In the Circle of Sense and Nonsense: Through a Mathematical Model of Meaning

The terms sense and nonsense indicate important phenomena in social environment. In this paper, we explore the relations and interactions between sense and nonsense with the following three-fold goal: 1. Elaborations of the formal definitions of these concepts. 2. Investigation of the processes of making sense from nonsense. 3. Establishment of the mathematical foundations for a sense-nonsense theory. On this way, we extend the dyad sense-nonsense by introducing the concept no-sense situated between sense and nonsense and by extending the concept sense to the concept poly-sense. In addition, we construct a mathematical model of these concepts and related processes using logical structures such as logical calculus, logical variety and logical prevariety. This approach correlates with the opinion of Leonardo da Vinci who wrote “No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof.”

Reasoning about Durations

A set of assertions is consistent provided they can all be true at the same time. Naive individuals could prove consistency using the formal rules of a logical calculus, but it calls for them to fail to prove the negation of one assertion from the remainder in the set. An alternative procedure is for them to use an intuitive system (System 1) to construct a mental model of all the assertions. The task should be easy in this case. However, some sets of consistent assertions have no intuitive models and call for a deliberative system (System 2) to construct an alternative model. Formal rules and mental models therefore make different predictions. We report three experiments that tested their respective merits. The participants assessed the consistency of temporal descriptions based on statements using “during” and “before.” They were more accurate for consistent problems with intuitive models than for those that called for deliberative models. There was no robust difference in accuracy between consistent and inconsistent problems. The results therefore corroborated the model theory.

A resolution calculus for MinSAT

The logical calculus for SAT are not valid for MaxSAT and MinSAT because they preserve satisfiability but not the number of unsatisfied clauses. To overcome this drawback, a MaxSAT resolution rule preserving the number of unsatisfied clauses was defined in the literature. This rule is complete for MaxSAT when it is applied following a certain strategy. In this paper we first prove that the MaxSAT resolution rule also provides a complete calculus for MinSAT if it is applied following the strategy proposed here. We then describe an exact variable elimination algorithm for MinSAT based on that rule. Finally, we show how the results for Boolean MinSAT can be extended to solve the MinSAT problem of the multiple-valued clausal forms known as signed conjunctive normal form formulas.

An algebraic analysis of categorical syllogisms by using Carroll’s diagrams

In this paper, we analyze the algebraic properties of categorical syllogisms by constructing a logical calculus system called Syllogistic Logic with Carroll Diagrams (SLCD).We prove that any categorical syllogism is valid if and only if it is provable in this system. For this purpose, we explain firstly the quantitative relation between two terms by means of bilateral diagrams and we clarify premises via bilateral diagrams. Afterwards, we input the data taken from bilateral diagrams, on the trilateral diagram. With the help of the elimination method, we obtain a conclusion that is transformed from trilateral diagram to bilateral diagram. Subsequently, we study a syllogistic conclusion mapping which gives us a conclusion obtained from premises. Finally, we allege valid forms of syllogisms using algebraic methods, and we examine their algebraic properties, and also by using syllogisms, we construct algebraic structures, such as lattices, Boolean algebras, Boolean rings, and many-valued algebras (MV-algebras).

Deductive Analysis in NTA

This chapter mostly describes implementation of logical inference by means of NTA. Besides the known logical calculus methods, NTA logical inference procedures can include new algebraic methods for checking correctness of a consequence or for finding corollaries to a given axiom system. Above feasibility of certain substitutions, these inference methods consider inner structure of knowledge to be processed thus providing faster solving of standard logical analysis tasks.