Lecture Notes on Logical Organization and Critical Thinking
⇐
Previous Lecture Notes
 ⇑ Up
Next Lecture Notes ⇒

Summary
MENU

1. Logical Ways of Thinking

In science & engineering, it is most important to think logically and present a process of thinking for reaching a conclusion clearly.
That should be based on formal logic such as propositional logic and predicate logic.
Thus, a curriculum of computer science commonly includes formal logic in the first year
(typically, as part of discrete mathematics).

A lot of practice of logical thinking is necessary for not only scientists and engineers but alo technical writers.
As a result, they can share information accurately.
Even if a conclusion was come across by insight, its truth must objectively be verified so that the other people are convinced.
Therefore, formal logic is the common means of analysis as well as verification necessary for sharing the conclusion.

To refresh your memory on formal logic,
important concepts and their corresponding English phrases are summarized below.
For more details, refer to Lecture Notes on Propositional Logic and Predicate Logic for my courses of Discrete Mathematics I and II at the University of Hawaii at Manoa.

The term sentence below always indicates a declarative sentence
that excludes an interrogative sentence (i.e., a question), an imperative sentence (i.e., a command), and an exclamatory sentence.
  1. Conjunction
    (AND) denoted ∧
    sentence and sentence      |       clause and clause
  2. Disjunction
    (OR) denoted ∨
    either sentence or sentence      |       sentence or sentence      |       clause or clause
  3. Negation
    (NOT) denoted ¬
    be not      |       adverb not verb      |       no subject verb
  4. Conditional (aka Implication)
    denoted →
    if sentence A, then sentence B      |       sentence A only if sentence B      |       sentence A implies sentence B
    when sentence A, sentence B      |       sentence B if sentence A      |       sentence B when sentence A      |       sentence B, provided sentence A
    Remark:
    The left-hand side A of a conditional is called a premise and the right-hand side B in a conditional is called a consequence.
  5. Universal Quantifier
    denoted ∀
    Every noun verb . . .      |       Each noun verb . . .      |       Any noun verb . . .
    for every variable, sentence (including a variable)      |       for each variable, sentence (including a variable)      |       for any variable, sentence (including a variable)
    sentence (including a variable) for every variable      |       sentence (including a variable) for each variable      |       sentence (including a variable) for any variable
    Remark:
    A variable is a noun or a pronoun or a symbol referring to a noun or a pronoun.
  6. Existential Quantifier
    denoted ∃
    There exists noun such that sentence      |       There is noun such that sentence
    Some noun verb . . .      |       For some variable, sentence (including a variable)      |       sentence (including a variable) for some variable
    Remark:
    The existential quantifier simply states the existence and does not state how many.
    Thus, there may be only one or may be two or more.
    If you intend to express the unique existence, then you should use the phrase
    "There is one and only one . . ." or "There uniquely exists . . ."

Example 1.1:
Logical Composition of a Sentence
  1. If an integer k is odd and k3 is positive, then k is greater than 0.
  2. There is at least one value of an integer variable m such that the equation m3 − 2m + 1 = 0 holds.
  3. For every bit string s, there exist two bit strings x and y such that s = xy holds.

Remark 1.1:
Disambiguation of the Word "or" (Inclusive OR)

The word "or" sometimes creates ambiguity regarding its scope in a sentence as shown below.
If an integer k is positive, then k is even or k2 is positive.

This compound sentence can be interpreted in two ways, depending on a scope of the word or.
To avoid such an ambiguity, the phrase "either . . . or . . . should be used instead of the single word "or".

There are two logical ways of thinking that are commonly used in science & engineering:
Deductive Reasoning
and
Inductive Reasoning.

Deductive Reasoning Inductive Reasoning
Theory
↓
Hypothesis
↓
Observation
↓
Confirmation
Theory
↑
Hypothesis
↑
Pattern
↑
Observation
Specialization Process Generalization Process

Example 1.2:
Deductive Reasoning
All men are mortal.
Joe is a man.
Therefore Joe is mortal.

