Categorical Syllogism

• Preface

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Note to students, similar materials are covered in the course Deductive Logic/Categorical Sentence Schemata. Students may find that study of the topic in one course reinforces the learning in the other course.

“ He is a true fugitive who flies from reason. ” — Marcus Aurelius

One needs the necessary knowledge in order to make the correct decisions. Likewise, one needs the necessary logic skills, in order to make a correct reasoning.

Aristotle is one of the forefathers of modern Logic and Philosophy. Aristotelian Logic, also known as Categorical Syllogism or Term Logic, may well be the earliest works of Formal Logic.

A Categorical Syllogism is modernly defined as

a particular kind of argument containing three categorical propositions, two of them premises, one a conclusion. 

Aristotle contemplating a bust of Homer by Rembrandt van Rijn

A categorical proposition is of the type "This S is P" and "This man is a man", no 'if', no 'but' and no 'either or'. There are other forms of syllogisms in use. Other examples include Disjunctive Syllogism, Hypothetical Syllogism and Polysyllogism. We will only be discussing on Categorical Syllogism in this article (unless otherwise mentioned).

The following is an example of a syllogism:

Socrates is a man.

All men are mortal.

Socrates is mortal.

A syllogism will be made up of 3 propositions. Each of the three propositions will have a truth value that is either true or false. No other values are allowed. Human awareness is NOT needed to make a proposition true or false. Truth value is the absolute value whether you know about it or not. That is, you might not know whether a proposition is true or false right now, but may find out later on its value.

Socrates is a man.	→ PROPOSITION 1
All men are mortal.	→ PROPOSITION 2
Socrates is mortal.	→ PROPOSITION 3

The first two of the three propositions are premises and the last one is a conclusion.

Socrates is a man.	→ PREMISE
All men are mortal.	→ PREMISE
Socrates is mortal.	→ CONCLUSION

Structure of a Categorical Proposition

A categorical proposition is an IDEA or concept expressed by a declarative sentence. Propositions are NOT sentences. We express ideas in our minds by sentences. In English grammar a declarative sentence is a type of sentence that can affirm or deny something about reality. Declarative sentences indicate something can be either true or false in reality. If two different sentences express the same idea there is only one proposition expressed. So if we can express the idea that dogs are mammals in English, we can ALSO use the same idea in other languages using different words and different sentences. Even in the English language we can express the same idea with different sentences: you are fired and you are terminated express the same idea and therefore are the same proposition. We don't count every sentence as a proposition if the same idea is expressed. You would be wrong to think propositions are literal sentences because now you would have to count each sentence as a different proposition each time a sentence uses different words-- even-though you express the same idea to another person. Would you have a new proposition to say a dog is a mammal in Spanish, French, German, etc.? No, in that case the SAME PROPOSITION is expressed regardless of the words or word order of a sentence. A declarative sentence, which is used to usually expresses a proposition, can be split up into 4 main grammatical parts: the Quantifier, Subject Term, the Copula and the Predicate Term.

The Subject term
The Subject is the "main noun" in a proposition. It is the main argument of the whole proposition, the actor in the sentence. The Subject can be thought of as the "What we are talking about".

Examples include the following:

"Socrates" in "Socrates is mortal"

"Throwers" in "All throwers throw something"

"Sparrows" in "Virtually all sparrows can fly"

The Predicate term
The Predicate tells us something about the Subject. The Predicate can be thought as the "What we are talking about of the Subject".

Examples include the following:

"Mortal" in "Socrates is mortal"

"Something" in "All throwers throw something"

"Fly" in "Virtually all sparrows can fly"

The Copula
Word or set of words that connect the Subject and Predicate There are only a few words that can be used as cupolas, and they are: are, is, and am.

Examples include the following:

"Is" in "Socrates is mortal"

"are" in "All throwers are people who throw something"

In the example "Virtually all sparrows can fly," "can" is not a cupola. We can add in "t.w." which stands for "that which" after the cupola we insert, which in this case is "are." This makes the word "are" fit smoothly into the sentence. We should also get rid of the word "virtually" to make it clear which type of proposition we are using, and because the word "virtually" means most but not all, we use the quantifier "some". The new example would be "Some sparrows are that which fly" of "Some sparrows are t.w. can fly"

The Quantifier
The extent or number of the subject. (E.g. All, some, none)

Examples include the following:

"All" in "All throwers throw something"

"No" in "No fish can fly"

As mentioned before in "Virtually all sparrows are t.w. fly," Virtually all is not a cupola. We make "Virtually all" into an I statement of "Some" because "some" is a quantifier.

