Table of Contents
Preface
Practical Logic
Chapter 1 Necessary Background: Biblical
Chapter 2 Necessary Background: Classical
Chapter 3 Definition of Logic: Purpose
Chapter 4 Definition of Logic: Limitations
Chapter 5 Foundational Patterns: Deduction and Induction
Chapter 6 Foundational Principles: Laws of Thought
Chapter 7 Patterns:
Premises and Conclusions
Chapter 8 Patterns Embodied in Language: Guidelines
Chapter 9 Patterns Embodied in Language: Premise Types
Chapter 10 Patterns Embodied in Language: Diagramming
Chapter 11 Principles: Evaluation of Arguments
Chapter 12 Principles:
Informal Fallacies: Composition
Chapter 13 Principles:
Informal Fallacies: Distraction
Chapter 14 Principles:
Informal Fallacies: Ambiguity
Chapter 15 Review of Practical Logic
Symbolic Logic
Chapter
16 Practical
and Symbolic Logic: Introduction
Chapter 17 Categorical Logic:
Translation
Chapter 18 Categorical Logic:
Distribution and Inference
Chapter 19 Categorical Logic:
Chapter 20 Categorical Logic:
Mood and Figure
Chapter 21 Categorical Logic:
Rules for Validity
Chapter 22 Propositional Logic:
Propositions
Chapter 23 Propositional Logic:
Truth Tables
Chapter 24 Propositional Logic:
Shorter Truth tables
Chapter 25 Propositional Logic:
Inference and Replacement
Chapter 26 Propositional Logic:
Basic Proofs
Chapter 27 Analogical
Reasoning: Definition or Description
Chapter 28 Analogical Reasoning:
Some Directions
Chapter 29 The Big Picture: Refocusing
Chapter 30 The Big Picture: Definition and Translation
Chapter 31 The Big Picture: Larger Arguments
Chapter 32 The Big Picture: Conclusion
Appendix I Categorical Logic Helps
Appendix II Propositional Logic Helps
Appendix III Imitation Reading
Glossary
Chapter 1: Necessary Background: Biblical
Come now, and let us reason together, saith the LORD: though your sins be as scarlet, they shall
be as white as snow; though they be red like crimson, they shall be as wool. Isaiah 1:18
Every reader of this book reasons even as he reads this sentence. Reasoning and thinking are inescapable. Thinking is a necessity to human existence that is comparable to breathing. Every thought that goes through your head is, in a sense, a use of your reason. In the study of logic, we have before us the difficult and sometimes confusing task of thinking about our thinking. Thoughts about thoughts can then become material for more thought about those thoughts. I introduce this text with this consideration to help you realize that the study of reasoning is not always easy, and is often fraught with the most difficult and interesting philosophical questions.
The history of the study of reasoning is, as a result, very philosophical. Since my goals for this course lie in a different direction, we will not spend much time discussing the ins and outs of philosophical systems. However, as some background is necessary, I will try to outline a Biblical perspective on human thought and reasoning, and then compare and contrast it with the classical or secular view. Forgive me if this seems cursory; I admit it is only that.
In this book you will learn about some of the intellectual tools for thinking about your thoughts. More particularly, you will learn to evaluate purposed thought (reasoning). The most common manifestations of purposed thought are found in “arguments” or, perhaps a bit more technically, “syllogisms.” In this kind of purposed thought there are two main parts: the premises, or reasons, and the conclusion, or result. I want you to be able to follow the reasons to the result, the premises to the conclusion. Most importantly, however, I want you to do this outside the classroom, as well as in the exercises in this textbook. I hope that the intellectual tools you pick up here will be handy enough for you to use them all your life. But before we get down to the nitty-gritty of these principles and techniques, let me give you at least a little bit of background.
Biblical View of Reason:
Men are created in God’s image. When Scripture teaches about God it teaches us also about ourselves, and when it teaches us about ourselves it teaches also about God. In this, Scripture is teaching us that purposive thought in ourselves can be seen as a reflection of a purposing God, and that the purposes of God provide a basis for purposively reasoning man. Man’s purpose is to glorify God and enjoy Him. Enjoying God means knowing Him and knowing who He is and what He is like. One of the first things that Scripture teaches us is that God is Truth—He is the fount of all that is true, good, or lovely. Pursuit of God, then, is a pursuit of ultimate Truth—this is at least one reason why God gave us the ability to reason. When men reject truth saying that they cannot know and that there is no truth, true (righteous) thought ceases (Romans 1).
