__Strategies
for Studying, Learning, and Researching__

__By
David Alderoty © 2014__

__Chapter 6)Types of
Reasoning: Visual, Verbal, Mathematical, and Deductive & Inductive Reasoning
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you want to go to the previous chapter,__

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Access Additional Information____ with Hyperlinks__

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__After I complete a writing task, I
select a number of websites from other authors, to provide additional
information, alternative points of view, and to support the material I wrote.__
These websites contain __articles__, __videos,__ and other useful
material. The websites can be accessed by clicking on the hyperlinks, which
are the **blue underlined words**, presented at the end of **some of the**
sections, subsections, and paragraphs. If a link fails, use the blue
underlined words as a search phrase, with www.Google.com
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__Reasoning: Introductory Concepts__

** **

A good understanding of logic and reasoning is helpful for mastering courses that involve writing, mathematics, science, critical thinking, and philosophy. In this chapter, I present some conventional and unconventional concepts on various types of reasoning. This includes visual, verbal, and mathematical reasoning, as well as a detailed discussion of deductive and inductive reasoning.

** **

** **

__Reasoning for Comprehension, and Reasoning
for Solving Problems, and Creating Entities__

Based
on the way am using the terminology, reasoning involves two concepts, which I
am calling ** reasoning input** and

__Reasoning input____ ____involves
comprehending information, and recognizing any logical relationships or
fallacies that it may contain.__ This process takes place
when we truly understand the material we are reading, listening to, and/or
viewing. This deep level of understanding facilitates learning and memory.

__Reasoning output____
involves creating and/or organizing entities, based on a set of rules or
the requirements to obtain a goal or solve a problem.__ The
entities can be physical structures, information, logical configurations,
words, mathematical concepts, visual elements, or anything else. The

Playing a game, especially if it involves logic

Creative thinking

Mathematical or logical concepts

Writing an essay, a term paper, a business report, poetry, a novel, or a computer program

Any type of mathematics, logic, or reasoning, including deductive or inductive reasoning

** **

** **

I
am dividing thinking and problem solving into three categories, which are __visual
reasoning__, __verbal reasoning__, and __mathematical reasoning__. There
is some overlap between the three categories. For example, certain types of
mathematical reasoning also involve visual and/or verbal reasoning.

If
we apply the concept presented in the previous subsection to the three
categories there is an ** input** and

__Visual Reasoning,
and Related Concepts__

__Visual Reasoning for Comprehension,
& Problem Solving__

Based
on the way I am using the terminology, visual reasoning involves two concepts,
which I am calling __visual reasoning input__, and __visual reasoning
output__.

__Visual reasoning input____
involves comprehending visual relationships, shapes, and logical configurations
presented in a visual format.__

__Visual reasoning output____
involves arranging, shaping, or building objects, based on a set of rules, or
on a specific objective.__ This includes modifying structures,
and problem solving that is based on visible entities.

People that have been blind since birth can carry out visual reasoning input and output, with the sense of touch. Individuals that became blind after they were born might have the ability to create visual images in their mind, based on the sense of touch. This involves interpreting visual relationships, and changes in visual configurations, with the sense of touch. Sighted people might also use the sense of touch under certain conditions, to assist them with visual reasoning.

The following are examples of visual reasoning, but some of the items in this list, might also require mathematical or verbal reasoning:

__EXAMPLES
OF VISUAL REASONING INPUT__

Comprehending the logical concepts and relationships in any type of diagram or blueprint

Inspecting and comprehending any visual entity, such as a work of art, a map, or any three dimensional structure

Examining and comprehending the components and related dynamics of a mechanical or electrical device

Recognizing any entity, with the aid of vision

__EXAMPLES
OF VISUAL REASONING OUTPUT__

Arranging toy blocks into a configuration that resembles a house

Creating a blueprint on paper, to build a real house

Playing a game of checkers, chess, or tic-tac-toe

Creating an organizational chart, with the lines of communication and authority

Creating a visual depiction of a flow pattern of electricity, data, or fluid, from one component to another

Creating pottery with clay

Creating a sculpture with marble

Any construction or engineering project (This includes building and/or designing an entity, such as an electronic circuit, an automobile, a jet plane, a house, or skyscraper.)

__Verbal Reasoning
and Related Concepts__

__Verbal Reasoning for Comprehension,
& Problem Solving__

Based
on the way I am using the terminology, verbal reasoning involves two concepts,
which I am calling __verbal reasoning input__, and __verbal reasoning
output__.

