THE THEOLOGICAL ANTHROPOLOGY OF RALPH WENDELL BURHOE

Abstract. A central aim of Ralph Wendell Burhoe's scientific theology is to define and interpret the meaning of human existence in relation to “ultimate reality.” As such, it can be understood as an exercise in theological anthropology. For Burhoe, this ultimate reality is “nature,” understood as the total reality system which is studied by the sciences and which the sciences are showing to be the sole determiner of the way things are. This article discusses various aspects of Burhoe's theological anthropology, as well as its value and credibility, and raises questions concerning his understanding of the value of the individual and the problems of evil and human sinfulness.     

The Incompleteness of Theories of Games

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.     

Partially Interpreted Relations and Partially Interpreted Quantifiers

Logics in which a relation R is semantically incomplete in a particular universe E, i.e. the union of the extension of R with its anti-extension does not exhaust the whole universe E, have been studied quite extensively in the last years. (Cf. van Benthem (1985), Blamey (1986), and Langholm (1988), for partial predicate logic; Muskens (1996), for the applications of partial predicates to formal semantics, and Doherty (1996) for applications to modal logic.) This is not so with semantically incomplete generalized quantifiers which constitute the subject of the present paper. The only systematic study of these quantifiers from a purely logical point of view, is, to the best of my knowledge, that by van Eijck (1995). We shall take here a different approach than that of van Eijck and mention some of the abstract properties of the resulting logic. Finally we shall prove that the two approaches are interdefinable.     

An Incomplete Relevant Modal Logic

The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley–Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that characterizes G. (iii) The class of frames for G characterizes the classically based logic T.     

Naming and Diagonalization, from Cantor to Godel to Kleene

We trace self-reference phenomena to the possibility of namingfunctions by names that belong to the domain over which thefunctions are defined. A naming system is a structure of theform (D, type( ),{ }), where D is a non-empty set; for everya D, which is a name of a k-ary function, {a}: D^k D is thefunction named by a, and type(a) is the type of a, which tellsus if a is a name and, if it is, the arity of the named function.Under quite general conditions we get a fixed point theorem,whose special cases include the fixed point theorem underlyingGödel's proof, Kleene's recursion theorem and many othertheorems of this nature, including the solution to simultaneousfixed point equations. Partial functions are accommodated byincluding "undefined" values; we investigate different systemsarising out of different ways of dealing with them. Many-sortednaming systems are suggested as a natural approach to generalcomputatability with many data types over arbitrary structures.The first part of the paper is a historical reconstruction ofthe way Gödel probably derived his proof from Cantor'sdiagonalization, through the semantic version of Richard. Theincompleteness proof–including the fixed point construction–resultfrom a natural line of thought, thereby dispelling the appearanceof a "magic trick". The analysis goes on to show how Kleene'srecursion theorem is obtained along the same lines.     

Mechanism, Truth, and Penrose's New Argument

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas–Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject.     

Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.     

The Sense of Incompleteness as a Motivator of Obsessive-Compulsive Symptoms: An Empirical Analysis of Concepts and Correlates

Contemporary models of obsessive-compulsive disorder emphasize the importance of harm avoidance (HA) and related dysfunctional beliefs as motivators of obsessive-compulsive (OC) symptoms. Recently, there has been a resurgence of interest in Janet’s (1908) concept of incompleteness (INC) as another potentially important motivator. Contemporary investigators define INC as the sense that one’s actions, intentions, or experiences have not been properly achieved. Janet defined INC more broadly to include alexithymia, depersonalization, derealization, and impaired psychological mindedness. We conducted two studies to address four issues: (a) the clinical correlates of INC; (b) whether INC and HA are distinguishable constructs; (c) whether INC predicts OC symptoms after controlling for HA; and (d) the relative merits of broad versus narrow conceptualizations of INC. Study 1 was a meta-analysis of the clinical correlates of narrowly defined INC (16 studies, N = 5,940). INC was correlated with all types of OC symptoms, and was more strongly correlated with OC symptoms than with general distress. Study 2 (N = 534 nonclinical participants) showed that (a) INC and HA were strongly correlated but factor analytically distinguishable; (b) INC statistically predicted all types of OC symptoms even after controlling for HA; and (c) narrow INC was most strongly correlated with OC symptoms whereas broad INC was most strongly correlated with general distress. Although the findings are limited by being correlational in nature, they support the hypothesis that INC, especially in its narrow form, is a motivator of OC symptoms.     

Visual,Tactile, and Auditory “Not Just Right” Experiences: Associations With Obsessive-Compulsive Symptoms and Perfectionism

“Not just right” experiences (NJREs), or uncomfortable sensations associated with the immediate environment not feeling “right,” are thought to contribute to obsessive-compulsive disorder (OCD) symptomatology. The literature suggests that NJREs are experienced across sensory modalities; however, existing in vivo measures have been restricted to visual inductions (e.g., viewing and/or rearranging a cluttered table). The present study used a large undergraduate sample (N = 284) to examine 4 in vivo tasks designed to elicit and assess NJREs across separate sensory modalities (i.e., visual, tactile, and auditory). Task ratings (discomfort evoked, and urge to counteract task-specific stimuli) were uniquely associated with self-report measures of NJREs, OC symptoms (ordering/arranging, checking, and washing), and certain maladaptive domains of perfectionism (doubts about actions, and organization). Findings have implications for experimental research and clinical work targeting NJREs specific to particular senses.