# Classical Logic

### From Logic

Classical Logic is the first formal logic system. Created by Aristotle, it is based on the syllogism. As Copi and Cohen write in "Introduction to Logic": "Because the syllogism was so widely employed as the most basic tool of scholarly argument, the set of Aristotelian writings devoted to the analysis of syllogisms received the venerable name: Organon - or "Instrument""

A syllogism is a two-premise deductive argument in which a conclusion is inferred from two premises. A categorical syllogism - a form of argument first inculcated by Aristotle - is a specific type of syllogism consisting of three categorical propositions that together contain exactly three terms or categories, each of which occurs in exactly two of the three propositions. A categorical syllogism is said to be in standard form when its premises and conclusion are all standard form Categorical Propositions (A, E, I and O, as defined in the section on categorical propositions) and when they are arranged in a specified standard order.

It is advised that readers first consult the section on Categorical Propositions before reading this entry.

Readers interested in the foundation of Classical Logic may consult the Laws of Classical Logic

## Major, Minor and Middle Terms

In order to grasp the workings of a syllogism, we must introduce some new terms: The **Major Term**, The **Minor Term** and the **Middle Term**. Let's make matters simple by using this example:

Premise 1: No heroes are cowards - (This is an E proposition - No S are P) Premise 2: Some soldiers are cowards (This is an I proposition - Some S are P ) Conclusion: Therefore, some heroes are not cowards (This is an O proposition, Some S are not P)

We can first break down propositions by looking at their **subject** and **predicate**: the first category in a proposition is called the **subject** and the second category is called the **predicate**. In our example, the subject of the first premise, *No heroes are cowards* would be the category of *heroes*. The second category *cowards*, would be the "predicate".

Now, in order to figure out which proposition is **major term** and which is the **minor term**, we look at the conclusion of the argument. The category referred to in the predicate of the conclusion is called the major term. So the major term in this argument is *cowards* This term appears in the first proposition, so the first premise in this argument is the **major premise**.

The standard form to syllogisms is to list the major premise first, but this need not be the case, so we cannot rely upon this as a clue as to which premise is the major premise. Another aspect of major premises is that they a generalization about some category. Examples would include: "All A's are B," "Either A or B," or "If A, then B"?

Now, for the minor term. The subject of the conclusion contains the minor term. In our example, the category of "heroes" would be the minor term. The second proposition of the categorical syllogism contains this term, so we will call it the '*minor premise*.

The 3rd item is the **middle term**. It occurs in both the major and the minor premise. It is the transitional device that connects the major premise to the minor premise. The middle term is the term that preserves truth along the premises, to the conclusion. It's like the equal sign in math. Can you identify the *middle term* in this argument? It is the category *soldiers*.

The categorical syllogism is said to appear in **standard form** when the propositions are ordered as we have ordered them here in our example: from major premise, minor premise, conclusion.

## The Mood and Figure of a Syllogism

### Let's get in the Mood

Oh yeah!

The mood of a syllogism is determined by the types of categorical propositions it contains (A, E, I, or O propositions, which again, are: A) All S are P (E) No S are P (I) Some S are P (O) Some S are not P)

Let's look at a specific example, and determine the mood, based on the types of categorical propositions it uses:

Premise 1: All cats are intelligent animals A Premise 2: All intelligent animals are cool A Conclusion: Therefore, All cats are cool A

Each categorical proposition in this categorical syllogism are are of the form A, meaning that the mood of this argument is AAA

### Let's Look at Some Figures

Now, we must examine some Figures

Oh yeah!

No, unfortunately, not that kind of figure.

The figure of a categorical syllogism can be one of the following four types, where: the Middle term is represented by m, the Major Premise is represented by P and the Minor Premise by S

form 1) form 2) form 3) form 4) m P P m m P P m S m S m m S m S S P S P S P S P

Any categorical syllogism can be rearranged so that the major Premise appears first, the minor second, and the conclusion last.

So, let's take our above example, switch the order of the premises so that we can have our predicate premise first, and we see that our AAA argument matches form 1:

Premise 1: All intelligent animals are cool A M - P Premise 2: All cats are intelligent animals A S - M Conclusion: Therefore, All cats are cool A ^ S-P

So our argument's mood and figure is AAA-1. As we will soon see, AAA-1 is a valid argument form. So cats are cool.

