Tag Archives: Categorical Logic

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!

Those weird categorical statements

Before studying categorical syllogisms, students learn to translate statements into standard categorical form. The first step is translating the statement such that it uses only the “to-be” verb, so the form becomes [Subject] [to-be verb] [Predicate nominative]. This standardizes the statements so that the arguments are more easily analyzed, which is beneficial when the arguments themselves get more complicated.

But it can result in some very strange statements, e.g. translating “The Apostle Paul rebuked Peter at Antioch” into

The Apostle Paul was a Peter-at-Antioch rebuker.

Most spell-checkers will mark “rebuker” with that squiggly red underline, and some students might balk at the goofy compound noun.

Also, if one is not careful to keep the meaning the same, some of the translations can get rather awkward, such as turning “Susan works hard to resist temptation” into (ahem),

Susan is a hard-to-resist temptation worker.

Most of my students have found the awkwardness of such translated categorical statements to be merely funny, and have just taken it in stride. But occasionally a student will be bothered by it, perhaps thinking that their answers (and thus they themselves) will be thought of as strange or weird. In a larger classroom setting, when everyone is saying the same strange statements, they get used to it pretty fast, but it might be different in a home school setting, or among a small set of students.

The awkwardness of the translations can often be reduced by simply adding a normal noun in a normal place, trying to make the statement sound as normal as possible. For example, rather than translating “The forests will echo with laughter” into

The forests will be with-laughter echoers,

an acceptable translation would be

The forests will be places that echo with laughter.

This requires the addition of a new noun (“places”), but it is perfectly correct. The two rather awkward statements from above could also be correctly translated

The Apostle Paul was a man who rebuked Peter at Antioch.

Susan is a girl who works hard to resist temptation.

This method usually results in long predicates, but more ordinary sounding statements. For more on this topic, read my earlier post, Common errors to avoid: The “to be” verb.

Common errors to avoid: The “to be” verb

Introductory Logic Lesson 11, “The One Basic Verb,” teaches the first step in translating categorical statements into standard form. This step is to translate the statement so that the main verb in the sentence is a verb of being: is, are, was, were, will be, and so on. Thus a statement like “Stars twinkle at night” gets translated into something like

Stars are nighttime twinklers. 

To do this correctly, the subject and predicate must both be nouns, and the verb must be the proper ‘to-be’ verb. The procedure outlined in the lesson is generally clear, but there are two errors I want to help you avoid.

One common error not mentioned in the textbook is the problem of the helping verb. Some students might try to translate the above sentence this way:

Stars are twinkling at night.

The student thinks, “I used the word are, which is a ‘to-be’ verb, so it must be correct.” The problem is that the whole verb here is “are twinkling,” the are being merely a helping verb. The way to fix this is to make sure that the predicate is a noun, usually formed by turning the main verb into a noun (e.g. twinkle –> twinklers).

Secondly, it is sometimes best to make the predicate a noun by adding a new noun, usually a genus of the subject. For example, you could translate the above statement as

Stars are bodies that twinkle at night.

For clarity’s sake, you may want to use a different noun than the one implied by the verb. For example, in translating “She’s got electric boots” it would be overly awkward to say,

She is an electric boots getter.

Much better to translate this as

She is an owner of electric boots

or

She is an electric-boot wearer.

Happy translating!

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

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.”

 

Sayers’ Helpful Summary of 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

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.

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

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 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.

A Brief History of Validity #1

Which forms of categorical syllogisms are valid? Logicians have disputed the answer for centuries, a dispute that can give us insight into the meaning of validity, the central concept of formal logic. This will be the first of a few posts in which I will briefly discuss the history of syllogistic validity.

Aristotle’s 16

It all started with Aristotle, who in his Prior Analytics, Book I, chapters 4-7, detailed sixteen valid forms:

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

If you read Prior Analytics (which is no trivial task), Aristotle presents only the first three figures as figures, omitting any mention of a fourth figure. But in chapter 7 he admits in passing the forms of EAO-4 and EIO-4 as valid, saying,

If A belongs to all or some B, and B belongs to no C … it is necessary that C does not belong to some A.

It is not difficult to see why Aristotle omits AAI-1, EAO-1, AEO-2, and EAO-2. These four forms are his AAA-1, EAE-1, AEE-2, and EAE-2 with the subimplication of the conclusion. Aristotle apparently saw no need to include syllogism forms with particular conclusions when the premises could imply the universal.

Aristotle and Figure 4

It is rather more difficult to understand why Aristotle does not admit the fourth figure, though logicians have argued that it has to do with how he defines a syllogism. We learn from Bertrand Russell, in his Cambridge Essays, that

The fourth figure…was added by Aristotle’s pupil Theophrastus and does not occur in Aristotle’s work, although there is evidence that Aristotle knew of fourth-figure syllogisms.

Theophrastus apparently recognized three more valid forms of figure 4: AAI, AEE, and IAI, bringing the total to 19. These were given Latin names by medieval scholars, but that will be the topic for my next post.