This is the well-known inference rule called hypothetical syllogism (or simply syllogism).
Refer to Lecture Notes on Inference Rules.

Remark 1.2:
A Fallacy in Deductive Reasoning
To get a Bachelor's degree, a student must have 120 credits.
David has more than 130 credits.
Therefore, David has a bachelor's degree.

Note that 120 credits are merely a necessary condition for a Bachelor's degree.
Having 120 credits does not necessarily imply a Bachelor's degree.

Example 1.2:
Inductive Reasoning

This horse is brown.
That horse is brown.
Another horse is brown.
Therefore, all horses are brown.

This example obviously shows that a conclusion drawn by inductive reasoning is not guaranteed to be true.

Since the truth of a conclusion drawn by deductive reasoning depends on truth of given hypotheses,
we need to apply both deductive reasoning and inductive reasoning for producing a true conclusion in the real world.

Review 1.1:
Assume that the following hypotheses are all true.
Then, deduce a meaningful conclusion from them.
  1. If today is Tuesday, then I have a test in either Computer Science or Chemistry.
  2. If my Chemistry professor is sick, then I don’t have a test in Chemistry.
  3. Today is Tuesday.
  4. My Chemistry professor is sick.

2. How to Construct Logical Arguments
Once we get an idea by a logical way of thinking,
we need to express the idea and a logical process of deriving it clearly.

There are at least two approaches to organizing an argument logically:
Top-Down and Bottom-Up.
In practice, we often apply both of them so that a global organization (e.g., at levels of sections and chapters) is constructed in a top-down mathod and local organizations (e.g., at levels of paragraphs and sentences) is constructed in a bottom-up method.

A. Top-Down Approach

  1. Decide the main theme and choose its appropriate title.
  2. Write an outline such as a table of contents.
  3. Put a heading to each component (such as a section) in the table of contents.
  4. Choose important topic sentences for each component.
  5. Repeatedly decompose each component hierarchically so that a component may correspond to a few to several paragraphs.
  6. Expand each topic sentence to compose a paragraph.

B. Bottom-Up Approach

  1. Compose sentences by using the phrases as shown above together with well-defined vocaburary (such as technical terms).
  2. Classify the composed sentences into facts, hypotheses, opinions, etc.
  3. Choose topic sentences that are important.
  4. Group sentences for each topic sentence.
  5. Form a paragraph consisting of a topic sentence and its supporting sentences.
  6. Form a section consisting of paragraphs.
  7. Arrange paragraphs in each section.
  8. Repeatedly aggregate components at a lower level into a component of a higher level.

How to Construct a Paragraph

A paragraph consists of the following sentences.

To group sentences (or paragraphs) and form a paragraph (or a section, respectively), there are at least two criteria.

Example 2.1:
Assume that we write a user's manual of a rice cooker,
where the anticipated readers are users of the rice cooker.
Consider the following outline of the manual.

  1. Hardware Configuration
  2. Features
  3. How to Use
  4. Maintenance
  5. Error Messages and Diagnosis
  6. Specifications

Then, each section will be decomposed further.
For example, the section 6 "Specifications" may consist of the following subsections.
Then, for each section or subsection, list important statements as topic sentences.
This will lead to paragraphs.
Arrange paragraphs logically and then proceed with composition of supporting sentences for a topic sentence of each paragraph.

To design structures of a technical document has similarities with the design of software structures.
In addition, software documentation is one of the major tasks in a process of software development.
Thus, exercises of technical writing help students to learn how to develop a software system.

Review 2.1:
Write a paragraph regarding "what is most important to learn a logical way of thinking."

3. Critical Thinking
Critical thinking is valued in the K-12 of U.S.A. very much
while it has received very little attention in curricula of Japanese educational systems.
In addition, a skill of critical thinking is highly demanded in the US job market.
For example, an article titled " Bosses Seek 'Critical Thinking,' but What Is That?" of Wall Street Journal (Oct. 21, 2014) states as follows.
Mentions of critical thinking in job postings have doubled since 2009,
according to an analysis by career-search site Indeed.com.

