Logic: A Science and Art

Classical Education, Logic, Trivium, Why learn logic?, , ,

Is logic a science or an art? Of course, a logician would answer Yes, and here is why.

A science is a systematic study of some aspect of the natural world that seeks to discover laws (regularities, principles) by which God governs His creation. Whereas botany studies plants, astronomy studies the sky, and anatomy studies the body, logic studies the mind as it reasons, as it draws conclusions from other information. Logic as a science seeks to discover rules that distinguish good reasoning from poor reasoning, rules that are then simplified and systematized. These would include the rules for validity, of inference and replacement, and so on.

For example, logic as a science could study the apostle Paul’s reasoning in 1 Cor. 15, “If there is no resurrection of the dead, then Christ has not been raised… But Christ has been raised, and is therefore the first fruits from among the dead.” It then simplifies this into a standard pattern: If not R then not C, C, therefore R. This rule can be further simplified, named, and organized in relation to other rules of logic.

An art is a creative application of the principles of nature for the production of works of beauty, skill, and practical use. The visual arts apply their principles to the production of paintings, sculptures, and pottery. The literary arts produce poems and stories. The performing arts produce operas, plays, and ballets.

Logic is one of the seven liberal arts, which include the Trivium of grammar, logic, and rhetoric. These arts are the skills which are essential for a free person (liberalis, “worthy of a free person”) to take an active part in daily life, for the benefit of others. Specifically, logic as an art seeks to apply the principles of reasoning to analyze and create arguments, proofs, and other chains of reasoning.

In summary:

Logic is the science and art of reasoning well. Logic as a science seeks to discover rules of reasoning; logic as an art seeks to apply those rules to rational discourse.

Synonyms, Antonyms & Scripture

Logic, Poetics,

While studying analogies and relationships between terms, I have been considering synonyms and antonyms, and I have come to some surprising realizations.

Defining Synonym and Antonym

A synonym is a word that has the same meaning as another word in the same language. If you were asked to think of several words and their synonyms, you would probably not have too much difficulty: rope & cord, huge & enormous, stone & rock, sleep & doze, etc. English has such an extensive vocabulary that most words have a synonym or near synonym. But if I asked you to think of words that have no synonym, that’s harder. Some possibilities are pencil, helmet, and elbow. But it takes some careful thought. In fact, can you think of a verb or adjective that has no synonym?

An antonym is a word that has the opposite meaning as another word in the same language. By its definition, it appears that antonym is the antonym of synonym. You can probably think up several antonym pairs without too much effort: freedom & slavery, large & small, clean & dirty, father & mother. But if you look around, you will see many things that have no antonym: bottle, brick, book, cabinet, keyboard. It seems about as difficult to think of things that have no synonym as it is to think of things that do have an antonym. Why is this?

Antonym Profundities

Synonyms say something about language and its development. But antonyms say something about the nature of the thing itself, that in someway it has a counterpart. If you develop a list of antonym pairs, they will likely be words that represent fundamental concepts. They seem to reflect something about how God made the world (light & darkness, evening & morning, male & female), or about the fallen nature (sin & righteousness, good & evil, freedom & slavery), or about kinds of separation or direction (present & absent, in & out, left & right).

There are also different species of antonyms. Some are complementary or binary, A and non-A, such as true & false, motion & rest, whole & part. In these cases there are only two options: if a statement is not true then it is false; if an object is moving then it is not at rest; the whole of something is not just a part; and vice versa for each of these.

Relational antonyms lie on a continuum, such as large & small, full & empty, rich & poor. These antonym pairs tend to be adjectives, and there are intermediate states. A house that is not large is not necessarily small; a pitcher can be neither full nor empty; if your uncle is not rich, it doesn’t mean he is poor.

Then there are opposites that are a compromise of these first two types: antonyms that have not a continual but a single intermediate state: positive, negative, & zero; above, below, & level.

Some antonym pairs exist in a relationship with a reversed direction or focus, such as husband & wife, lend & borrow, employer & employee. In such pairs, one can usually not exist without the other: if there is a husband there is a wife; if one lends another borrows; a person with no employees is not an employer. These are called converse antonyms.

