On the Syntax of Logic and Set Theory

We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.Comment: 34 pages, accepted, to appear in the Review of Symbolic Logi

On Existential First Order Queries Inference on Knowledge Graphs

Reasoning on knowledge graphs is a challenging task because it utilizes observed information to predict the missing one. Specifically, answering first-order logic formulas is of particular interest because of its clear syntax and semantics. Recently, the query embedding method has been proposed which learns the embedding of a set of entities and treats logic operations as set operations. Though there has been much research following the same methodology, it lacks a systematic inspection from the standpoint of logic. In this paper, we characterize the scope of queries investigated previously and precisely identify the gap between it and the whole family of existential formulas. Moreover, we develop a new dataset containing ten new formulas and discuss the new challenges coming simultaneously. Finally, we propose a new search algorithm from fuzzy logic theory which is capable of solving new formulas and outperforming the previous methods in existing formulas

Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models

The $\epsilon$-logic (which is called $\epsilon$E-logic in this paper) of Kuyper and Terwijn is a variant of first order logic with the same syntax, in which the models are equipped with probability measures and in which the $\forall x$ quantifier is interpreted as "there exists a set $A$ of measure $\ge 1 - \epsilon$ such that for each $x \in A$, ...." Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational $\epsilon \in (0, 1)$, respectively $\Sigma^1_1$-complete and $\Pi^1_1$-hard, and ii) for $\epsilon = 0$, respectively decidable and $\Sigma^0_1$-complete. The adjective "general" here means "uniformly over all languages." We extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability by and validity over finite models in $\epsilon$E-logic are, i) for rational $\epsilon \in (0, 1)$, respectively $\Sigma^0_1$- and $\Pi^0_1$-complete, and ii) for $\epsilon = 0$, respectively decidable and $\Pi^0_1$-complete. Although partial results toward the countable case are also achieved, the computability of $\epsilon$E-logic over countable models still remains largely unsolved. In addition, most of the results, of this paper and of Kuyper and Terwijn, do not apply to individual languages with a finite number of unary predicates. Reducing this requirement continues to be a major point of research. On the positive side, we derive the decidability of the corresponding problems for monadic relational languages --- equality- and function-free languages with finitely many unary and zero other predicates. This result holds for all three of the unrestricted, the countable, and the finite model cases. Applications in computational learning theory, weighted graphs, and neural networks are discussed in the context of these decidability and undecidability results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger Kuype

Discursive and Non-Discursive Design Processes

This research study investigates the hypothesis that Space Syntax plays a role in enhancing architectural design as a knowledge-based process by bringing the nondiscursive design process onto a discursive level, and by making explicit the logic of processing, evaluating, and reasoning about design. In order to establish an evidencebased argument for this hypothesis the study will scrutinize the performances and outcomes of architects solving a well-defined problem. The paper constructs the study on a literature background exploring the different theories which were concerned with the analysis and evaluation of design processes and outcomes. The analysis of design processes was investigated on micro and macro scales and the evaluation of solutions was considered in terms of spatial configurations and the social organization embodied in space. The research then goes on to apply some of these analytical studies to a set of design tasks made by architects who have a background in Space Syntax theory, and architects with other architectural backgrounds. The question then turns to the influence of Space Syntax theory on the strategies and cognitive actions of the design processes and the observational study will attempt to prove whether the knowledge of Space Syntax can have a positive effect on architects during their design process, taking into consideration that Space Syntax, as a morphic language, can render the non-discursive discursive of architecture. In the following step the design solutions are evaluated in terms of qualities regarding social organization, and in terms of quantities measuring the values of their spatial configurations. The analysis of the design processes and outcomes will show differences between the two groups of architects, in addition to some individual differences between the architects. Thus this research proves that the knowledge of space syntax may partially enhance the productivity of design process by making it more explicit