__Verbal reasoning input____
involves comprehending written and/or spoken language, and identifying logical
relationships, and fallacies contained in a statement.__
This includes evaluating arguments, based on deductive reasoning.

__Verbal reasoning output____ ____involves
arranging words into arguments, phrases, sentences, or paragraphs, based
on a set of rules, and/or based on one or more goals.__ In
general, it also includes using written or spoken language, to convey
information, or to describe a concept or entity. This includes writing essays,
poetry, or term papers. This also includes creating arguments with deductive
reasoning, and proving or evaluating logical relationships with premises.

__Verbal Reasoning, and Logical
Statements__

Verbal reasoning can involve various types of logical statements, which are used in written and spoken language. Many of these statements are quite simple, and most six-year-olds will have no difficulty understanding them. However, the simple statements can be connected together in very complex ways. These statements can be converted to various types of symbolic formats, and used in electronics and computer technology. See the following examples:

__IF THEN STATEMENTS: ____The general example is IF
A, then B.__

__A specific example
is if it rains, the game is canceled.__

__AND STATEMENTS:____ The general example is A
and B.__

__A specific example
is Mike and Susan will help you tomorrow.__

__OR STATEMENTS:____ The general example is A
or B.__

__A specific example
is Mike or Susan will help you tomorrow.__

Additional examples with premises are presented below:

__IF THEN STATEMENT:____ If John is in Washington,
then his cell phone is in Washington. PREMISE: John always takes his
cell phone wherever he goes.__

__IF THEN STATEMENT:____ If Susan is in
Connecticut, so are her legs. PREMISE: Susan’s legs are attached to her
body__

__IF THEN STATEMENT:____ If the display light is
on, so is the radio. PREMISE: The display light is part of the radio,
and is connected to the on off switch.__

__IF THEN STATEMENT ____and OR STATEMENT:____ If the display light is
off, the computer is not on, or the display light burnt out. PREMISE: The
display light is designed to turn on, when the computer is on, but if it is
burnt out, it cannot turn on, even if the computer is on.__

__Verbal Reasoning: Two Categories of
Logical Statements__

Logical
statements can be placed into two general categories. The first category involves
logical statements that are based on factual principles of nature,
circumstances, or wording. For example, if the mayor is in Washington, then
his arms and legs are also in Washington. I am calling this category, statements
** based on natural rules**. The truth or validity of statements in
this category, are

__The second category__ involves logical
statements that are based on rules created by human beings. This essentially
involves manufactured logical relationships, which do not exist in nature. This
concept is important because it is used in law, electronics, computer science,
and many other fields. This is illustrated with the four examples presented
below. **Note, the manufactured logical relationships are highlighted in
yellow.**

__Example
1, relates to law)____ If
you are in New York City, and your automobile speedometer indicates 95 miles an
hour, you are breaking the law, and guilty of speeding. __

__Example
2, involves electronics)____ A LCD light is connected
to the on off switch of an electronic device. This represents a simple logical
statement, which is, if the LCD light is
glowing, the device is turned on.__

Examples 3 and 4 involve computer technology, and they are based on simple logical statements, connected together into complex configurations. These examples are illustrated in more detail with online software, presented on the websites indicated in the examples. I created this software in 2012, for my undergraduate studies, initially with Microsoft Excel. The Excel version was electronically converted to JavaScript code, to create the online version. The examples are illustrated with the logical statements in the Excel format.

__Example 3) __www.TechForText.com/Using-One-Symbol-to-Represent-a-Set-in-Programming__ The
software on this website demonstrates that one
symbol can be used to represent a set of symbols, such as the following:
__

__A={Set
of words you entered in input Box-One}__

__B={Set
of words you entered in input Box-Two}__

__C={Set
of words you entered in input Box-Three}____ __

__I
created this software with a number of manufactured logical statements,
involving if then. I translated the statements to a symbolic format
that a computer running Microsoft Excel can comprehend. See the following:__

__=IF(B30="",B31,B30)____,
=IF(B39="",B40,B39), =IF(B48="",B49,B48), =IF(B57="",B58,B57 ), =IF(B66="",B67,B66),
=IF(B75="",B76,B75 ), =IF(B84="",B85,B84),
=IF(B93="",B94,B93), =IF(B102="",B103,B102), =IF(B111="",B112,B111)__