All
truth and knowledge depend on God’s being, as does all of the creation. Scripture is unequivocal about this. This means that we cannot seek truth without
seeking God; for man, there is no such thing as discovering truth on his
own. All of our thoughts depend on
God. This is strikingly illustrated in
the language Scripture uses concerning truth.
Where autonomous human thinking tends to draw an impersonal line between
true and false, Scripture speaks of truth and lies. God’s Word is truth; Jesus Christ himself is
Truth; if it is not God’s Word (or consistent with God’s Word) it is a
lie. Scripture teaches us that truth
is ethical and personal, not merely intellectual.
This ethical and personal approach to truth is demonstrated further in the Bible. Scripture does not recognize all “reasoning” as good and legitimate. Unrighteous human reasoning exists, as does righteous human reasoning. Far from being amoral, reasoning is always ethically charged. While there are examples of a positive reference to reasoning in Scripture (Paul’s defenses of the faith in Acts, etc.) we often find that the scribes and Pharisees are the ones “reasoning among themselves.” Their reasoning is not necessarily invalid, but it is unsound and leads not to truth but to lies. Reason is not a means for automatic ascension to knowledge, truth, and God; rather, it is a tool given to us by God, which we can use either to ascend to knowledge of Him or to descend to hell.
The epistles of the Apostle Paul are great examples of holy reasoning. “Shall we continue in sin that grace may abound? Certainly not! How shall we who have died to sin live any longer in it?” This reasoning is phrased as a question and is also rhetorically fashioned, but Paul is clearly engaging his God-given reason to explain salvation in Christ. The argument is simple: if you die to something, you leave it behind; if you died to sin in Christ, you left it behind. Therefore, you cannot continue to live in something you died to. He goes on to explain that we have died to sin in Christ, and if we died with Christ then we will also be raised with Him. This is why we are to walk in newness of life. There is much food for thought in these passages.
The book of Job presents another interesting example of reasoning in Scripture. Job’s friends reasoned with him at length about God’s purpose in allowing him to suffer, only to find in the end that they were wrong. Again, their reasoning was not necessarily invalid. Most of their premises were even legitimate, so how did they go wrong? At least one of the things that caused their error was their certainty that they could parse out the purposes of God. Their error is in effect a warning to those who do seek knowledge, truth, and God Himself with reason, reminding them that while they can and will know what God reveals, they must not think they can know more. Reason can tempt us to assume we have more knowledge than we actually have.
Reason, as a good gift of God, is subject (like all of the created order) to use or misuse by man. One common misuse of reason is idolatry—worshipping the created thing instead of the Creator. Most of the unbelieving world wants to use human reason, in one way or another, as an ultimate standard. Christians, just as the Israelites were, are susceptible to the temptation to idolatry; to bring things to Reason as an ultimate standard is nothing less. No matter how cogent our reasoning, we must hold our thought in submission to God and His Word. When reason joins the worshipping chorus of creation and Scripture, it becomes a means to understanding and applying revealed truth; but when it is an ultimate standard, it becomes an idol. I want to make this very clear. I do not want you to study logic if you think that by it you can determine truth. Truth is revealed. Logic and reason are designed to help you understand revealed truth, and, by God’s grace, for that task they are very capable.
The
Proverbs speak often of wisdom, understanding, and knowledge as prize
possessions. Knowledge, understanding,
and wisdom deliver man from sin and folly.
Did you know that if you learn well the discipline of logic and subject
it to God’s Word, it will give you treasure and safety? In Proverbs 8 and 9 Wisdom, personified,
explains all of the bounty that is hers.
Right use of reason is at least one of the aspects of Godly wisdom. Throughout the Proverbs the wise man is
pictured as the one who sees and has understanding. Reason, as we mentioned above, seeks
understanding. However, reasoning is not
something done in a vacuum; it is not a system without premises. If you do not have the first premise right,
in fact, you cannot get any further. The
fear of the Lord is the beginning of wisdom.
Colossians 2:3 says that all the treasures of wisdom and knowledge are in Christ. In Christ we understand the mysteries and we know God. But even beyond Who Christ is and what He accomplishes in His person (treasure beyond imagination), we have His specific example. In His life on this earth Christ was not only a teacher of truth but a refuter of error. He often confounds the scribes and Pharisees with His answers. In His example, Christ shows us the right use of reason: to glorify God and leave the unbeliever without an excuse. Christ’s use of reason and language is our first example; following in His steps, other holy men of the Scriptures provided examples for us, as well.
