Introduction

As an Intuitive Practical Introduction, This Course Is Easy

I call this course an "intuitive, practical introduction," because it is thoroughly based in the practice of logical techniques that leverage what the student already knows. The main text of the course briefly introduces each technique. However, the instructional core of the course material is found in the exercises and quizzes. The exercises elaborate upon the techniques, and train the student in those techniques. Then the quizzes test the student, and give definitive feedback on their progress. When memorization is needed, flashcards are provided. The exercises and quizzes are sequenced in a very precise manner. At each stage the techniques are developed in a subtly incremental way, so that the learning is quite natural and intuitive—almost uneventful. That means that this course is easy. All you have to do is work through the exercises and quizzes in sequence. If you do this, then you will learn the techniques. And it is the techniques that are of utmost importance at this very beginning level of logic. 

Streamlined Explanations

The course material here will provide explanations of everything you need to know. Effort has been made to keep all explanations as streamlined as possible. 

Techniques Learned through the Exercises

Again, it is the techniques that are of utmost importance, and the techniques are actually learned through careful review of the exercises. Make sure that you work the exercises according to the instructions given. 

Few Trick Questions

There are few trick questions in this course of study. The explanations prepare you for the exercises. In the exercises there will be some content not covered in the explanations, but the answers to the exercise problems are provided. Using these answers, you will be able to easily figure out the problem types. The quizzes are almost identical to the exercises. And the exams are almost identical to the quizzes. The quizzes simply test how well you have worked the exercises. And the exams simply test how well you have worked the quizzes. 

Zeroth-Order Logic for the Absolute Beginner

In this course, we will be studying "propositional" logic, also known as "sentential" logic, also known as "zeroth-order" logic. Zeroth-order logic is the foundation of all other forms of logic. It is like the ground floor of a building. We are starting with the very basic beginnings of logic, and then working up from there. There is no need for you to know anything about logic at this point. This course is made for the absolute beginner. 

Scientific Concepts Based in Techniques

Because this course is rooted in technique, and because I will guide you step-by-step through each technique, the course is very easy to complete. But that should not be taken to mean that the course is anything less than rigorous. In the process of completing this practical introduction you will come to understand the fundamental concepts of logic. However, it is important to put the techniques first, because the scientific concepts are understood in relation to the techniques. Some concepts are best understood—most precisely understood—as the very point of the techniques. And some concepts are only understood once we begin to see the limitations inherent in the techniques. 

This course will thoroughly prepare you for the study of first-order logic and beyond. For more explanation (unnecessary at this point), see Wikipedia, "Zeroth-Order Logic," "First-Order Logic," and "Second-Order Logic." 

The Triviality of Logic

Some will wonder why we study logic. At the risk of being too clever by half, I want to answer that we study logic because it is a trivial matter. —Let me explain. 

The western tradition of education goes back to the Academy, the first school to exist in the west, the school founded by Plato in about 385 BC (about 2,400 years ago). Since then, whenever someone would go to school (in Europe, north Africa, or the Americas—but many other places around the world as well) they would do so in the fashion first established by Plato.

In his Socratic dialogue the Republic, Plato arranged all “academic” subjects into seven major fields, with three being designated as elementary, and four as higher studies—and these seven were all merely preparatory for the dialectics of philosophy. Plato’s taxonomy was slightly altered over the centuries, but well into modern times it remained essentially the same.

In late antiquity throughout Europe, during the decline of the Roman empire, classical education became very standardized around the seven “liberal arts” as they were known—those "arts" that would "liberate" the mind. And, as with Plato, there were three designated as elementary, and four as higher studies. The following is the classical arrangement.

The Seven Liberal Arts

Trivium

1. Grammar

2. Logic

3. Rhetoric

Quadrivium

4. Arithmetic

5. Geometry

6. Music

7. Astronomy

The trivium is the collection of three ("tri") elementary subjects. And the quadrivium is the collection of four ("quad") more advanced subjects.

In Anglo-American Victorian culture, the very basics of a primary school education were considered to be reading, writing and arithmetic (the so-called three "R"s). Throughout Europe, the Americas and beyond—for millennia before that—the basics of a high school education were grammar (a rigorous study of reading), rhetoric (a rigorous study of writing), and logic (a rigorous study of human reasoning). A young adult was not prepared to properly read, write and reason until they had mastered grammar, rhetoric and logic. And these were prerequisites for the more advanced studies of the quadrivium. Logic was a prerequisite for the more advanced studies of the quadrivium just as much as were grammar and rhetoric. These were the basic skills a young person needed in order to be successful in academic pursuits.

Furthermore, placing logic at the basic level of education for young adults made a lot of sense in the classic approach to the quadrivium, because logic was essential to the study of arithmetic and geometry. Up until the late twentieth century, arithmetic and geometry were taught along the method of Euclid, who was active as a professor in about 300 BC (just a few decades after Plato). In fact, until the late twentieth century most geometry textbooks were directly derived from Euclid’s Elements. And in this treatise Euclid deductively proves—largely using elementary logic—the foundational theorems of two-dimensional and three-dimensional geometry, as well as the theorems of arithmetic. Music (mathematical harmonics) and astronomy (the study of three-dimensional geometric figures in motion) were then an extension of Euclid’s Elements.

The studies of the quadrivium were then themselves considered prerequisites for any further philosophical research—natural philosophy (biology and physics), philosophy of the mind (psychology), et cetera. To be successful in the academic pursuit of what we now think of as a college major, a young person had to first be proficient at the level of the quadrivium. And to be proficient at the level of the quadrivium, a young person had to first be proficient at the rudimentary level of the trivium.

Be assured that there is a connection between the words "trivial" and "trivium." That is why I began by saying that logic is a "trivial" matter. Classically, it has always been placed at the most rudimentary level of any serious academic pursuit. 

Your Brain Is Made for This

In this introduction I have tried to emphasize that the course is very easy. Be assured that it is. This is because your brain is made for this. You already think logically. Logic is an essential characteristic of human beings. All we want to do is systematically exercise and strengthen the logic that you already use on a day-to-day basis. Honestly, I have literally never met a student that is incapable of excelling in this course. Just follow the program of exercises I have put together for you, and observe the unfolding of the logic of your mind.

Navigating the Course

Underlined Text a URL Link

Any underlined text on this website is a URL link. Click on the link to navigate to the content indicated. Please do actually click on the links. 

Anonymous Google Account Required

To complete the exercise, quizzes and exams, you will need to use a Google account. Please create a new google account specifically for this course, and choose an email address that is anonymous. For the email address, don't use your name, student number or other identifying information. Other than that, anything will do. But keep the address as short as possible so that it is easy for me to read. If you can come up with something clever or even humorous, that helps me. 

The use of an anonymous Google account allows for useful features of Google Forms, while providing fool proof security. 

Follow the instructions in the following link. 

Google, Create a Google Account

Enter the email address when prompted to do so in the exercises and quizzes. I will track your performance by your anonymous Google email address.  

