Category Archives: Introductory Logic

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 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.

Logic with James B Nance

Introductory Logic Prerequisite for Intermediate 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

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

Fallacies in Comics

Comic strips are a great place to find examples of informal fallacies. It seems that we tend to find improper reasoning funny. In the “Peanuts” comic strip below, Lucy is ad baculum incarnate.

lucy-ad-baculum

Note that the fallacy is not really made by Lucy making the threat, but by Charlie Brown, who is convinced by her “argument.”

Here is another example, where Lucy persuades Linus to memorize his lines using “five good reasons”: Continue reading Fallacies in Comics

Help With Establishing Conclusions

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?

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

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