Genetically complete yet authorless artworks seem possible, yet it is hard to understand how they might really be possible. A natural way to try to resolve this puzzle is by constructing an account of artwork completion on the model of accounts of artwork meaning that are compatible with meaningful yet authorless artworks. However, I argue that such an account of artwork completion is implausible. Therefore, I leave the puzzle unresolved.

Given a conditional sentence “${\varphi}\Rightarrow \psi$" (if ${\varphi}$ then $\psi$) and respective facts, four different types of inferences are observed in human reasoning: Affirming the antecedent (AA) (or modus ponens) reasons $\psi$ from ${\varphi}$; affirming the consequent (AC) reasons ${\varphi}$ from $\psi$; denying the antecedent (DA) reasons $\neg\psi$ from $\neg{\varphi}$; and denying the consequent (DC) (or modus tollens) reasons $\neg{\varphi}$ from $\neg\psi$. Among them, AA and DC are logically valid, while AC and DA are logically invalid and often called logical fallacies. Nevertheless, humans often perform AC or DA as pragmatic inference in daily life. In this paper, we realize AC, DA and DC inferences in answer set programming. Eight different types of completion are introduced, and their semantics are given by answer sets. We investigate formal properties and characterize human reasoning tasks in cognitive psychology. Those completions are also applied to commonsense reasoning in AI.

The use of συντɛλέω to speak of God's ‘completion’ of the new covenant (Heb 8.8) has generated various explanations. Yet none of them factor in an important clue in Hebrews, namely, the rest discourse. By establishing literary and theological connections between Heb 3.7–4.13 and 8.8–12, this study argues that the promise of the completion of the new covenant evokes the completion of creation and its ensuing sabbath rest. Such an evocation brings to surface a logic of Christology and new creation embedded in Hebrews.

Theory of stable models is the mathematical basis of answer set programming. Several results in that theory refer to the concept of the positive dependency graph of a logic program. We describe a modification of that concept and show that the new understanding of positive dependency makes it possible to strengthen some of these results.

In this paper we consider the algebraic crossed product ${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field with involution and $C_{K}(X)$ denotes the $K$-algebra of locally constant $K$-valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on ${\mathcal{A}}$ by means of full ergodic $T$-invariant probability measures $\unicode[STIX]{x1D707}$ on $X$. To do so, we present a general construction of an approximating sequence of $\ast$-subalgebras ${\mathcal{A}}_{n}$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $\ast$-algebra ${\mathcal{A}}$ into ${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on ${\mathcal{A}}$ by restricting the unique one defined on ${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on ${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure $\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset $U\subseteq X$ must equal the measure of $U$.

For a division ring $D$, denote by ${{\mathcal{M}}_{D}}$ the $D$-ring obtained as the completion of the direct limit $\underset{\to n}{\mathop \lim }\,{{M}_{{{2}^{n}}}}(D)$ with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial $D$-ring $B$ and any non-discrete extremal pseudo-rank function $N$ on $B$, there is an isomorphism of $D$-rings $\overline{B}\,\cong \,{{\mathcal{M}}_{D}}$, where $\overline{B}$ stands for the completion of $B$ with respect to the pseudo-metric induced by $N$. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for $*$-algebras over fields $\text{F}$ with positive definite involution, where the algebra ${{\mathcal{M}}_{\text{F}}}$ is endowed with its natural involution coming from the $*$-transpose involution on each of the factors ${{M}_{{{2}^{n}}}}\,(F)$.

For a prime $p$, let $\hat{F}_{p}$ be a finitely generated free pro-$p$-group of rank at least $2$. We show that the second discrete homology group $H_{2}(\hat{F}_{p},\mathbb{Z}/p)$ is an uncountable $\mathbb{Z}/p$-vector space. This answers a problem of A. K. Bousfield.

Based on the metrisation of $b$-metric spaces of Paluszyński and Stempak [‘On quasi-metric and metric spaces’, Proc. Amer. Math. Soc.137(12) (2009), 4307–4312], we prove that every $b$-metric space has a completion. Our approach resolves the limitation in using the quotient space of equivalence classes of Cauchy sequences to obtain a completion of a $b$-metric space.