What is critical thinking?

Robert Ennis (1989) defined critical thinking as follows.
"Critical thinking is reasonable, reflective thinking that is focused on deciding what to believe or do."
Alec Fisher and Michael Scriven (1997) defined it as follows.
"Critical thinking is skilled and active interpretation and evaluation of observations and communications, information and argumentation."
For more definitions and a history behind critical thinking, refer to Alec Fisher, Critical Thinking An Introduction, Cambridge University Press, 2001.

Peter Facione gave the following intuitive definition of critical thinking
(see Peter A. Facione, " Critical Thinking: What It Is and Why It Counts," Insight Assessment, 2006).
This definition suggests us that critical thinking excludes not only prejudice, but also wishful, sloppy thinking.
That is why a critical thinking skill is a MUST for professionals in fields of STEM (Science, Technology, Engineering and Mathematics).

An article titled " Critical Thinking Is Best Taught Outside the Classroom" of Scientific American (March 1, 2013) introduced a very interesting episode on research conducted by cognitive scientists at Vanderbilt University.
[Excerpt below]

An assignment of creating a recovery plan to protect bald eagles from extinction was given to a group of fifth graders and another group of college studnts.
Surprisingly, the two groups came up with plans of similar quality
(although the college students had better spelling skills).
This observation indicated that education for about 8 years had failed to improve students' skills of problem solving on a major scientific issue.

Then, the researcher decided to give another assignment of generating questions about important issues needed to create recovery plans.
In this assignment, the two groups showed big differences.
College students focused on critical issues of interdependence between eagles and their habitats
("What type of eco-system supports eagles?" and
"What different kinds of specialists are needed for different recovery areas?").
Fifth graders tended to focus on features of individual eagles
("How big are they?" and "What do they eat?").
The college students had cultivated the ability to ask questions, the cornerstone of critical thinking.
They had learned how to learn.

Three Major Benefits of Critical Thinking in Science & Engineering

  1. Ability to Raise Questions
    →
    This also leads to creative thinking.
  2. Ability to Thoroughly Verify a Claim Objectively
    →
    This also leads to reflective thinking.
  3. Ability to Think an Issue from Different Viewpoints
    →
    This also leads to innovative thinking.

California Critical Thinking Skills Test (CCTST) proposes the following five-step problem solving process (IDEAS) by critical thinking.
  1. I =
    Identify the Problem and Set Priorities
  2. D =
    Determine Relevant Information and Deepen Understanding
  3. E =
    Enumerate Options and Anticipate Consequence
  4. A =
    Assess the Situation and Make a Preliminary Decision
  5. S =
    Scrutinize the Process and Self-Correct as Needed
This is quite generic so that it can be applied to a wide spectrum of problems.

Example 3.1:
Assume that you wrote a Java program for computing the power 2k and successfully compiled it without any error.
When you get its output 103,520,048 for input k = 30, do you accept the output value without any question?

You should start with questioning yourself whether the output is really accurate even if you are an experienced Java programmer and confident with the Java program.

Note that 210 = 1024 ≈ 103.
With basic knowledge of exponentiation, we can get a ballpark estimate of 230 as follows.
230 = (210)3 ≈ (103)3 = 109 = 1,000,000,000
Thus, it is easy to conclude that the output is erroneous.

Remark 3.1:
Assume that you conduct a Black Box Testing of the Java program in the above Example 3.1
(for more information about black box testing, refer to Testing Overview and Black-Box Testing Techniques).
What kind of a test data set do you use to demonstrate its correctness?
How do you generate such a test data set?
These are very crucial questions in software engineering.

Review 3.1:
Consider the 2nd assignment given to students in the episode in the article of Scientific American.
What are questions that you raise for creating a recovery plan to protect bald eagles from extinction?

4. Exercises
  1. Let p and q be the propositions "The election is decided" and "The votes have been counted," respectively.
    Express the compound proposition ¬q ∨ (¬p ∧ q) as an English sentence without any ambiguity.