The Apostle Paul also reasoned with unbelievers. Notice that Paul didn’t just appeal to his personal experience, or to his faith, but he reasoned with them. He became positively didactic when rooting out error in unbelievers and in the church. He did see the Christian message as more than a set of true propositions, but he definitely didn’t see it as less than that. “If there is no resurrection, we are worse than fools.” In other words, “If there isn’t a truth to be argued for, there isn’t any good news.” So an important use for reason is the defense of the faith against unbelief. Those who worship God use reason; those who worship Reason are used by it.
As in all things, God comes both first and last. The fear of the Lord is both the beginning and the end of right thinking. When you subject your thoughts to the Lord, He will give you the desires of your heart. He will show you truth and knowledge, even beyond your years. Scripture’s attitude toward human reason is ethical and personal. Reason is a tool, and used well it serves God in beauty and truth; used sinfully, it condemns the one who uses it and all who listen.
Summary: Reasoning is inescapable, and it involves many philosophical questions. Though this is not my focus in the course, this chapter provides some necessary background to the study of reason. The Biblical perspective on truth and reason is personal and ethical, and it reveals the fact that reason is a tool which can be used or abused. Reason is abused when it is worshipped, or used by man to justify rebellion against God.
Exercise
1: Theory
a. Summarize the purpose of
this book as stated in this chapter.
b. What is another word for
“syllogism”?
c. Is reason a gift from God
or a temptation to idolatry, or both?
d. What is the Biblical view
of reason?
e. What, according to the
Bible, is truth?
Exercise
2: Imitation
Read the first half of the book
of Proverbs. How does the writer
reason? Can you follow his reasoning?
Exercise
3: Practice
Use a concordance or Bible
search program to search for passages about reasoning, truth, and wisdom. List the characteristics or qualities given
for reasoning, truth, and wisdom.
Example:
Reasoning Truth Wisdom
1. Another name 1.God gives truth 1. God gives spirit of wisdom
for pleading with mercy; Gen for making priestly garments
Job 13:6 24:27 & 32:10 Exodus 28:3
2. continue 2. etc 2.
etc
Chapter
3: Definition of Logic: Purpose
“Language is
the dress of thought.” Samuel Johnson
Logic, then, is the discipline you will learn in the following pages. But what is logic? As it turns out, the answer to this question is not as simple as you might expect. The way you define the word reflects what you hope to accomplish with logic, and suggests its limitations as well. While most definitions of logic are similar, there is some amount of variation from one author to another. In this chapter I will set forth my answer to this question.
As I mentioned in the last chapter, there is a sense in which when we practice logic, we are thinking about thought. Logic is not indiscriminately concerned, however, with all kinds of thought. Logic teaches you how to evaluate thought processes such as those involved in defending a position or solving a problem; that is to say, logic is about purposive thought. By this I mean the kind of thinking that sets out specifically to accomplish something. I do not mean random thoughts that cross a person’s mind. So logic is “the study of purposive thought.” This definition is workable, but let me refine it a bit more.
The word logic comes from the Greek word logike or logikos,
indicating something belonging to speech or to the reason. The fact that speech
and language are part of the etymology gives us a clue about how we will
actually use logic. When you analyze
purposive thought with logic, you are not going to be looking at arguments as they
form themselves in someone’s head—that is impossible. Rather, you will look at arguments embodied
in language, whether spoken or written.
So we should add this to our definition.
Logic is “the study of purposive thought, as embodied in the spoken or
written word.” This definition is
better, but I predict objections from students of logic who have been taught
that logic is “the science of necessary inference.” For them our definition is far too
general. There are a couple of ways to
answer this objection, but instead let us refine the purposive-thought clause a
bit first. Logic applies common and
standard forms and certain criteria of judgment to the purposive thought it
evaluates. In other words, there are principles
and patterns specific to the discipline of Logic. Logic is the study of the principles
and patterns of purposive thought, as embodied in language spoken or written.
This will be the operative definition of logic for this course. It includes what we will study, and excludes what we won’t. It will still be too broad a definition for some people’s tastes, but I believe even they must admit it as acceptable since it meets the requirements for definition set forth by the principles of logic itself. The definition is neither too broad (including other disciplines besides logic) nor too narrow (excluding legitimate kinds of inference). It is not negative, and it includes the essential attributes of the referent without using metaphorical or vague language. This definition also has the advantage of being fairly close to the conventional use of the word. The logic of computer programming, for example, does fit the definition, although it is not our specific focus in this book.
