Category Archives: Logic

Not both v Both not, again

Mr. Nance,

In Copi’s 14th edition of Introduction to Logic, one problem reads, “Iran and Libya both do not raise the price of oil.” The symbolic translation is ~I • ~L. I thought it might also be translated as ~(I • L). However, using a truth table to check for equivalence, I found the two are NOT equivalent.

Later in the exercise there is a problem that reads, “Either Iran raises the price of oil and Egypt’s food shortage worsens, or it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” The symbolic translation is (I • E) ∨ ~(J • S). I’m confused by reading “…it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes” as ~(J • S). That seems a lot like saying “It is not the case both that Iran and Libya do not raise the price of oil,” which I thought might be translated ~(I • L).

Can you explain how to read this correctly? That is, why are they not logically equivalent? Or did I just mess up royally?

Thanks so much.

You are correct in saying that ~(p • q) is not equivalent to ~p • ~q. How then do we determine the correct form for statements that use “both” and “not”?

Fundamentally, we must use the forms that reflect the meaning of the statements. The form ~(p • q) means “not both p and q”, as in “Tom and Jim are not both from Idaho.” The form ~p • ~q means “both not p and not q” which is equivalent to “neither p nor q”, as in “Tom and Jim are both not from China.”

Practically, the first thing to ask when symbolizing statements like this is, “Which comes first in the statement, the ‘not’ or the ‘both’?” If it is ‘not both’ then it is probably the form ~(p • q). If it is ‘both not’ then is is probably the form ~p • ~q. Let’s apply this to the statements in question.

1. “Iran and Libya both do not raise the price of oil.” This is correctly symbolized ~I • ~L. The meaning is that neither Iran nor Libya raise the price of oil.
2. “It is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” This is correctly symbolized ~(J • S).

You have too many nots in your second to last paragraph, which is confusing the issue. But I trust that my explanation clears things up.

For more on this issue, read this EARLIER POST.


Those weird categorical statements

Before studying categorical syllogisms, students learn to translate statements into standard categorical form. The first step is translating the statement such that it uses only the “to-be” verb, so the form becomes [Subject] [to-be verb] [Predicate nominative]. This standardizes the statements so that the arguments are more easily analyzed, which is beneficial when the arguments themselves get more complicated.

But it can result in some very strange statements, e.g. translating “The Apostle Paul rebuked Peter at Antioch” into

The Apostle Paul was a Peter-at-Antioch rebuker.

Most spell-checkers will mark “rebuker” with that squiggly red underline, and some students might balk at the goofy compound noun.

Also, if one is not careful to keep the meaning the same, some of the translations can get rather awkward, such as turning “Susan works hard to resist temptation” into (ahem),

Susan is a hard-to-resist temptation worker.

Most of my students have found the awkwardness of such translated categorical statements to be merely funny, and have just taken it in stride. But occasionally a student will be bothered by it, perhaps thinking that their answers (and thus they themselves) will be thought of as strange or weird. In a larger classroom setting, when everyone is saying the same strange statements, they get used to it pretty fast, but it might be different in a home school setting, or among a small set of students.

The awkwardness of the translations can often be reduced by simply adding a normal noun in a normal place, trying to make the statement sound as normal as possible. For example, rather than translating “The forests will echo with laughter” into

The forests will be with-laughter echoers,

an acceptable translation would be

The forests will be places that echo with laughter.

This requires the addition of a new noun (“places”), but it is perfectly correct. The two rather awkward statements from above could also be correctly translated

The Apostle Paul was a man who rebuked Peter at Antioch.

Susan is a girl who works hard to resist temptation.

This method usually results in long predicates, but more ordinary sounding statements. For more on this topic, read my earlier post, Common errors to avoid: The “to be” verb.

Common errors to avoid: The “to be” verb

Introductory Logic Lesson 11, “The One Basic Verb,” teaches the first step in translating categorical statements into standard form. This step is to translate the statement so that the main verb in the sentence is a verb of being: is, are, was, were, will be, and so on. Thus a statement like “Stars twinkle at night” gets translated into something like

Stars are nighttime twinklers. 