Some words have more than one antonym, depending on how you think about them. What is the antonym of father? Is it mother? Or is it son? The definition of father is ‘male parent.’ The opposite of male is female, and a female parent is a mother. On the other hand, the opposite of parent is child, and a male child is a son.  Other examples are possible.

Synonyms and Antonyms in Scripture

Biblical authors make regular use of synonyms and antonyms. A quick glance through Proverbs will reveal this. Consider all the antonyms in this passage:

For the perverse person is an abomination to the Lord, but His secret counsel is with the upright. The curse of the Lord is on the house of the wicked, but He blesses the home of the just. Surely He scorns the scornful, but gives grace to the humble. The wise shall inherit glory, but shame shall be the legacy of fools. (Prov. 3:32-35)

Proverbs also include synonym pairs for poetic purposes:

Does not wisdom cry out, and understanding lift up her voice? She takes her stand on the top of the high hill, beside the way, where the paths meet. She cries out by the gates, at the entry of the city, at the entrance of the doors: “To you, O men, I call, and my voice is to the sons of men. O you simple ones, understand prudence, and you fools, be of an understanding heart.” (Prov. 8:1-5)

Ecclesiastes 3:2-8 has a poetic list of fourteen verbal antonyms:

A time to be born, and a time to die; A time to plant, and a time to pluck what is planted; A time to kill, and a time to heal; A time to break down, and a time to build up; A time to weep, and a time to laugh; A time to mourn, and a time to dance; A time to cast away stones, and a time to gather stones; A time to embrace, and a time to refrain from embracing; A time to gain, and a time to lose; A time to keep, and a time to throw away; A time to tear, and a time to sew; A time to keep silence, and a time to speak; A time to love, and a time to hate; A time of war, and a time of peace.

Can you identify the synonyms and antonyms in Matthew 7:13-14?

Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. But small is the gate and narrow the road that leads to life, and only a few find it.

How many examples of synonyms and antonyms in the Bible can you find?

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?

What would you advise? Thank you in advance.

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!

Minecraft Logic

Intermediate 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?

Eureka! A Discovery of Proportions

Classical Education, Logic,

I have been making a study of analogies and analogical reasoning, and recently saw a connection that I had not seen before. That connection is between what is called ordered-pair analogies, i.e. A is to B as C is to D (or more briefly A : B :: C : D) and mathematical fractions. I was fascinated by what I found. Let me explain.

Re-arranging analogy pairs

I first noticed that, in an ordered-pair analogy, corresponding parts had to be the same part of speech (noun, verb, adjective, etc). Either A & B and C & D had to be the same part of speech, or A & C and B & D had to be the same. For example, this is a good analogy:

drink : eat :: liquid : solid.

Here we have “verb is to verb as noun is to noun.” But an equally valid analogy is

drink : liquid :: eat : solid.

This is “verb is to noun as verb is to noun.” If the first analogy is A : B :: C : D, this second one is A : C :: B : D. Similarly, we can invert both pairs to get valid analogies, as in these examples:

eat : drink :: solid : liquid

liquid : drink :: solid : eat

These would be B : A :: D : C, and C : A :: D : B. We could also switch each pair around the double colon. All these work as good analogies.

The connection

Now, those of you reading closely who remember your basic fractions probably see the connection already. If this is a true equality,

A/B = C/D

then so are all these:

A/C = B/D

B/A = D/C

C/A = D/B.

These equalities follow the same patterns as the analogies above. You might see it clearer with specific numbers. If the first equality is true (and it is), then all the rest must be true:

16/24 = 6/9
16/6 = 24/9
24/16 = 9/6
6/16 = 9/24.

The question

Do you see it? Every re-arrangement that is valid for verbal analogies is equally valid for mathematical fractions, and vice versa. But why should this be so? What is the connection between these two very different kinds of proportions?