So how does my definition indicate what I think logic should do? The phrase “embodied in language spoken or written” reflects the fact that I want you to learn to use logic practically. Logic is a tool that needs to be used to be learned well, and to use it you must (obviously) have opportunity to use it. By emphasizing the fact that arguments are embodied in language, and in language which we use every day, I hope to help you realize that you have many opportunities to use logic practically. If your study of logic never meets the ground, you will be an expert in rarified reasoning and nothing else. That is why we begin with practical logic in this book. You will learn how to identify and diagram English arguments and give them a general evaluation, before you go on to the more abstract and precise science of symbolic logic.
The phrase “of purposive thought” points to a couple of ways in which I expect you to use logic. First, as we discussed earlier in this chapter, the study you will do here is not just of thought in general but rather of thought that is going somewhere and doing something. Second, “purposive thought” is a rather general phrase but it helps us to include in the study of logic legitimate inferences that are sometimes excluded. For example, those who define logic as “the science of necessary inference” exclude both induction and analogous reasoning, neither of which deal in necessary inferences[1]. I want you to be able to see both of these as legitimate and important rational processes, and to be able to evaluate them.
The phrase “principles and patterns” points to the systematic and evaluative aspects of logic. Logic helps you to identify and distinguish sound reasoning from unsound. In order to do this effectively, you will have to learn the patterns and the principles of sound reasoning. Equipped with these tools you will be able to analyze reasons and arguments—both your own, and those of others. The patterns will help you quickly classify arguments, so that they can be analyzed according to the principles appropriate to their form. Principles and patterns are at the heart of logical analysis.
The phrase “the study of” indicates what you will have to do to learn and to practice this discipline. This study is an ongoing process which demands constant maintenance—in other words, it is a discipline. The principles and patterns of purposive thought will eventually become second nature to you. You have to think, and if you habitually use logic to help you in this process, eventually it will not be hard to maintain.
Summary: The definition of Logic is
foundational to the rest of this course:
Logic is the study of the principles and patterns of purposive
thought as embodied in language spoken or written.
Exercise
1: Theory
a. Memorize the definition of
logic given in this chapter.
b. Outline how the definition
helps us understand the purpose of logic.
c. Why is this definition
better than some of the common narrower definitions?
d. What is purposive thought?
e. What does “embodied in
language” mean?
Exercise
2: Imitation
Read the first quarter of the
book of Job. What is Satan’s argument
concerning Job? What are Job’s friends
trying to convince him of?
Exercise
3: Practice
Think of five hypothetical
examples of purposive thought.
Example:
I am trying to think of examples of purposive thought. Purposive thought is thought that has a particular end in mind—it tries to accomplish something. So it seems that trying to think of examples of purposive thought is an example of purposive thought.
Or,
As a Christian, I love Christ. Christ says that if I love Him I will want to keep his commandments. So, you can see, I want to keep his commandments.
Chapter
6: Foundational Principles: Laws of Thought
There are some who…assert that it is possible for the
same thing to be and not be…. We can,
however, demonstrate negatively even that this view is impossible, if our
opponent will only say something; and if he says nothing, it is absurd to seek
to give an account of our views to one who cannot give an account of anything,
in so far as he cannot do so. For such a
man, as such, is from the start not better than a vegetable. Aristotle, Metaphysics
In this chapter we come to some foundational principles of “purposive thought.” Some of you are probably already familiar with the phrase “the laws of logic.” The meaning of this phrase is not always clear, but when it is, it most often refers to the three laws of thought. These laws originated at the time of Plato and Aristotle in the schools of the Stoics and in their own writings. They have traditionally been considered the three laws necessary to rational thought; that is to say, if these were not true there would be no thinking. In recent years this has been questioned by science, and also by those who hold observation to be a higher court of appeal than reason. In this case we see two different camps of unbelief holding up conflicting standards for truth. One camp uses reason founded on the laws of thought as the ultimate standard; the other wants to use the scientific method, founded on the laws of physics and observation of physical phenomena, as the ultimate standard of reality. Notice that unbelieving worldviews still need to have an ultimate point of reference if they want to keep thinking. These two extremes are the result. For the Christian, science and reason are not in conflict. They are both legitimate thought processes and are both subject to the Word and will of God. In this chapter I hope to show how the laws of thought and scientific or inductive reason can be reconciled, and what role each plays in forming a basis for logic.
- The Law of Identity: A thing is itself and not something
else. If a thing is A then it is
A.
- The Law of Non-contradiction: A thing and its
opposite cannot both be true at the same time and in the same
respect. If a thing is A then it is not not-A.
- The Law of Excluded Middle: A thing is either true or false but not
between. Either A is true or A is
false.
