# Sayers’ Helpful Summary of Logic

#### Sayers’ Vision for Logic

In her seminal essay “The Lost Tools of Learning,” the author Dorothy Sayers describes her understanding of the medieval scheme of education, specifically the Trivium — the three liberal arts of grammar, logic, and rhetoric. She argues that students in the Middle Ages were taught the proper use of the tools of learning by means of these arts. Of logic she says,

“Second, he learned how to use language; how to define his terms and make accurate statements; how to construct an argument and how to detect fallacies in argument.”

As I have taught logic in the classroom, written logic texts (and blog posts), and spoken on logic and classical education around the world, I have regularly returned to this quote. It is for me perhaps the most useful sentence (of the 238 sentences) in the essay.

#### A Proper Pedagogical Progression

In this sentence Sayers explains what logic is for: logic teaches us how to use language. This reminds us that the liberal arts of the Trivium are language arts (whereas the Quadrivium are mathematical arts). Specifically, logic teaches us how to use the language of reasoning, of disputation and proof.

This sentence also describes a proper pedagogical progression of logic:

1. We must start with terms: how to define them, relate them, and work with them, including understanding the value of defining terms.
2. Terms are related in statements (categorical statements connect subject terms with the predicate terms). Logic teaches us “how to make accurate statements”; that is, how to make statements that are true and applicable, as well as understanding how we know that they are true, and how they relate to each other. It teaches how to do this with many different types of statements: simple and compound, categorical and hypothetical, immediate inferences, and so on. Terms are the building blocks of statements.
3. Statements are the building blocks of arguments, as we connect premises together to draw conclusions. So logic teaches us “how to construct an argument”; that is, how to write a valid argument to establish a desired conclusion.  It teaches how to do this with many types of arguments: categorical and propositional, conditional and disjunctive, symbolic arguments and arguments in normal English.
4. Finally, logic teaches us “how to detect fallacies in argument,” both the formal fallacies from the rules of validity for categorical syllogisms and propositional arguments, and the informal fallacies of ordinary discourse, like circular reasoning and ad hominem. Logic teaches us not only to detect them, but to name them, and to expose them by means of counterexamples to those untrained in logic.

Were I to add one element to Sayers’ list, it would be “to construct a proof in a step-by-step, justified manner.” With this addition, every page, every concept of both Introductory and Intermediate Logic is covered in Sayers’ helpful description of what is encompassed in learning logic.

# Logic: A Science and Art

Is logic a science or an art? Of course, a logician would answer Yes, and here is why.

A science is a systematic study of some aspect of the natural world that seeks to discover laws (regularities, principles) by which God governs His creation. Whereas botany studies plants, astronomy studies the sky, and anatomy studies the body, logic studies the mind as it reasons, as it draws conclusions from other information. Logic as a science seeks to discover rules that distinguish good reasoning from poor reasoning, rules that are then simplified and systematized. These would include the rules for validity, of inference and replacement, and so on.

For example, logic as a science could study the apostle Paul’s reasoning in 1 Cor. 15, “If there is no resurrection of the dead, then Christ has not been raised… But Christ has been raised, and is therefore the first fruits from among the dead.” It then simplifies this into a standard pattern: If not R then not C, C, therefore R. This rule can be further simplified, named, and organized in relation to other rules of logic.

An art is a creative application of the principles of nature for the production of works of beauty, skill, and practical use. The visual arts apply their principles to the production of paintings, sculptures, and pottery. The literary arts produce poems and stories. The performing arts produce operas, plays, and ballets.

Logic is one of the seven liberal arts, which include the Trivium of grammar, logic, and rhetoric. These arts are the skills which are essential for a free person (liberalis, “worthy of a free person”) to take an active part in daily life, for the benefit of others. Specifically, logic as an art seeks to apply the principles of reasoning to analyze and create arguments, proofs, and other chains of reasoning.

In summary:

Logic is the science and art of reasoning well. Logic as a science seeks to discover rules of reasoning; logic as an art seeks to apply those rules to rational discourse.

# Eureka! A Discovery of Proportions

I have been making a study of analogies and analogical reasoning, and recently saw a connection that I had not seen before. That connection is between what is called ordered-pair analogies, i.e. A is to B as C is to D (or more briefly A : B :: C : D) and mathematical fractions. I was fascinated by what I found. Let me explain.

#### Re-arranging analogy pairs

I first noticed that, in an ordered-pair analogy, corresponding parts had to be the same part of speech (noun, verb, adjective, etc). Either A & B and C & D had to be the same part of speech, or A & C and B & D had to be the same. For example, this is a good analogy:

drink : eat :: liquid : solid.

Here we have “verb is to verb as noun is to noun.” But an equally valid analogy is

drink : liquid :: eat : solid.