A note on reformatting arguments to fit the standard forms:

Often, for the sake of good writing, arguments appear in differing order from this standard format, or only hint at a premise or conclusion. So a person wishing to examine the argument must first reformat the argument into a standard syllogism. There are two main types of non standard formats for arguments:

#### Enthymemes

The defining characteristics of an enthymeme is that either it is missing at least one of the parts of a syllogism or that the conclusion is not certain, given the premises. So an enthymeme does not fit into the syllogistic form, as it stands:

Example: "All dogs are mammals... so they (all dogs) are warm blooded."

The minor premise of "all mammals are warm blooded" is missing.

Or it may fit the form, but, failing to reach a necessary conclusion, cannot be considered a proper syllogism:

Example 2: Most instructors at HSU are excellent teachers. Jay is an instructor at HSU. Therefore, Jay is probably a good teacher.

#### Sorties

A sequence of categorical syllogisms in which the intermediate conclusions have been omitted because they can be safely assumed. A sorties is in standard form when each of the component propositions is a standard form categorical proposition, when the predicate term of the conclusion is in the first premise, when each term occurs exactly twice, and when each premise after the first has a term in common with the preceding one. To evaluate a sorties, express it in standard form, supply the intermediate conclusions, and then break each into separate component syllogism.

Ok, let's return to our discussion on valid logical forms...

## The Formal Nature of Syllogistic Logic

The mood and figure of a syllogism uniquely determine its form, and the form of an argument, from the viewpoint of logic, is its most important aspect. The validity or invalidity of a syllogism, (whose constituent propositions are all contingent, we will discuss this is more detail later) depends exclusively on its form, completely independent of its content. In fact, I've always held that this fact is one of the most impor..

Excuse me, my man, my man...

Yes?

You gotta hit me with another picture.

Oh yeah!

Ok. Once the task of forming a categorical syllogism in standard form is accomplished, validity of the form can be ascertained according to the following table. You don't need to work it out on your own, it's already been done for you: thousands of years ago! Of the 256 possible permutations of mood and figure (64 types of mood X 4 types of figure), only 15 of the possible forms are valid. Here they are:

Figure 1: AAA, EAE, AII, EIO Figure 2: EAE, AEE, EIO, AOO Figure 3: IAI, AII, OAL, EIO Figure 4: AEE, IAI, EIO

We will return to discussing WHY only these forms are valid, after we go over the rules for valid categorical syllogisms.

Note: In the traditional interpretation of categorical propositions from the square of opposition, more valid forms exist. I present here only the modern forms, based on the rules of the modern square of opposition (the Boolean interpretation of the square), which are far more stringent.

Now, here are some interesting facts about what you've just learned: First, a valid syllogism is valid by virtue of its form alone. The content is immaterial. This means that if a given syllogism is valid, any other syllogism that uses the same form is also valid. So you can plug in an argument into a valid form, in order to ensure that you have a valid argument!

The converse also follows: if the form of an argument is invalid, then any other syllogism that uses that form is automatically invalid, no matter the content. What this means is that you can use this knowledge to form logical analogies in order to prove that your argument is valid, or your opponents argument is invalid!

Ok, so let's try it out! Let's deconstruct an argument, by showing that it has an invalid form, meaning that your opponent has no grounds to hold to his conclusion.

Let's imagine you are offered the following, rather mind numbing argument, that you wish to destroy on simple grounds of validity...

Original argument:

"All atheists are nonbelievers, so all atheists are materialists, since all materialists are nonbelievers."

The first thing to do is to work out what the conclusion is. The word "so" is a tip off that "all atheists are materialists" is the conclusion. It labels the preceding statement as a support for it, and since the last statement is prefaced by the word "since", it too is being labeled as a support for the statement "all atheists are materialists". We can now isolate the term "materialists', as the predicate of the conclusion, giving us our Major term. The subject of the conclusion gives us the minor term "atheists" and the middle term, used in both premises, to preserve truth, is "nonbelievers".

P = materialists (the predicate of the conclusion) S = atheists (the subject of the conclusion) m = nonbelievers - occurs in both premises, used to preserve truth across the premises to the conclusion.

