Category Archives: Logic

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.

Analogy in Proverbs

In my first post of this series on analogies, I explained that one typical analogy form is the ordered-pairA is to B as C is to D, or more briefly A : B :: C : D. This is how most people think about analogies, having seen them in the vocabulary or reasoning sections of standardized tests. But in reading through Proverbs recently, I uncovered about fifty analogies, most of which can be reduced to ordered-pair form.

One Ordered Pair Illuminates the Other

For example, Proverbs 3:12 says,

“For whom the Lord loves He corrects, just as a father the son in whom he delights.”

This analogy can be reduced to an ordered pair:

The Lord : His beloved :: a father : his delighted son

The common concept between these analogous pairs is that the first corrects or disciplines the second. The comparison is helpful because the familiar, concrete image of a father correcting the son in whom he delights illuminates the less familiar, more abstract idea of the Lord correcting His beloved.

Here are two more examples:

“As a dog returns to his own vomit, so a fool repeats his folly.” (Prov. 26:11)

“Where there is no wood, the fire goes out; and where there is no talebearer, strife ceases.” (Prov. 26:20)

In those examples, the first analogous pair illuminates the second.

Synonymous Pairs

Rather than using an analogy to illuminate the less familiar by means of the more, many proverbs simply restate the main point using synonymous pairs. Here are several examples in which the ordered pairs are synonyms:

“Then they will call on me, but I will not answer; They will seek me diligently, but they will not find me.” (Prov. 1:28, cf. Mt. 7:7)

“I have not obeyed the voice of my teachers, nor inclined my ear to those who instructed me!” (Prov. 5:13)

“Does not wisdom cry out, and understanding lift up her voice?” (Prov. 8:1)

“For a harlot is a deep pit, and a seductress is a narrow well.” (Prov. 23:27)

Antithetical Pairs

Many other proverbs set up an antithesis, using antonyms in ordered pairs:

“The curse of the Lord is on the house of the wicked, but He blesses the home of the just.” (Prov. 3:33)

“A wise son makes a glad father, but a foolish son is the grief of his mother.” (Prov. 10:1)

“The hand of the diligent will rule, but the lazy man will be put to forced labor.” (Prov. 12:24)

Analogies by means of Hebrew parallelism are employed throughout Scripture. In my next post, we will consider analogies in the New Testament.

Constructing Illustrative Parallels

In my last post, I claimed that there are three typical ways we use analogies: basic comparisons, ordered-pairs, and illustrative parallels. In this post I will explain how to construct an illustrative parallel, which is a powerful means of proof.

The Pattern

An illustrative parallel reasons from a particular example (the source) to a particular conclusion (the target). The process combines inductive reasoning (from the particular example to a general statement) and deductive reasoning (from the general statement to the particular conclusion) as shown:

I am fascinated by the inductive-deductive process that the mind goes through when reasoning by analogy, such as in the parables. For example, Jesus teaches in Matthew 5:14-15, “You are the light of the world. A city that is set on a hill cannot be hidden. Nor do they light a lamp and put it under a basket, but on a lampstand, and it gives light to all who are in the house.” The source (“no one lights a lamp to put it under a basket, but to give light to the house”) inductively implies the general intermediate conclusion that what is meant to illuminate something should not be covered, and that it is uncovered not in order to display itself, but something else. So when he deductively makes the particular conclusion in verse 16, “Let your light so shine before men, that they may see your good works and glorify your Father in heaven,” we understand that we should do good works, not to shine a light on ourselves, but that men might glorify God.


Inventing good analogies can be difficult, but we can be helped using the pattern above. Say that you want to use an analogy to respond to this challenge: “Why study formal logic? Everyone can already reason!” You could argue that the study of formal logic helps to improve our reasoning skills by providing standards to distinguish between good and bad reasoning. This is your target. It can be deduced from the general statement that studying a language art can provide standards by which we distinguish between the proper and improper use of that art. Given this, we must then invent a source, a different example of the general statement, and one that is preferably more familiar that the target. What familiar language art provides us with such standards? English is a good example; the study of English helps us improve our speaking and writing skills by providing standards to distinguish proper English from improper. The basic analogy could then be simply stated: “‘Why study formal logic? Everyone can reason.’ That’s like arguing, ‘Why study formal English? Everyone can speak!’”

