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?
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.
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
Here I answer a specific question about filling out truth values in shorter truth tables for validity.
YouTube version HERE.
Shorter truth tables take some time to learn. Do not rush through them. Students need lots of examples to see how they work.
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 invariably to a contradiction, then the argument cannot be invalid, so it must be valid. But if you can assume the argument is invalid and 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: 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 would result in the following:
~[(p • q) ⊃ r]
T T T T F F
Mr. Nance,
I was doing the exercises for Intermediate Logic Lesson 6 and got stumped by #2. In the part of the proposition that is ~q ⊃ ~p, when I was doing the defining truth table for the variables, I assumed the first variable, though out of order alphabetically, would get the TTFF pattern. But in the answer key, the letter that comes first in the alphabet (p), though the consequent, got the TTFF pattern. Why is that? Continue reading The Pattern of T & F in Truth Tables
Lesson 3 states the order for completing truth tables is: standard variables, negated variables, propositions in (). Yet I noticed while watching the Lesson 3 session on the DVD and in the example at the bottom of page 23, the operations are not always followed in that precise order; sometimes the parenthesis are completed prior to the negation. How important is this order? Does it matter only if there is a negation within the parenthesis, as in the example at the top of p 23? Continue reading Negations and Parentheses
Mr. Nance,
I am having a hard time with problem 3 in Exercise 3 of Intermediate Logic. For the first proposition, ~(J ⋅ R), the answer key says “Joe and Rachel are not both students.” For the second proposition, ~J ⋅ ~R, the answer key says “Both Joe and Rachel are not students.” Those sound the same to me. Continue reading “Not both” v. “Both not”
Mr. Nance,
Regarding question no. 10 on page 252: What is the reason for its invalidity? Does the pure hypothetical syllogism also use the five rules of validity? The argument is:
If he is the Antichrist, then he opposes God’s people.
If he is the Beast, then he opposes Gods people.
Therefore, if he is the Antichrist, then he is the Beast.
Is the Beast major term? And is the Antichrist the minor term? How do we make it into valid categorical syllogism? Continue reading The Antichrist and the Beast Syllogism
Mr. Nance,
I am working on the 256 forms of syllogisms right now and I’m confused why one of them is invalid. I’m on the AAE-2 and each way I work it out, it comes out valid. Could you please tell me why it’s invalid.
Thank you! Continue reading Why is AAE-2 Invalid?