Online Dictionaries and Encyclopedias 

As you work through the material in the course, if you have difficulty understanding any of the terminology, use a dictionary or an encyclopedia. If it seems like the difficulty is simply with the level of English vocabulary, use a dictionary. If the difficulty seems to be some reference to a technical subject, use an encyclopedia. Usually, I will explain anything germane to elementary logic, but sometimes I will make reference to subjects beyond the scope of this course that it would not be instructive to describe. I do assume that you have a general knowledge of the world from the north American cultural perspective. I also assume that you are working online, and have access to dictionaries and encyclopedias right at your fingertips.

The Merriam-Webster dictionary and Wikipedia are usually appropriate for students taking this course. They provide explanation sufficient for mastery of the course material. 

Merriam-Webster Dictionary

Wikipedia—The Free Encyclopedia

If you desire more in-depth and sophisticated explanation, the following resources are good.

Online Etymology Dictionary

Stanford Encyclopedia of Philosophy  

PART 1—EVOLVING OUR SYMBOLIC APPARATUS

Chapter 1—Lookup Tables

This course is about zeroth order logic. However, before we begin the proper study of zeroth-order, propositional logic, we will do some preparatory work. In this chapter we will practice the use of lookup tables, and explore how lookup tables can allow us to efficiently understand some basic arithmetic. At first we will not be doing genuine logic, but by starting with some familiar concepts and techniques of simple arithmetic we will prepare the brain to absorb the basic principles of symbolic logic. 

Students come to this course with a wide variety of experience, knowledge and skill. Without some expertise in the manipulation of the symbolic apparatus of logic, many of the concepts of the subject can be elusive. The first priority, then, is to learn how to properly handle the symbolic system. This requires attention to detail, and the precise placement of symbols on the page—but it doesn’t require much experience or knowledge. If you find yourself unable to grasp logic at a more abstract and conceptual level, focus on the symbolic system. Learn how to properly place the symbols on the page. Once you have some comfort with this, the concepts will naturally become apparent. 

Some Familiar Lookup Tables

In the first portion of the course, I will emphasize the use of lookup tables. Lookup tables are found every day in ordinary life. We want to leverage our experience with familiar lookup tables in order to understand how to use the lookup tables of logic. To get us in the right frame of mind, we will start with an exercise in looking up information on a schedule of yoga classes. 

Demonstration

The following is a video demonstrating how to complete the exercise further below. 

How to Complete Exercise 1.1—Yoga Schedule 

Practice 

Remember that this is a practical introduction to logic. Practice is of paramount importance. The exercises are where practice is most emphasized. Keep in mind that you will learn new things as you work through the exercises. In order to learn all that is being taught by the exerices, it is absolutely necessary that you use the following procedure. 

1. Attempt the exercise without looking at the answers provided. 

2. Review the feedback provided, noting any differences between your answers and the correct answers. 

3. Reattempt the exercise without looking at the correct answers as many times as it takes to get your answers to match the correct answers. 

Conscientiously do this for all exercises. 

You will need a Google account to complete following exercise. For explanation, see the Introduction. 

Click on the following link to access the exercise. 

Exercise 1.1—Yoga Schedule 

Having worked through this first exercise, you will notice that you are required to enter the answers in a very specific way. After each attempt the correct answers are displayed when you click on the "View score" button. Use that feedback to reattempt until you get the text entry exact. Precise text entry will be very important throughout the course. 

This first exercise, and the exercises through the course are designed to function something like exercises in a hardcopy textbook where the answers are in the back of the book. Try each exercise without looking at the answers, but refer to the answers as much as needed to understand what is required. 

Demonstration

Now we’re going to continue to put into practice familiar lookup table skills. 

How to Complete Exercise 1.2 

Practice

Keep reworking the exercise until you get a 100% score. 

Exercise 1.2—Easy Multiplication Lookup Table 

Demonstration

How to Complete Exercise 1.3 

Practice

Rework the exericse until you get 100%. 

Exercise 1.3—Difficult Multiplication Lookup Table 

The Answers Are in the Lookup Tables

Maybe with the Easy Multiplication Lookup Table you found yourself not needing to reference the table, because you have the information in your head. But with the Difficult Multiplication Lookup Table we went beyond the standard 12 times 12 limit. So, in that case you might have found the need to actually look up the answers. That is perfectly fine. 

Since we are studying rather familiar types of lookup tables, my hope is that you are not finding any of this all that difficult. Assuming that is the case, I hope you will be comforted by the fact that you have one of the key skills to thoroughly comprehend logic. Until we get to proofs at the end of the course, most of the time you will be able to look up answers in lookup tables. The answers are in the lookup tables. Refer to the lookup tables as much as needed. —Let me say it one more time: Lookup the answers in the lookup tables, because the answers are found in the lookup tables. 

Chapter 2—Arithmetic of Evens and Odds

In this chapter we will leverage our lookup table skills to explore the arithmetic of even and odd numbers. We will still not yet begin logic proper, but in the next chapter we will smoothly transition from the arithmetic of evens and odds into logic—because many of the concepts of this arithmetic of even and odds are essential to zeroth-level logic. 

Commutative and Noncommutative Binary Operations

In arithmetic, addition is an "operation." And it is a "commutative" operation. For example, 

When adding two numbers, it doesn't matter which comes before the operation (addition) and which comes after. The numbers can "commute" (move across) the operation without changing the sum. 

And notice that addition only takes two numerical inputs. For example, if we are summing up several numbers, such as

we logically need to add two at a time. 

We can associate the first sum and do that before adding the final number, or we can associate the second sum and do that before adding the first number. A single instance of addition only has two numerical inputs. We call such two-numbered operations "binary" (bi-, two; and -nary, numbered). 

And, again, with addition the binary inputs can be switched without changing the result. 

Subtraction, however, is not commutative. For example

is not equal to 

Subtraction is a binary operation, but its two inputs cannot be switched without changing the difference. 

Consider a simple subtraction table. 

Notice that this table has to be interpreted so that the numbers in the far left column are treated as the left binary input of the operation, while the numbers horizontally along the header are treated as the right binary input of the operation. This is because subtraction is not commutative. 

Because we are very familiar with subtraction of small integers, the intent of the table is clear, but there is an inherent ambiguity in the arrangement. We could eliminate the ambiguity and make the intent very clear by listing the order of the binary inputs explicitly. 

To solidify in our mind the importance of the order of the binary inputs, L is for the “left” binary input, and R is for the “right” binary input. 

Reduced Subtraction of Even and Odd Numbers

We’re now going to tackle some mathematical concepts that are slightly more abstract. For those of us who are very comfortable with mathematics, this will make the transition into logic very smooth. For those of us who struggle with math, the coming exercises should empress upon you the importance of using the lookup tables provided. Remember: the answers are in the lookup tables. 

Now, the mathematical concepts we are going to tackle are not all that difficult. We’re going to take a look at some basic arithmetic involving even and odd numbers. From your studies of basic arithmetic, recall that the "integers" are all the counting numbers, 1, 2, 3, ..., along with zero, and all the negative counterparts, -1, -2, -3, .... Also recall that an even number is an integer that is "divisible" by two. That means that two divides "evenly" into an even number. An odd number is an integer that is not divisible by two. If you want a more in-depth explanation, now would be a good time to use an encyclopedia to get a brief and standard explanation. I will simply assume that you have a grasp of the standard treatment of even and odd numbers at a very basic level. We are interested in some aspects of the arithmetic of evens and odds that are not so standard. 