There may be some connection between reducing the numerical fraction and finding the fundamental relationship in the verbal analogy. Just as 16/24 = 6/9 because they both equal 2/3, so ‘eat : solid :: drink : liquid’ because they share the relationship of ‘mode of consuming : state of matter of what is consumed.’

I am confident that there is something deeper going on here. Can you find any other connections between verbal analogies and numerical fractions?

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

Ask-a-question, Logic, Why learn 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.

A Tool and Gift

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.

Analogies of Jesus

Logic, Rhetoric,

A parable is a type of analogy. Consequently, most of the recorded words of Christ are teachings by means of analogy. Many of the parables take the form of short stories, such as the story of the Prodigal Son, the Unforgiving Servant, or the Workers in the Vineyard. Some of the analogies used by Jesus, however, are not in story form, but simply use comparison (ordered-pairs and illustrative parallels) to illuminate or emphasize the point. Several of these appear in the Sermon on the Mount, from Matthew 5-7.

Ordered-Pair Analogies

Ordered-pair analogies, which take the form A is to B as C is to D, or simply A : B :: C : D, often appear in standardized tests.  But as we saw in my earlier post, Analogy in Proverbs, they often appear in the Bible as well in slightly more ordinary language. Here are two examples from the Sermon on the Mount.

“He makes His sun rise on the evil and on the good, and sends rain on the just and on the unjust.” (Matt. 5:45)

We can simplify this analogy in standard form: ‘making the sun rise’ is to ‘sending rain’ as ‘evil and good’ is to ‘just and unjust.’ The first pair are similar, the second synonymous.

“But whoever slaps you on your right cheek, turn the other to him also. If anyone wants to sue you and take away your tunic, let him have your cloak also. And whoever compels you to go one mile, go with him two.” (Mat. 5:39-41)

These three ordered pairs are applications of the general principle, “Do not resist an evil person. If he forces you to do something painful but otherwise not sinful, do even more.” This is such a familiar analogy that the phrase go the second mile has become a cliché.

Synonymous Pairs

Some ordered-pair analogies use terms that are basically synonyms. Here Jesus uses synonymous repetition to emphasize His point:

“But I say to you, love your enemies, bless those who curse you, do good to those who hate you, and pray for those who spitefully use you and persecute you” (Matt. 5:44)

“Ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you.” (Matt. 7:7)

Antithetical Pairs

Other ordered pairs set up an antithesis using antonyms, as in these examples from the Sermon:

“Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of heaven; but whoever does and teaches them, he shall be called great in the kingdom of heaven.” (Matt. 5:19)

“Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. But small is the gate and narrow the road that leads to life, and only a few find it.” (Matt. 7:13-14)

Illustrative parallels

Jesus also persuades by means of illustrative parallels, which were explained in detail in my earlier post, Constructing Illustrative Parallels. Here are three examples, one from each chapter of the Sermon on the Mount.

“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. Let your light so shine before men, that they may see your good works and glorify your Father in heaven.” (Matt. 5:14-16)

The intermediate conclusion is that light is not meant to be hidden, nor to illuminate itself, but to shed light on something else. Thus we should neither hide our good works, nor use them to glorify ourselves, but to glorify our Father.

“So why do you worry about clothing? Consider the lilies of the field, how they grow: they neither toil nor spin; and yet I say to you that even Solomon in all his glory was not arrayed like one of these. Now if God so clothes the grass of the field, which today is, and tomorrow is thrown into the oven, will He not much more clothe you, O you of little faith?” (Matt. 6:28-30)

Here the intermediate conclusion is, of course, that God clothes all of His creatures. Thus he will clothe you, His child.

“Beware of false prophets, who come to you in sheep’s clothing, but inwardly they are ravenous wolves.” (Matt. 7:15)

The intermediate conclusion here is that some creatures who look harmless on the outside are concealing their true, dangerous nature. So be wary!

Logic Video Session Info

Intermediate Logic, Introductory Logic, Logic, , ,

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

Classical Education, Logic, Poetics, Rhetoric,

In my first post of this series on analogies, I explained that one typical analogy form is the ordered-pair,  A 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.