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. Consider compound propositions (Such as those we learn about in Intermediate Logic). We know, for example, that “She is a princess and a queen” implies that “She is a princess” — that is, p • q implies p. Also, contradiction is equivalent to negation, so the contradiction of these statements would be ~(p • q) and ~p, resulting in this square of opposition:


You can work through the different relationships and see how they apply. For example, this says that propositions of the form p • q are contrary to statements of the form ~p. This means that the statements “She is a princess and a queen” and “She is not a princess” cannot both be true, but can both be false. You can look at all the other relationships this way.

Try to come up with your own square of opposition for non-categorical statements.


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

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.


Why Logic Test A and B?

Hello, Mr Nance!

Could you please explain to me why the tests and quizzes pack for the Introductory Logic has a test form A and a test form B for the same chapters? I am sure it is logical (pun intended) but I am not getting it and just want to be sure I am administering them correctly.

Thanks in advance!

I included two tests per section to give the instructor some options:

1) The instructor may use one test as a practice test, the other as the graded test. (This is how I use them: my online students take the B version of each test for practice, then watch me work through the B version to correct any errors or misunderstandings before they take the A version).

2) If a student or a class does poorly on a test the first time through, the second test allows them to take it again after further review.

3) The instructor may like the format or the questions of one test over the other, and thus they have a choice which one they wish to administer.


Invalidity and truth

Mr. Nance,

I have a question on Intermediate Logic, Test 2, Form B, Problem 4. The question says: “An invalid argument can have true premises and a true conclusion, is this true or false?” The answer book says it’s true but the definition of an invalid argument would prove that statement to be false. Is there a typo or is that correct? Continue reading Invalidity and truth


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. Continue reading Common errors to avoid: The “to be” verb


I Know London is the Capital of France

I teach logic online. In addition to my regular logic students I have several Classical Conversations tutors who audit my logic course. After I finish the lesson and my students leave, the auditors join the class live, turning on camera and mic, and we discuss the lesson. I appreciate these discussions, because I often learn as much from them as they do from me. Continue reading I Know London is the Capital of France

A Logic Tool of Learning

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 have used this tool in every course I have taught: Logic, Rhetoric, Calculus, Physics, and Doctrine.

For example, in my rhetoric course, my students and I construct the genus and species chart shown below, to organize the seven causes of actions presented by Aristotle in his Rhetoric I.10: Continue reading A Logic Tool of Learning