To get started, let's take a closer look at the last table presented above (Lookup Table—Simple Subtraction with Binary Input Order Made Explicit). Notice certain patterns that emerge in the examples contained in the subtraction table. An odd number minus an odd number gives an even number. An odd minus an even is odd. An even minus an odd is odd. And an even minus an even is even. These patterns hold for all integers. We can organize this information in the following table. 

Notice that we indicate "even" with 

The number

is an even number, and we let it stand for any even number. For example, in the arithmetic of odds and evens we are developing, we say

The integer

is even. So, it "reduces to"

Likewise,

is an odd number, and we let it stand for any odd number. For example,

Another example—

Here we have "reduced" the left and right binary inputs of subtraction. 

reduces to

and 

to

We can then complete the calculation in reduced form.

An odd minus an even is odd (compare Lookup Table—Subtraction of Evens and Odds above). 

Sometimes it might make sense to first reduce the left and right binary inputs, and then compute the operation in a standard way before finally reducing the output. 

But if you use the lookup table, you can skip the negative one step. 

We have an even minus an odd. That is represented in the third row of the table above. 

The table tells us that the final result is odd. 

Demonstration

How to Complete Exercise 2.1 

Practice 

Use the lookup table for the subtraction of evens and odds to complete the following exercise. The answers are in the lookup table. Look them up. 

Exercise 2.1—Reduced Subtraction of Even and Odd Numbers 

Demonstration

How to Complete Quiz 2.1 

Test

WARNING: for the quiz that follows, you will only be allowed one attempt. Once you have used the above exercise through multiple attempts to understand what is required, then attempt the quiz. Again, for quizzes you only get one attempt.

Quiz 2.1—Reduced Subtraction of Even and Odd Numbers 

Make Sure You Are Getting at Least a 75% on Quizzes

Do your best on each quiz. If you are getting less than a 75% on any quiz, that means you are not properly working the exercises. Rework each exercise until you get a 100%—without looking at the answers provided in the feedback. 

Reduced Addition and Subtraction of Evens and Odds

Similar to subtraction, we can reduce when adding integers. 

All possibilities of reduced addition are captured in the following table. 

And we can combine addition and subtraction into one table. 

With this combined table we can complete calculations involving both addition and subtraction. For example, 

Calculate the value within the parentheses first. 

One more example: 

Use the lookup table when completing the exercise. Look up the answers. The answers are in the table. All you have to do is figure out how to use the lookup table to get the right answers. 

Demonstration

How to Complete Exercise 2.2 

Practice

Exercise 2.2—Reduced Addition and Subtraction of Even and Odd Numbers 

Demonstration

How to Complete Quiz 2.2 

Test

Quiz 2.2—Reduced Addition and Subtraction of Even and Odd Numbers 

Chapter 3—Interpretations

Interpretations Introduced

We will continue to use the lookup table for addition and subtraction of evens and odds from the last chapter. Restricting ourselves to addition and subtraction, we will now explore techniques in connection with "interpretations." For example, suppose I ask you compute the following expression.

Is it even or odd? We don't know. For now we will assume that any variable represents an integer, and is therefore either even or odd. But we can't calculate the sum above, because we don't know whether each variable is either even or odd. We need an "interpretation" of the variables in order to make sense of the expression. 

Let's take the interpretation 

Then the expression 

In other words, on the interpretation 

the expression 

is odd. 

We can represent this interpretation in table form. 

The interpretation proper is the set of values assigned to the variables in the far left of the table. However, I will sometimes refer to the entire table form as an interpretation. 

Example 3.1

As an exercise problem, an interpretation will look like this: 

Notice that the question mark is aligned vertically under the subtraction sign. This is because in evaluating the expression, you compute what is in parentheses first. Then you compute the remaining operation. We want to know the final result—when we complete the last of the operations. 

Based on the interpretation given to the variables, 

Thus, 

On the given interpretation of the variables, the entire expression evaluates to odd. In table form— 

Example 3.2

With interpretations we can have any number of variables. For example, we may want to compute

We need an interpretation—say, 

Completing the parentheses first, and looking up the answers in our lookup table, we see that on the given interpretation, the expression evaluates to even.

Example 3.3

But the expression could be given a different interpretation. 

Notice the difference in the far-left columns. On this interpretation we have

The expression is now odd—under the alternative interpretation. 

Demos

Example 3.1

Example 3.2

Example 3.3   

Practice and Test

Exercise 3.1—Interpretations Introduced 

Quiz 3.1—Interpretations Introduced 

Make Sure You Are Getting a 75% on Quizzes

If your quiz score is below 75%, that means you are not properly working the exercises. Slow down, and be more thorough. 

Interpretations with Addition, Subtraction and Multiplication

We can also multiply even and odd numbers, expanding our lookup table. 

An even number multiplied by any integer is even. The only way for a product of two integers to be odd is if both binary inputs are odd. You may want to explore some examples on your own to convince yourself of this. 

Now, using the expanded lookup table, we can complete interpretations that include the operation of multiplication. 

Example 3.4

For example, 

receives the answer

Example 3.5

And 

would be answered with

Work from the inner parentheses outward.

Example 3.6

One more example: 

The answer is

Demos

Example 3.4

Example 3.5

Example 3.6   

Practice and Test

Exercise 3.2—Interpretations with Addition, Subtraction and Multiplication 

Quiz 3.2—Interpretations with Addition, Subtraction and Multiplication 

Interpretations Forward and Backward

We will now do something that requires more critical thinking compared to previous exercises. 

Example 3.7

Complete the interpretation. 

This is doing an interpretation "forward," just like we've been doing it.  

The answer is

Example 3.8

Now let's work it "backward"—using a little bit of "forward" action as needed.   

Complete the interpretation. 

On the interpretation that the entire expression 

evaluates to an even number, we are going to determine the value of 

Since the addition evaluates to even, we can eliminate the two middle rows of our lookup table. 

On the given interpretation, we must be in either the first or the last row of the lookup table. Furthermore, since the interpretation gives 

as odd, and

as odd, we know that 

that it is odd. This is the little bit of "forward action" that we need. 

So, in the expression 

the right-hand input of the addition is odd. That means we can eliminate the first row of our lookup table, because the right-hand input of the addition is even in that first row. 

This leaves us with the last row, where the left-hand input of the addition is odd. Thus, under the given interpretation, 

in the expression 

must be odd. 

Again, this is more challenging than exclusively "forward" interpretations. In the following exercise, give yourself time to think it through, and keep reworking the problems until you get the idea of how to read the lookup table both "forward" and "backward." 

Demos

Example 3.7

Example 3.8

Exercise 3.3   

Practice and Test

Exercise 3.3—Interpretations Forward and Backward 

Quiz 3.3—Interpretations Forward and Backward 

Chapter 4—Answer Strings

We are going to continue working on interpretations. However, we are going to follow a new convention. We will evaluate, where possible, every character of the expression. I will demonstrate in a step-by-step fashion.