Imitating the Masters

Jesus is, of course, the Master of analogies, as of all other forms of argument. But there are also many lesser masters from whom we can learn this art. My favorites include C. S. Lewis, G. K. Chesterton, Mark Twain, and Doug Wilson. Here are some of my favorites:

“I believe in Christianity as I believe that the Sun has risen; not only because I see it but because by it I see everything else.” ― C. S. Lewis

“The object of opening the mind, as of opening the mouth, is to shut it again on something solid.” ― G.K. Chesterton

“Laws are sand, customs are rock. Laws can be evaded and punishment escaped, but an openly transgressed custom brings sure punishment.” ― Mark Twain

“We have no structure any more. We have no shared creed. We do not know what we are here for. It makes no sense to speak of our inherited ‘shared values,’ or better yet, ‘core values.’ If they are arbitrary, shared values are worthless. If they are arbitrary, core values are simply located where our intestines are, and are full of the same thing.” ― Doug Wilson

“If we had no winter, the spring would not be so pleasant: if we did not sometimes taste of adversity, prosperity would not be so welcome.” ― Anne Bradstreet

What are some of your favorite analogies? Leave a comment!

Reasoning by Analogy

I have been thinking about analogies lately, and finding them fascinating. There appear to be three basic uses for the term analogy.


First, almost any comparison, especially one in which a familiar, simpler, or concrete thing is used to clarify or illuminate something that is unfamiliar, complex, or abstract, can be called an analogy. For example, this excerpt from George Orwell’s essay “A Hanging” is considered an analogy:

They crowded very close about him, with their hands always on him in a careful, caressing grip, as though all the while feeling him to make sure he was there. It was like men handling a fish which is still alive and may jump back into the water.

The manner in which the guards handled the prisoner is compared to men handling a fish. Most people have tried to handle a live fish just pulled from the water that wants back in, so this comparison gives the reader a vivid mental picture of the less familiar situation Orwell is describing.


Second, we see analogies in what can be called ordered-pair form: A is to B as C is to D, or more briefly A : B :: C : D. Typically these appear in the vocabulary or reasoning section of standardized tests, like this sample question from the GRE. Choose the analogous pair:

A. player : coach
B. child : parent
C. student : teacher
D. intern : doctor

The best answer is D. Just as an apprentice is training to be a plumber, so an intern is training to be a doctor. A child does not formally study to become a parent, and a player or student is not necessarily studying to become a coach or teacher (respectively).

Illustrative parallels

Third, we see analogies being used for the purposes of persuasion, called arguments by analogy, or what Aristotle calls illustrative parallels. Here is an example from Aristotle’s Rhetoric II.20:

Public officials ought not to be selected by lot. That is like using the lot to select athletes, instead of choosing those who are fit for the contest; or using the lot to select a steersman from among a ship’s crew, as if we ought to take the man on whom the lot falls, and not the man who knows most about it.

Illustrative parallels use both inductive and deductive reasoning. We use inductive reasoning to mentally move from the source (e.g. we ought not use the lot to select athletes) to a more general, unspoken intermediate conclusion (we ought not randomly select someone for a skilled position). We then use deductive reasoning to move from this intermediate conclusion to our specific conclusion, the target (we ought not select public officials by lot).

In my next post, I will explain how to construct illustrative parallels.

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?


A Brief History of Validity #3

We read in my last post on this subject about the nineteen traditional valid forms of syllogisms named by medieval scholars, plus the five forms which can be deduced by subimplication of those with a universal conclusion. These comprise the twenty-four forms of syllogisms identified as valid in my Introductory Logic text.

Statements with Existential Import

However, since the time of George Boole, a 19th-century mathematician, only fifteen of those twenty-four forms are recognized as valid. Why is this? Boole argued that the truth of a particular statement cannot be inferred from the truth of its corresponding universal, because a particular statement asserts the existence of its subject, but a universal statement does not. That is, to say, “Some athletes are dedicated people” is to assert that at least one athlete exists, but to say that “All athletes are dedicated people” is only to say that if one is an athlete then one is a dedicated person. According to Boole, the four categorical statements should be interpreted this way:

All S is P = If S exists then it is P
No S is P = If S exists then it is not P
Some S is P = There exists at least one S that is P
Some S is not P = There exists at least one S that is not P

Particular statements are said to have existential import; they claim that the terms in the statement exist. Universal statements, however, do not have existential import; they are considered as material conditionals.

The Existential Fallacy

In this interpretation, no particular statement can be inferred from a universal statement, or from universal premises. One could not validly argue, for instance,

All grandfathers are fathers.
All fathers are men.
∴ Some men are grandfathers.

This AAI-4 (Bramantip) syllogism is said to make the existential fallacy, which is based on this sixth rule of validity: “A valid syllogism cannot have universal premises and a particular conclusion.” By the modern interpretation, the premises only say this:

If grandfathers exist then they are fathers.
If fathers exist then they are men.
∴ There exists at least one man who is a grandfather.

These premises do not claim that grandfathers or fathers or men exist, but the conclusion does. Thus the conclusion claims more than is contained within the premises, which means the syllogism is invalid.

