Tag Archives: Formal proofs

Logic & the Resurrection

Intermediate Logic Unit 4 teaches how to apply the tools we have learned in logic to real-life arguments. One such argument is contained in 1 Cor. 15:12-20, in which Paul argues that Christ has been raised from the dead, and as such He is the firstfruits of the general resurrection to come.

There are many points to Paul’s argument, but the main one is from verses 13, 16, and 20:

“If there is no resurrection of the dead, then Christ is not risen…But now Christ is risen from the dead, and has become the firstfruits of those who have fallen asleep.”

This argument can be symbolized as follows (C = Christ is risen, R = There is a resurrection of the dead):

~R ⊃ ~C    C     ∴   R

You can use the tools of truth table, truth tree, or formal proof to demonstrate that Paul’s argument is valid.

There are several other arguments in 1 Cor. 15:12-20, but the others leave premises assumed, so they take additional effort to analyze. But it is a beneficial exercise to work through them. Look at Exercise 28b.

Have a blessed Good Friday!

Where does the CPA come from?

Mr. Nance,

I’m stumped on Logic lesson 18 #5. We got same answers as answer key until line 7…I can see from line 8 why line 7 is important, but how did we deduce a consequent that was not the original consequent of line 1 (from which we assumed the antecedent in line 3)?

Hope that makes sense! Continue reading Where does the CPA come from?

Rule of Commutation

Mr. Nance,

I have a question on Intermediate Logic, Exercise 17a, problem 5. To justify the conclusion (L • M) ⊃ N, the answer key says to use the rule of commutation from (M • L) ⊃ N. But the rule of commutation says (p • q) ≡ (q • p). How can I use that rule without switching the propositions, but switching the letters inside of a proposition? For example, in step 3, they are switching the propositions and not the letters inside the parentheses. Continue reading Rule of Commutation

Tackling More Difficult Proofs

Mr. Nance,

In our logic studies, my son and I wrestle to work through the proofs, generally together. When we get stuck, really stuck, we go to the answer key, cover the answer, and move through the proof step-by-step until we find where we veered off-track. Then we use that one step to get us back where we need to be; and then, hopefully, we finish the proof. My question is, is this a reasonable approach? Continue reading Tackling More Difficult Proofs

Improving in Proofs

If you are studying Intermediate Logic, Unit Two, and you are having trouble writing formal proofs of validity (especially if you are in either Lesson 15 or 17), here are my last two suggestions (after reading THIS and THIS):

  1.  Work together with someone who can help you. If you do not have a study partner, shoot me a question on my Facebook logic page. I would be happy to answer specific questions about solving any of the proofs. Or give me a call during work hours. My phone number is on my personal Facebook page. I’m serious.
  2. If you and a friend finally get through the proofs in the exercise after a lot of struggle and effort together, then do this. Take a break, go have lunch. Then return to the exercise, and re-do it, looking back at the answers if you need to. Repeat until you can complete all of the proofs without looking at the answers.

Logicians hate this trick because it lets you solve proofs without effort!

Sorry. No magic pill. No “Logicians hate this trick because it lets you solve proofs without effort!” Just hard work and practice.

Rules for Proofs

Two types of rules can be used to justify steps in formal proofs: rules of inference and rules of replacement. In order to use these properly, you should understand the differences between them.

The main difference is that rules of inference are forms of valid arguments (that’s why they have a therefore ∴  symbol), but rules of replacement are forms of equivalent propositions (which is why they have the equivalence sign  ≡  between the two parts).  This fundamental distinction is the cause of all other differences in how they are applied in proofs. Continue reading Rules for Proofs

An exhortation to teachers regarding formal proofs

Formal proofs are hard, like many other things worth learning!
In this video, I talk through the difficulties of formal proofs of validity, and why it’s worth enduring the hardship to learn them.

YouTube version HERE.

Re: Formal Proofs

Formal proofs of validity are challenging. Unlike truth tables (longer and shorter), completing formal proofs is not merely a question of following all the steps correctly; they require some creativity. Consequently, students may have more difficulty solving them. But some students enjoy the challenge of figuring out how to prove the conclusion. It is very much like solving puzzles, and can be an enjoyable challenge. This is how the instructor should present them. I have found that many of my students over the years have risen to the challenge, done exceptionally well with formal proofs, and enjoyed them.

Formal proofs of validity give students practice thinking in a straight line. The process teaches them how to connect premises in a proper way in order to reach the desired conclusion. For example, consider this argument:

If we want to send a manned mission to Mars then it must be either funded by taxpayers or privately funded. We want to send a manned mission to Mars and other planets. It should not be funded by taxpayers. Therefore a Mars program must be privately funded.

How do I get to that conclusion? The argument can be symbolized as follows:

M ⊃ (T v P)    M • O    ~T    ∴ P

This can be shown to be valid by truth table, but how to we prove the conclusion by connecting the premises? In the video below, I work through the proof, showing how to connect the premises using the rules of inference to reach the desired conclusion.

Trouble with video? YouTube version HERE.