# 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.

# Mr. Nance the Younger

“A wise son makes a father glad” (Prov. 15:20)

It is my sincere pleasure to announce that this year for Roman Roads Classrooms, the teacher of Logic and Rhetoric will be my son Josiah. He will also be teaching an Astronomy course (his favorite subject).

Josiah is an experienced teacher, having taught several math and science courses for the last three years at Sequitur Classical Academy in Baton Rouge, before returning to his roots in Idaho this year. He was a very popular teacher at Sequitur among both students and parents.  Roman Roads Classroom is blessed to have him as a teacher.

In his 12  years at Logos School, Josiah was one of my best students. He has a sharp mind, true artistic talent, a fun sense of humor, and a mature sense of the seriousness of life firmly founded on godly joy. He received his bachelor of arts in liberal arts and culture from New St. Andrews College in 2014, before being hired by Sequitur to help develop their math and science program.

If you want to learn logic, rhetoric, or astronomy, I highly recommend this young man!

# Don’t Let Your Child Cheat

Dear homeschool parent,

This is something I feel compelled to share. In my 25+ years teaching in a classroom, and my 3+ years teaching online classes for homeschoolers, I have seen a disproportionately large number of homeschooled students cheating in online classes. I have spoken with several other teachers of online courses for homeschoolers, and some have reported cheating by as many as 25% of their students. Most incidents of cheating involve the student either plagiarizing a paper, or looking at the answer key while taking a test unsupervised at home.

Plagiarizing a paper is harder for parents to catch and prevent. But stopping your child from cheating on tests should be easy.

Believe me, I understand how it can happen. An otherwise good child is taking an online course. He is struggling, the other students seem to be understanding the material better than he does, and unlike most of his previous courses, he is (perhaps for the first time) not allowed to learn at his own pace. He is being required to complete a test over material he knows he has not mastered. He also knows where the answer key is kept, and can look at it because his mom is busy and she trusts him. The temptation becomes too strong, he caves and decides to copy the answers, and does not tell anyone.

Thankfully, he is probably bad at it. When children cheat in this way, they tend to use words and phrases directly from the answer key, even when the answers could have been worded in many different ways. They also aim at 100%, of course, not realizing that a perfect score from a struggling student will set off a warning light to the teacher.

Parents, when the teacher claims that your child has apparently cheated on a test or assignment, listen to him! No good teacher will make such an accusation lightly. It is very uncomfortable to charge a student with cheating, and most teachers will err on the side of giving the student the benefit of the doubt.

When I discover a student has cheated on a test, I first send an email to the parent. I express my concern as a possibility, along with the evidence of the cheating. The child is allowed to make a defense, which is usually pretty flimsy. Almost every time, the child admits to cheating. Some confess everything immediately, some at first confess to cheating to a certain degree (“I only looked at the answers for the first few questions”), and some initially deny it, only to admit it later when they realize that the evidence is overwhelming, or (Lord willing) when their conscience smites them. According to other teachers I have spoken with, the only students who steadfastly refuse to confess in the face of overwhelming evidence are those students whose parents “can’t believe Jimmy would do such a thing.” When they are caught and confess their sin, they should be forgiven, fellowship should be restored on all sides, and they should be given a zero on the test or paper.

Here are three easy rules to follow when your child needs to take a test for their online class:

Be in the room with them as they are taking a test, and pay attention to what they are doing.

Have them give you the test to send in.

Getting caught cheating is a hard blessing on both parent and child. The child learns that the truth behind the biblical proverb, “Be sure your sin will find you out.” He also learns that there are consequences to cheating. And the parents learn that their child is a sinner who can fall to temptation, just like the rest of us.

# 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.

# Synonyms, Antonyms & Scripture

While studying analogies and relationships between terms, I have been considering synonyms and antonyms, and I have come to some surprising realizations.

#### Defining Synonym and Antonym

A synonym is a word that has the same meaning as another word in the same language. If you were asked to think of several words and their synonyms, you would probably not have too much difficulty: rope & cord, huge & enormous, stone & rock, sleep & doze, etc. English has such an extensive vocabulary that most words have a synonym or near synonym. But if I asked you to think of words that have no synonym, that’s harder. Some possibilities are pencil, helmet, and elbow. But it takes some careful thought. In fact, can you think of a verb or adjective that has no synonym?