** **

** **

__Example
4)____ __www.TechForText.com/Computing-Devices-Relativity-of-Meaning__ This software demonstrates the relativity of the
meaning of symbols in computing devices. It
has three mechanisms that interpret the meaning of red, blue, and green, in
three different ways. __

__I
created this software with a number of manufactured logical statements,
involving if then. I translated the statements to a symbolic format
that a computer running Microsoft Excel can comprehend. See the following:__

__=IF(I4="","",I4)____,
=IF(D5="red", C20,
"")&IF(D5="Blue", C21,
"")&IF(D5="Green", C22, ""), =IF(J5="Red",I20,""), =IF(J5="Blue",I21,""), =IF(J5="Green",I22,""), __

__=IF(O4="red",O20,
"")&IF(O4="Blue", O21, "") &IF(O4="Green",O22,"")__

__Mathematical
Reasoning, and Related Concepts__

** **

__Mathematical Reasoning for
Comprehension, And Problem Solving__

** **

Based
on the way I am using the terminology, mathematical reasoning involves two
concepts, which I am calling__ mathematical reasoning input__, and__
mathematical reasoning output.__

** Mathematical reasoning input**
involves

** Mathematical reasoning output**
involves

__Reading and Learning Mathematics__

** Mathematical
reasoning input,** includes examining, or reading
mathematical concepts and expressions. Unlike conventional reading, this
process must be carried out at a relatively slow rate. For example, reading a
page from a social science textbook might require five minutes. If you are
studying unfamiliar social science concepts, it might require 15 minutes per
page. However, if you are reading a page with mathematical principles, or
instructions for solving math problems, it might require anywhere from 20 to 40
minutes for each page. If you are unfamiliar with the mathematical concepts or
formulas, it might require

Many people do not understand the above, and they may attempt to read mathematical material in a way they would read a book on sociology. People that do this usually think they are terrible in math. However, the problem is they are using the wrong reading and studying strategies, for subjects that involve mathematics. This can make it difficult, or impossible, to successfully master courses involving mathematics.

The
concepts presented above are well known by anyone that is reasonably successful
with mathematics. For additional information, see the following three
websites: **1)** How
to Read Mathematics **2)** HOW
TO LEARN FROM A MATH BOOK** 3)** Tips
for Reading Your Mathematics Textbook

__Mathematical Reasoning in General__

Mathematical reasoning often involves verbal and/or visual reasoning, with numbers, letters that represent numbers, graphs, or diagrams. Verbal reasoning is most apparent with mathematical problems presented in written language. This requires a translation to a mathematical format, to solve the problem. Mathematical proofs that involve deductive reasoning with axioms, postulates, and theorems, also involve verbal reasoning. The use of visual reasoning is most obvious with geometric calculations and proofs.

** **

__Deductive &
Inductive Reasoning, with Related Concepts__

__Introductory Note on Variations in
the Descriptions of Deductive and Inductive Reasoning__

Deductive
and inductive reasoning are used in mathematics, science, and engineering. The
websites that I encountered, with a philosophical focus, described deductive
and inductive reasoning, with an emphasis that was somewhat different from the
conventional scientific perspective. They emphasize the precision and accuracy
of deductive reasoning, and describe inductive reasoning as less precise. **This
is only true, if all the premises are correct in a deductive argument.**

Inductive reasoning is used in the sciences, and it can be extremely precise, especially if conclusions are evaluated experimentally. This is obvious, from the extremely precise and powerful technologies that were developed from the sciences.

For
the philosophical point of view, see the following three websites: **1)*** *“Internet
Encyclopedia of Philosophy (IEP)”

The websites that did not have a philosophical focus described deductive and inductive reasoning, in a manner that is consistent with the conventional descriptions presented in science and mathematics. My descriptions are based on the conventional descriptions, because they are more relevant for the material in this e-book.

See
the following four websites for typical delineations of inductive and deductive
reasoning: **1)** Deductive
Reasoning Versus Inductive Reasoning **2)**** **Deduction
and Induction **3) **Inductive
vs. Deductive Reasoning **4) **Induction
Vs. Deduction Economics My description of deductive and
inductive reasoning is presented under the following subheadings.

