Not both v Both not, again

Mr. Nance,

In Copi’s 14th edition of Introduction to Logic, one problem reads, “Iran and Libya both do not raise the price of oil.” The symbolic translation is ~I • ~L. I thought it might also be translated as ~(I • L). However, using a truth table to check for equivalence, I found the two are NOT equivalent.

Later in the exercise there is a problem that reads, “Either Iran raises the price of oil and Egypt’s food shortage worsens, or it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” The symbolic translation is (I • E) ∨ ~(J • S). I’m confused by reading “…it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes” as ~(J • S). That seems a lot like saying “It is not the case both that Iran and Libya do not raise the price of oil,” which I thought might be translated ~(I • L).

Can you explain how to read this correctly? That is, why are they not logically equivalent? Or did I just mess up royally?

Thanks so much.

You are correct in saying that ~(p • q) is not equivalent to ~p • ~q. How then do we determine the correct form for statements that use “both” and “not”?

Fundamentally, we must use the forms that reflect the meaning of the statements. The form ~(p • q) means “not both p and q”, as in “Tom and Jim are not both from Idaho.” The form ~p • ~q means “both not p and not q” which is equivalent to “neither p nor q”, as in “Tom and Jim are both not from China.”

Practically, the first thing to ask when symbolizing statements like this is, “Which comes first in the statement, the ‘not’ or the ‘both’?” If it is ‘not both’ then it is probably the form ~(p • q). If it is ‘both not’ then is is probably the form ~p • ~q. Let’s apply this to the statements in question.

1. “Iran and Libya both do not raise the price of oil.” This is correctly symbolized ~I • ~L. The meaning is that neither Iran nor Libya raise the price of oil.
2. “It is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” This is correctly symbolized ~(J • S).

You have too many nots in your second to last paragraph, which is confusing the issue. But I trust that my explanation clears things up.

For more on this issue, read this EARLIER POST.

Blessings!

Preparing Younger Children for Logic

Mr. Nance,

Do you know of resources to better gradually prepare our younger Foundations students for formal logic? Anything ages 4-11?

I have often said that the best preparation for the study of logic is the study of truth. Most children don’t need to experience much of what we could call “formal pre-logic”. Rather, they would do well to concentrate on learning other topics common to upper elementary (Latin, literature, arithmetic/pre-algebra), as these provide plenty of material to prepare their minds for the study of formal logic. If you do want some specific pre-logic books, I like the The Fallacy Detective by the Bluedorns. Also, Learning Logic by Dr. William Craig looks good. These would be best just before the study of formal logic.

But consider what the guys at Trivium Pursuit say: “We suggest that formal academics should be the focus after age ten, hence the focus before age ten should be to build a good foundation for the later academics. The way to accomplish this is to exercise the mind so as to develop those parts of the mind which are appropriate for the specific age of the child. The early years are the time to sow the seeds of honoring God and parents, developing the capacity for language and the appetite for learning, enriching the memory, encouraging creativity, and instilling a work and service ethic. These are the kind of things which will lay a good foundation for the formal academics later. First things come first.” Read more from this article HERE.

Another good idea is to challenge younger children with puzzles. Teach them to solve a Rubik’s Cube. Play Twenty Questions, Mastermind, Chess, and Situational Games. This will be a fun way to get their minds tuned to thinking in a straight line. And ask them challenging questions at the dinner table. “Billy, you have two legs. Gorillas have two legs. Are you a gorilla?” Get them thinking, and keep them thinking. Eventually they will be hungry to know the proper rules of thinking. Then they are ready for logic.

Knowing the Truth of Statements

Mr. Nance,

Introductory Logic Exercise 8, Question # 9 asks what type of statement this is:

“Jesus is God, and He is man.”

The answer key says “supported, by authority.” Could definition also be a possible answer?

I see what you are thinking. Jesus is both God and man by nature, and definitions are (to a certain extent) trying to get at the nature of the term. But the question is basically asking, “How would you know that this statement is true?” Ask anyone how they know that Jesus is both God and man, and they will point to some authority: the Bible, or a creed, or their pastor tells them, etc. Besides, we don’t really define people.

Blessings!

Differences in the new versions of the Logic texts?

Mr. Nance,

How different is the 2006 version of introductory & intermediate logic to the most recently published version? My friend is selling it, but I need it for Classical Conversations next year and want to make sure it’s close enough to being the same.

