Category Archives: Logic

Common errors to avoid: I’s don’t imply O’s

Introductory Logic,

Logic students who are first learning about categorical statements may mistakenly think that any I statement, Some S is P, necessarily implies the O statement, Some S is not P. This is a reasonable error, since it seems to accord with our common use. For example, if I say “Some astronauts are men,” it is reasonable for you to think I also believe that some astronauts are not men.

But this is not always the case. Statements of the form Some S is P logically allow for the possibility that All S is P. When a theology student first learns that some books of the Old Testament speak about Jesus, he may not be surprised to later discover that all books of the Old Testament speak about Jesus (Luke 24:27). Or when a physics student first learns that some forms of usable energy end up as thermal energy, she is well on her way to acknowledging that eventually all usable energy ends up as thermal energy. Astronomers once knew only that some gas giants in the solar system are ringed planets (e.g. Saturn). They eventually discovered that all gas giants in the solar system are ringed planets.

These examples show that Some S is P does not necessarily imply that Some S is not P. Everyone would agree that “Some songs are poems” is a true statement, but it is reasonable still to argue that “All songs are poems.”

 

Do You Smangle?

Introductory Logic, Logic

The first lesson in Introductory Logic discusses several different purposes for defining terms, one of which is to “increase vocabulary.” This is meant in two or three senses.

First, when a student first learns the meaning of a word, such as learning that apiary means ‘a bee house’, his vocabulary has been increased. He has added a new word to the thousands he has access to. Increasing a child’s vocabulary like this is an essential part of his education, in every subject he studies.

Second, when a new word (or a new meaning to an existing word) is added to a language it is given a stipulative definition, until such a word gets generally adopted. This can happen in many ways, such as when an author introduces a new word in his book, and stipulates a definition for it. For example, in his book The Abolition of Man, C. S. Lewis takes the Chinese word Tao and gives it this stipulative definition: “The doctrine of objective value, the belief that certain attitudes are really true, and others really false.” This word has become part of the vocabulary of many people who have read and discussed Lewis’s book.

Here are ten new words that have recently been added into English (and perhaps into your own personal vocabulary):

Afterparty : Social gathering which takes place after a party, concert, or other event
App : Computer program designed for use on a mobile digital device
Brexit : Departure of the United Kingdom from the European Union
Crowdfunding : Raising money by getting many people to make a small contribution
Emoji : Small digital image used to express an idea or emotion
Meh : Interjection used to express indifference
Photobomb : Intrude into the background of a photograph just before it is taken
Selfie : Photograph that one has taken of oneself
Troll : Person who is provocatively rude or insulting on the Internet
Unfriend : Remove (someone) from a list of friends or contacts on a social networking site.

It can be fun for students to invent their own words and definitions, or to share words that are used within the confines of their immediately family. In our house, a “ninker” is a small, difficult to remove item that prevents the opening of a drawer.

My favorite stipulated word from a student is “to smangle,” meaning to rub the top of someone’s head with an open palm (especially if they have a crew cut). This would mean that smangle and noogie are species of the genus, “to rub someone’s head”!

Do you have any stipulated words to share from your students or your family? Share in the comments!

 

The Genus & Species Tool

Introductory Logic

The purpose of classical education is to provide students with tools of learning. One of the most useful tools is the genus and species chart. I used this tool in every course I taught, including Logic, Rhetoric, Calculus, Physics, and Doctrine.

For example, when studying judicial rhetoric in Aristotle, I would follow his descriptions to construct the genus and species chart shown below, which shows the relationships between the seven causes of human actions:

Causes of actions chart

Aristotle argues that every human action is the result of one or more of these seven causes: habit, rational craving, anger, appetite (all voluntary actions – used for prosecution); chance, compulsion, nature (all involuntary actions – used for defense). This visual aid is much clearer than the wordy paragraph given in Aristotle’s Rhetoric text.

I used a similar chart in Calculus to show the arrangement between the types of elementary functions, in Physics for the various branches of physics, and in Doctrine for the “Liar, Lunatic, Lord” argument for the deity of Jesus. For example, the chart for the types of elementary functions looked like this:

When teaching this tool in Logic, one should insist on a clear dividing principle between species, to avoid species overlapping or being placed at the wrong level. In the above chart, the top dividing principle is “whether or not the action is due to oneself.” Under involuntary actions, the dividing principle between chance and necessity is “whether or not the cause is fixed and determined”; under necessity, the dividing principle between compulsion and nature is “whether it is external or internal.” Aristotle’s dividing principles between habit and craving or between anger and appetite are less clear, though the dividing principle under craving is obvious.

The Logic teacher not only presents this tool for use in other subjects, but also in teaching Logic itself. Formal Logic is the “master faculty” of the dialectic stage, and as such it not only teaches the tools of logic, but demonstrates how to use them in teaching. For example, I used the tool of genus and species in my Logic class when I taught the difference between supported  and self-supporting statements. The dividing principle is “how the truth value is determined.”

I would encourage logic teachers to use this tool often, both to present the lesson clearly and to train the students in its proper use. Logic teachers should also encourage their colleagues to use this tools for their students at this stage.

 

Preparing Younger Children for Logic

Ask-a-question, 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!

Sayers’ Helpful Summary of Logic

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

Sayers’ Vision for Logic

In her seminal essay “The Lost Tools of Learning,” the author Dorothy Sayers describes her understanding of the medieval scheme of education, specifically the Trivium — the three liberal arts of grammar, logic, and rhetoric. She argues that students in the Middle Ages were taught the proper use of the tools of learning by means of these arts. Of logic she says,

dorothy[1]“Second, he learned how to use language; how to define his terms and make accurate statements; how to construct an argument and how to detect fallacies in argument.”

As I have taught logic in the classroom, written logic texts (and blog posts), and spoken on logic and classical education around the world, I have regularly returned to this quote. It is for me perhaps the most useful sentence (of the 238 sentences) in the essay.

A Proper Pedagogical Progression

In this sentence Sayers explains what logic is for: logic teaches us how to use language. This reminds us that the liberal arts of the Trivium are language arts (whereas the Quadrivium are mathematical arts). Specifically, logic teaches us how to use the language of reasoning, of disputation and proof.

This sentence also describes a proper pedagogical progression of logic:

  1. We must start with terms: how to define them, relate them, and work with them, including understanding the value of defining terms.
  2. Terms are related in statements (categorical statements connect subject terms with the predicate terms). Logic teaches us “how to make accurate statements”; that is, how to make statements that are true and applicable, as well as understanding how we know that they are true, and how they relate to each other. It teaches how to do this with many different types of statements: simple and compound, categorical and hypothetical, immediate inferences, and so on. Terms are the building blocks of statements.
  3. Statements are the building blocks of arguments, as we connect premises together to draw conclusions. So logic teaches us “how to construct an argument”; that is, how to write a valid argument to establish a desired conclusion.  It teaches how to do this with many types of arguments: categorical and propositional, conditional and disjunctive, symbolic arguments and arguments in normal English.
  4. Finally, logic teaches us “how to detect fallacies in argument,” both the formal fallacies from the rules of validity for categorical syllogisms and propositional arguments, and the informal fallacies of ordinary discourse, like circular reasoning and ad hominem. Logic teaches us not only to detect them, but to name them, and to expose them by means of counterexamples to those untrained in logic.

Were I to add one element to Sayers’ list, it would be “to construct a proof in a step-by-step, justified manner.” With this addition, every page, every concept of both Introductory and Intermediate Logic is covered in Sayers’ helpful description of what is encompassed in learning logic.

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?

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?