Deductive reasoning
usually starts with a general concept, which is narrowed down to
a specific case. **For a simplified example**, __let us assume that
insects are six legged creatures, with a three-segment body structure, with
wings__. This is a **general statement**,
which in this case happens to be a definition, (which is somewhat simplified).
Let us assume you see a butterfly, and you notice that it is a __six legged
creature, with a three-segment body structure, with wings,__ which is an
example of a **specific case**. Your **conclusion** is the butterfly is an insect, which is
true by definition.

The above is a simplified example of deductive reasoning. Below there is a more complex explanation.

Deductive
reasoning involves a **proposition,** which
is a general statement that may be **true or false.**
This is followed by one or more **premises**
that lead to a **conclusion**. The **premises** are __statements that are known to be
true, or are assumed to be true__. Premises can be __axioms__, __postulates__,
__theorems__, __scientifically verified theories__, and __definitions.__
The ***conclusion** indicates whether the proposition
is **true** or **false**. Deductive reasoning is sometimes described as
top-down reasoning. This is because it starts with a **proposition**, followed by one or more **premises** that logically lead to a **conclusion**.

*****Note, sometimes deductive reasoning
can involve statements that are conditionally true, or other types of
conclusions, such as butterflies are insects.

Sometimes the premises in deductive reasoning are not stated, but they are implied by the wording and logical structure of the argument. When the premises are stated, the reasoning is likely to be very precise. Deductive reasoning, with stated premises, is especially useful for proving mathematical and geometric concepts.

__Deductive Reasoning, with Verbal
and/or Visual Reasoning__

Most deductive reasoning is based on verbal reasoning. However, in some cases there may be both verbal and visual reasoning involved. For example, geometric proofs usually involve visual reasoning with diagrams, and verbal reasoning with postulates and theorems. In many cases, it is possible to convert deductive reasoning to a visual format, using various types of diagrams. When this is done, visual reasoning is involved. See the following example.

The
above is called a __Venn Diagram,__ and it indicates the conclusion, which
in this case is butterflies are insects. This can also be stated as insects represent
a __set____ of six legged creatures,
with a three-segment body structure, with wings__. This **set** is represented by the yellow circle. __Butterflies
are a subset of the above, because they are six legged creatures, with a
three-segment body structure, with wings.__ A ** subset**
is a smaller set, contained within a larger set. The small green circle is the

Diagrams do not always clarify or
simplify deductive reasoning, and in some cases, it may make it more
confusing. However, if you want more information see the following six
websites: **1)** Venn
Diagram Solution, **2) **Venn
Diagrams and Logic, **3) **DEDUCTIVE LOGIC
AND VENN DIAGRAMS, **4) **With Venn Diagrams
Optional, **5) **Solving
Word Problems with Venn Diagrams, part 1, **6) **Solving
Word Problems with Venn Diagrams, part 2

__Writing, with an Informal Deductive Reasoning Structure__

An
***informal** type of deductive reasoning is frequently used in writing. For
example, a college essay often starts with a thesis, which is analogous to a
proposition in deductive reasoning. The thesis is supported or proved in the
body of the text, with premises. This leads to a conclusion, which can be more
or less similar to a conclusion from deductive reasoning. The same general
idea applies to an academic thesis, and even some business reports.

*****However, arguments in written language
are often **based on evidence**, which leads
to a conclusion, which does not fit the requirements of ** formal logic**. This is, because evidence
usually does not consist of indisputable premises. Evidence suggests or
indicates the possibility or likelihood that an argument and related conclusion
might be correct.

In
addition, the premises in formal logic are logically related to the conclusion,
and if **one of the premises turns out to be false**, the **entire argument
is invalid**. This is **not** the case
with evidence. For an example, let us assume a murder suspect was arrested
because his __blood__, __DNA__, __hair__, and __fingerprints__, was
found on the crime scene, and video surveillance shows him committing the
murder. If it turned out that the blood, hair, and fingerprints, was not from
the suspect, the video surveillance, and DNA would be adequate evidence for
conviction.

Inductive
reasoning is sometimes described as the opposite of deductive reasoning, or as ** bottom-up**
thinking. This is because it starts with observation of specific cases, to
create a general concept, or a hypothesis, which is the conclusion. That is,
inductive reasoning, usually involves examining a relatively small number of
entities in a specific category, and devising a general principle that applies
to all entities in the category. The following example will clarify this
concept.

