Tag Archives: Categorical Logic

A Brief History of Validity #1

Introductory Logic, , , ,

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.

After Intermediate Logic?

Logic, Rhetoric, Trivium, ,

What is recommended after Intermediate Logic? The short answer is: Rhetoric! But let me give you a bit more than that.

Introductory and Intermediate Logic together provide a complete foundational logic curriculum. Informal, categorical, and modern propositional logic are all included. The next step in a student’s classical education is to begin to apply what they have learned in logic to effective speaking and writing. This means that the student should move on to study formal rhetoric. Rhetoric applies the tools of logic: defining terms, declaring truth, arguing to valid conclusions, and refuting invalid ones. Indeed, of the modes of rhetorical persuasion – ethos, pathos, and logos – one-third is applied logic.

With this in mind, Roman Roads has released a new curriculum, Fitting Words: Classical Rhetoric for the Christian Student. I am the author of this text, and in Fitting Words I work to apply in rhetoric much of what the student has learned in logic. I am very excited about this project, because one significant reason that I wrote this text was to provide a satisfying answer the question of where to go next!

Take a look HERE for the most up-to-date information about Fitting Words.

Introductory Logic Prerequisite for Intermediate Logic?

Intermediate Logic, Introductory Logic, , , ,

It is certainly possible for a student who has not taken (or not completed) Introductory Logic to take and successfully complete Intermediate Logic. Though the Intermediate Logic text is designed as a continuation to Introductory Logic, it does not assume a mastery of the concepts in it. Almost all of the concepts from Introductory Logic that are essential for Intermediate Logic are re-taught (the only exceptions being the definitions of logical argument, premise, and conclusion; definitions assumed in Intermediate Logic, Lesson 7, but taught explicitly in Introductory Logic, Lesson 19).

That being said, a new Intermediate Logic student who is familiar with Introductory Logic will have an advantage over a student who is not. The following concepts from Introductory Logic are repeated and re-taught in Intermediate Logic (the concepts are first taught in the respective given lesson numbers): Continue reading Introductory Logic Prerequisite for Intermediate Logic? →

Truth Tables for Validity

Intermediate Logic, Introductory Logic, ,

Truth tables can be used to determine the validity of propositional arguments. In a valid argument, if the premises are true, then the conclusion must be true. The truth table for a valid argument will not have any rows in which the premises are true and the conclusion is false. For example, here is a truth table of a modus tollens argument, with the final columns, showing it to be valid:

TT1

The fourth row down is the only row with true premises, and in that row it also has a true conclusion. So this argument is valid.

An argument is invalid when there is at least one row with true premises and a false conclusion, such as in this affirming the consequent truth table: Continue reading Truth Tables for Validity →

Help With Establishing Conclusions

Introductory Logic

One of the most practical lessons in Introductory Logic is Lesson 32, “Establishing Conclusions.” Here you are no longer analyzing someone else’s arguments; you are now writing your own. The hardest part of this lesson is developing an argument for a conclusion while being allowed to use any valid form. In the video for this lesson, I encourage you to find a middle term that connects to the major and minor terms in the conclusion. Let me suggest another way to continue this process.

If you understood hypothetical syllogisms well in the lesson prior, you may use them to help you develop a valid argument. For example, in Exercise 35, question 2, you are asked to establish this conclusion (straight out of Calvin and Hobbes):

bats-arent-bugs1

Ask yourself why bats aren’t bugs. You might say, “Because mammals are not bugs.” Turn that into a hypothetical statement, and complete the modus ponens: Continue reading Help With Establishing Conclusions →

Exercise 30, problem 4: Is my answer correct?

Ask-a-question, Introductory Logic,

Mr. Nance,

As I am reviewing Exercise 30, I am confused  if my answer works , since its different from the original answer. I put:

No logic is a tangible study
No chemistry is logic
∴ All chemistry is a tangible study

As opposed to the answer key:

All non-logic sciences are tangible studies
All chemistry is a non-logic science
∴  All chemistry is a tangible study

Which one is right? Continue reading Exercise 30, problem 4: Is my answer correct? →

Gospel Enthymemes

Introductory Logic, Why learn logic?, , ,

Arguments in which one statement is left assumed are called enthymemes. Most logical arguments encountered in daily life are enthymemes. We can use the tools of logic to determine the assumption being made in an enthymeme.

Let’s examine three enthymemes in the Bible, all on the topic of Gospel salvation. Continue reading Gospel Enthymemes →

Exclusive help

Introductory Logic

Unit 4: Arguments in Normal English in my Introductory Logic text is a difficult section, primarily because of the ambiguities within English. But if we want to be able to apply the tools for analyzing syllogisms to everyday arguments, it is essential that we understand it.

One of the more difficult parts of this difficult section ideals with translating exclusive statements into categorical form. Exclusives are statements that exclude all or part of the predicate of the subject, statements that use words like only, unless, except. Let me give some suggestions that may help. Continue reading Exclusive help →

Alternate Answers for Exercise 27

Introductory Logic, ,

I have received several inquiries regarding other possible solutions to the syllogism translations in Introductory Logic Exercise 27. Though the Teacher’s Edition offers only one solution per problem, there are in fact many possible correct answers to each question.

Here is one more reasonable possibility for each: Continue reading Alternate Answers for Exercise 27 →