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
Mr. Nance,
I have a question Lesson 11 of Intermediate Logic. I have all of the right answers to the exercises, but I noticed that on a few of the questions, I wrote more lines in my answers because I thought I had to be exhaustive in my efforts to find no contradiction. A specific example would be #5. I agree that there is only one way for the conditional to be false. But, there are multiple ways the conditional can be true. Why didn’t you try lines with I as true and C false, or C false and I true? Am I misinterpreting the fourth instruction from p.70 to “switch the assigned truth values and try again”? Is this directive leveled at each whole proposition, or at each constant? (I hope this is making sense). Continue reading Shorter Truth Tables for Equivalence
Mr. Nance,
Question regarding the ten rules of replacement. Earlier we learned that a conditional can be rewritten as ∼(p • ∼q). Why is this not included as a replacement for a conditional? Both p ⊃ q and ∼(p • ∼q) are equivalent. Continue reading A New Material Implication?
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):
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.
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.
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.
Here is a chart that summarizes the shorter truth table methods for validity, consistency, and equivalence, taught in Intermediate Logic Lessons 8 through 11.
Mr. Nance,
I lead a group of young men in Intermediate Logic and they wanted to put their skills to use. They used an argument they pulled from chapter 1 of the book Defeating Darwinism. We tried several different ways of representing the propositions but always came up with an invalid argument. We really want it to be valid! What do we need to do? The original argument is: “If God created space and time, then He is outside of time. Therefore, He is not affected by time.” Any helpful hints will be very appreciated. Continue reading Analyzing a real argument
If you are going through Intermediate Logic Lesson 9 and want some help understanding shorter truth tables for validity, here is a flowchart that shows the decision path:
Here I answer a specific question about filling out truth values in shorter truth tables for validity.
YouTube version HERE.