And here is how it looks like when we put them together:

Quantifier		Subject		Copula		Predicate
All		S		is		P
All		throwers		are t.w.		throw something

And that will be the structure of a simple Categorical Proposition.

Minor/Major Premise and Term

As explained above, a syllogism is made up of 3 propositions, with 2 being premises and 1 as the conclusion. Of the two premises, one will be the minor premise, whereas the other will be a major premise.

In order to differentiate a minor premise from a major premise, we shall first take a look at the conclusion. It has been defined in the last section that a conclusion, being a proposition, will have a subject and a predicate. Although we have already defined a Subject and Predicate, we must also be aware that the Subject and Predicate are also known as terms.

We can think of a term as a boundary. In Deductive Logic, written by St. George, he defined a term as

the same thing as a name or noun. A name is a word, or collection of words, which serves as a mark to recall or transmit the idea of a thing, either in itself or through some of its attributes.

As we have seen, there will always be 2 terms in a Categorical Proposition (Subject and Predicate). Therefore, the conclusion of a syllogism will have a Subject and a Predicate as well. Here are two rules to take note of:

1. The Subject of a conclusion will be the Minor Term of the syllogism.
2. The Predicate of a conclusion will be the Major Term of the syllogism.

A syllogism is made up of 2 premises and 1 conclusion. So how do we differentiate between one premise from the other? Simple, take a look at that following two rules:

3. The Premise in which the Minor Term appears will be called the Minor Premise.
4. The Premise in which the Major Term appears will be called the Major Premise.

But that's not all. A syllogism is actually made up of 3 terms. The third term, or the Middle Term, can be thought of as a term used to link the two premises together in forming the conclusion. Here is how Britannica Online Encyclopedia define the 3 terms.

The subject and predicate of the conclusion each occur in one of the premises, together with a third term (the middle) that is found in both premises but not in the conclusion.

This brings us to a fifth and final rule.

5. The Middle Term will appear in both premises but not in the conclusion.

The following table attempts to summarize the above.

			Minor Term		Middle Term		Major Term
	Minor Premise		O		O
	Major Premise				O		O
	Conclusion		O				O

Let us the following syllogism as an example:

Socrates is a man.

All men are mortal.

Socrates is mortal.

The conclusion of this syllogism is "Socrates is mortal".
The subject here is "Socrates", which is also the minor term. "Socrates" appeared within the premise "Socrates is a man" making it the minor premise in this syllogism.

Socrates is a man. ← Minor premise

All men are mortal.

Socrates is mortal.

The predicate of the conclusion will be "mortal", thus the second proposition, "All men are mortal", will be the major premise.

Socrates is a man.

All men are mortal. ← Major premise

Socrates is mortal.

The middle term will appear in both the minor and major premises, but not in the conclusion. Therefore, the middle term in this syllogism will be "man/men".

Socrates is a man.

All men are mortal.

Socrates is mortal.

To summarize:

		Minor Term	Middle Term	Major Term
Minor Premise	"Socrates is a man"	Socrates	Man	-
Major Premise	"All men are mortal"	-	Man	Mortal
Conclusion	"Socrates is mortal"	Socrates	-	Mortal

One must take note that the middle term will not always come after the minor term and before the major term (as in the example above). Rather that the placement of the middle term in the example is only one of the many figures. Figure refers to placement of the middle term in the premises. There are four distinct figures (1-4). Mood is a concept that describes the premise --this is discussed later down below in detail. For now we need to know there are four distinct MOODS (A, E, I and O). Every standard form categorical syllogism has a MOOD and Figure. Argument validity can be determined by MOOD and FIGURE also because this is a FORMAL aspect regardless what the argument is about. This is how the name FORMAL LOGIC came about. Please note an argument may be valid in one mood and figure and that same argument is INVALID in a different mood and figure. Mood and Figure of a syllogism can make a difference between a syllogism being valid or invalid.

Term Distribution

Quick reference: The standard form to represent an A type of proposition is to use the letters S (for the subject term), P (for the predicate term) and M (for the middle term). So we can shorten this to types of propositions:

    A type propositions are shortened to "All S are P."
    E type propositions are shortened to "No S are P."
    I-type propositions are shortened to "Some S are P." 
    O type propositions are shortened to "Some S are not P".

Obviously you will need to substitute S, P and M for real words to make grammatical sentences out of this. The form does not change even-though the subject matter of the sentences can change.