Now, let's put the argument in standard form. In standard form, this argument becomes:

All materialists (P) are nonbelievers (m) - P-m A All atheists (S) are non believers (m) - S-m E Therefore, all atheists (S) are materialists (P) - E

This form is figure number 2 , specifically AAA-2

A quick check on the list tells us that AAA-2 does not appear on the list as a valid form! In fact, as we will see, the form commits the specific formal fallacy of "undistributed middle".

So there's no need for you to struggle with your opponents argument any further. His argument is invalid. He has no justification for holding to his conclusion. QED.

Ok, now let's take a look

....at that fine lady again?

Ok, one more...

Oh yeah!

## Syllogistic Rules and Syllogistic Fallacies

The first three rules have to do with the **distribution of terms**, and the last three have to do with the **quality** and **quality** of the propositions in the syllogisms. When any of these rules are broken, truth fails to be preserved across the premises to the conclusion, and a corresponding **formal fallacy** is committed. This leads to an invalid argument, and again, using the analogy of a road map, invalid arguments are dead ends - the truth is lost somewhere before the destination.

Before we cover rules 1, 2, and 3, let's review how the four types of categorical propositions **distribute** their terms.

**Distribute**: To use (a term) so as to include all individuals or entities of a given class.

In an A proposition, the S is distributed, the P is not In an E proposition, S and P are both distributed In an I proposition, S and P are both undistributed In an O proposition, S in undistributed, P is distributed

Now, because some find the concept of distribution complex, I want to delve into it further.

In the A proposition, "All S are P", S is extended completely to P, but we cannot assume that the reverse is true. Think of the famous example of rectangles and squares. We know all squares are rectangles, but we also know that all rectangles are not squares. So we see that to make the assumption that All S are P is equal to All P is S is a faulty one. So, in "All S are P", S is distributed, whereas P is undistributed. Got it?

In and E statement, "No S is P", we can make the assumption of reversibility. Once we know that all pigs are not sheep, we can state that all sheep in turn, are not pigs. So in a universal negative statement, both terms are distributed. Easy, right?

In an I statement, "Some S is P", we know by definition that we are not talking about all cases of S, so we know that neither of the terms is distributed.

Finally, in an O statement, "Some S is not P", as per an I proposition, we do not distribute the subject S, but in denying that this subclass is included in the predicate, we rule out the predicate term entirely, so an O proposition does distribute, through exclusion, its predicate term.

Finally the rule for conclusions - One premise should be negative (E or O), if the conclusion is negative. If the conclusion is affirmative (A or I), however, both premises should be affirmative.

If you go on to check out the Syllogistic Machine in the **Further Resources** section below, you'll see that the algorithm that evaluates the different syllogisms in the machine makes use of this property of premises and conclusion. Given the status of distribution of the terms in the conclusion, it makes certain demands on the status of distribution of the premises. If a term is distributed in the conclusion, it should also be distributed in the premise in which it occurs. Also the middle term should be distributed at least once. Violation of these rules are called "illicit premise" (major or minor) and "undistributed middle" respectively.

Here are the rules clearly laid out for you:

Recall: In an A proposition, the S is distributed, the P is not In an E proposition, S and P are both distributed In an I proposition, S and P are both undistributed In an O proposition, S in undistributed, P is distributed

### Rule 1 and The Fallacy of Four Terms

A valid, standard form categorical syllogism must contain *exactly three terms*, each of which is used in the same grammatical sense throughout the argument.

A categorical syllogism with four terms commits the **fallacy of four terms**.

Example: Sometimes, it is not readily apparent that four terms are being used. One way a fourth term can be snuck in is through a fallacy of equivocation - a term with two possible meanings is used in two different ways. Here's an easy example:

All rivers have banks All banks have money Therefore, all rivers have money

Since the word "banks" is being used in two different senses, as the side of a river, and as a place where money is kept, this argument commits the informal fallacy of equivocation, and the formal fallacy of four terms.

### Rule 2 - And the Fallacy of The Undistributed Middle

The middle term must be distributed in one of the premises, or the fallacy of the undistributed middle occurs. The middle term is what connects the major and the minor term. If the middle term is never distributed, then the major and minor terms might be related to different parts of the M class, thus giving no common ground to relate S and P.

Example:

All P are m All S are m All S are P

Here, the m is undistributed in both premises (appears in same position) AAA-2. A common error.

