THE DIVINE CONJECTURES: A CONTEMPORARY ACCOUNT OF HUMAN ORIGINS AND DESTINY

Six “divine conjectures” frame the place of Theóne (The One to Whom we pray) in the creation of our universe and for its continuing development in five subsequent stages into a loving universe. The first stage, the cosmological universe, establishes the laws of nature, understood by scientists as the “standard model”. The second stage introduces life and death into the universe by a process we are only now beginning to understand. Stage 3 requires certain life forms to become conscious with a subset of those life‐forms acquiring language that results in that subset becoming self‐conscious. The next stage, Conjecture 4, identifies certain persons who become addicted to learning in their unrelenting effort to learn as much of what can be known as possible. The fifth conjecture requires individual persons to act as agents of Theóne in achieving Conjecture 6—a universe that is both loving and lawful. During the course of the exposition subsidiary discussions of the concepts of conjecture and hypothesis explicate the function of each in the advancement of knowledge and understanding. There are brief discussions of prayer and purpose in relation to the Divine.     

A counterexample to Fishburn's conjecture on finite linear qualitative probability

Kraft, Pratt and Seidenberg (Ann. Math. Statist. 30 (1959) 408) provided an infinite set of axioms which, when taken together with de Finetti's axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C₄,…,C_k is representable. By the work of Kraft, Pratt, and Seidenberg (1959) and Fishburn (J. Math. Psychol. 40 (1996) 64; J. Combin. Design 5 (1997) 353), it is known that n-1?f(n)?n+1 for all n?5. Also Fishburn proved that f(5)=4, and conjectured that f(n)=n-1 for all n?5. In this paper we confirm that f(6)=5, but give counter-examples to Fishburn's conjecture for n=7, showing that f(7)?7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn's conjecture.     

Naïve Comprehension and Contracting Implications

In his paper [6], Greg Restall conjectured that a logic supports a naïve comprehension scheme if and only if it is robustly contraction free, that is, if and only if no contracting connective is definable in terms of the primitive connectives of the logic. In this paper, we present infinitely many counterexamples to Restall's conjecture, in the form of purely implicational logics which are robustly contraction free, but which trivialize naïve comprehension.     

Symmetry, direct measurement, and Torgerson's conjecture

Two forms of direct measurement are considered in the article: a strong form in which ratio productions named by number words are interpreted veridically as the numerical ratios they name; and a weak form in which the ratio productions named by number words may have interpretations as ratios that are different from numerical ratios they name. Both forms assume that the responses to instructions to produce ratios are represented numerically by ratios, and thus the word “ratio”—and supposedly the participants concept associated with it—is being “directly” represented. The strong form additionally “directly represents” the number mentioned in the instruction by itself. The article provides an axiomatic theory for the numerical representations produced by both forms. This theory eliminates the need for assuming anything is being “directly represented,” allowing for a purely behavioral approach to ratio production data. It isolates two critical axioms for empirical testing. An measurement-theoretic explanation is provided for the puzzling empirical phenomenon that subjects do not distinguish between ratios and differences in a variety of direct measurement tasks.     

Complete Infinitary Type Logics

For each regular cardinal , we set up three systems of infinitary type logic, in which the length of the types and the length of the typed syntactical constructs are < . For a fixed , these three versions are, in the order of increasing strength: the local system ₍₎, the global system g₍₎ (the difference concerns the conditions on eigenvariables) and the -system ₍₎ (which has anti-selection terms or Hilbertian -terms, and no conditions on eigenvariables). A full cut elimination theorem is proved for the local systems, and about the -systems we prove that they admit cut-free proofs for sequents in the -free language common to the local and global systems. These two results follow from semantic completeness proofs. Thus every sequent provable in a global system has a cut-free proof in the corresponding -systems. It is, however, an open question whether the global systems in themselves admit cut elimination.