Let
us assume, you examine 10 species of spiders, and they all have eight legs.
Then your conclusion is a **general statement**, which is __all spiders
have eight legs__. How accurate is this conclusion? Conclusions derived
with inductive reasoning are not considered 100% accurate, because they are
usually based on a limited number of samples. Using the example with spiders,
there is a possibility that if you examine more spiders, you may find that
there are spiders with six legs.

__Conclusions From Deductive Reasoning
Can Sometimes Be Converted To Definitions That are Indisputable__

Sometimes
conclusions from deductive reasoning can be changed into **indisputable definitions.** To clarify this idea, I will
return to the example presented in the previous subsection, which involve the
conclusion (or hypothesis)

The above essentially involves defining the observed or experimental sample with precise terminology. The following example will provide further clarification of this idea, in terms of social science.

Let
us assume that a scientifically designed survey is carried out on an __ethnic
group X,__ and the results indicate 97% of the sample was exceptionally
self-disciplined. Instead of saying that people in __ethnic group X are
extraordinarily self-disciplined, you can create a definition for highly self
disciplined people in ethnic group X, such as disciplined-Xs.__ Then
you can state that based on your survey 97% of the

With
the above technique, further study can be carried out based on the category
defined by the definition. For example, additional studies can be carried out
on the ** disciplined-Xs,** such as to determine how they became
highly self-disciplined, and to determine how this affects their achievements
in school and employment.

__Checking the Validity of Inductive
Reasoning__

The
hypotheses derived with deductive reasoning, usually cannot be converted to
indisputable definitions. **In such a case, the hypothesis must be evaluated
experimentally.** With published work, this usually involves a number of
experiments, carried out by different researchers, over a period of months or
years to assess the validity of a hypothesis. With unpublished work, a
hypothesis can be evaluated with a few experiments, which can be informal in
structure.

__Checking Inductive Reasoning in
Everyday Life__

Most
people derive assumptions or hypotheses, about __other individuals__, __products__,
__service providers__, and __vendors__. People also derive assumptions
about their own capabilities as well as the abilities of others. These
assumptions are conclusions derived with an informal version of inductive
reasoning. I am calling conclusions in this category, personal assumptions, or
personal hypotheses. The ***relative degree of
validity** of these assumptions should be tested with informal
experimentation, when feasible. Sometimes-ongoing experience with a person,
product or other entity, can determine the relative degree of validity of your
personal assumptions.

*****I used the words **relative
degree of validity**, because many of the conclusions we derive in everyday
life are partly true and partly false, or conditionally true. For example, we
may find that a service provider is very angry and unfriendly at the end of the
day, when he is overwhelmed with too many clients. At other times, he may be
quite friendly. Thus, the __assumption that he is a hostile person__ would
be only partly correct, or conditionally true.

It
is important, and sometimes difficult to recognize personal hypotheses that are
incorrect, and to abandon them. However, this process is likely to be much
easier, if you assume that your personal hypotheses might be __erroneous__, __partly
correct__, __conditionally true__, or __perfectly correct__. In
addition, you should never stake your reputation on any unproven hypotheses.
If you do, it may be embarrassing, and a risk to your reputation, if the
hypothesis is determined to be incorrect.

__Can Inductive Reasoning Be as
Precise as Deductive Reasoning?__

There
are **certain types** of inductive reasoning that can be as precise as
deductive reasoning. In the previous subsection, one of these examples was
presented, which involved inductive conclusions that can be converted to
definitions. There are other variations of inductive reasoning that have a
similar level of precision, presented with the following four examples.

______ EXAMPLE 1______

When
a problem involves a precise set of entities forming a pattern, or sequence of
numbers, it is usually possible to devise a related hypothesis, based on
observation that is certain. __For example, what is the missing number in the
following sequence: 0, 4, 8, 12, 16, ?__ The solution is presented below:

**The hypothesis I derive
from observation** **is
any number in the sequence is represented by the formula N+4=Next number in sequence. Thus, when** **N=16, thus,** **16+4=20**
**Conclusion: the missing
number is** **20** **That is
0, 4, 8, 12, 16, 20**

______ EXAMPLE 2______

The
following is an example of a pattern comprised of *****:
(),(***),(******),(*********),(**?**) __Find
the pattern that belongs in the parentheses with the question mark.__

**The hypothesis derived by observation is the number of *
in any section of the sequence, can be determined with the following formula:** **X*+3*=Next number of * in a following pattern.**