Use a Monospaced Font

Notice that with each table earlier in the chapter the characters in the header portion of the table match up one-to-one in vertical alignment with the characters within the body of the table. This is because a “monospaced” font is being used. Using a monospaced font is essential to our evaluation procedure. Please do not attempt to follow this procedure without employing a monospaced font. See Wikipedia, "Monospaced Font." 

Example 4.1—Step-by-Step Full Evaluation of an Interpretation

Step 1. Work from left to right, and attempt to evaluate each character of the expression as you come to it. The first character in the expression is an open parenthesis. That does not receive an even or odd value. So, we will put an underscore directly beneath the the open parenthesis. 

Step 2. Continue working left to right in the expression being evaluated. Next we come to the variable

which will receive an even or odd value. For now, use the lowercase letters “o” and “i”—with “o” standing for zero, and “i” standing for one. 

It is possible to evaluate this variable within the expression being evaluated, because the value is stipulated in the interpretation in the far left of the table. We simply copy the far-left value under

in the given interpretation to the position under

in the expression being evaluated. But we use an

instead of a

Step 3. Continue to work left to right. Leave a blank under the multiplication operator, because we will come back and evaluate it once we have evaluated the left input. 

And place an

beneath 

Step 4. Now go back to the multiplication, because it can now be evaluated. 

Following our lookup table, an odd times an odd is odd. So, we put an "i" under the multiplication operation. 

Step 5. We continue working left to right, and put an underscore underneath the closing parenthesis.

Step 6. We continue working left to right, skipping the addition, because it cannot be evaluated without it right side input

And we copy the zero value for the last variable in the expression. 

Step 7. We can now evaluate the addition. 

Notice that here we use the number one, because addition is the last character we come to in the evaluation procedure. The last value we come to is the "dominant" operation in the expression. The value of the dominant operation is the value of the entire expression. 

Also notice that within the parentheses, multiplication is the last element that we came to. Multiplication is the "dominant" operation of the expression enclosed within the parentheses. The value appearing under this dominant operator is the value for the entire expression enclosed within the parentheses. So, the left input of the addition is odd. 

Generally, notice that we have simply copied the stipulated values for the variables to their positions within the expression, leaving blanks for elements that are yet to be evaluated, and placing an underscore under any element that will not be evaluated. But we do so working left to right as much as logically possible. 

We have essentially completed the problem. But there is another new convention that we need to adhere to.

The Answer String

Going forward, we will typically respond to an exercise question with an “answer string.” Your answer must precisely match that programmed into the quiz mechanism—character-for-character. This convention is inspired by XML style, and some other standard programming conventions, but is deliberately distinct from XML so as to not interfere with XML programming, which is becoming quite ubiquitous. 

We continue on with the step-by-step procedure. 

Step 8.

Add the opening and closing "table" tags. 

Notice the slash in the closing tag.

Step 9. Put a semicolon at the end of each line of the interpretation (only within the tags).

Step 10. Eliminate all line breaks within the tags.

This is the "answer string" that should be entered into the quiz mechanism. Notice that there are no spaces in the answer string. Make sure that you only include what is within the tags. Avoid any leading or trailing spaces. Your answer has to match that programmed into the quiz mechanism character-for-character, because the procedure above produces a precise and unique string.   

Keys to Success in Constructing Answer Strings

Work in a text editor or word processor (students often choose Google docs).   

Use a monospaced font (for example, Consolas). 

In your word processor, turn off any autocorrection for double hyphens. 

Copy and paste as much as possible. 

Copy and paste the given partial table into your word processor. 

Work left to right as much as logically possible. 

Make sure there are no spaces in the answer string. 

Copy and paste your answer string into the quiz mechanism.

Avoid copying any leading or trailing blank spaces.

Notice after each attempt the correct answer programmed into the mechanism.

Pay attention to lowercase letters versus the proper numeral for the dominant operation.

Use text-compare (notice the hyphen) to find minute differences between your answer and the programmed answer. 

Keep in mind that much instruction is achieved when you compare answer strings using text-compare. 

Example 4.2—Full Evaluation of an Interpretation with a Trailing Parenthesis

Here is a similar example, but with the parentheses placed differently. 

In table form, the answer is

Notice that there is no underscore under the closing parenthesis. We do not want to include trailing underscores in the evaluation line. An underscore would not add any information. It would be superfluous. So, we just drop it. 

The answer string to be entered into the quiz mechanism is 

Demonstrations

Use a Monospaced Font

Example 4.1—Step-by-Step Full Evaluation of an Interpretation

The Answer String

Example 4.2—Full Evaluation of an Interpretation with a Trailing Parenthesis    

Practice and Test

Exercise 4.1a—Full Evaluation Procedure and Answer Strings 

The following checklist is helpful for proper answer string formatting.

Exercise 4.1b—More Interpretations as Answer Strings 

Quiz 4.1—More Interpretations as Answer Strings 

[}x*x{]

Chapter 5—Logical Operations

The use of tables we have made in the previous chapters has set us up for success as we move into logic proper in this chapter. Tables play a big part in zeroth-level logic. We use lookup tables to understand operations, and we create tables to evaluate expressions. This is true of any introductory logic course. The answer string technique we are using, however, is peculiar to my approach. The technique is a little difficult to get the hang of at first, but once you do, it will serve us well. Answer strings provide a very efficient means of communicating the content of logic. With this method of communication in place you can receive timely and very precise feedback on the progress you are making. 

Even and Odd Values as Truth Values

To make the transition to logic, we need to interpret the values even and odd somewhat differently. We will continue to treat them arithmetically as we have been doing. But additionally we will also think of them as "truth values." There are two truth values—true and false. We will treat odd as true, and even as false. 

Addition Is Exclusive Disjunction

Let's consider our working lookup table. 

Thinking of odd as true, we see that 

is true when either the left input or the right input is true, but is false when both inputs are true or when neither input is true. This is called "exclusive disjunction." 

Exclusive disjunction is something that we often use in normal conversation. For example, my daughter and I are walking to the ice cream shop, she'll tell me that her two favorite flavors are rainbow sherbet and cotton candy. And I tell her that she can have a scoop of rainbow sherbet or a scoop of cotton candy. I mean that she can have one or the other, but not both. That's exclusive disjunction. We are thinking of the variable 

as standing for the statement, 

And the variable

stands for, 

Then the combined expression, 

means, 

Sometimes exclusive disjunction is called the "exclusive or." 

Subtraction Is Superfluous

Notice in the lookup table that subtraction is also exclusive disjunction. With our reduced arithmetic of evens and odds, subtraction is logically the same operation as addition. So, subtraction is not a distinct operation. We can drop it. 

Multiplication Is Conjunction

Now, let's consider the logical interpretation of multiplication. Taking odd to be true, multiplication is true when both binary inputs are true, and otherwise it is false. So, if we assume that 

is true, then we are assuming that the left input and the right input are true. Any other truth value configuration is ruled out. This is called "conjunction." 

