Introduction to Logical Structures

© 2001, 2013 Willem Esterhuyse. All rights reserved.

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Published by AuthorHouse 12/28/2012

ISBN: 978-1-4817-8080-3 (e)

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CONTENTS

1. Introduction

2. Basic Structure Elements

3. Information, Ideas, Concepts, Propositions and Structure Conventions

4. Knowledge, Logical Processes and Operations

5. Structure Composition and Validity Rules

6. Arguments, Rules of inference for Object-Concept Structures

7. Proofs

8. Using Structural Logic

Appendix 1: Enclosure list

Appendix 2: Relatiom list

Appendix 3: Analysis 1

Appendix 4: Puzzles Solutions

Appendix 5: Relation properties

Bibliography

1. Introduction

The logical symbols introduced in this book may be used to construct logical structures. It can also be used to solve logical problems. It is also useful as a means to record the steps you used to solve a logical problem and can be used to express advanced mind maps that includes logical relations between information and to express knowledge structures (see chapter 4, essential for every student) or to express contexts of concepts. The symbols also serve as a starting point for philosophical reasoning.

Classical Logic (CL) can be translated into logical structures with the symbols introduced here. In this system it is no longer necessary to reserve letters of the alphabet for certain logical elements. The specification of variables, connectives and quantifiers are not of the same type.

Structural Logic (SL) is more suited to the organization of data, data types and links between data. SL is easier to follow because it is not represented in a single series (relevant to natural language sentences and propositional logic formulas). It is also easier to not lose the thread/structure of what is being expressed in SL especially when the formulas become complex. SL is less restrictive than CL.

The organizational ability of CL is limited and unrecognized. CL is also compartmentalized and this fact is not explicit. CL is like building a pyramid with bricks, where the bottom bricks don’t touch yet. The bottom bricks of the pyramid are at a low level of organization while the top of the pyramid is at a high level of organization. SL is definitely superior to CL with regard to its informational organization capability.

SL starts building at the top of the organizational pyramid. SL starts with symbols and through application of consistency breaks down the logical structure/arrangement into more detail. As soon as a method of application is fixed it can be used and applied. The key method of SL is for fixing structure types and the inferences on them and then testing their usefulness.

Of course logical structures must make sense, be free from trivialities, ambiguousness and circularity and must be accurate and consistent. SL assumes and is capable of expressing a high level of organization of information.

The symbols for relations introduced in this book, except three, are original.

All except two of the symbols for enclosures are original here.

Every chapter introduces enclosures and links/relations that can be used with them.

2. Basic Structure Elements

Starting at the most fundamental level requires the definition of the elements in terms of the known symbol (=).

The first few enclosures are:

Figure 2.1

A conceptual class is a class that can be equivalent to a set of enclosure of the same type. Class Enclosures may contain names relatable to its elements. I use the words class and set interchangedly. All enclosures whether they have names or not are assumed to be non-empty.

See Structure 3.1 for how these enclosures are used.

Any enclosure may be empty when its content is irrelevant. An enclosure may contain words (constants), other symbols, a variable or a reference name or a formula (in CL or another format). When naming an enclosure with contents, a Label Enclosure may be used. An enclosure obviously can’t contain a name when it is attached to a label. A name to an enclosure or structure is only referable to inside a single structure, or in the paragraph with the reference to the structure, unless it starts with an acronym followed by a colon, followed by a name e.g. KN: NAME. For what is regarded as a single structure see Chapter 3.

No link is also a data container (enclosure).

The relations in SL (with symbols, developed in this text) are mainly mental or informational, not physical.

A Label Enclosure looks like this:

Figure 2.2

An example of labeled Element Of . . . Enclosure, with contents looks like this:

Figure 2.3

Any enclosure may be labeled by drawing the enclosure, with the Label Enclosure touching the enclosure on any side or linked by a Relevance link (figure 2.4). When labeling an entire structure, simply put the label inside the region of the structure. For an enclosure with no contents the name may be written inside the enclosure (as in figure 2.6).

An enclosure is to be thought of as a packet that may contain data defining its properties to belong to a certain logical mode, each enclosure with its own set of usage rules. In other words the enclosure type specifies what properties may be coupled to its contents.

The basic links are:

Figure 2.4

In this system links represents the relations between enclosures and their contents. These links take the place of predicate letters in Classical Logic. Links may be drawn horizontally or vertically.

A Relevance link is used between enclosures plus contents that is relevant to each other. A structure with only empty enclosures and relevance links simply states that some contents (for all enclosures) making the structure useful is findable i.e. the structure may exist.

A Relevance link is symmetrical (relation holds in both directions). More generally any link with no arrow on it or with two arrows pointing both ways on it is symmetrical and those with one arrow (filled or unfilled) on it is not. A Relevance link is negatable. It is