These probably seem intuitively obvious—which is why they have been almost universally accepted throughout the years. And, as long as you talk about deductive reasoning (moving from generals to a particular conclusion) they pose no problems. The questions concerning their ultimacy have arisen with the increased interest in the scientific method and induction (reasoning from particulars to a general conclusion). The deductive method involves putting general principles together to create a more particular conclusion. An everyday example would be the following argument: “Light switches turn on lights; this is a light switch; this turns on a light.” Although your mind is able to skip most of these stages and go directly from the observation to the conclusion, at one point in time you were applying the deductive method to this situation in order to figure out how light switches worked. However, before that process, you needed to get the general principle that light switches turn on lights from somewhere. There are a couple of ways you could come up with this general principle. First of all, someone could tell you that it is the case that these kinds of switches turn on lights. You would then try this principle out and flip as many light switches as you could. These particular instances of turning on a light switch and seeing a light come on begin to form an inductive process of reasoning in your thought. You already have the conclusion “light switches turn on lights” in mind, but the more you test the principle and find it true the more certain you are of the reliability of the general rule, and the stronger your argument. Here you see the difference that leads to the disagreement between those who deduct and those who induct. If the premises of the deductive argument are true, and the reasoning valid, then the conclusion must be true. If the premises of the inductive argument are true and the reasoning correct, the conclusion is merely strong. Inductive arguments can seem to be inconsistent with the Law of Excluded Middle, since a strong or weak conclusion is not specifically identified as either true or false. There is a potential for the principle to be false in one case (when you turn on the switch for the garbage disposal) while it is true in all the rest. And while we aren’t saying that the same thing is true and false at the same time in the same respect, we are saying that it can be true at one time and false at another. This doesn’t technically break the Law of Non-contradiction, but a system of reasoning that regularly makes arguments like this also isn’t the most perfect match for the rule. So you see why induction (and science) and the three Laws of Thought can seem to be at odds.
But why are we discussing this, anyway? As a logician, you will have to consider these questions. What role do the laws of thought play? Are they ultimate? Are they practical? Especially, how can we make induction jive with the laws of thought? Should one of these methods of reasoning be considered supreme and the other subject to it? My answer to that last question is that, since the Word of God governs both induction and deduction, we as Christians can reconcile these two seemingly disparate points of view as both being useful, without necessarily making one subordinate to the other. The answer to the seeming dilemma is that deduction and induction deal with the world in different ways. Deduction is essentially abstract and theoretical, dealing with universals or point-in-time questions. Induction is concrete and practical, trying to establish patterns and general principles among particular examples. Don’t misunderstand; there are good and necessary purposes for both abstract and concrete purposive thought. They actually build on one another. And we find (and use) both in reasoning every day.
The
Biblical basis for both is clear.
Scripture everywhere upholds the idea that we learn from our interaction
with the created order around us. From
the illustrations in Proverbs that tell us to observe the ants, to the heavens
that declare the glory of God, Scripture upholds the idea that the creation
gives us many particular reasons to conclude that God is good and great. In addition to the example of Scripture, we
have the principle from God’s law that from the mouth of two or three witnesses
a matter is established. If you consider
the Scriptures with this in mind, you will find that there are almost always at
least two accounts of every event recorded in Scripture. In the Old Testament we have four books that
tell the story of
The deductive method of reasoning is also supported throughout Scripture. Especially obvious are the deductive arguments of the New Testament epistles. When you are reading these passages, one of the most valuable questions you can ask is, “What is the ‘therefore’ there for?” In the next chapters we will learn to look for sign words that indicate the conclusion of an argument; we find many of them in Scripture. There is another argument for deductive reasoning that is even stronger than examples of it from Scripture: God’s nature itself is the foundation for the Laws of Thought. God is Himself, the same yesterday, today, and forever (Law of Identity). We also know that He doesn’t lie (Law of Non-contradiction). Finally, God and His words are Truth, and all else is a lie (Law of Excluded Middle). And so, the foundational principles of deduction are a reflection of God’s nature as well.
So what do we do with the seeming conflict between these two methods? The short answer is, “nothing.” God’s Word is our ultimate standard. Reason, both deductive and inductive, is used in our understanding of Scripture, but a systematic philosophy of each is, luckily, not necessary to understand the meaning of Scripture which is “plain,” according to the creeds. To satisfy the curious, however, we can point to the differences of perspective and purpose between the abstracting tendencies of deductive reason and the practical tendencies of inductive reason. The practical inductive process is the one we most naturally use to form opinions and general rules for our everyday living (as in the light switch example). We all know there will be exceptions, but we also count on a relative amount of consistency from one time to another and throughout time. Time, then, is a factor in the inductive process. Inductive principles are considered not at only one moment in time, or as being stationary in time. They are fluid. This is why we can only refer to the relationship between induction and deduction as a seeming contradiction. The seeming contradiction between the Law of Excluded Middle and induction arises from the fact that induction is an over-time process, and deduction is a moment-in-time or outside of time process. Induction doesn’t claim that a statement can be true and false at the same time in the same way. It means that, over time, due to change, the object of knowledge can change, as can human perception. The laws of thought state clearly that they are to deal with things that are not in motion or subject to change (i.e. “in the same place and in the same respect”). They are used to apply timeless or general truths to a particular situation.