  2. The following puzzle called Knights and Knaves was posed by Raymond Smullyan.
    Inhabitants of a mysterious island consists of two tribes:
    Knights who always tell the truth and knaves who always lie.
    You met two inhabitants A and B.
    One inhabitant A told you "Both of us are knights."
    Another inhabitant B told you "A is a knave."
    Can you determine whether each of them is a knight or a knave?
    If it is possible to determine which tribe each inhabitant belongs to, elaborate how you determine it.
    Otherwise, justify why you cannot.

  3. You were called in to solve a baffling murder mystery.
    The following facts were given to you.
    If you think it is possible to deduce the identity of the murderer from the facts, give the murderer and show how to deduce it.
    Otherwise, prove that it is impossible.
    1. Lord Hazelton, the murdered man, was killed by a blow on the head with a brass candlestick.
    2. Either Lady Hazelton or a maid, Sara, was in the dining room at the time of the murder.
    3. If the cook was in the kitchen at the time of the murder, then the butler killed Lord Hazelton with a fatal dose of strychnine.
    4. If Lady Hazelton was in the dining room at the time of the murder, then the chauffeur killed Lord Hazelton.
    5. If the cook was not in the kitchen at the time of the murder, then Sara was not in the dining room when the murder was committed.
    6. If Sara was in the dining room at the time the murder was committed, then the wine steward killed Lord Hazelton.

  4. The police have three suspects for the murder of Mr. Cooper:
    Mr. Smith, Mr. Jones, and Mr. Williams.
    Smith, Jones, and Williams each declare that they did not kill Cooper.
    Smith also states that Cooper was a friend of Jones and that Williams disliked him.
    Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed.
    Williams also states that he saw both Smith and Jones with Cooper the day of killing and that either Smith or Jones must have killed him.
    Suppose that one of the three men is guilty, the two innocent men are telling the truth, but the statements of the guilty man may or may not be true.
    Show how you can logically deduce who the murderer was.

  5. We plan to arrange a party and want to choose guests from Alice, Eric, John and Tina.
    Knowing the following facts, by using propositional logic,
    present your logical reasoning rigorously to determine which of the four persons should be invited so as not to make someone unhappy.
    When Alice attends a party, she is unhappy if Eric is in the party.
    When Eric attends a party, he is happy only when Tina attends it.
    Tina does not want to attend a party unless Alice also attends it.
    John does not want to attend a party, provided that both Alice and Tina attend it.

  6. A man who was captured by savages was promised his freedom if he could determine with a single "yes or no" question the color of the tribe's idol.
    He knew that the idol was either white or black.
    Unfortunately, the tribe consists of two types of individuals:
    liars, who always gave the wrong answer to any "yes or no" question they were asked, and
    truth-tellers, who always gave the right answer.
    Fortunately, the victim got A+ for a discrete math course and
    could derive a question to any individual whose answer determines the color of the idol.
    What is the question?
    By using propositional logic, logically explain why and how the man could determine the color.

  7. Write a paragraph regarding "what is most important to pass this course on technical communication."

  8. Assume that you work for Robot Taxi and are assigned to a task of writing a manual about a driverless taxi.
    What kind of readers do you anticipate?
    Then, design a logical organization of the manual.

  9. Assume that you need to make an oral presentation on the progress of your project (development of a new rice cooker) to executives.
    Create an outline of your presentation (15 min. for a talk and Q&A).

  10. How do you put a dinosaur into a refrigerator?
    What about the next question "How do you put a giraffe into the refrigerator?"?

  11. Assume that you are assigned to a project for improving a rice cooker and designing its new model.
    To determine what to be improved, what will you do?

  12. You know a controversial incident on STAP cells that a researcher of Riken claimed to discover.
    The community of scientists in biology and relevant fields criticized the researcher's claim.
    Then, Riken concluded that the claim was false and withdrew papers published by the researcher.
    Do you agree with the Riken's conclusion that STAP cells do not exist?
    Write a paragraph about your opinion on this issue.