To do this correctly, the subject and predicate must both be nouns, and the verb must be the proper ‘to-be’ verb. The procedure outlined in the lesson is generally clear, but there are two errors I want to help you avoid.

One common error not mentioned in the textbook is the problem of the helping verb. Some students might try to translate the above sentence this way:

Stars are twinkling at night.

The student thinks, “I used the word are, which is a ‘to-be’ verb, so it must be correct.” The problem is that the whole verb here is “are twinkling,” the are being merely a helping verb. The way to fix this is to make sure that the predicate is a noun, usually formed by turning the main verb into a noun (e.g. twinkle –> twinklers).

Secondly, it is sometimes best to make the predicate a noun by adding a new noun, usually a genus of the subject. For example, you could translate the above statement as

Stars are bodies that twinkle at night.

For clarity’s sake, you may want to use a different noun than the one implied by the verb. For example, in translating “She’s got electric boots” it would be overly awkward to say,

She is an electric boots getter.

Much better to translate this as

She is an owner of electric boots


She is an electric-boot wearer.

Happy translating!

Common errors to avoid: I’s don’t imply O’s

Logic students who are first learning about categorical statements may mistakenly think that any I statement, Some S is P, necessarily implies the O statement, Some S is not P. This is a reasonable error, since it seems to accord with our common use. For example, if I say “Some astronauts are men,” it is reasonable for you to think I also believe that some astronauts are not men.

But this is not always the case. Statements of the form Some S is P logically allow for the possibility that All S is P. When a theology student first learns that some books of the Old Testament speak about Jesus, he may not be surprised to later discover that all books of the Old Testament speak about Jesus (Luke 24:27). Or when a physics student first learns that some forms of usable energy end up as thermal energy, she is well on her way to acknowledging that eventually all usable energy ends up as thermal energy. Astronomers once knew only that some gas giants in the solar system are ringed planets (e.g. Saturn). They eventually discovered that all gas giants in the solar system are ringed planets.

These examples show that Some S is P does not necessarily imply that Some S is not P. Everyone would agree that “Some songs are poems” is a true statement, but it is reasonable still to argue that “All songs are poems.”


Do You Smangle?

The first lesson in Introductory Logic discusses several different purposes for defining terms, one of which is to “increase vocabulary.” This is meant in two or three senses.

First, when a student first learns the meaning of a word, such as learning that apiary means ‘a bee house’, his vocabulary has been increased. He has added a new word to the thousands he has access to. Increasing a child’s vocabulary like this is an essential part of his education, in every subject he studies.

Second, when a new word (or a new meaning to an existing word) is added to a language it is given a stipulative definition, until such a word gets generally adopted. This can happen in many ways, such as when an author introduces a new word in his book, and stipulates a definition for it. For example, in his book The Abolition of Man, C. S. Lewis takes the Chinese word Tao and gives it this stipulative definition: “The doctrine of objective value, the belief that certain attitudes are really true, and others really false.” This word has become part of the vocabulary of many people who have read and discussed Lewis’s book.

Here are ten new words that have recently been added into English (and perhaps into your own personal vocabulary):

Afterparty : Social gathering which takes place after a party, concert, or other event
App : Computer program designed for use on a mobile digital device
Brexit : Departure of the United Kingdom from the European Union
CrowdfundingRaising money by getting many people to make a small contribution
EmojiSmall digital image used to express an idea or emotion
MehInterjection used to express indifference
PhotobombIntrude into the background of a photograph just before it is taken
Selfie : Photograph that one has taken of oneself
Troll : Person who is provocatively rude or insulting on the Internet
Unfriend : Remove (someone) from a list of friends or contacts on a social networking site.

It can be fun for students to invent their own words and definitions, or to share words that are used within the confines of their immediately family. In our house, a “ninker” is a small, difficult to remove item that prevents the opening of a drawer.

My favorite stipulated word from a student is “to smangle,” meaning to rub the top of someone’s head with an open palm (especially if they have a crew cut). This would mean that smangle and noogie are species of the genus, “to rub someone’s head”!

Do you have any stipulated words to share from your students or your family? Share in the comments!