The inductive method collects data according to a hunch or hypothesis, which then becomes a principle when the argument is strong enough. The inductive method relies on the laws of thought, as well. If it were possible for a thing to be both itself and not itself at the same time and in the same way, no one would ever be able to collect the particular data for inductive proof. Unfortunately, I have to leave this discussion here. If your interest is piqued by this topic, more information on epistemology can be found in the bibliography or on the web site.
As we begin in the next chapter to look for arguments in English, keep your eyes open for arguments that seem inductive versus those that seem deductive. Both will appear in everyday reasoning, and both are important to the overall processes of purposive thought. As Christians, we can easily reconcile what we learn from experience to what is ultimately true. Truth is revealed to us both progressively, through time and space, and ultimately unchanging in God and His Word.
Summary: The laws of thought, (1) the Law of Identity, (2) the Law of Excluded Middle, and (3) the Law of Non-contradiction, are basic to all reasoning, although they have been questioned in recent years by proponents of the scientific method. Although there are some initial seeming inconsistencies between induction (scientific reasoning) and deduction (traditional logic), the two methods are actually complementary. The apparent inconsistency is the result of a difference in the way they interact with the world.
Exercise
1: Theory
a. List, define, and give
examples illustrating the three laws of thought.
b. What makes the inductive
process and the laws of thought seem incompatible?
c. How can you reconcile the
seeming conflict between the laws of thought and induction?
d. How are the laws of thought
foundational to both induction and deduction?
e. Give a brief Biblical
defense of both induction and deduction.
Exercise
2: Imitation
Read the last quarter of
Job. God reasons by asking
questions. What is His point? How does this book illustrate that the fear
of the Lord is the beginning of wisdom?
Discuss Job 42:7.
Exercise
3: Practice
Think of five real-life
situations in which the laws of thought make rational communication
possible. If you get stuck, take a break
and come back to it later.
For example:
At the grocery store they said that the total for all the items I wanted was $5.25. The Law of Identity assures me that $5.25 is $5.25.
When my dad says he wants what is best for me I know that doesn’t mean he doesn’t want what is best for me, since the Law of Non-contradiction would then be violated.
If it is false that pigs are whales, I don’t have to worry that it might be partially true, since the Law of Excluded Middle says that things are either true or false but not in between.
Chapter
10: Patterns Embodied in Language: Diagramming
If the English language had been properly
organized…then there would be a word which meant both “he” and “she,” and I
could write, “If John or Mary comes heesh will want
to play tennis,” which would save a lot of trouble.
A.A.
Milne, The Christopher Robin Birthday Book
This is the final chapter on the patterns of purposive thought in language. In it you will learn two ways of diagramming the premises and conclusions which you are already learning to identify. Diagrams are helpful tools in evaluation and understanding since their standard form can help to reveal differences, similarities, and irregularities of arguments. They also help to systematize the connection between premises and conclusions.
The first method of diagramming is the simple listing method. When you begin to set up a diagram it will be necessary to change and distill the wording of the argument out of the language in which it occurs. That means that sometimes a statement, as it occurs in English, needs to be changed a little to fit the diagrammatic system. We’ve seen this at work already in chapters 5 and 7. As we’ve also discussed in previous chapters, you have to identify premises and conclusions in the original language first. Then you can begin the process of standardizing your premises. Here is a basic three-step process for doing so: First, turn the language into indicative statements, removing as much of the “padding” language as you can, and arrange the argument so that all the premises come first and the conclusion last (whether or not this is the order in English). Second, take terms[2] that refer to the same thing in English but are different in word order or detail, and change the word order and detail to make them alike. Third, order and standardize—as much as possible—the tense and mood and type of verb in each statement. The goal is not to change the English any more than necessary, so be careful in each of these steps. Let’s look at the passage from C.S. Lewis again to illustrate this process:
If you asked twenty good men today what they thought
the highest of the virtues, nineteen of them would reply, Unselfishness. But if you asked almost any of the great
Christians of old, he would have replied, Love.
