful. On the other hand, if it is, then an EBL will only have been useful during
the intervening time frame, but a rather long period involving certainly thousands of
years, one would suspect, ending when we come to attain that processible workably
representable complete understanding of R and R is in fact two-valued. Hence, in any case
an EBL would seem a good investment as a base logic for our march toward fathoming R.
3. Evidence Logic (EL)
To illustrate some of the possibilities for such an explorationist base logic, let us
consider Evidence Logic (EL) [J. Symbolic Logic 1994, pp. 347-348 (abstract)]. In EL,
predications are evidential: they are subscripted, with either c or r depending
respectively on whether the predication is confirmatory or refutatory, and annotated with
an evidence level e where e is in an Evidence Space En = {i/(n-1): i =
1,…,n-1} of evidence levels (n fixed, n>1). In practice, two competing factors in
the application domain primarily determine the choice of n: first, data with finer
granularity, the packeting of which determines evidence valuations, tend to require a
larger value of n; and second, capacities of the computer-based implementation of the
logic impose an upper bound on n. In addition, added to any usual set of logical axioms
are axioms which assure that ‘stronger evidence strictly entails weaker
evidence’. Thus, for example, where P is a 0-ary predicate symbol, Pc:e
asserts that there is evidence at level e which is confirmatory of P while Pr:e
asserts evidence at level e refutatory of P. Note the increased primitive expressivity of
EL over classical logic. For example, distinction between "absence of evidence"
and "evidence of absence" is realized in EL: NOT Pc:e is an assertion
of the former type, Pr:e an assertion of the latter type. Models of EL are
similarly equipped, providing annotated confirmatory and refutatory relations interpreting
each predicate symbol. Clearly EL exhibits features one would expect to find in an
explorationist base logic, allowing for example crisp representation of strongly
conflicting evidence while maintaining noncontradiction. The syntactic machinery in EL
allows us to represent just that type of knowledge we have as we grope toward R, namely
evidential knowledge. Further, the semantic machinery, unlike that of classical logic,
provides intermediate, evidential structures helpful to us as we iteratively form our
partial and tentative approximative models of R.
In spite of this increased expressivity, it is interesting to note, for Quineans
desiring simplicity, that EL has a Boolean algebraic structure. In fact [loc. cit.] the
Boolean Sentence Algebras of EL, which vary according to the number and arities of the
various predicate and function symbols stipulated, can be completely characterized in
terms of isomorphism types of languages of classical logic by making use of powerful
techniques developed by Bill Hanf and Dale Myers for such analyses of classical logic.
Further, one can find [B. Symbolic Logic (two abstracts, to appear)], in axiomatizable
extensions of EL, a variety of logics which reach out to grapple with many complexities,
for example those involved in the concept of negation, including logics recently designed
to address the knowledge representation and knowledge processing problem area of
Artificial Intelligence (AI). That is, EL is indeed foundational in the explorationist
sense, in that it provides a framework wherein axiomatizable extensions provide a mosaic
of important, although often even pairwise incompatible, systems; in such a framework much
that is in need of study, for example the commonalities and differences between these
systems, is precisely laid out and, due to the common underlying framework, susceptible to
analysis. For example, a logic which models the logic of privatives, which Aristotle found
so fruitful in ferreting out some of the finer distinctions involved in the concept of
negation, is realizeable as a simple axiomatizable extension of EL. Similarly,
Dempster-Shafer logics and other logics which model some of the important distinctions
being made in logics addressing AI knowledge representation problems can also be realized
as axiomatizable extensions of EL. In fact, an analysis made possible by the unifying
framework of EL shows in a precise way that part of the Dempster-Shafer work is a
generalization of part of the work of Aristotle on privatives.
4. Conclusion
We have argued that our substantial ignorance of R implies the inadequacy of our
theories of truth and systems of logic: they are surely only partial and tentative. A
viewpoint of Explorationism is therefore seen as appropriate, which emphasizes this
partial and tentative nature of our theories and systems. Explorationism further
maintains that a change in base logic is needed, to one which goes beyond classical logic
in including machinery for the representation and processing of evidential confirmatory
and refutatory knowledge. Finally, the example explorationist base logic of Section 3,
Evidence Logic (EL), shows that it is indeed possible to construct such a logic which
allows the representation and processing of strongly conflicting evidence while retaining
a simple Boolean algebraic structure and maintaining noncontradiction.
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