Again, this is a notion used in ordinary conversation. If, on the way to the ice cream shop, I tell my daughter that she can have a scoop of rainbow sherbet and a scoop of cotton candy, then she is going to expect both. Anything else will not be acceptable. In this scenario, the symbolic expression

is taken to mean, 

The logical notion of conjunction captures the primary way in which we use the word "and." For this reason, we often use the ampersand symbol

to express conjunction. 

Inclusive Disjunction

Above, I described exclusive disjunction—the "exclusive or." There is another way that we use the word "or"—in the "inclusive" sense. Suppose I ask my daughter, "Are you going to have a scoop of rainbow sherbet or a scoop of cotton candy?" She says, "I'm going to have both." She interpreted my question as meaning, "You can have a scoop of rainbow sherbet or you can have a scoop of cotton candy" in the inclusive sense, where the truth of both disjuncts (the binary inputs) is allowed. 

In logic, this use of "inclusive disjunction" is captured in the following lookup table. 

We use the lowercase letter 

to symbolically represent inclusive disjunction. 

Now, thinking in terms of even and odd numerical values, we can define inclusive disjunction arithmetically. 

Naturally, since inclusive disjunction is a possible interpretation of the English usage of "or," my daughter would get both scoops of ice cream. I would have to reward her for her logical perspicacity. 

Negation

Negation is an essential operation for logic. When thinking in terms of truth values, we often want to say that something is not true. For example, my daughter might say, 

Symbolically, we can represent this as,

The tilde symbol,

is the negation operation. It is an operation—but it is not a binary operation. It only takes one input. It is a "unary" (unus-, one; -nary, numbered) operation. 

Notice that negation only takes a right side input. 

Arithmetically, we can define negation as follows. 

There are only two possibilities for the value of 

Described in lookup table form, we have, 

You might be temped to think that negation is the same as adding a negative sign to an integer, especially since we have been doing arithmetic with integers. But logical negation is different than negating an integer. The negative of an even integer is still even, and the negative of an odd integer is still odd. In terms of our arithmetic of evens and odds, logical negation is quite different. It switches an even to an odd, or an odd to an even. And in terms of truth values, negation changes a false value to a true value, or a true to a false. 

Material Consequence

Another common logical operation is "material consequence." We can define material consequence as follows. 

Described in lookup table form, 

Material consequence is true in every binary case except when the right input is true and the left input is false. 

Notice that we are using the symbol

to mean something quite different from "greater than." And if you think about it, "greater than" doesn't have an obvious usage in our reduced arithmetic of evens and odds. We are repurposing the symbol. The ability to interpret symbols flexibly in this way is an essential skill that we learn through logic. 

Material consequence maps to the "if ..., then ... " construction in English. For example, I might tell my daughter, "If you clean up your room, we will go get a scoop of ice cream." Here we are thinking 

Sometimes the combined expression is symbolized as follows.

The symbol 

is conceived of as an arrow going to the right. We are trading upon this usage, but reducing it down to one character

The use of a single character is essential to our monospaced table methodology. 

Antecedent and Consequent

What comes before material consequence is called the "antecedent" (from ante, "before"; and cedere, "to go, yield"). The antecedent is the left input of material consequence. What follows material consequence is called the "consequent" (from com, "with, together"; and sequi, "to follow"). The consequent is the right input of material consequence. We use this terminology only in connection with material consequence. 

Two Common Logical Errors

When applied to ordinary language, the formal symbolic logic of material consequence is not as intuitive as the other operations we have covered so far. This leads to two logical fallacies that people often fall into. 

When considering my deal with my daughter you might think that if my daughter does not clean up her room, then we won't go get a scoop of ice cream—symbolically, 

This is a logical error. To see the error, consult the lookup table for material consequence. 

The first and third rows are those cases where the antecedent (left input) is false. In both cases

is true. And in the third row, not only is 

true, but also 

is true, even though 

is false. On the interpretation that my "if ..., then ... " construction is captured by material consequence, it is possible that my daughter does not clean up her room, and nonetheless we still go to get a scoop of ice cream. Maybe an emergency comes up, and we have to rush over to her grandmother's house. Once the crisis is over, we decide to get a scoop of ice cream. There is no logical contradiction with our deal in this. To think that there is is a misunderstanding of the logic. 

Another common misunderstanding is to think that if we end up getting a scoop of ice cream, then my daughter must have cleaned up her room—symbolically, 

Again, in the third row of the lookup table, we see that this is also a logical error. 

The expression 

can be true when the consequent (right input) is true and the antecedent is false. 

Root of the Errors

The root of both of these errors is that often when we say something like, "If you clean up your room, then we will go get a scoop of ice cream," we intend the phrasing to mean that if you don't clean up your room, then we will not go get a scoop of ice cream, and we will get a scoop of ice cream only if you clean up your room. But using a simple "if ..., then ..." construction, when something else is intended is just sloppy logic. It is both unclear and ambiguous—because frequently we intend to use the "if ..., then ..." construction precisely according to the logic of material consequence as described above. 

Material Equivalence

There is a logical operation—and a corresponding English construction—that is perfectly clear and unambiguous—that I could have explicitly and properly used when I made the deal with my daughter—if that is what I intended. 

"Material equivalence" is defined arithmetically as follows. 

The lookup table is

The English construction is "if and only if." If I meant that if my daughter doesn't clean up her room, then we are not going to get a scoop of ice cream—and that if we do go to get a scoop of ice cream, then she must clean up her room—then I could have said, and probably should have said, "We will go get a scoop of ice cream if and only if you clean up your room."  

Notice in the lookup table that the third row is significantly different from the problematic third row of material consequence. 

Translation between English and Symbolic Logic

One of the most challenging aspects of elementary logic is learning to translate back and forth between English and symbolic logic. I've given examples above to start the process going. However, I don't want you to worry about translation just at the moment. For now, we will focus on the manipulation of the symbolic apparatus. Once this is solidified we can begin working on translation techniques. 

Validity of English Arguments

Although I don't want you to worry now about being able to translate between English and symbolic notation, I do want you to keep in mind that such translation is crucial for the application of the symbolic apparatus. In philosophy we are primarily concerned with the evaluation of arguments. An argument is a set of propositional statements in English (or any other natural language). And once we translate the English statements into symbolic expressions, formal logic allows us to mechanically test whether an argument is "valid" or not. A full explanation of "validity" will come later in the course. For now we just want to be aware that a "valid" argument has something good about it. And an "invalid" argument is simply a bad argument. And all of this is wrapped up in the grammar of English, which we use each and every day. 

Symbolic Manipulation and English Grammar Simultaneously

One challenge when being introduced to logic is that you need to do abstract symbolic manipulation (normally associated with mathematics) at the same time as you are parsing English grammar. I don't think this is naturally difficult for human beings. I tend to believe that the human mind is very much made for this. However, our current education system tends to keep these activities strictly segregated. So, there is some need to unlearn this segregation, which is not really all that difficult to do. But at first you may find it a little challenging to—expressing it metaphorically—get your English brain to talk with your math brain. 

Although this is typically somewhat challenging at first, I have never had a student who could not overcome this obstacle. Keep working the exercises, and it will all come together. In reality, your brain is very much built for this. 

