Situation-Theoretic Account of Valid Reasoning with Venn Diagrams

1996 ◽  
Author(s):  
Sun-Joo Shin
Keyword(s):  
Semantic Analysis ◽  
Formal System ◽  
Venn Diagram ◽  
Formal Proofs ◽  
Representation System ◽  
Venn Diagrams ◽  
Linguistic Representation ◽  
Deductive Systems ◽  
Representation Systems ◽  
Mathematical Tradition

Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. Since elementary school we have been taught how to draw Venn diagrams for a problem, how to manipulate them, how to interpret the resulting diagrams, and so on. However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs. This is just a reflection of a general prejudice against visualization which resides in the mathematical tradition. With this bias for linguistic representation systems, little attempt has been made to analyze any nonlinguistic representation system despite the fact that many forms of visualization are used to help our reasoning. The purpose of this chapter is to give a semantic analysis for a visual representation system—the Venn diagram representation system. We were mainly motivated to undertake this project by the discussion of multiple forms of representation presented in Chapter I More specifically, we will clarify the following passage in that chapter, by presenting Venn diagrams as a formal system of representations equipped with its own syntax and semantics:. . . As the preceding demonstration illustrated, Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules of manipulating them. . . . We think it should be possible to give an informationtheoretic analysis of this system, . . . . In the following, the formal system of Venn diagrams is named VENN. The analysis of VENN will lead to interesting issues which have their ana logues in other deductive systems. An interesting point is that VENN, whose primitive objects are diagrammatic, not linguistic, casts these issues in a different light from linguistic representation systems. Accordingly, this VENN system helps us to realize what we take for granted in other more familiar deductive systems. Through comparison with symbolic logic, we hope the presentation of VENN contributes some support to the idea that valid reasoning should be thought of in terms of manipulation of information, not just in terms of manipulation of linguistic symbols.

scholarly journals Experiencer as quasi-agent: «from meaning to the linguistic form»

2020 ◽  
Vol 40 ◽  
pp. 112-136
Author(s):  
М.А. Fomina ◽  
Keyword(s):  
Semantic Analysis ◽  
Linguistic Analysis ◽  
Functional Approach ◽  
Conceptual Category ◽  
Linguistic Representation ◽  
Linguistic Form ◽  
Grammatical Subject

The paper focuses on the category of semantic subject within the framework of a functional approach to linguistics. The variety of roles subject may have in a sentence accounts for the radially structured category of subject. With the agent subject being the center of the category, other members – Possessor, Experiencer, Neutral, etc. – appear to be scattered within the syntactical category of subject being more central or peripheral. The paper deals with the Experiencer subject. The author stresses the key role of a well-elaborated metalanguage in linguistic analysis and assumes that a thorough analysis of the relevant conceptual category, its structure and content, should precede the stage of developing a metalanguage. The paper 1) differentiates between similar though not interchangeable notions such as semantic subject, grammatical subject, and the bearer of predicative feature, 2) features the peripheral status of the Experiencer within the category of semantic subject, 3) reveals the means of its linguistic representation, 4) makes a structural and semantic analysis of the models with the Experiencer.

Sets and Venn diagrams.

2021 ◽  
pp. 85-94
Author(s):  
Donald Quicke ◽  
Buntika A. Butcher ◽  
Rachel Kruft Welton
Keyword(s):  
Venn Diagram ◽  
Venn Diagrams

Abstract This chapter focuses on sets and Venn diagrams. Venn diagrams, also known as set diagrams, are commonly used to represent the overlap between sets. However, there is no in-built Venn diagram function in R so packages are used.

scholarly journals Novel Binary Signed-Digit Addition Algorithm for FPGA Implementation

2019 ◽  
Vol 29 (09) ◽  
pp. 2050136
Author(s):  
Yuuki Tanaka ◽  
Yuuki Suzuki ◽  
Shugang Wei
Keyword(s):  
Field Programmable Gate Array ◽  
High Speed ◽  
Number System ◽  
Logic Circuit ◽  
Fpga Implementation ◽  
Number Representation ◽  
Representation System ◽  
Representation Systems ◽  
Field Programmable ◽  
Gate Array

Signed-digit (SD) number representation systems have been studied for high-speed arithmetic. One important property of the SD number system is the possibility of performing addition without long carry chain. However, many numbers of logic elements are required when the number representation system and such an adder are realized on a logic circuit. In this study, we propose a new adder on the binary SD number system. The proposed adder uses more circuit area than the conventional SD adders when those adders are realized on ASIC. However, the proposed adder uses 20% less number of logic elements than the conventional SD adder when those adders are realized on a field-programmable gate array (FPGA) which is made up of 4-input 1-output LUT such as Intel Cyclone IV FPGA.