An antonym is a word that has the opposite meaning as another word in the same language. By its definition, it appears that antonym is the antonym of synonym. You can probably think up several antonym pairs without too much effort: freedom & slavery, large & small, clean & dirty, father & mother. But if you look around, you will see many things that have no antonym: bottle, brick, book, cabinet, keyboard. It seems about as difficult to think of things that have no synonym as it is to think of things that do have an antonym. Why is this?

#### Antonym Profundities

Synonyms say something about language and its development. But antonyms say something about the nature of the thing itself, that in someway it has a counterpart. If you develop a list of antonym pairs, they will likely be words that represent fundamental concepts. They seem to reflect something about how God made the world (light & darkness, evening & morning, male & female), or about the fallen nature (sin & righteousness, good & evil, freedom & slavery), or about kinds of separation or direction (present & absent, in & out, left & right).

There are also different species of antonyms. Some are complementary or binary, A and non-A, such as true & false, motion & rest, whole & part. In these cases there are only two options: if a statement is not true then it is false; if an object is moving then it is not at rest; the whole of something is not just a part; and vice versa for each of these.

Relational antonyms lie on a continuum, such as large & small, full & empty, rich & poor. These antonym pairs tend to be adjectives, and there are intermediate states. A house that is not large is not necessarily small; a pitcher can be neither full nor empty; if your uncle is not rich, it doesn’t mean he is poor.

Then there are opposites that are a compromise of these first two types: antonyms that have not a continual but a single intermediate state: positive, negative, & zero; above, below, & level.

Some antonym pairs exist in a relationship with a reversed direction or focus, such as husband & wife, lend & borrow, employer & employee. In such pairs, one can usually not exist without the other: if there is a husband there is a wife; if one lends another borrows; a person with no employees is not an employer. These are called converse antonyms.

Some words have more than one antonym, depending on how you think about them. What is the antonym of father? Is it mother? Or is it son? The definition of father is ‘male parent.’ The opposite of male is female, and a female parent is a mother. On the other hand, the opposite of parent is child, and a male child is a son.  Other examples are possible.

#### Synonyms and Antonyms in Scripture

Biblical authors make regular use of synonyms and antonyms. A quick glance through Proverbs will reveal this. Consider all the antonyms in this passage:

For the perverse person is an abomination to the Lord, but His secret counsel is with the upright. The curse of the Lord is on the house of the wicked, but He blesses the home of the just. Surely He scorns the scornful, but gives grace to the humble. The wise shall inherit glory, but shame shall be the legacy of fools. (Prov. 3:32-35)

Proverbs also include synonym pairs for poetic purposes:

Does not wisdom cry out, and understanding lift up her voice? She takes her stand on the top of the high hill, beside the way, where the paths meet. She cries out by the gates, at the entry of the city, at the entrance of the doors: “To you, O men, I call, and my voice is to the sons of men. O you simple ones, understand prudence, and you fools, be of an understanding heart.” (Prov. 8:1-5)

Ecclesiastes 3:2-8 has a poetic list of fourteen verbal antonyms:

A time to be born, and a time to die; A time to plant, and a time to pluck what is planted; A time to kill, and a time to heal; A time to break down, and a time to build up; A time to weep, and a time to laugh; A time to mourn, and a time to dance; A time to cast away stones, and a time to gather stones; A time to embrace, and a time to refrain from embracing; A time to gain, and a time to lose; A time to keep, and a time to throw away; A time to tear, and a time to sew; A time to keep silence, and a time to speak; A time to love, and a time to hate; A time of war, and a time of peace.

Can you identify the synonyms and antonyms in Matthew 7:13-14?

Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. But small is the gate and narrow the road that leads to life, and only a few find it.

How many examples of synonyms and antonyms in the Bible can you find?