The Introductory Logic student text just has cosmetic differences between the fourth and fifth editions: a nicer cover, reformatted style. Most of the improvements are in the additional materials: suggested lesson plans, notes to the teacher, additional tests and quizzes. But the student texts are essentially the same.

Not so with Intermediate Logic. In addition to similar cosmetic changes to the student text and similar additional materials, the student text also has two completely new units, both showing how the tools learned can be applied to real-life, but in different ways. One new unit is designed to teach how to use the tools to analyze chains of reasoning found in real writings: Boethius, Augustine, the Bible, and so on. Another new unit introduces Digital Logic, the logic of electronic devices, as a ubiquitous and powerful application of the tools of Intermediate Logic.

The DVDs are quite different, significantly improved, and professionally filmed. The new DVD lessons are much easier to follow and to navigate. After comparing the two, I am frankly embarrassed by the older version of the DVDs. Look HERE for a side-by-side comparison of the videos on YouTube.

Blessings!

Suggested Rhetoric Schedule?

Mr. Nance,

My son is in high school, and we want to use your Fitting Words curriculum. If he works through the Fitting Words text in tenth grade, would it be appropriate for him to work through it again as a review in eleventh and/or twelfth grade?

Fitting Words: Classical Rhetoric for the Christian Student was designed to be taught as either a one-year intensive course, or a two-year regular course. In the front of the Fitting Words Answer Key are one- and two-year schedules. I would suggest that your son work through the curriculum over two years, tenth and eleventh grade.

In the first year the topics covered are:

Unit 1: Foundations of Rhetoric
Unit 2: Invention and Arrangement
Unit 3: Understanding Emotions: Ethos and Pathos
Unit 4: Fitting Words to the Topic: Special Lines of Argument

In the second year the topics are:

Unit 5: General lines of Argument (a review of logic)
Unit 6: Fitting Words to the Audience: Style and Ornament
Unit 7: Memory and Delivery

For twelfth grade, I would suggest a thesis paper (or papers, perhaps including a research paper) and defense, applying what they have learned in the first two years.

Blessings!

Re: Nonsense and Self-reports

Mr. Nance,

I am using your Introductory Logic course to teach an informal class in logic to four young people in my church. Thank you for creating a rigorous, explicitly-Christian logic textbook!

During a recent class (working through Lessons 6-8), two questions came up. Can I get your thoughts on them?

(1) Nonsense Statements

On page 57 you give the example of the nonsense sentence “The round square sweetly kicked the green yesterday.” A few students began waxing philosophical about what precisely rendered this sentence nonsense. One asked if it was nonsense in virtue of the fact that squares, by definition, cannot be round. If so, they asked, wouldn’t the sentence, “The square sweetly kicked the green yesterday” be eligible for statement-hood? Sure, squares aren’t known to kick, but that only means that the sentence is likely a false statement. Further, “green” might refer metaphorically to the green grass or a public common grassy land.

I love having such inquisitive students, but I’m afraid I wasn’t able to give them a tidy answer to these questions: Instead, I suggested that we take statements like “this statement is false” as clear examples of nonsense and leave the rest for an epistemology class. What would you have said?

(2) Self-supporting statements

There was some consternation about the notion that self-supporting statements are true (p. 61). One student gave the example of James 2:14 where the self-report “I have faith” is false. I answered by saying that, in general, we should give self-reports the benefit of the doubt. That is, we should judge a self-report true, until or unless we have some good reasons or arguments for thinking it false. Of course, this doesn’t mean that all self-reports are true. Categorizing self-reports as self-supporting, I told them, is more a point of intellectual decency and doing-as-you-would-be-done by than of hard logical categories.

I also pointed out that many self-reports fall into the category of incorrigible statements—that is, for some self-reports, we simply will never have any means, whether by authority, experience, or deduction, of proving them false. Most self-reports about mental states fall into this category—for example, “I wish I had purchased Apple Stock five years ago.”

If you can give any general pointers here, I would be grateful. Thank you. Continue reading Re: Nonsense and Self-reports

Christian Logic

I was recently asked the question, is there a distinctly Christian view of logic? I offer here the beginning of an answer to that question. (I am not trying to be original here. These thoughts are from many sources. Just trying to be faithful.)