Generalized Euler-Venn Diagrams for Fuzzy Sets

2020 ◽  
Vol 7 (4) ◽  
pp. 34-43
Author(s):  
Yu. Mironova
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Venn Diagram ◽  
Linguistic Variables ◽  
Venn Diagrams ◽  
Time Of Year ◽  
Special Case ◽  
Three Dimensional Space

The fuzzy set concept is often used in solution of problems in which the initial data is difficult or impossible to represent in the form of specific numbers or sets. Geo-information objects are distinguished by their uncertainty, their characteristics are often vague and have some error. Therefore, in the study of such objects is introduced the concept of "fuzziness" — fuzzy sets, fuzzy logic, linguistic variables, etc. The fuzzy set concept is given in the form of membership function. An ordinary set is a special case of a fuzzy one. If we consider a fuzzy object on the map, for example, a lake that changes its shape depending on the time of year, we can build up for it a characteristic function from two variables (the object’s points coordinates) and put a certain number in accordance with each point of the object. That is, we can describe a fuzzy set using its two-dimensional graphical image. Thus, we obtain an approximate view of a surface z = μ(x, y) in three-dimensional space. Let us now draw planes parallel to the plane. We’ll obtain intersections of our surface with these planes at 0 ≤ z ≤ 1. Let's call them as isolines. By projecting these isolines on the OXY plane, we’ll obtain an image of our fuzzy set with an indication of intermediate values μ(x, y) linked to the set’s points coordinates. So we’ll construct generalized Euler — Venn diagrams which are a generalization of well-known Euler — Venn diagrams for ordinary sets. Let's consider representations of operations on fuzzy sets A a n d B. Th e y u s u a l l y t a k e : μA B = min (μA,μB ), μA B = max (μA,μB ), μA = 1 − μA. Algebraic operations on fuzzy sets are defined as follows: μ A B x μ A x μ B x ( ) = ( ) + ( ) − −μ A (x)μ B (x), μ A B x μ A x μ B x ( ) = ( ) ( ), μ A (x) = 1 − μ A (x). Let's construct for a particular problem a generalized Euler — Venn diagram corresponding to it, and solve subtasks graphically, using operations on fuzzy sets, operations of intersection and integrating of the diagram’s bars.

scholarly journals SEMANTIC POLLUTION AND SYNTACTIC PURITY

2015 ◽  
Vol 8 (4) ◽  
pp. 649-661 ◽  
Author(s):  
STEPHEN READ
Keyword(s):  
Modal Logic ◽  
Formal System ◽  
Logical Constants ◽  
Deductive Systems ◽  
Labelled Deductive Systems

AbstractLogical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically pure as any formal system in which the rules define the meanings of the logical constants.

scholarly journals Venn Diagrams with Few Vertices

10.37236/1382 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Bette Bultena ◽  
Frank Ruskey
Keyword(s):  
Lower Bound ◽  
Planar Graph ◽  
Connected Region ◽  
Venn Diagram ◽  
Closed Curves ◽  
Venn Diagrams ◽  
Intersection Points ◽  
Few Vertices

An $n$-Venn diagram is a collection of $n$ finitely-intersecting simple closed curves in the plane, such that each of the $2^n$ sets $X_1 \cap X_2 \cap \cdots \cap X_n$, where each $X_i$ is the open interior or exterior of the $i$-th curve, is a non-empty connected region. The weight of a region is the number of curves that contain it. A region of weight $k$ is a $k$-region. A monotone Venn diagram with $n$ curves has the property that every $k$-region, where $0 < k < n$, is adjacent to at least one $(k-1)$-region and at least one $(k+1)$-region. Monotone diagrams are precisely those that can be drawn with all curves convex. An $n$-Venn diagram can be interpreted as a planar graph in which the intersection points of the curves are the vertices. For general Venn diagrams, the number of vertices is at least $ \lceil {2^n-2 \over n-1} \rceil$. Examples are given that demonstrate that this bound can be attained for $1 < n \le 7$. We show that each monotone Venn diagram has at least ${n \choose {\lfloor n/2 \rfloor}}$ vertices, and that this lower bound can be attained for all $n > 1$.

scholarly journals ggVennDiagram: An Intuitive, Easy-to-Use, and Highly Customizable R Package to Generate Venn Diagram

