Category Archives: Ask-a-question

Two Strange Proofs

Mr. Nance,

Could you give real-world examples of the arguments to prove in Intermediate Logic Lesson 18, number 7) U / ∴ W ⊃ W, and number 8) X / ∴ Y ⊃ X, showing how they would be used, or explain them a bit? Thank you.

Thanks for the great question! These two arguments are unusual, so I am not surprised that you are asking about them.

A real-world example for #7 might be Esther 4:16, “I will go to the king which is against the law; if I perish, then I perish!” This argument form basically shows that any proposition implies a tautology.

An example for #8 could be, “God created all things. So even if evolution can be used to explain some fossils, it’s still true that God created all things.” The form of this argument shows that if a proposition is given, any other proposition implies it.

To be honest, my purposes for including those two problems were: 1) to show how very strange the conditional proof is, and 2) to show how this method can be used to simplify otherwise difficult proofs.


May Proofs Use the Same Line Twice?

Mr. Nance,

In the answer to Exercise 17a, problem #12, is there a typo? It has row 5 twice.

There is no mistake there. A given line may be used more than once in a proof, as I say at the end of Lesson 15, “Usually, though by no means always, every step in a proof is used and used once.” Line 5 is used twice, once to simplify to get ~L, and once to commute and simplify to get ~M. 


Guessing w/ Shorter Truth Tables for Consistency

Mr. Nance,

Two questions:
1.  If you guess on consistency must you guess again as in validity and equivalence?
2.  If you find it inconsistent once does that trump consistency no matter what?

Thanks so much in advance for your help.


1.  If you guess on consistency and get no contradiction, then you do not have to guess again. This is also true for validity and equivalence. In all three, if you guess, fill in truth values, and get no contradiction, then the question is answered (either that they are consistent, as in this case, or that the argument is invalid, or that the propositions are not equivalent).

2.  No, if you guess and get a contradiction (which looks like an inconsistency), then you must guess again, because the contradiction may mean that you just guessed wrong.

If you think about what consistency means, then all this makes sense. Consistency means the propositions can all be true. So if you assume they are true and can fill in the truth values without contradiction (even when you need to guess), then you have shown that they can be true.

Below is a short video in which I explain how shorter truth tables are used to determine consistency. 


Parentheses in Conditionals

Mr. Nance,

Could you please explain how the placement of the parentheses is determined in Test 1, Form A, #12 of Intermediate Logic? My student wrote “(M ⊃ P) ⊃ ~C,” but the answer key says “M ⊃ (P ⊃ ~C).”

The original proposition is,

“If we see a movie then if we eat popcorn then we do not eat candy.”

This proposition has the overall form pq, where p is the antecedent, “We see a movie” (abbreviated M) and q is the consequent. This consequent is another complete conditional: “If we eat popcorn then we do not eat candy.” This is the (P ⊃ ~C). Because it is a complete proposition in itself, this consequent gets placed in parentheses.

It will be important later to note that propositions of the form p ⊃ (qr) are equivalent to propositions of the form (p • q) ⊃ r. The given proposition could be understood in this way. “If we see a movie and eat popcorn, then we do not eat candy.” Notice in this form, “We see a movie and eat candy” is the antecedent, and it is a complete proposition in itself, and thus gets placed in parentheses.


Shorter Truth Tables for Validity

Mr. Nance,

As I am teaching shorter truth tables for validity, I noticed that sometimes (on a valid argument) I get the contradiction in a different place than the answer key does. Is that okay, or am I making a mistake?

You are probably not making a mistake.

The shorter truth table method used on a valid argument will always result in a contradiction, but where that contradiction appears depends on the order of the propositions you work with, which can certainly vary.

For example, on Exercise 8, problem #1, the answer key shows the contradiction in two places, which happens if you find all of the truth values in the conclusion first, before going back to the premises.


But you might start by getting the truth values for S and W from the antecedent of the conclusion first, and then going directly to the premises. That would bring you to this point:STT2

Now, for the premises to be true, the consequents of each (P and F) must be true as well. That gives the contradiction in the conclusion, instead of in the premises as before:STT1

This is a perfectly legitimate answer. In the answer key, I tried to place the truth values in the positions I thought most likely for other who did the problem correctly. Typically, after making the premises true and the conclusion false, I try to start on the right side (the conclusion) and work my way left.

Here are a few more thoughts.

Shorter truth tables take some time to learn. Do not rush through them. Students need lots of examples to see how they work. Also, make sure you and they understand the concept behind them. You are assuming that the argument is invalid (by making the premises true and the conclusion false). If this assumption leads to an unavoidable contradiction, then the argument cannot be invalid, so it must be valid. But if you assume the argument is invalid and can fill out all the truth values without any contradiction, you have shown that the premises can be true and the conclusion false, i.e. you have shown it to be invalid.

Keep this in mind also: For each proposition (premise or conclusion), you must place the truth values under the main logical operator. The main logical operator is the operator in the column that would be the last to be filled out in the larger truth table. For example, consider this compound proposition:

~(p • q) ⊃ r

If this were a premise of an argument, the T would be placed under the conditional. But for the proposition

~[(p • q) ⊃ r]

the T would be placed under the negation. Working the truth values all the way out for this proposition would result in the truth values shown here:

~[(p • q) ⊃ r]
T   T T T  F  F

Feel free to comment if you have any questions.

Question about Conditionals

Mr. Nance,

My student has a question on Exercise 4 number 14.  Her answer for was ~C ⊃ S instead of S ⊃ ~C. Can the statements “I will go swimming only if the water is not cold” be considered logically equivalent to “If the water is not cold, I will go swimming”?

Also, how can I explain the difference between “If the water is not cold I will go swimming” and “I will go swimming unless the water is cold”?

Thank you! Continue reading Question about Conditionals

Exercise 30, problem 4: Is my answer correct?

Mr. Nance,

As I am reviewing Exercise 30, I am confused  if my answer works , since its different from the original answer. I put:

No logic is a tangible study
No chemistry is logic
∴ All chemistry is a tangible study

As opposed to the answer key:

All non-logic sciences are tangible studies
All chemistry is a non-logic science
∴  All chemistry is a tangible study

Which one is right? Continue reading Exercise 30, problem 4: Is my answer correct?

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