This is “verb is to noun as verb is to noun.” If the first analogy is A : B :: C : D, this second one is A : C :: B : D. Similarly, we can invert both pairs to get valid analogies, as in these examples:

eat : drink :: solid : liquid

liquid : drink :: solid : eat

These would be B : A :: D : C, and C : A :: D : B. We could also switch each pair around the double colon. All these work as good analogies.

#### The connection

Now, those of you reading closely who remember your basic fractions probably see the connection already. If this is a true equality,

A/B = C/D

then so are all these:

A/C = B/D

B/A = D/C

C/A = D/B.

These equalities follow the same patterns as the analogies above. You might see it clearer with specific numbers. If the first equality is true (and it is), then all the rest must be true:

16/24 = 6/9
16/6 = 24/9
24/16 = 9/6
6/16 = 9/24.

#### The question

Do you see it? Every re-arrangement that is valid for verbal analogies is equally valid for mathematical fractions, and vice versa. But why should this be so? What is the connection between these two very different kinds of proportions?

There may be some connection between reducing the numerical fraction and finding the fundamental relationship in the verbal analogy. Just as 16/24 = 6/9 because they both equal 2/3, so ‘eat : solid :: drink : liquid’ because they share the relationship of ‘mode of consuming : state of matter of what is consumed.’

I am confident that there is something deeper going on here. Can you find any other connections between verbal analogies and numerical fractions?

# Analogy in Proverbs

In my first post of this series on analogies, I explained that one typical analogy form is the ordered-pairA is to B as C is to D, or more briefly A : B :: C : D. This is how most people think about analogies, having seen them in the vocabulary or reasoning sections of standardized tests. But in reading through Proverbs recently, I uncovered about fifty analogies, most of which can be reduced to ordered-pair form.

#### One Ordered Pair Illuminates the Other

For example, Proverbs 3:12 says,

“For whom the Lord loves He corrects, just as a father the son in whom he delights.”

This analogy can be reduced to an ordered pair:

The Lord : His beloved :: a father : his delighted son

The common concept between these analogous pairs is that the first corrects or disciplines the second. The comparison is helpful because the familiar, concrete image of a father correcting the son in whom he delights illuminates the less familiar, more abstract idea of the Lord correcting His beloved.

Here are two more examples:

“As a dog returns to his own vomit, so a fool repeats his folly.” (Prov. 26:11)

“Where there is no wood, the fire goes out; and where there is no talebearer, strife ceases.” (Prov. 26:20)

In those examples, the first analogous pair illuminates the second.

#### Synonymous Pairs

Rather than using an analogy to illuminate the less familiar by means of the more, many proverbs simply restate the main point using synonymous pairs. Here are several examples in which the ordered pairs are synonyms:

“Then they will call on me, but I will not answer; They will seek me diligently, but they will not find me.” (Prov. 1:28, cf. Mt. 7:7)

“I have not obeyed the voice of my teachers, nor inclined my ear to those who instructed me!” (Prov. 5:13)

“Does not wisdom cry out, and understanding lift up her voice?” (Prov. 8:1)

“For a harlot is a deep pit, and a seductress is a narrow well.” (Prov. 23:27)

#### Antithetical Pairs

Many other proverbs set up an antithesis, using antonyms in ordered pairs:

“The curse of the Lord is on the house of the wicked, but He blesses the home of the just.” (Prov. 3:33)

“A wise son makes a glad father, but a foolish son is the grief of his mother.” (Prov. 10:1)

“The hand of the diligent will rule, but the lazy man will be put to forced labor.” (Prov. 12:24)

Analogies by means of Hebrew parallelism are employed throughout Scripture. In my next post, we will consider analogies in the New Testament.

# Reasoning by Analogy

I have been thinking about analogies lately, and finding them fascinating. There appear to be three basic uses for the term analogy.

#### Comparisons

First, almost any comparison, especially one in which a familiar, simpler, or concrete thing is used to clarify or illuminate something that is unfamiliar, complex, or abstract, can be called an analogy. For example, this excerpt from George Orwell’s essay “A Hanging” is considered an analogy:

They crowded very close about him, with their hands always on him in a careful, caressing grip, as though all the while feeling him to make sure he was there. It was like men handling a fish which is still alive and may jump back into the water.

The manner in which the guards handled the prisoner is compared to men handling a fish. Most people have tried to handle a live fish just pulled from the water that wants back in, so this comparison gives the reader a vivid mental picture of the less familiar situation Orwell is describing.

#### Ordered-pairs

Second, we see analogies in what can be called ordered-pair form: A is to B as C is to D, or more briefly A : B :: C : D. Typically these appear in the vocabulary or reasoning section of standardized tests, like this sample question from the GRE. Choose the analogous pair:

APPRENTICE : PLUMBER ::
A. player : coach
B. child : parent
C. student : teacher
D. intern : doctor

The best answer is D. Just as an apprentice is training to be a plumber, so an intern is training to be a doctor. A child does not formally study to become a parent, and a player or student is not necessarily studying to become a coach or teacher (respectively).

