Tag Archives: counterexamples

Logic with James B Nance

Introductory Logic Prerequisite for Intermediate Logic?

It is certainly possible for a student who has not taken (or not completed) Introductory Logic to take and successfully complete Intermediate Logic. Though the Intermediate Logic text is designed as a continuation to Introductory Logic, it does not assume a mastery of the concepts in it. Almost all of the concepts from Introductory Logic that are essential for Intermediate Logic are re-taught (the only exceptions being the definitions of logical argument, premise, and conclusion; definitions assumed in Intermediate Logic, Lesson 7, but taught explicitly in Introductory Logic, Lesson 19).

That being said, a new Intermediate Logic student who is familiar with Introductory Logic will have an advantage over a student who is not. The following concepts from Introductory Logic are repeated and re-taught in Intermediate Logic (the concepts are first taught in the respective given lesson numbers): Continue reading Introductory Logic Prerequisite for Intermediate Logic?

Truth Tables for Validity

Truth tables can be used to determine the validity of propositional arguments. In a valid argument, if the premises are true, then the conclusion must be true. The truth table for a valid argument will not have any rows in which the premises are true and the conclusion is false. For example, here is a truth table of a modus tollens argument, with the final columns, showing it to be valid:


The fourth row down is the only row with true premises, and in that row it also has a true conclusion. So this argument is valid.

An argument is invalid when there is at least one row with true premises and a false conclusion, such as in this affirming the consequent truth table: Continue reading Truth Tables for Validity

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

Counterexample challenge

Mr. Nance,

My question has to do with Lesson 24. Exercise 22 has the students do a challenge, testing the validity of all 256 forms. I understand it’s to practice counterexample. Is there another reason not to put this challenge after lesson 26 after they’ve learned all the rules? (Which I think would make it easier.) Continue reading Counterexample challenge

Counterexample Challenge

Introductory Logic formally teaches two methods for determining the validity of a syllogism: rules of validity, and counterexamples. The rule that tells us that any AAO-4 syllogism is invalid is this: “A valid syllogism cannot have two affirmative premises and a negative conclusion.” But can we show the invalidity of AAO-4 with a counterexample? Here is the schema:

All P is M
All M is S
∴ Some S is not P

I contend that there is only one way to write a counterexample for a syllogism of that form. I challenge you to write a counterexample to AAO-4. Remember that a counterexample must be the same form, and have true premises and a false conclusion.

Good luck!

Help with Hypotheticals

One of the more practical parts of Introductory Logic is Lesson 31 on Hypothetical Syllogisms. Hypothetical syllogisms of all kinds are a very common form of reasoning, so we should not only be able to identify them quickly, but we should also learn to use the valid forms confidently.

A hypothetical statement is an “if/then” statement, such as this one: Continue reading Help with Hypotheticals

More than a switching

Mr. Nance,

Tomorrow morning I will be explaining why EIO-2 is valid but IEO-2 is invalid. The only problem is that I think both ought to be valid. I do not understand why reversing the order of the premises invalidates the syllogism, especially when the placement of the middle term remains the same. Thank you in advance. Continue reading More than a switching

Fixing a counterexample

Mr. Nance,

One of my students came up with a counter-example for OAO-1 (#6 Quiz 9) in class yesterday:

Some fish are not cats.
All catfish are fish.
∴ Some catfish are not cats.

Because of subimplication “NO catfish are cats” is true, would this counterexample be incorrect since “some catfish are not cats” is implied to be true as well? We had several class examples so by the end of the class, we were all a bit bogged down.

Thanks for your help! Continue reading Fixing a counterexample