Laws of Logic

The laws of logic are universal (applicable everywhere), abstract (immaterial, grasped by thought), invariant (not changing), and authoritative (they must be accepted). A non-Christian worldview has a difficult time accounting for such laws. The laws of logic cannot be denied with any kind of consistency, since a denial of logic is tantamount to a denial of truth and reason. But if it is affirmed that the laws of logic are universal, abstract, invariant, and authoritative, yet not “from God,” how can they be justified? Where do such laws come from? They are not invented by men, because they would not then be universal, invariant, or authoritative. They are not material, because they would not then be abstract.

Rather, logic is an expression of God’s unchanging, orderly, truthful, authoritative character.

The Character of God

God Himself is logical; He is a reasoning being: “Come, let us reason together” (Isa. 1:18).  As the ultimate lawgiver He orders His cosmos in a logical way. “God is not a God of disorder” (1 Cor. 14:33). God is orderly, and order implies reason. Where there is no reason, there is only chaos. God’s word is truth (Jn. 17:17), and He would have us be truth tellers (Eph. 4:15). God Himself is non-contradictory. He is truthful (Jn. 3:33), He cannot lie (Heb. 6:18). He does not deny Himself (2 Tim. 2:13). John Frame, in his book The Doctrine of the Knowledge of God, identifies these attributes of God, and then adds: “Does God, then, observe the law of noncontradiction? Not in the sense that the law is somehow higher than God Himself. Rather, God is Himself noncontradictory and is therefore Himself the criterion of logical consistency and implication. Logic is an attribute of God, as are justice, mercy, wisdom, and knowledge.”

The Christian worldview does account for the properties of logical laws. The laws are universal because God is omnipresent; His character is expressed throughout His creation. The laws are abstract, needing no created, material foundation, because they existed before the creation, being attributes within God. The laws are invariant, because God does not change, and neither do His attributes. If the laws of what is true and rational could change, then how could God be trustworthy? How could He keep His covenant promises if truth could be non-truth? He can and does keep His promises, because Christ, the logos, is the same yesterday, today, and forever. The laws are authoritative, because God is the ultimate authority.

God has communicated logic to man as a tool by which we can come to truth. God made us in His image with the ability to reason. We are created as rational beings, and God uses our reasoning ability to speak to us. For example, the giving of law presupposes an ability to reason. Laws are given in the form of universal propositions. “God has commanded all men everywhere to repent.” To obey this, we finish the syllogism: I am a man, therefore I must repent. Without logic, the command could not be applied to particulars. A denial of logic opens the door for disobedience, for without it we cannot obey.

Logic is presupposed, not only in law, but in all revelations of God to men. God gives us minds that reason just as He has given us eyes that see, in order that we may receive His revelation to us. Cornelius Van Til said, “The gift of logical reason was given by God to man in order that he might order the revelation of God for himself.” In order to comprehend any doctrine, we must use logic. The truth that there is one God, eternally existent in three Persons, though clearly contained in the Bible, is not found in any one place in scripture. To see the truth of the Trinity requires a godly, submissive use of logic. If a truth is truly and logically derived from the scripture, we have a divine obligation to believe whatever it is. This is what the Westminster Confession is referring to where it says, “The whole counsel of God concerning all things necessary for His own glory, man’s salvation, faith and life, is either expressly set down in scripture or by good and necessary consequence may be deduced from scripture.” Isaac Watts, the great hymn writer and logician, said it this way in his book on logic: “It was a saying among the ancients, Veritas in puteo, Truth lies in a well; and, to carry on this metaphor, we may very justly say, that logic does, as it were, supply us with steps whereby we may go down to reach the water…. The power of reasoning was given us by our Maker, for this very end, to pursue truth.”

Logic is thus a tool which God has given us in order to understand and obey Him. Like other tools, our grasp of it as humans is no doubt incomplete and imperfect, but it is sufficient for the task for which it is given. And like any other tool, we need to be careful how we use it.

Rhetoric Interview

The following is a slightly edited version of a survey given me by Joshua Butcher – rhetoric instructor at Trinitas Christian School in Pensacola, Florida – regarding the teaching of rhetoric in a classical, Christian setting.

Josh:  How long have you taught rhetoric in a classical education setting?
Jim:  I taught Classical Rhetoric for 18 years at Logos School to 11th graders. I have also written a rhetoric text – Fitting Words: Classical Rhetoric for the Christian Student – and lectured through it.

Josh:  What are the essentials of rhetoric that every classically educated student should have?
Jim:  Do you mean, “What are the essential rhetorical skills that every classically educated student should seek to master?” Continue reading Rhetoric Interview

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!

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!