Logical Operations of Primary Importance

Since our ultimate aim is to evaluate English arguments (or arguments in any other natural language), we want to focus on the logical operations that are most intuitively applicable to English grammar. In principle there are 16 binary operations, and at least two unary operations. But it is customary to begin the study of zeroth-level logic with reference to a set of five, because these five operations in combination lead most naturally to a basic analysis of English arguments. Here they are in a lookup table. 

We will only use these five operations throughout the rest of the course. 

With negation, notice that only the first and third rows are filled in. This is because rows two and four would be redundant. 

Notice that exclusive disjunction (addition) is missing here. In zeroth-level logic we favor inclusive disjunction, and we simply call it "disjunction." 

Also notice that we are using the ampersand sign

instead of the multiplication sign

This is because we are emphasizing what was previously multiplication in the reduced arithmetic of evens and odds as being logically equivalent to conjunction, which naturally comes across in English with the use of the word "and." 

Identifying the Operations

The flash card set here will teach you the names of the operations. Please study the flash cards before attempting the exercise. Some new information is in the flash cards. For example, the material consequence symbol

not only maps to the English construction "if ..., then ...," but also can be read as "only if." And we favor "only if" when reading symbolic expressions. 

Notice that there is a "Choose a Study Mode" dropdown list in the lower right corner of the flash card frame. You may want to start in "Flashcards" mode. But the "Learn" mode works very well, even if you are just wildly guessing the first few times through. 

Practice and Test

Flash Cards—Identifying the Operations

Exercise 5.1—Identifying the Operations 

Quiz 5.1—Identifying the Operations 

Reading and Writing Symbolic Expressions

There are conventions for reading symbolic expressions. The flash cards will teach you how to read and write symbols according to the conventions. 

Flash Cards—Reading and Writing Symbolic Expressions 1 

Flash Cards—Reading and Writing Symbolic Expressions 2 

Flash Cards—Reading and Writing Symbolic Expressions 3 

Flash Cards—Reading and Writing Symbolic Expressions 4 

Exercise 5.2—Reading Symbolic Expressions 

Quiz 5.2—Reading Symbolic Expressions 

Boolean Algebra

This is not essential, and you won't be tested on it. However, it is interesting to compare our current symbolic apparatus with Boolean algebra. 

Our Symbolic Apparatus Compared to Alternative Systems

This also is something of interest, but not essential. The symbolic apparatus that we are using is not the only possible system. Other authors will use different symbols for our five logical operations. And there is not any strict conformity among any of the authors. However, there are typically limited options. 

Focusing just on the logical operations from this chapter, we have the following comparison. 

There are two unconventional symbols in our system—

and 

=

Although they are unconventional, I hope that you can see their similarity to 

⊃ 

and 

respectively. And we might also think of 

as similar to the right-arrow, 

For more information (if you like), see Wikipedia, "List of Logic Symbols."

Goofy Symbols

The symbols used need not be conventional at all — as demonstrated this short video: "Goofy Symbols Defined." 

Meanings of Symbols Are Context Dependent

An important concept of logic is that the meanings of symbols are context dependent. The meaning of any given symbol depends upon the context in which it is found. In connection with this are a couple of important skills—the ability to recognize context, and the ability to assign appropriate context-dependent meaning to symbols. 

As I've mentioned, different authors employ different symbolic systems. And it even goes so far as to be problem specific. A single author may use symbols in a particular way to solve one problem, and then use those same symbols in a different way to solve other problems. So, always pay attention to the scope of the context in which symbols are being used. 

In this course I try to be very consistent. We are still evolving our symbolic apparatus, but as we reach each new stage, we will leave behind any previous usages. And once we leave something behind, it will not reappear. 

Advantages of Our Symbolic Apparatus

There are two advantages to our symbolic apparatus. The first advantage is that we are using symbols readily accessible on a standard QWERTY computer keyboard. The non-QWERTY symbols often used in logic require complicated keyboard inputs, or other work-arounds. At our introductory level it is best to avoid those complications. The second advantage is that we can strictly use a monospaced font when completing interpretations and other tables. The non-QWERTY symbols do not readily conform to monospacing. Remember, as you complete tables throughout the course always use a monospaced font. 

(The monospaced methodology and choice of characters in our system is inspired by Richard Jeffrey, Formal Logic—Its Scope and Limits, Fourth Edition, edited by John P. Burgess (Indianapolis: Hackett Publishing, 2006). See especially chapter 7, "Uncomputability.")

Example 5.1—An Interpretation for One Expression Using the Logical Operations

An interpretation for one expression using our logical operations is very similar to the interpretations at the end of the previous chapter. 

Example 5.2—An Interpretation for Two Expressions

Multiple expressions will be separated by a comma (no space). Place an underscore beneath the comma. Each expression has a dominant operation. So, each should have a one or zero underneath the main, or dominant, operation in the expression. 

There can be any number of expressions in an interpretation. 

Demonstrations

Example 5.1—An Interpretation for One Expression Using the Logical Operations

Example 5.2—An Interpretation for Two Expressions  

Practice and Test 

Exercise 5.3—Interpretations with the Logical Operations

Quiz 5.3—Interpretations with the Logical Operations  

 Chapter 6—Truth Value Notation

"F" and "T" Instead of "0" and "1"

There is one more adjustment to our symbolic apparatus that I want to make. Instead of using 

for "even" or "false," we will now use

And instead of 

for "odd" or "true," we will use

From here on out we will think solely in terms of truth value, and we will not make reference to even and odd values. 

Lookup Table for the Operations

Moving forward, use the following table to lookup truth values. 

Notice that in the far left columns the truth value cases are arranged differently. In this conventional presentation the value "true" is given primacy, I suppose because it is intuitive to think "true or false" (in that order). 

Example 6.1—Interpretation with "T" and "F"

In the following exercise and quiz—and in the first exam—complete interpretations using 

and 

We will use lowercase letters for non-dominant operations, and capital letters for the dominant operations. 

Demonstrations

Example 6.1—Interpretation with "T" and "F"

Exercise 6.1 and Using Text-Compare.com  

Practice and Test

Exercise 6.1—Interpretations with "T" and "F" 

Quiz 6.1—Interpretations with "T" and "F" 

Chapter 7—Interpretations Backwards and Forwards

Now we will again do some more challenging interpretations that require you to read the lookup table "backwards" and "forwards." 

Example 7.1—Interpretation Backwards and Forwards

First, we will use the value indicated for 

We place a lowercase 

below the first variable in the first expression. 

Now looking at the lookup table for material consequence, we know that the conjunction in the first expression must be false, 

because we know we are in the second row of the table. 

Since we know the conjunction is false, one or the other (maybe both) of its inputs must be false. 

By considering the first expression of our interpretation alone we cannot determine the values of 

and

We have to consider the second expression. We were given that the material consequence in the second expression is false. Just like with the first expression, this tells us that in the second expression the left input must be true; and the right false. We're in the second row of the lookup table. This means that 

is true, and 

is false. 

That allows us to fill out most of the missing pieces. 

And, of course, the last expression, simply

is true. 