The Genus & Species Tool

The purpose of classical education is to provide students with tools of learning. One of the most useful tools is the genus and species chart. I used this tool in every course I taught, including Logic, Rhetoric, Calculus, Physics, and Doctrine.

For example, when studying judicial rhetoric in Aristotle, I would follow his descriptions to construct the genus and species chart shown below, which shows the relationships between the seven causes of human actions:

Causes of actions chart

Aristotle argues that every human action is the result of one or more of these seven causes: habit, rational craving, anger, appetite (all voluntary actions – used for prosecution); chance, compulsion, nature (all involuntary actions – used for defense). This visual aid is much clearer than the wordy paragraph given in Aristotle’s Rhetoric text.

I used a similar chart in Calculus to show the arrangement between the types of elementary functions, in Physics for the various branches of physics, and in Doctrine for the “Liar, Lunatic, Lord” argument for the deity of Jesus. For example, the chart for the types of elementary functions looked like this:

When teaching this tool in Logic, one should insist on a clear dividing principle between species, to avoid species overlapping or being placed at the wrong level. In the above chart, the top dividing principle is “whether or not the action is due to oneself.” Under involuntary actions, the dividing principle between chance and necessity is “whether or not the cause is fixed and determined”; under necessity, the dividing principle between compulsion and nature is “whether it is external or internal.” Aristotle’s dividing principles between habit and craving or between anger and appetite are less clear, though the dividing principle under craving is obvious.

The Logic teacher not only presents this tool for use in other subjects, but also in teaching Logic itself. Formal Logic is the “master faculty” of the dialectic stage, and as such it not only teaches the tools of logic, but demonstrates how to use them in teaching. For example, I used the tool of genus and species in my Logic class when I taught the difference between supported  and self-supporting statements. The dividing principle is “how the truth value is determined.”

I would encourage logic teachers to use this tool often, both to present the lesson clearly and to train the students in its proper use. Logic teachers should also encourage their colleagues to use this tools for their students at this stage.


Preparing Younger Children for Logic

Mr. Nance,

Do you know of resources to better gradually prepare our younger Foundations students for formal logic? Anything ages 4-11?

I have often said that the best preparation for the study of logic is the study of truth. Most children don’t need to experience much of what we could call “formal pre-logic”. Rather, they would do well to concentrate on learning other topics common to upper elementary (Latin, literature, arithmetic/pre-algebra), as these provide plenty of material to prepare their minds for the study of formal logic. If you do want some specific pre-logic books, I like the The Fallacy Detective by the Bluedorns. Also, Learning Logic by Dr. William Craig looks good. These would be best just before the study of formal logic. 

But consider what the guys at Trivium Pursuit say: “We suggest that formal academics should be the focus after age ten, hence the focus before age ten should be to build a good foundation for the later academics. The way to accomplish this is to exercise the mind so as to develop those parts of the mind which are appropriate for the specific age of the child. The early years are the time to sow the seeds of honoring God and parents, developing the capacity for language and the appetite for learning, enriching the memory, encouraging creativity, and instilling a work and service ethic. These are the kind of things which will lay a good foundation for the formal academics later. First things come first.” Read more from this article HERE.

Another good idea is to challenge younger children with puzzles. Teach them to solve a Rubik’s Cube. Play Twenty Questions, Mastermind, Chess, and Situational Games. This will be a fun way to get their minds tuned to thinking in a straight line. And ask them challenging questions at the dinner table. “Billy, you have two legs. Gorillas have two legs. Are you a gorilla?” Get them thinking, and keep them thinking. Eventually they will be hungry to know the proper rules of thinking. Then they are ready for logic.

Knowing the Truth of Statements

Mr. Nance,

Introductory Logic Exercise 8, Question # 9 asks what type of statement this is:

“Jesus is God, and He is man.”

The answer key says “supported, by authority.” Could definition also be a possible answer?

I see what you are thinking. Jesus is both God and man by nature, and definitions are (to a certain extent) trying to get at the nature of the term. But the question is basically asking, “How would you know that this statement is true?” Ask anyone how they know that Jesus is both God and man, and they will point to some authority: the Bible, or a creed, or their pastor tells them, etc. Besides, we don’t really define people.


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,

dorothy[1]“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!