______ EXAMPLE 3______

In
some unusual cases, the same solution can be derived ** inductively**,
and

**The inductive solution is based on observation, which resulted in the**,**
following hypothesis: ** **Thus,** **S=16, and** **Y=next number
in sequence. **** ****The
Conclusion is** **Y=256** or__ 2, 4, 16, 256__

The
above solution can be devised with deductive reasoning by using algebra, as
follows. **The given is the sum of all the numbers in the sequence, including
Y, is equal to 278.
Thus, 2+4+16+Y=278**. **Then 22+Y=278, and Y=278‑22**
**Thus, the conclusion is** **Y=256. Thus the sequences is:
2, 4, 16, 256**

______ EXAMPLE 4______

Usually, deductive reasoning is based on the observation or evaluation of a relatively small number of samples of a very large set, or population. However, if you are dealing with a set, or population, that consists of a relatively small number of entities, you can observe or evaluate every entity, to derive your conclusions. In such a case, the conclusion should be as accurate as deductive reasoning. This applies to observations or evaluations of small groups, such as students in a classroom, or the employees of specific organization.

__See the Following Websites From
Other Authors for Additional Information, and Alternative Perspectives on____ ____Logical
Thinking and Related Concepts__

__TEN VIDEOS__

**1)** Video
“Inductive Reasoning - Concept”,
**2)** Video “Episode 1.3:
Deductive and Inductive Arguments“, **3)** Video “Deductive and
Inductive Arguments“, **4)** Video
“1.1 Basic Concepts: Arguments, Premises, &
Conclusions, **5)** Video,
The limits of computer logic, Morten Rand-Hendriksen,
**6)** Video,
Fuzzy Logic: An Introduction, IEEE Computational Intelligence, Society IEEE
Computational Intelligence Society, **7)** What is Fuzzy Logic? -
Professor Bob John, **8)** Lotfi Zadeh and Fuzzy Logic,
**9) **Fuzzy Logic and Beyond - A New Look by Lotfi Zadeh -
Workshop on Fuzzy Logic, **10)** Logic and Set Theory

__ARTICLES ON WEBSITE__

**1)** Ontologies
and Logic Reasoning as Tools in Humanities?, **2) **The psychology of moral
reasoning, Monica Bucciarelli, University of Turin,
**3) **Some
uses and limitations of fuzzy logic in artificial intelligence reasoning for
reactor control, Michael A.S. Guth, **4) **Logical
Varieties in Normative Reasoning, Mark Burgin Kees (C.N.J.) de Vey Mestdagh,
**5) **Abductive
reasoning: Logic, visual thinking, and coherence, Paul Thagard and Cameron
Shelley, **6)** The Limits Of Logic, Brian
D. Rude, 1974 ,
**7)** WHAT
IS THE DIFFERENCE BETWEEN REASON AND LOGIC? HOW RELIABLE IS INDUCTIVE
REASONING? ARE WE PREDICTABLY IRRATIONAL?, **8)** Tutorial
L01] What is logic?, , **9)** DEFINITION

fuzzy logic, **10)** THE
FUZZY WORLD

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click on these words.**

Below
is the hyperlink table of contents of this e-book. If you Left click on a
specific: chapter, section, or subsection, it will appear on your computer
screen. Note the red chapter headings, the yellow highlighted sections, and
the blue subheadings are **all active links.**

Reasoning: Introductory Concepts

Reasoning for Comprehension, and Reasoning for Solving Problems, and Creating Entities

Visual Reasoning, and Related Concepts

Visual Reasoning for Comprehension, & Problem Solving

Verbal Reasoning and Related Concepts

Verbal Reasoning for Comprehension, & Problem Solving

Verbal Reasoning, and Logical Statements

Verbal Reasoning: Two Categories of Logical Statements

Mathematical Reasoning, and Related Concepts

Mathematical Reasoning for Comprehension, And Problem Solving

Reading and Learning Mathematics

Mathematical Reasoning in General

Deductive & Inductive Reasoning, with Related Concepts

Introductory Note on Variations in the Descriptions of Deductive and Inductive Reasoning

Deductive Reasoning, with Verbal and/or Visual Reasoning.

Writing, with an Informal Deductive Reasoning Structure

Conclusions From Deductive Reasoning Can Sometimes Be Converted To Definitions That Are Indisputable

Checking the Validity of Inductive Reasoning

Checking Inductive Reasoning in Everyday Life

Can Inductive Reasoning Be as Precise as Deductive Reasoning?

Logical Thinking and Related Concepts

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