Conflict without Contradiction: Don Faust

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 wellknown 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 Tscheme 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 nary 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. 9192]: "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 objectpredicate 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 twovalued semantics declare a sentence ‘true’ if what it says is the case and ‘false’ if what it says is not the case, and this isthecaseness 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 EhrenfeuchtFraisse games for elementary equivalence. Two structures are nelementarily equivalent if no nsentence (a sentence with at most n alternations of quantifiers) distinguishes them, that is if no nsentence holds in one but not the other. If m>n, an msentence potentially penetrates a structure somewhat further than does an nsentence, uncovering more of the structure’s nature / uniqueness. However, it is well to keep in mind that the msentence’s penetration may yet be quite superficial, for example in cases where the structure is of sufficient depth that there are in fact ksentences 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 evergrowing mass of evidence about that beast we wish to understand. Temporal, multivalued, fuzzy, DempsterShafer, 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 longterm 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 twovalued. 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 twovalued. 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. 347348 (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 E_{n} = {i/(n1): i = 1,…,n1} 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 computerbased 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 0ary predicate symbol, P_{c}:e asserts that there is evidence at level e which is confirmatory of P while P_{r}: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 P_{c}:e is an assertion of the former type, P_{r}: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, DempsterShafer 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 DempsterShafer 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. 