Tag Archives: square of opposition

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.”


What will I learn in Fitting Words – 2nd half?

Fitting Words: Classical Rhetoric for the Christian Student is arranged around the five canons of rhetoric: invention, arrangement, style, memory, and delivery. In the first half of this course, after laying the Christian philosophical and historical foundation of the subject, we concentrated on constructing the first two canons: invention, and arrangement (primarily the six parts of a discourse). We also studied the three artistic modes of persuasion: ethos, pathos, and logos (including the special lines of argument: forensic, political, and ceremonial oratory).

In the second half of this course, we will continue to learn about logos by constructing general lines of argument. In Unit 5 we will review the applicable parts of logic: defining terms, determining truth, employing maxims, and using inductive and deductive arguments. We will also considering the destruction of our opponents’ arguments in refutation, including identifying informal fallacies.

In Unit 6 we will learn about Style: understanding the nature of the soul, speaking with clarity and elegance, the levels of style, and figures of speech and thought. In Unit 7 we will learn the essential skills of memory and delivery.

We will continue to see examples of all of these concepts in historical and biblical speeches and other discourse. Click HERE to learn more.


Two questions on Intro Logic Exercise 22

Mr. Nance,

I am loving Logic, and have understood the lessons up until now, but the syllogisms and validity has me a bit overwhelmed.

You lost me in the 256 challenge when you started using the same term (dogs) for the major, minor and middle terms. I thought we needed to use different terms when testing for validity. I went back and tried putting dogs into Exercise 22 to see how that worked, and now I’m even more confused. It looks to me that it doesn’t prove the syllogisms to be valid or invalid (when using only one term).

Could you also let me know if I am on the right track on somethings else? Can you test for validity by using the relationships between statements when going from the minor premise into the conclusion? For example, in exercise 22, #1 would be false by contradiction, #2 would be false by contrariety, #3 would be false by superimplication etc…#5 would be true by subimplication.

Thank you for considering my questions! Continue reading Two questions on Intro Logic Exercise 22

On Logical Independence

Mr. Nance,

On Test 2b there were two questions on the issue of statements being logically independent that I found myself tripping on a little. Can you help me understand them more clearly?

The first is Test 2b, 11a: “It is later than 1:00 pm. / It is later than 2:00 pm.”
The next is Test 2b, 11c: “Some siblings are twins. / Some siblings are not twins.”

Both are said to be not logically independent. I would appreciate if you could help me see that more clearly than I do.

Thank you! Continue reading On Logical Independence

Why Standard Form?

Teaching students how to translate syllogisms into standard categorical form occupies several lessons in Introductory Logic. Lessons 11 and 12 explain how to translate categorical statements into standard form, which is then emphasized while learning about the Square of Opposition in Lessons 13-18. Lesson 19 teaches students how to distinguish premises and conclusions, in part so that in Lesson 20 they will understand how to identify the major and minor premises, so that they may know how to arrange a syllogism in standard order. Finally, Lessons 21 and 22 teach them how to identify the syllogism form using mood and figure. All this occurs before the students begin to learn how to determine the validity of a syllogism. Continue reading Why Standard Form?

The Square of Opposition (for logic nerds only!)

The Square of Opposition is a useful tool for representing and understanding the relationships between categorical statements with the same subject and predicate:


The relationships are defined this way:

Contrariety: The statements cannot both be true, but can both be false.
Contradiction: The statements cannot both be true, and cannot both be false.
Subcontrariety: The statements can both be true, but cannot both be false.
Subimplication: If the universal is true, the particular must be true.
Superimplication: If the particular is false, the universal must be false.

Students might be interested to discover that the square of opposition can be created for non-categorical statements as well. Continue reading The Square of Opposition (for logic nerds only!)

Distributed Terms

A term is distributed in a statement when the statement makes some claim about the entire extension of the term. For the four types of categorical statements, the highlighted terms are distributed, as shown in this simplified square of opposition:

   All S is P              No S is P

Some S is P     Some S is not P

You should discern two patterns to help you remember which terms are distributed: Continue reading Distributed Terms