2021 ◽  
Vol 12 ◽  
Author(s):  
Chun-Hui Gao ◽  
Guangchuang Yu ◽  
Peng Cai
Keyword(s):  
Open Source ◽  
Open Source Software ◽  
Source Code ◽  
R Package ◽  
Venn Diagram ◽  
Free Access ◽  
High Quality ◽  
Venn Diagrams ◽  
Label Edge ◽  
The Cost

Venn diagrams are widely used diagrams to show the set relationships in biomedical studies. In this study, we developed ggVennDiagram, an R package that could automatically generate high-quality Venn diagrams with two to seven sets. The ggVennDiagram is built based on ggplot2, and it integrates the advantages of existing packages, such as venn, RVenn, VennDiagram, and sf. Satisfactory results can be obtained with minimal configurations. Furthermore, we designed comprehensive objects to store the entire data of the Venn diagram, which allowed free access to both intersection values and Venn plot sub-elements, such as set label/edge and region label/filling. Therefore, high customization of every Venn plot sub-element can be fulfilled without increasing the cost of learning when the user is familiar with ggplot2 methods. To date, ggVennDiagram has been cited in more than 10 publications, and its source code repository has been starred by more than 140 GitHub users, suggesting a great potential in applications. The package is an open-source software released under the GPL-3 license, and it is freely available through CRAN (https://cran.r-project.org/package=ggVennDiagram).

Undergraduate Students’ Combinatorial Proof of Binomial Identities

2021 ◽  
Vol 52 (5) ◽  
pp. 539-580
Author(s):  
Elise Lockwood ◽  
Zackery Reed ◽  
Sarah Erickson
Keyword(s):  
Undergraduate Students ◽  
Teaching Experiment ◽  
Combinatorial Proof ◽  
Representation System ◽  
Representation Systems ◽  
Symbolic Reasoning ◽  
Binomial Identities

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

Towards a Model Theory of Venn Diagrams

1996 ◽  
Author(s):  
Eric Hammer ◽  
Norman Danner
Keyword(s):  
Deductive Reasoning ◽  
Logical Consequence ◽  
Formal System ◽  
Semantic Content ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Venn Diagrams ◽  
Linguistic Representations ◽  
Finite Set

One of the goals of logical analysis is to construct mathematical models of various practices of deductive inference. Traditionally, this is done by means of giving semantics and rules of inference for carefully specified formal languages. While this has proved to be an extremely fruitful line of analysis, some facets of actual inference are not accurately modeled by these techniques. The example we have in mind concerns the diversity of types of external representations employed in actual deductive reasoning. Besides language, these include diagrams, charts, tables, graphs, and so on. When the semantic content of such non-linguistic representations is made clear, they can be used in perfectly rigorous proofs. A simple example of this is the use of Venn diagrams in deductive reasoning. If used correctly, valid inferences can be made with these diagrams, and if used incorrectly, they can be the source of invalid inferences; there are standards for their correct use. To analyze such standards, one might construct a formal system of Venn diagrams where the syntax, rules of inference, and notion of logical consequence have all been made precise and explicit, as is done in the case of first-order logic. In this chapter, we will study such a system of Venn diagrams, a variation of Shin’s system VENN formulated and studied in Shin [1991] and Shin [1991a] (see Chapter IV of this book). Shin proves a soundness theorem and a finite completeness theorem (if ∆ is a finite set of diagrams, D is a diagram, and D is a logical consequence of ∆ , then D is provable from ∆ ). We extend Shin’s completeness theorem to the general case: if ∆ is any set of diagrams, D is a, diagram, and D is a logical consequence of ∆. then D is provable from ∆. We hope that the fairly simple diagrammatic system discussed here will help motivate closer study of the use of more complicated diagrams in actual inference.

scholarly journals Proving infinitary formulas

2016 ◽  
Vol 16 (5-6) ◽  
pp. 787-799 ◽  
Author(s):  
AMELIA HARRISON ◽  
VLADIMIR LIFSCHITZ ◽  
JULIAN MICHAEL
Keyword(s):  
Propositional Logic ◽  
Formal System ◽  
Answer Set Programming ◽  
Logic Programs ◽  
Deductive Systems ◽  
Answer Set

AbstractThe infinitary propositional logic of here-and-there is important for the theory of answer set programming in view of its relation to strongly equivalent transformations of logic programs. We know a formal system axiomatizing this logic exists, but a proof in that system may include infinitely many formulas. In this note we describe a relationship between the validity of infinitary formulas in the logic of here-and-there and the provability of formulas in some finite deductive systems. This relationship allows us to use finite proofs to justify the validity of infinitary formulas.

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