Category Archives: Intermediate Logic

Minecraft Logic

So apparently, creating digital logic circuits on the game Minecraft redstone is a thing.

I was recently sent some screen-captures of the answer to Exercise 34, problem 4. You can create the circuit in the game, and it will give you the outputs for the various inputs. It appears to use an SPST switch for the inputs.

Anyone else out there use the Minecraft game for their Digital Logic studies?

Rules for Guessing

Shorter truth tables can help us find if an argument is valid, or a set of propositions are consistent, or if two propositions are equivalent. However, when completing a shorter truth table, we must sometimes guess a truth value for a variable. This occurs when there are no “forced” truth values — that is, when there exists more than one way to complete the current truth value for every remaining proposition.

Here are two rules to keep in mind when you must guess a truth value:

  1. If guessing allows you to complete the shorter truth table without contradiction, then stop; your question is answered. Either you have shown the argument is invalid, or the given propositions are consistent, or the two propositions are not equivalent.
  2. If the guess leads to an unavoidable contradiction, then you must guess the opposite truth value for that variable and continue, because the contradiction just might be showing that your guess was wrong.

Take a look at this post for a flowchart for guessing with validity.

 

The Value of Propositional Logic over Categorical Logic

Logic Video Session Info

If you are using the Introductory or Intermediate Logic videos to teach your students, you may want to know the duration of the sessions. That information is now available on this printable document: Video Session durations.

Here are some quick facts:

Introductory Logic Videos
Total duration 10 hrs, 52 min, 33 sec.
Average session 13 min, 7 sec. (excluding the test and optional sessions)
Longest session 33 min, 21 sec.
Shortest session 5 min, 33 sec.
Total number of sessions 45.

Intermediate Logic Videos
Total duration 12 hrs, 30 min, 42 sec.
Average session 14 min, 14 sec. (excluding the test sessions)
Longest session 49 min, 54 sec.
Shortest session 4 min, 35 sec.
Total number of sessions 51.

You’re welcome.

What comes after Logic? Rhetoric!

Introductory and Intermediate Logic together provide a complete foundational logic curriculum. Informal, categorical, and modern propositional logic are all included. The next step in your student’s classical education is to begin to apply what he has learned in logic to effective speaking and writing. This means your student should move on to the study of formal rhetoric, the capstone of a classical education. Rhetoric applies the tools of logic – defining terms, declaring truth, arguing to valid conclusions, and refuting invalid ones – to the persuasion of people. Rhetoric puts flesh onto the bones of logical analysis, that we may breathe arguments into life through the wise use of fitting words.

Fitting Words: Classical Rhetoric for the Christian Student is a complete formal rhetoric curriculum. Presented from a thoroughly Christian perspective, Fitting Words provides students with tools for speaking that will equip them for life. Drawing from Aristotle, Quintilian, Augustine, and others, and using examples from the greatest speeches from history and scripture, this robust curriculum guides Christian students in the theory and practice of persuasive communication.

The complete curriculum includes:

  • Student text with 30 detailed lessons
  • Student workbook with exercises for every lesson
  • Answer key for the exercises and tests
  • Test packet with nine tests, review sheets for every test, and speech judging sheets
  • Video course in which the author introduces and teaches through every lesson

Digital Logic Q & A

Mr. Nance,

I have an overload of questions on digital logic. Hope that is okay!

1. Truth-functional completeness often makes circuits more complicated than they have to be. Is there anything besides cost-effectiveness that is beneficial about truth-functional completeness?

I assume you are asking “Why do we learn how to use NOR gates or NAND gates exclusively in a circuit?” Primarily, it is just to teach students how the gates work. But in practice, if you are constructing an electronic device, you may not have all the gates available (e.g. Radio Shack ran out and will not get them in for ten days), and so need to use a couple of NAND gates to do the job of one AND gate. Also, you might only use one NOR gate in the circuit, but a chip might contain four NOR gates, so why not just use three of them to replace an AND gate instead of buying one?

2. Since the symbol for NOR is the upside-down triangle, is there a symbol for NAND?

The triangle symbol is largely my own convention. See the Wikipedia page for the standard ways of expressing NOR. I know of no special symbol for NAND.

3. Is there a conditional gate (P ⊃ Q)?

Not that I am aware of. You can make a conditional using other gates.