Demonstration

Example 7.1—Interpretation Backwards and Forwards 

Practice and Test

Exercise 7.1—Interpretations Backwards and Forwards 

Quiz 7.1—Interpretations Backwards and Forwards 

PART 2—TRANSLATION

Chapter 8—Pseudo-English

The whole purpose of our investigation of symbolic logic is to better understand logic expressed in English. So, we need to bridge the gap between symbolic expressions and logically equivalent English sentences. The easiest way to do this is by using what I call “pseudo-English.” 

Example 8.1—Translating Symbolic Expressions into Pseudo-English

Translate the following symbolic expression into pseudo-English. 

We will complete the translation character-by-character in order as they appear in the symbolic expression. 

From the lookup table we know that the tilde 

is 

in pseudo-English. 

So far, we have 

The variable 

is the clause

Now we have 

The ampersand

is 

So, we have

The variable 

is the clause

Thus, we get 

Finally, we put a capital letter at the beginning and a period at the end. 

And that is the translation from symbols into pseudo-English. 

It is “pseudo” (phony, artificial), because it is neither elegant nor natural. But it does conform to the rules of proper English grammar. 

Notice that we are not using any commas in the pseudo-English sentences. Pseudo-English translation is primarily a matter of using the lookup table, replacing each symbolic character with its corresponding phrase, keeping the order of the original symbolic expression. 

Propositions 

Intuitively it might help to think of a proposition as a complete thought. It doesn't leave the mind hanging. Consider the following. 

This is does not leave the mind hanging. But now, consider the following. 

This does leave the mind hanging—anticipating more. 

Grammatically, a proposition is a phrase that can be written as a complete sentence, and that makes a claim of fact. It can be true or false. 

This is a complete sentence, and it could be true or it could be false. It is a proposition. 

Atomic Propositions 

An "atomic" proposition is a proposition that contains no other proposition within itself. The atomic propositions in the section above are as follows. 

Each is a complete sentence, makes a claim of fact, and does not contain any other proposition within itself. In other words, an atomic proposition is a proposition that cannot be broken down into smaller propositions. 

Example 8.2—Translating Pseudo-English Statements into Symbolic Expression

Keeping in mind the notion of an atomic proposition, we can now translate from pseudo-English into symbolic expression. We will use the same lookup table as before. 

Translate the following pseudo-English statement into symbolic expression. 

Remember that pseudo-English follows the order of symbolic expression. So, we simply reverse-translate each component into the symbolic equivalent. Working left to right, the first component we come to is the phrase 

The lookup table tells us that the phrase translates into the tilde

So, we have 

The next component of the pseudo-English is an atomic proposition. The phrase

can be formulated as a complete sentence that makes of claim of fact, and it does not decompose into smaller propositions. Each atomic proposition we come to, we designate with a variable, working in alphabetical order. So, our partial translation is 

The phrase 

is in the lookup table, giving

And finally, we have second atomic proposition. Thus, 

is our complete translation from the given pseudo-English sentence into symbolic representation. 

Propositional Logic

Recall that zeroth-level logic is also called "propositional" logic. This is because symbolic statements in propositional logic are propositions, and they are made up of atomic propositions (variables), logical operations, and parentheses. 

Pseudo-English Translation Is Straightforward

Pseudo-English translation is, as we have seen, straightforward—because all that is needed is one-for-one replacements that maintain the same order of arrangement. The key characteristics of pseudo-English are (1) that each pseudo-English phrase maps one-to-one to a symbol, and (2) the order of proper symbolic expression is adhered to. 

Demonstrations

Example 8.1—Translating Symbolic Expressions into Pseudo-English

Example 8.2—Translating Pseudo-English Statements into Symbolic Expression

Exercise 8.1—Translating Symbolic Expressions into Pseudo-English

Exercise 8.2—Translating Pseudo-English into Symbolic Expression  

Practice and Test

Exercise 8.1—Translating Symbolic Expressions into Pseudo-English 

Quiz 8.1—Translating Symbolic Expressions into Pseudo-English 

Exercise 8.2—Translating Pseudo-English into Symbolic Expression 

Quiz 8.2—Translating Pseudo-English into Symbolic Expression 

Chapter 9—Natural English

Translation of natural English is not nearly as straightforward as pseudo-English translation. However, pseudo-English can serve as a bridge between natural English and it symbolic representation. 

Again, the whole point of symbolic logic is to give us some insight into reasoning that takes place in English—and not pseudo-English, but natural English. We need some way to get English propositions into symbolic expression. One way to do this is to translate natural English into pseudo-English. Then the pseudo-English can easily be translated into symbols. 

We will only deal here with some simple preliminary examples, because translating from natural English into pseudo-English is not nearly as mechanical as our previous translation exercises. 

I’ve expanded our previous table to include some natural English alternatives to the pseudo-English phrases. 

This table is not comprehensive. It merely gives you some indication of the variations that we can see in natural English. 

Example 9.1—Translating Natural English into Pseudo-English

Translate the following into pseudo-English. 

How do we capture the meaning of this in pseudo-English? We see in the table that implication, 

has associated with it the natural English construction, 

which seems to match up with the natural English sentence we were given. And the table also has the construction, 

Let’s use this second construction to rearrange the natural English. 

It is this ordering of the atomic propositions that is used with the 

But before we translate into pseudo-English, we need to deal with the negation. In the natural English we have 

This needs to be unpacked. First, we can eliminate the contraction, 

Notice that the 

is inserted into the middle of the first atomic proposition. The ordering so far is far from the ordering of the corresponding symbolic representation using a tilde,

And recall that the key characteristic of pseudo-English is that it follows the order of symbolic representation. So, to convert the natural English, we want to bring the negation out in front of the atomic proposition in which it is embedded. We can do that using the phrase

Using that phrase we get

We now have something that is close to pseudo-English. We just need to translate the 

construction into 

The implication goes in between the negated proposition and the affirmative proposition. 

Notice that the pseudo-English uses no commas. This is the complete translation. 

Okay, it sounds strange. That is why I call it “pseudo” English. And the meaning may not be entirely clear, because this is not the way that we normally use 

But the meaning of the pseudo-English claim is technically the same as the original natural English.

There are subtleties of meaning here that are quite important when reasoning logically. The following example will clarify. 

Example 9.2—Alternative Natural English Proposition

Example There is a quite natural way of using 

Suppose the person from before instead said, 

This is translated into pseudo-English quite easily, simply by handling the negation. 

Notice that the claim here means something quite different from the claim of the first example. In this second example we have a very honest claim coming from someone who occasionally rides their bike to work. It says that whenever they ride their bike it isn’t raining. They don’t ride their bike when it is raining. 

The Meaning of the First Example

The first example is different, and it is a quite extraordinary claim, coming from someone who allegedly rides their bike to work every time it isn’t raining. The meaning is perhaps most clear in the following formulation. 

Thursday it isn’t raining. They ride their bike to work. Friday it is raining. Maybe they don’t ride their bike to work. —Or maybe they do. There is nothing in the claim to rule out riding to work in the rain. Saturday, their day off work, it isn’t raining. They ride their bike to work anyway. This is at least the logical meaning of the claim. 