You see what has happened? A
negative term has been substituted for a positive, and this is of more than
philological importance. The negative
idea of Unselfishness carries with it the suggestion not of securing good
things for others, but of going without them ourselves, as if our abstinence
and not their happiness was the important point. I do not think this is the Christian virtue
of Love.
First step: there are a number of arguments in this passage, as we considered in Chapter 8. Since we already set forth the hypothetical inductive arguments in that chapter, let us turn to the third argument. Arguments in this passage are already arranged according to the “premises first, conclusion last” pattern. All we have to do is identify and list the premises and conclusion minus the non-essential padding language.
Premise: Good men today say the
highest of the virtues is Unselfishness.
Premise: Good men (Christians) of
old say the highest of the virtues is Love.
Conclusion: A negative term has been
substituted for a positive. Premise: Of twenty good men today,
nineteen of them would say the highest of the virtues is
Unselfishness. Premise: The great Christians of
old would have replied, Love. Conclusion: A negative term has been
substituted for a positive.
Second and Third Steps
First Step
You can see that the wording of the terms in the first box is different in each premise. When we compare the two premises together and look at the rest of the passage we see that Lewis is making a comparison between good men today and good men of the past (specifically Christians). Since this is the case, we are justified in standardizing the first term in each premise as “good men.” The “highest of virtues” phrase is clearly implied in the second premise, so we can standardize the second term as well. Then the third step in this argument is not that difficult. We merely change the second premise’s verb from “reply” to “say,” which it was already obvious we needed to do.
There is one further thing that we can do to make our analysis even more compact and easy to use—abbreviate some of the terms. Like this:
T = good men today, V =
highest virtue, U = unselfishness, O = good men of old, L = love, N = negative term, P =
positive term Premise: T
say V is U Premise: O
say V is L Conclusion: N
has been substituted for P
Now once we do this, a terminology gap appears between the premises and the conclusion. The terms in the conclusion are nowhere in the premises. Have we identified and reproduced the argument correctly? After looking at the original passage again, indeed, it seems that we have. So the most likely solution to the problem is that there are assumed premises.
T = good men today, V =
highest virtue, U = unselfishness, O = good men of old, L = love, N = negative term, P =
positive term Premise: T
say V is U Premise: O
say V is L Assumed Premise: U
is N Assumed Premise: L
is P Conclusion: N
has been substituted for P
The assumed premises are easy to overlook because they seem so intuitively obvious when you read the original text. Lewis didn’t include them because it would have been unnecessary and it would have weakened his writing style. However, when we want to look at the argument in detail it is important to see how the premises fit together and point to the conclusion, and that is why we need to make them explicit here.
The second method of diagramming, the arrow method, is specifically concerned with how the premises point to the conclusion (i.e., dependently or independently). Think back to our discussion of dependent and independent premises in the last chapter. The second method of diagramming is based on this distinction. First, you follow all the steps above and abbreviate your terms, just as in the first method. But then you add one step. That is, you examine the premises and conclusion to see if the premises are dependent or independent and then you show the relationship in the diagram. Here are a couple of examples.
Hypothetical Inductive (#2) Argument: Lewis O1 says L is V O2 says L is V O3
says L is V Etc. Therefore, O generally say L
is V T = good men today, V =
highest virtue, U = unselfishness, O =
good men of old, L = love, N = negative
term, P = positive term (T say
V is U) + (O say V is L) + [U is N] + [L is P] N has been substituted
for P
In the diagram to the left you see that the deductive argument we’ve been diagramming throughout this lesson (brackets indicating assumed premises) uses all dependent premises. None of the premises by itself points to the conclusion, but all of them, taken together, do. In the hypothetical inductive argument from the same passage in Lewis we see that each points, independently of all the others, to the conclusion that good men of old, as a group, say that love is the highest virtue. The inductive conclusion grows stronger with each added premise, but none of the premises relies on any others in order to point to the conclusion. Let’s look at two more examples of this diagramming method:
I am bored; therefore
I am bored or it is Tuesday. B = I am bored; T = it is
Tuesday B
Therefore: B or T Jesus is the Christ:
the Christ came to redeem His people, therefore Jesus came to redeem His
people. J = Jesus; C = Christ, R =
came to redeem His people J
is C + C is R Therefore: J is R
On the left we see an independent premise, and on the right, dependent premises. These are, in fact, the examples from last chapter. They illustrate, perhaps in a simpler way, how the arrow diagrams work.
The arrow diagrams have one added benefit which is worth considering. Look back to Chapter 7 and consider the whale diagram. This diagram illustrates the way in which an arrow diagram will allow you very flexibly to show relationships, not only between premises and conclusions, but also between individual supporting arguments in a passage. It will also allow you to represent a situation in which an author presents both inductive and deductive reasons for a particular conclusion. We will discuss this in more detail in Chapter 31.