4. Why do we write the names of logic gates in all caps? ( AND instead of and or And)

Just to distinguish them from the words in a sentence. It would be confusing to read “Take an or gate and an and gate…”

5. Why in K-maps do we circle in groups in powers of two?

Because that’s how they work to correctly simplify propositions. Draw yourself a K-map with 0111 across the top four cells, and 1110 across the bottom four cells. If you made two circles with groups of three across and ask, “What variable stays the same (negated or unnegated)?” the answer is that nothing stays the same, so no proposition can be identified. To get the simplest proposition, you must circle the middle four square, and the two on the top right and bottom left. Spend some time thinking through exactly what the K-map is doing when you circle groups and determine the proposition from the circled group. (See the next question.)

6. Finally, do K-maps eliminate the need for the Algebraic identities? I found that doing the Digital Logic Project didn’t require using them.

Yes, that is their primary benefit. Consider the proposition (p • q) ∨ (p • ~q). This simplifies this way:

  1. (p • q) ∨ (p • ~q)  Given
  2. p • (q ∨ ~q)  Distribution
  3. p • 1  Tautology
  4. p  Alg. identity

Now do a 2×2 K-map for this proposition:

 

 

 

See how it does the same thing in a faster, easier way?

Blessings!

A Simpler Truth Tree

In this video, I decompose a set of propositions from Intermediate Logic, Additional Exercises for Lesson 24. I first decompose the truth tree in the order of the given propositions. I contrast this with a second truth tree that uses the simplifying techniques from Lesson 24.

This shows first, how to use a truth tree to determine consistency, and second, how the techniques from Lesson 24 make the truth tree simpler.

Equivalence w/ Shorter Truth Tables

Mr. Nance,

Within Intermediate Logic Lesson 11, what would keep us from setting up the propositions both being true at the same time, and if there were a contradiction they would not be equivalent? Instead of setting them up one true and one false and if there’s a contradiction then they are equivalent?

That would be checking for consistency, not equivalence. If you set them both as true, and get a contradiction, then they are not consistent (which of course also means they are not equivalent, nor related by implication, per the chart in Introductory Logic, p. 71). But if you get no contradiction, all you have shown is that they can both be true, which is the meaning of consistency. To show equivalence, you have to show that they cannot have opposite truth values: the first cannot be true while the second is false, and vice versa.

Blessings!

Formal Proof Challenge!

Several years ago I was teaching a logic course, and we were learning about formal proofs of validity. I enjoy proofs, and to keep myself sharp I was working through a practice quiz in David Kelley’s The Art of Reasoning, when I came across this argument:

D ⊃ (E ⊃ F)
D ⊃ (F ⊃ G)
∴ D ⊃ (E ⊃ G)

I was in a quiet library with plenty of time, but despite all my efforts I could not solve this (without using the Conditional Proof). The next day in class some students were finishing their assignment early, so I  challenged them with this proof, thinking to myself, “That ought to keep them busy,” but not really expecting anyone to succeed. Before the end of class, Caroline Jones came forward and said, “I solved it, Mr. Nance.” I scoffed inwardly at first, only to be pleasantly surprised by her correct solution.

Since that time I have called this “The Caroline Jones” proof, and have challenged my logic students to solve it using only the regular rules of inference and replacement. The most elegant proof I have seen requires twelve total steps.

Anyone up to the challenge?

Reductio Challenge

In formal proofs of validity, the reductio ad absurdum method can be used to make some proofs easier, and even some shorter. For example, consider this argument:

(~P ⊃ R) • (~Q ⊃ S)    ~(R S)    ∴ P • Q

The proof for this valid argument is 14 steps without the reductio (which I will let you try to solve on your own), but only 7 steps with the reductio, as shown here:

  1. (~P ⊃ R) • (~Q ⊃ S)
  2. ~(R ∨ S)   /  ∴  P • Q
  3. ~(P • Q)                     R.A.A.
  4. ~P ∨ ~Q                    3 De M.
  5. R ∨ S                         1, 4 C.D.
  6. (R ∨ S) • ~(R ∨ S)   5, 2 Conj.
  7. P • Q                          3-6 R.A.

The reasoning behind the reductio method is this: If assuming that a proposition is false leads to a self-contradiction, then the proposition must be true. This reasoning can itself be written as a propositional argument:

~P ⊃ (Q • ~Q)   ∴  P

This is a valid argument, as a shorter truth table will show. But the proof for this argument (if you are not allowed to use reductio) requires 13 steps, and it is rather difficult to solve. Any takers?