Maybe what they actually meant was the claim in the second example. As we work through our translation exercises, try to think about the differences in meaning among the various examples given. 

Pseudo-English as a Bridge to Symbolic Expression

Once natural English propositions have been translated into pseudo-English, further translating them into symbolic expression is quite easy. Thus, pseudo-English serves as a bridge between natural English and symbolic expression. 

The pseudo-English of the first example above was

Separating it into its pseudo-English components we have 

Each component is translated into symbolic expression in a one-for-one manner, and in the same order. 

The first atomic proposition is designated with the first letter of the alphabet; and the second with the second letter of the alphabet. 

Putting the symbolic components together we get

The pseudo-English of the second example, 

similarly comes directly over into symbolic expression. However, for comparisons sake. We will maintain the variable equivalencies from the first example. 

A Helpful Example

I have found that this example is very helpful. Keep in mind that if we are to use our symbolic apparatus to analyze propositions and arguments given in natural English, then we must get the translation from natural English to symbolic expression right. As emphasized above, pseudo-English is a bridge in the process of this crucial translation process. Translating from pseudo-English to symbolic expression is very mechanical. The trick is getting a pseudo-English translation that conveys the meaning of the original natural English. We have to get the grammar of the original natural English phrasing "stuffed into the box" of the symbolic syntax. And if we can get a pseudo-English representation that seems to work, then we have made the natural English conform in the necessary way. 

Natural English. Suppose we want to analyze the following natural English sentence. 

First Pass. As a first pass, we might rephrase the natural English as follows. 

Compare Table--Translation among Symbolic Expressions, Pseudo-English and Natural English above.

Second Pass. As a second pass, we might have—

Third Pass.

Pseudo-English. Finally, we can convert this into pseudo-English. 

Symbolic Expression. Once we have the pseudo-English, getting to the proper symbolic expression is just a matter of one-for-one substitution. 

Demonstrations

Example 9.1—Translating Natural English into Pseudo-English

Example 9.2—Alternative Natural English Proposition  

Practice and Test

Exercise 9.1—Translating Natural English into Pseudo-English 

Quiz 9.1—Translating Natural English into Pseudo-English 

EXAM 1 SAMPLE QUIZ 

The sample exam below is programmed a bit differently from our exercises and quizzes. It is somewhat like an exercise in that you will be allowed to edit your response an unlimited number of times. However, the correct answers are not provided. Rework the problems as needed until you feel you have achieved your best result. Ideally, you should be able to rework the problems until you have them all correct. If you diligently work to get a 100% grade on the sample quiz, then it functions as a logical puzzle. The feedback will tell you each time you get an answer exactly right. 

Exam 1 Sample Quiz 

PART 3—TRUTH TABLE ANALYSIS

Chapter 10—Truth Tables

In this chapter we are going to delve into one of the key components of any introductory logic course—truth table methodology. A truth table is very similar to the interpretations we have been doing already. However, whereas an interpretation includes only one set of values for the relevant variables, a truth table covers all possible value assignments—or, in other words, all possible "cases." 

Example 10.1—Single Variable Truth Table

Suppose that we want to analyze a statement that has only one variable. Then there are two values that the variable can take. 

The single variable can be either true or false. This covers all truth value possibilities—all cases, or all interpretations. 

Then we, of course, fill out the table, doing so much in the same way that we complete a single interpretation. 

We work left to right as much as possible, filling columns vertically. The first element of the expression that we can evaluate is 

So, we evaluate that all the way down the column underneath it. Then we continue working left to right, and the only thing we can evaluate is again the variable. 

Now we can backtrack and evaluate the negation.

And finally we can evaluate the dominant operation with capital letters. 

Example 10.2—Two Variable Truth Table

If we have two variables in the expression to be analyzed, then there are more cases to consider. There are four possible interpretations for the two variables considered jointly. 

Notice that we work horizontally left to right as much as possible, filling each column vertically. 

Branching Tree Logic

One way to conceive of all possible interpretations is through a branching tree logic. 

Two Variables

The first variable can be either true or false. If it is true, then additionally the second variable can be either true or false. If the second is true, then we are moving along the topmost branch and end up with 

If the first variable is true, and the second false, then we get 

in the second branch. 

The third branch is

And the fourth branch is

There are no other truth value possibilities. We have covered all possible cases. 

Three Variables

We can carry out the same procedure for three variables. 

—Eight possibilities. With one variable there were two possibilities. With two variables there were four possibilities. Now, with three variables there are eight possibilities. With the addition of each variable, the number of cases doubles. 

Formula for the Number of Rows

There is a formula for this. 

The number of rows in a truth table, covering all possible cases, is two raised to the number of variables. 

is the number of rows. And 

is the number of variables. The caret,

indicates exponentiation. 

Example 10.3—Three Variable Truth Table

Again, in the following example notice how we work left to right as much as possible, but fill in each column vertically as we come to it. 

Demonstrations

Turn Off Auto Substitution in Google Docs

Example 10.1—Single Variable Truth Table

Example 10.2—Two Variable Truth Table 

The Minimum Number of Socks I Must Take Out of the Drawer to Get a Matching Pair 

Example 10.3—Three Variable Truth Table  

Chapter 11—Memorization of Operations

As demonstrated in the previous chapter, a single truth table can require quite a large number of computations. So, before we go any further, it is a good idea to memorize the lookup table for the logical operations. This will allow you to complete truth tables much more efficiently. 

Use the following flash cards to achieve the memorization. 

In "Match" mode, challenge yourself to clear everything in under two minutes. Once you achieve that, see if you can do it in under one minute. How low can you go?

Flash Cards 11.1—Operations Memorization 

Demonstration

Pointers on Memorization of Operations 

Chapter 12—Modality of a Single Proposition

For a single symbolic proposition, truth table analysis will tell us the "modality" of the proposition. In other words, the truth table for the statement will indicate whether it is necessarily true, possibly true, or necessarily false. 

Example 12.1—Tautological Statement

A tautological statement is necessarily true, because it is true in every possible case. 

In this example there are two cases. And in all two cases the dominant operation is true. 

Example 12.2—Contingent Statement

A contingent statement is possibly true, but not necessarily true. It is true in some cases, and false in others. And when we say "some" cases, we mean at least one. So, a contingent proposition is true under at least one interpretation, and also false under at least one interpretation. 

In the first, third and fourth cases the proposition is true. But in the second case it is false. It is "contingently" true, depending on the interpretation given to the variables. 

Example 12.3—Contradictory Statement

A contradictory statement is necessarily false, because it is false under every possible interpretation. Sometimes we say that such a statement is "self" contradictory.

Demonstrations

Example 12.1—Tautological Statement

Example 12.2—Contingent Statement

Example 12.3—Contradictory Statement     

Practice

Exercise 12.1a—Fill in Modality of Statements to Complete Truth Tables 

Example 12.4—Truth Table Analysis Given a Single Statement

Consider the following statement. 

Let's complete the truth table for the statement. 

We see that the statement contains two variables, and so construct a two variable set up. Then we complete the body of the table working left to right and vertically by columns.