### Rule 3 - and The Fallacy of Illicit Major and Illicit Minor

If a term in distributed in the conclusion, it must be distributed in the premise. When a term is distributed in the conclusion, (letâ��s say that P is distributed), then that term is saying something about every member of the P class. If that same term is not distributed in the major premise, then the major premise is saying something about only some members of the P class. Remember that the minor premise says nothing about the P class. Therefore, the conclusion contains information that is not contained in the premises, making the argument invalid.

If the rule is broken, the fallacy committed is either illicit major or illicit minor, depending on whether the Predicate or Subject is undistributed.

Here P is distributed in the conclusion (E) but undistributed in the major Premise (A) this commits the fallacy of **illicit major**:

All m are P (A) All S are m (A) No S are P (E)

Here S is distributed in the conclusion (A) but undistributed in the minor Premise (E) this commits the fallacy of **illicit minor**:

All P are m (A) All m are S (A) No S are P (E)

### Rule 4 - and The Fallacy of Exclusive Premises

Two negative premises are not allowed. If the premises are both negative, then the relationship between S and P is **denied**. The conclusion cannot, therefore, say anything in a positive fashion. That information goes beyond what is contained in the premises.

No P are m No S are m No S are P

### Rule 5a - And The Fallacy of Drawing an Affirmative Conclusion from a Negative Premise

A negative premise requires a negative conclusion, and a negative conclusion requires at least one negative premise. Otherwise, you've committed the fallacy of drawing a negative conclusion from affirmative premises

Example of Drawing an affirmative conclusion from a negative premise:

2) No P are m Some S are m Some S are P

The conclusion states that the S class is either wholly or partially contained in the P class. The only way that this can happen is if the S class is either partially or fully contained in the M class (remember, the middle term relates the two) and the M class fully contained in the P class. Negative statements cannot establish this relationship, so a valid conclusion cannot follow.

### Rule 5b - And The Fallacy of Drawing a Negative Conclusion from a Positive Premise

The same holds true for a positive conclusion. It must have at least one positive premise.

Drawing a negative conclusion from affirmative premises:

1) All P are m All S are m No S are P

The conclusion asserts that the S class is separated in whole or in part from the P class. If both premises are affirmative, no separation can be established, only connections. Thus, a negative conclusion cannot follow from positive premises.

Note: These last four rules working together indicate that any syllogism with two particular premises is invalid.

### Rule 6 - And The Existential Fallacy

Are both premises universal? Then, the conclusion cannot be particular, otherwise, you commit the existential fallacy! Recall that Universal statements can be made about hypothetical entities that may not actually exist, while particular statements imply existence. Ergo derving from *All unicorns have horns' that *This unicorn has a horn* commits an existential error.*

Example:

All P are m All S are m Some S are P

However, these claims are considered valid from the Aristotelian standpoint.

These six rules apply only to standard form categorical propositions. To examine other logical forms and their validity and invalidity, you will need to read the section on Propositional Logic.

## Exposition of the Valid Syllogistic Forms

For now, let's use are knowledge of these rules to uncover why *there are only 15 valid categorical syllogisms*. The easiest manner to perform this process is to at the four possible figures for an argument, and to see which of the four possible classical propositions: A, E, I or O can be a valid conclusion given the figure.

### Conclusion with an A proposition

There is only one valid form with an A conclusion: **AAA-1**

Neither premise in a syllogism with an A conclusion can be E or an O proposition, as per rule 5. The minor premise cannot be I, as per rule 3. The two premises, major and minor, cannot be IA, because if they were, either rule 3 would be violated because the distributed subject of the conclusion would not be distributed in the premise, or there would be a violation of rule 2, since the middle term of the syllogism would not distributed in either premise. Therefore the only possible valid mood with an A conclusion is AAA. But in the second figure, AAA would violate rule 2, and in both the third and fourth figure AAA would violate rule 3.