We compute coherent presentations of Artin monoids, that is, presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier’s and Knuth–Bendix’s completions into a homotopical completion–reduction, applied to Artin’s and Garside’s presentations. The main result of the paper states that the so-called Tits–Zamolodchikov 3-cells extend Artin’s presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.

We investigate the relationship between the generalization of program completion defined in 1984 by Lloyd and Topor and the generalization of the stable model semantics introduced recently by Ferraris et al. The main theorem can be used to characterize, in some cases, the general stable models of a logic program by a first-order formula. The proof uses Truszczynski's stable model semantics of infinitary propositional formulas.

This paper is the first step in thesolution of the problem of finite completion of comma-free codes.We show that every finite comma-free code is included in afinite comma-free code of particular kind, which we called, for lack of a better term, canonical comma-free code. Certainly, finite maximal comma-free codesare always canonical. The final step of the solution which consistsin proving further that every canonical comma-free code is completedto a finite maximal comma-free code, is intended to be published in a forthcomingpaper.

This paper is a sequel to anearlier paper of the present author, in which it was proved thatevery finite comma-free code is embedded into a so-called (finite)canonical comma-free code. In this paper, it is proved that every(finite) canonical comma-free code is embedded into a finite maximal comma-freecode, which thus achieves the conclusion that every finite comma-freecode has finite completions.


A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.

As shown by the author in {\em Proc.\ Amer.\ Math.\ Soc.} 115 (1992) 345–352, for every metric space $(K,d)$ with compact closed balls one has $(\mbox{lip}\,\varphi(K))^{**} = \mbox{Lip} \,\varphi (K),$ where $\varphi$ is any majorant (that is, non-decreasing function on ${\Bbb R}_+$ with $\varphi (0+) = \varphi (0) = 0$) such that $\varphi(t)/t$ monotonically tends to $+\infty$ as $t\rightarrow 0.$ Here $\mbox{Lip}\,\varphi(K)$ is the Lipschitz space on $K$ with respect to the metric $\varphi(d),\ \mbox{lip}\,\varphi(K)$ is the corresponding ’little‘ Lipschitz space of functions vanishing ’at infinity‘, and ’=‘ means ’canonically isometrically isomorphic‘. The main idea of the proof consisted of finding a normed space $M$ such that $M^* = \mbox{Lip}\,\varphi(K)$ and $M^c = (\mbox{lip}\,\varphi(K))^*,$ where ’$c$‘ stands for the completion, and identifying $M$ with the space of Borel measures on $K$ equipped with the Kantorovich norm. In the present paper, this argument is carried over to generalized Lipschitz spaces on ${\Bbb R}^n$ defined in terms of higher order differences. For an integer $k$ and for a majorant $\varphi$ with $\lim_{t \rightarrow 0}\varphi(t)/t ^k = +\infty,$ define $\Lambda ^k_\varphi$ to be the space of all bounded functions $f$ on ${\Bbb R}^n$ such that for some constant $C,\ \omega_k(f\,;t) \leq C\varphi(t)$ for all $t\geq 0$, where $\omega_k(f\,;\cdot)$ is the $k$th modulus of continuity of $f$. Let $\lambda ^k_\varphi$ be the (closed linear separable) subspace in $\Lambda ^k_\varphi$ which consists of functions $f$ vanishing at ’infinity' and such that $\lim_{t \rightarrow 0}\omega _k(f\,; t)/\varphi (t) = 0.$ We introduce an appropriate analogue $\Vert \cdot \Vert_{k,\varphi}$ of the Kantorovich norm on the space $M$ of finite Borel measures on ${\Bbb R}^n$ with compact support. The properties of this norm for $k \geq 2$ are significantly different from those of the Kantorovich norm ($k = 1$) which reflects the difference in analytic nature of generalized Lipschitz spaces versus classical ones. However, the core of the duality theory survives, and it is shown that $(M, \Vert\cdot \Vert _{k, \varphi})^* = \Lambda ^k _\varphi,\ \ (M, \Vert \cdot \Vert _{k,\varphi})^c = (\lambda ^k _\varphi)^*$ and consequently, $(\lambda ^k _\varphi )^{**} = \Lambda ^k _\varphi$. Several applications of these results are discussed, and a few open problems are formulated.

1991 Mathematics Subject Classification: 46E15, 46E35.