There is much to commend the modern view of interpreting categorical statements. For example, in the traditional view, this is a valid chain of reasoning:

No athletes are people that breathe underwater ⇒ (by converse)
No people that breathe underwater are athletes ⇒ (by obverse)
All people that breathe underwater are non-athletes ⇒ (by subimplication)
Some people that breathe underwater are non-athletes.

Everyone would agree that the first statement is true, but most people would say that the last statement is false, because it seems to imply that there are people that breathe underwater.

Rethinking the Modern Interpretation

However, I think that the modern interpretation of categorical statements is potentially flawed. Does not the statement “No people that breathe underwater are athletes” seem to imply that there are people that breathe underwater? And why insist that particular statements have existential import? Consider these particular statements:

Some hobbits are not Shire dwellers.
Some black holes are members of binary stars.
Some of your sons will be the king’s horsemen.

Most people would argue that, in the sub-created world of Tolkien, the first statement is true, even though (in our world) hobbits do not exist. Most astronomers would argue that the second is almost certainly true, even though the existence of black holes is still in doubt. The last statement was uttered by Samuel to the people in the hope that it would not be true, that such sons would not exist.

Much more could be said, but the logic student should at least be aware that this debate exists. I would appreciate your thoughts.

Immediate Inference Cheat Sheet

Equivalent Immediate Inferences of the four Categorical Statements:

All S is P
=  No S is non-P  (obverse)
=  All non-P is non-S  (contrapositive)

No S is P
=  All S is non-P  (obverse)
=  No P is S  (converse)

Some S is P
= Some S is not non-P  (obverse)
= Some P is S  (converse)

Some S is not P
= Some S is non-P  (obverse)
= Some non-P is not non-S  (contrapositive)

Immediate inferences work in reverse:

All S is non-P
= No S is P  (obverse)

All non-S is non-P
= All P is S  (contrapositive)

No S is non-P
= All S is P  (obverse)

Some S is non-P
= Some S is not P  (obverse)

Some S is not non-P
= Some S is P  (obverse)

Some non-S is not non-P
= Some P is not S  (contrapositive)

Immediate inferences can be combined:

No non-S is P
= No P is non-S = All P is S  (converse, obverse)

Some non-S is P
= Some P is non-S = Some P is not S  (converse, obverse)

Other translations:

All non-S is P
= All non-P is S  (contrapositive)

No non-S is non-P
= All non-S is P  (obverse)

Some non-S is not P
= Some non-P is not S  (contrapositive)

Some non-S is non-P
= Some non-S is not P  (obverse)

All of this and more is included in this complete Immediate Inference Chart.

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.

A Brief History of Validity #2

The 19 Traditional Forms

In the first post in this series, we saw that Aristotle identified 16 valid forms of categorical syllogisms (though he formally acknowledged only the first three figures). Some thirteenth-century logicians such as William of Sherwood and Peter of Spain recognized nineteen valid forms, giving them Latin names as a mnemonic device for ease of memorizing:

Barbara, Celarent, Darii, Ferioque prioris.
Cesare, Camestres, Festino, Baroco secundae.
Tertia Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison habet.
Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The vowels in each name correspond with the mood, such that “Barbara” is AAA-1, “Cesare” is EAE-2, and so on. Thus the medievals recognized these valid forms:

Figure 1: AAA, EAE, AII, EIO
Figure 2: EAE, AEE, EIO, AOO
Figure 3: AAI, IAI, AII, EAO, OAO, EIO
Figure 4: AAI, AEE, IAI, EAO, EIO

The five forms not included in this list are AAI-1, EAO-1, EAO-2, AEO-2, and AEO-4. Why were these five not included? They are the forms in which the conclusion is the subimplication of moods with all universal statements, namely AAA-1, EAE-1, EAE-2, AEE-2,  and AEE-4. Thus they were seen as “weaker” forms of the syllogisms (why bother concluding the particular “Some S is not P”when you can conclude the universal “No S is P”?).

Defending the Missing Five

Interestingly, these five omitted forms can readily be shown to be equivalent to Bramantip (AAI-4) using immediate inferences, as follows:

AAI-4 (given)
All P is M
All M is S

∴ Some S is P

AAI-1 (taking the converse of the conclusion, correcting the premise order)
All M is S
All P is M

∴ Some P is S

EAO-1 (taking the obverse of the major premise and conclusion of the AAI-1)
No M is non-S
All P is M

∴ Some P is not non-S

EAO-2 (taking the converse of the major premise of the EAO-1)
No non-S is M
All P is M

∴ Some P is not non-S

AEO-2 (From the AAI-1, take the contrapositive of the major premise, obverse of the minor premise and conclusion)
All non-S is non-M
No P is non-M

∴ Some P is not non-S

AEO-4 (From the AEO-2, take the converse of the minor premise)
All non-S is non-M
No non-M is P

∴ Some P is not non-S.

This is one practical application of the immediate inferences learned in Lesson 27 of Introductory Logic.