#### Illustrative parallels

Third, we see analogies being used for the purposes of persuasion, called arguments by analogy, or what Aristotle calls illustrative parallels. Here is an example from Aristotle’s Rhetoric II.20:

Public officials ought not to be selected by lot. That is like using the lot to select athletes, instead of choosing those who are fit for the contest; or using the lot to select a steersman from among a ship’s crew, as if we ought to take the man on whom the lot falls, and not the man who knows most about it.

Illustrative parallels use both inductive and deductive reasoning. We use inductive reasoning to mentally move from the source (e.g. we ought not use the lot to select athletes) to a more general, unspoken intermediate conclusion (we ought not randomly select someone for a skilled position). We then use deductive reasoning to move from this intermediate conclusion to our specific conclusion, the target (we ought not select public officials by lot).

In my next post, I will explain how to construct illustrative parallels.

# What comes after Logic? Rhetoric!

Introductory and Intermediate Logic together provide a complete foundational logic curriculum. Informal, categorical, and modern propositional logic are all included. The next step in your student’s classical education is to begin to apply what he has learned in logic to effective speaking and writing. This means your student should move on to the study of formal rhetoric, the capstone of a classical education. Rhetoric applies the tools of logic – defining terms, declaring truth, arguing to valid conclusions, and refuting invalid ones – to the persuasion of people. Rhetoric puts flesh onto the bones of logical analysis, that we may breathe arguments into life through the wise use of fitting words.

Fitting Words: Classical Rhetoric for the Christian Student is a complete formal rhetoric curriculum. Presented from a thoroughly Christian perspective, Fitting Words provides students with tools for speaking that will equip them for life. Drawing from Aristotle, Quintilian, Augustine, and others, and using examples from the greatest speeches from history and scripture, this robust curriculum guides Christian students in the theory and practice of persuasive communication.

The complete curriculum includes:

• Student text with 30 detailed lessons
• Student workbook with exercises for every lesson
• Answer key for the exercises and tests
• Test packet with nine tests, review sheets for every test, and speech judging sheets
• Video course in which the author introduces and teaches through every lesson

# Rhetoric Interview

The following is a slightly edited version of a survey given me by Joshua Butcher – rhetoric instructor at Trinitas Christian School in Pensacola, Florida – regarding the teaching of rhetoric in a classical, Christian setting.

Josh:  How long have you taught rhetoric in a classical education setting?
Jim:  I taught Classical Rhetoric for 18 years at Logos School to 11th graders. I have also written a rhetoric text – Fitting Words: Classical Rhetoric for the Christian Student – and lectured through it.

Josh:  What are the essentials of rhetoric that every classically educated student should have?
Jim:  Do you mean, “What are the essential rhetorical skills that every classically educated student should seek to master?” Continue reading Rhetoric Interview

# Hard Words for Homeschool Moms

While teaching my classes online, speaking with customers at conventions, and swapping stories with my son who is also a teacher of homeschoolers, I have observed a troubling tendency among homeschooling mothers of teenage sons, a tendency pervasive enough that I feel compelled to say something. I do not want to unduly offend, but I want to speak to the problem as I see it. Here it is: Continue reading Hard Words for Homeschool Moms

# Books I read my children

I recently had the pleasure of listening to Andrew Pudewa, the director of the Institute for Excellence in Writing and a father of seven, speak at an education conference on the importance of reading aloud to your children. As he spoke, I recalled with delight the many hours I spent reading to my four children before they went to sleep. My eldest is about seven years older than the youngest. I would first read the Bible, making sure I at least turned the page every night. We read it all the way through, Genesis to Revelation, over and over again (once we finished it in an airport), occasionally changing the translation. We would pray and sometimes sing, and then I would read a story, alternating each night between the boys’ room and the girls’ room (I tried to keep track by which room my chair was left in, only later learning that my girls would move my chair to the boys’ room during the day so that I would move it back to their room so they could be in their own beds as I read).

I am thankful that I kept a record of the books I read to them. I have listed them below, for anyone who might somehow benefit from knowing the stories that shaped my children (and me). They are listed roughly in the order that I first read them over the years. The underlined books are favorites that I read more than once. No doubt some books went unrecorded (I’m pretty sure I read some missionary biographies, but I cannot recall which ones). You can see that I would sometimes get on a theme. I do not offer these as the best books, and I would not even offer all of them as suggestions (were I to do it over, for instance, I would not read Pollyanna to the girls). But they are all worth considering. Anyway, here they are. Continue reading Books I read my children