### Conclusion with an E proposition

There are four valid forms with an E conclusion: **AEE-2'**, **AEE-4**, **EAE-1**, and **EAE-2**

Since both the subject and the predicate of an E proposition are distributed, all three terms in the premises in a syllogism with an E conclusion must be distributed, and this is possible only if one of the premises is also an E proposition. But both premises cannot be E, as per rule 4 (two negative premises), nor can the other premise be an O proposition, by the same rule. And the other proposition cannot be I, as this would violate rule 3. Hence, all we are left with is A and E, so the only moods are AEE or EAE. AEE cannot work in the first or third figure, since again, this would lead to a term distributed in the conclusion not being distributed in the premises (rule 3). EAE cannot work in figures 3 or 4, again because of rule 3.

### Conclusion with an I proposition

There are only four valid forms with an I conclusion: **AII-1**, **AII-3**, **IAI-3** and **IAI-4**

Neither premise can be E or O, as per rule 5 concerning negative premises. The premises cannot be AA, by rule 6 (Existential fallacy!), nor can they be II, by rule 2 (middle term distribution) So the premises must either be AI or IA and the only possible moods with an I conclusion are AII or IAI. AII is invalid in the second and fourth figures by rule 2. The same rule invalidates figures 1 and 2 for the mood IAI.

### Conclusion with an O proposition

There are six valid syllogisms with an O conclusion. **AOO-2**, **EIO-1**, **EIO-2**, EIO-3, **EIO-4** and **OAO-3**.

The major case cannot be an I proposition, by rule 3. Supposing that the major premise were A, the minor premise could not be either A or an E, by rule 6 (existential fallacy), nor could the minor premise then be an I, because in that case either the middle term would not be distributed (violation of rule 2) or a term distributed in the conclusion would not be distributed in the premises (rule 3). So if the major premise were an A, the minor premise must be O. But AOO-4 is not possibly valid, since then also the middle term would not be distributed; and neither is AOO-1 or AOO-3, since they would have terms distributed in the conclusion that were not distributed in the premises, (rule 3). This leaves only AOO-2.

Suppose that the major premise were an E. In that case, the minor premise could not be an E or an O, by rule 4 concerning 2 negative premises, nor cold the minor premise be an A, by rule 6 (existential fallacy). This leaves EIO which is valid in all four figures.

Finally, suppose that the major premise were also an O proposition. The minor premise could not be an E or an O, as per rule 4 concerning 2 negative premises, nor could the minor premise be an I premise without violating either rule 2 or rule 3 for that matter. Therefore, if the major premise is an O proposition, the minor premise must be an A, and the mood must be OAO. OAO-1 is eliminated by rule 2 (distribution of the middle term) and OAO-2 and OAO-4 are eliminated by rule 3 (a term distributed in the conclusion would not be distributed in the premises). This leaves OAO-3.

### Nicknames for the Valid Forms

This leaves us with 15 valid syllogisms. Classical logicians gave each of these 15 valid syllogisms a name.

In figure 1, in which the middle term is the subject of the major premise and the predicate of the minor premise: AAA-1, **Barbara**, EAE-1 **Celarent**, AII-1 **Darii**, and EIO-1, **Ferio**.

In figure 2, the middle term is the predicate of both premises: **AEE-2**, **Cametres**, EAE-2, **Cesare**, AOO-2, **Baroko**, EIO-2, **Festino**.

In figure 3, the middle term is the subject of both premises: AII-3, **Datisi**, IAI-3 **Disamis**, EIO-3 **Ferison**, AOA-3 **Bokardo**.

In figure 4, the middle term is the predicate of the major premise and the subject of the minor premise: AEE-4, **Camenes**, IAI-4, **Dimaris**, and EIO-4, **Fresison**.

If you think it's daunting to remember this list, remember that this is the list for the more limited Modern Square of Opposition. classical logicians memorized all the forms that were considered valid under the Traditional Square of Opposition! For many centuries, it was common practice to present formal disputations in classical syllogistic form, and it was considered a sign of being a learned man if you could use the mnenumonic "on the fly". *Datisi! My form is valid!*

However, it is actually not necessary to remember this list today. Today, we simply look back at it to understand the beginnings of formal logic, and to appreciate how classical scholars approached the subject.

## Further Resources

If you feel you have mastered this section, then first try out this Categorical Syllogism Program

http://www.roninabox.com/venndiagram.html

Then further test your knowledge of syllogisms here:

http://duniho.com/fergus/sillysyllogisms.html

Those following the Course in Logic 101 should proceed to the next section: Disjunctive and Hypothetical Syllogisms

## References

- Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.