These two methods of diagramming are a middle way between the language itself and the more precise systems of symbolic logic which we will consider later. As a middle way it is subject to some of the dangers and difficulties of a middle way. There is always a tendency to drift into one extreme or the other—either not changing the language enough for it to be diagrammed effectively, or changing it so eagerly into a systematic form that the meaning of the language is treated carelessly. It is also important to note that this system of diagramming is not meant to work in the same way or do the same things as the symbolic logic you will learn in the second half of this course. The purpose of these systems of diagramming is to help you see two things. First you will see how premises are related to conclusions and how the basic structure of an argument works. In this way it is actually similar to the more precise symbolic methods we will consider later. The second purpose of practical logic is to help the student see arguments in everyday language, and understand how the argument he sees in the diagram relates to the argument he just read in a book or newspaper. The diagrams of practical logic can only treat the structures and functions of reasoning in a general way, compared to that of symbolic logic, but they deal with the language more flexibly.
Summary: This chapter outlines two basic methods for diagramming arguments. One is the “list method,” in which the premises and conclusion are listed with abbreviated standardized terms. The second is the “arrow method,” in which arrows are used to represent the relationship between premises and conclusion. Both of these methods are a middle way between the language itself and the more precise systems of symbolic logic you will learn about later. Both help to clarify the relationship between premises and conclusions.
Exercise 1: Theory
a. Give the three steps for
the listing method of diagramming.
b.
In the second and third steps, what is the balance that must be maintained?
c. What further information
does the arrow method of diagramming give you than the listing method?
d. What, in practical terms,
does “order and standardize—as much as possible—the tense and mood and type of
verb in each statement” mean?
Exercise
2: Imitation
Read the first half of The
Hound of the Baskervilles by Sir Arthur Conan Doyle. What kind of logic does Sherlock Holmes
use?
Exercise
3: Practice
Diagram each of the following arguments, using both the
listing and the arrow method, on a separate sheet of paper.
1. The Lord is my strong tower and my deliverer, so God is my help in distress.
Assumed Premises: a tower gives protection, and protection and deliverance are help in distress.
2. All my enemies have turned against me, but the Lord is my protector; therefore I shall not be put to shame (be overcome by my enemies). Assumed Premise: My enemies cannot overcome the Lord.
3. Abel made his offering to the Lord by faith. Enoch walked in faith and was taken. Noah built the ark by faith. Abraham and Sarah lived by faith. By faith Isaac, Jacob, and Joseph blessed their children. Moses, saved by the faith of his parents who didn’t obey Pharaoh, lived by faith. Rahab acted by faith, and so did Gideon, Barak, Samson, Jephthah, David and Samuel, not to mention the prophets. The saints who went before us lived by faith!
4. This product will make your life better. You want a better life. Therefore, you want this product.
5. How excellent is Thy loving-kindness, O God! therefore the children of men put their trust under the shadow of Thy wings.
6. If a brother or
sister is destitute and you send him away empty, merely wishing him well, it is
good for nothing. Therefore, faith
without works is dead.
Chapter 16: Practical and Symbolic Logic: Introduction
Obstinate people may be subdivided into the
opinionated, the ignorant, and the boorish. Aristotle, Nicomachean
Ethics, VII
Thus far in this course, I have emphasized the differences between practical and symbolic logic. Now it is time to introduce symbolic logic and turn to the similarities between the two. Symbolic logic does not differ in kind from practical logic, merely in degree. The main differences are that practical logic is broader and less logically precise, and that symbolic logic does not include inductive reasoning and requires more translation. Now for the similarities.
First of all, both types of logic deal with argumentation in language. And, as a result, both deal with the translation of language into a peculiar logical format. The first steps of eliminating padding words, regularizing the language, and listing out the premises and conclusion of an argument are very much the same. In fact, if you diagram an argument according to the systems I set forth in practical logic (listing or arrow methods), your translation for symbolic logic is at least half done.
I will introduce two logical languages in this second half of the book. The first is the categorical calculus. This system is firmly grounded on the conceptual system of Greek philosophy. The philosophical systems of both Plato and Aristotle were based on the essential reality of ideas and the “idea realm.” Plato saw the essence of a thing as existing even apart from the physical object related to it (e.g., “treeness” exists, even apart from trees). Aristotle found the essence of things in physical objects, but still distinct from them. These “essences” were then conceived of in a certain logical hierarchy. </