Mr. Nance,

My son is in high school, and we want to use your Fitting Words curriculum. If he works through the Fitting Words text in tenth grade, would it be appropriate for him to work through it again as a review in eleventh and/or twelfth grade?

Fitting Words: Classical Rhetoric for the Christian Student was designed to be taught as either a one-year intensive course, or a two-year regular course. In the front of the Fitting Words Answer Key are one- and two-year schedules. I would suggest that your son work through the curriculum over two years, tenth and eleventh grade.

In the first year the topics covered are:

Unit 1: Foundations of Rhetoric
Unit 2: Invention and Arrangement
Unit 3: Understanding Emotions: Ethos and Pathos
Unit 4: Fitting Words to the Topic: Special Lines of Argument

In the second year the topics are:

Unit 5: General lines of Argument (a review of logic)
Unit 6: Fitting Words to the Audience: Style and Ornament
Unit 7: Memory and Delivery

For twelfth grade, I would suggest a thesis paper (or papers, perhaps including a research paper) and defense, applying what they have learned in the first two years.

Blessings!

# Minecraft Logic

So apparently, creating digital logic circuits on the game Minecraft redstone is a thing.

I was recently sent some screen-captures of the answer to Exercise 34, problem 4. You can create the circuit in the game, and it will give you the outputs for the various inputs. It appears to use an SPST switch for the inputs.

Anyone else out there use the Minecraft game for their Digital Logic studies?

# 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?

# Rules for Guessing

Shorter truth tables can help us find if an argument is valid, or a set of propositions are consistent, or if two propositions are equivalent. However, when completing a shorter truth table, we must sometimes guess a truth value for a variable. This occurs when there are no “forced” truth values — that is, when there exists more than one way to complete the current truth value for every remaining proposition.

Here are two rules to keep in mind when you must guess a truth value:

1. If guessing allows you to complete the shorter truth table without contradiction, then stop; your question is answered. Either you have shown the argument is invalid, or the given propositions are consistent, or the two propositions are not equivalent.
2. If the guess leads to an unavoidable contradiction, then you must guess the opposite truth value for that variable and continue, because the contradiction just might be showing that your guess was wrong.

Take a look at this post for a flowchart for guessing with validity.

Mr. Nance,

I am using your Introductory Logic course to teach an informal class in logic to four young people in my church. Thank you for creating a rigorous, explicitly-Christian logic textbook!

During a recent class (working through Lessons 6-8), two questions came up. Can I get your thoughts on them?

(1) Nonsense Statements

On page 57 you give the example of the nonsense sentence “The round square sweetly kicked the green yesterday.” A few students began waxing philosophical about what precisely rendered this sentence nonsense. One asked if it was nonsense in virtue of the fact that squares, by definition, cannot be round. If so, they asked, wouldn’t the sentence, “The square sweetly kicked the green yesterday” be eligible for statement-hood? Sure, squares aren’t known to kick, but that only means that the sentence is likely a false statement. Further, “green” might refer metaphorically to the green grass or a public common grassy land.

I love having such inquisitive students, but I’m afraid I wasn’t able to give them a tidy answer to these questions: Instead, I suggested that we take statements like “this statement is false” as clear examples of nonsense and leave the rest for an epistemology class. What would you have said?

(2) Self-supporting statements

There was some consternation about the notion that self-supporting statements are true (p. 61). One student gave the example of James 2:14 where the self-report “I have faith” is false. I answered by saying that, in general, we should give self-reports the benefit of the doubt. That is, we should judge a self-report true, until or unless we have some good reasons or arguments for thinking it false. Of course, this doesn’t mean that all self-reports are true. Categorizing self-reports as self-supporting, I told them, is more a point of intellectual decency and doing-as-you-would-be-done by than of hard logical categories.

I also pointed out that many self-reports fall into the category of incorrigible statements—that is, for some self-reports, we simply will never have any means, whether by authority, experience, or deduction, of proving them false. Most self-reports about mental states fall into this category—for example, “I wish I had purchased Apple Stock five years ago.”

If you can give any general pointers here, I would be grateful. Thank you. Continue reading Re: Nonsense and Self-reports