With that said, the distribution of a term refers to if an entire class or set is being referred to or not. In each case of a proposition we either have distributed terms or we don't. Here is a run down of what is distributed among the four types of propositions:

    A type propositions can distribute only the subject term.  A type propositions distribute their subject term whatever that happens to be.

for example, All Women are human beings indicates the entire classification of the subject is a member of the classification human being. It would be wrong to say all human beings are women. The order in which the terms are in may matter. Another example, all dogs are mammals. This says all dogs fall into the classification of the predicate mammals. Nothing is said of the entire class of the predicate. The predicate may have other members besides what is listed in a premise in front of us. Be warned not to jump to conclusions here.

    An E type of proposition distributes both the subject term and the predicate terms.

for example, No cats are dogs. Here we are referring to two classes and we are told both classes are not compatible. That is, the proposition expresses one class is not inside the other class. There are no parts of the class that intersect if we were to diagram this with a picture. No S is P eliminates two classes from joining together and here the order of the subject and predicate do not make a difference. I can swap the predicate and subject term positions and still have the same truth value: No dogs are cats is still a true proposition; No P are S is just as true as No S are P.

    An I type of proposition does not distribute any terms.

for example, Some men are married. no entire class is referred to exclude or include an entire set of subjects or predicate. All we know in this context is that Some means AT LEAST one S exists and that S is also in the predicate class whatever that might be. The middle term can't be distributed in this type of proposition. Therefore, a syllogism with two I type propositions and a conclusion is automatically invalid.

    An O type proposition only distributes the predicate term.

for example, Some animals are not reptiles. This proposition excludes the possibility of certain types of subjects. The proposition does not exclude All [of a certain subject] animals from being reptiles but at least one. Some here means AT LEAST ONE as it does in an I-type proposition. So an O type proposition excludes the subject term from an entire set of attributes described by the predicate. So we have distribution of the predicate, but not of the subject term. We don't have enough details to also make inferences about the subject. We only go by what we have available in the premise.

Rules of Categorical Syllogisms

The following guidelines can determine validity without using a diagram for a Categorical Syllogism:

1) There must exactly three terms in a syllogism where all terms are used in the same respect & context. That is, no using one word in two different contexts in the same argument. For instance using the term man to represent all humans in one premise and man to represent a single individual later on in a different premise. This would be the fallacy of four terms.

2) The subject term and the predicate term ought to be a noun or a noun clause. The predicate should not be an adjective or an adverb. That means the sentence should be detailed as possible. No withholding information being purposely vauge or ambigious to cause a person to draw the wrong conclusion on purpose. No ending premises with adverbs or adjectives. When that occurs you must make those words into a noun or noun clause by adding details. This is usually done when arguments are not in standard categorical form. To place a non-formal argument in standard categorical form, you need to add details; you should turn adjective or adverb into concrete nouns or noun clauses. This eliminates ambiguity,vagueness and a person from backing out of a claim. This is called technically adding PARAMETERS. This helps expose deceptive tactics in the argument. This step is ignored when you get to Mathematical logic (symbolic logic).

3) The middle term must be distributed at least once in the premises or the argument is invalid. The fallacy of Undistributed Middle. For example, no conclusion follows from two affirmative particular premises because the middle term is not distributed. This means the link between the premises we have used cannot guarantee our conclusion.

4) If a term is distributed in the conclusion that same term must be distributed in a premise or the argument is invalid. This would be the fallacy of Illicit major or Illicit minor depending on the premise where the term in the conclusion is distributed but not in the premise. So if the term is not distributed in the major premise but the same term is distributed in the conclusion this would be the fallacy of illicit Major.

5) If the conclusion is negative then there must also be a negative premise.

6) No conclusion follows two negative premises. This would be the fallacy of exclusive premises.

7) No negative conclusions follows from two affirmative premises.

Types of Proposition

All forms of Propositions can exist in one of 4 different types. These 4 types are denoted by the code letters A,E,I,O. These code letters are derived from the 2 Latin vowels affirmo and nego.

Code Type	Name	English	Example
Type A	Universal Affirmative	All S is P	All birds have wings
Type E	Universal Negative	No S is P	No birds have gills
Type I	Particular Affirmative	Some S is P	Some birds can fly
Type O	Particular Negative	Some S is not P	Some birds cannot fly

Type A proposition

Type A - Universal Affirmative proposition
All of the subject will be distributed in the class defined by the predicate.

Example:

All birds have wings

Type E proposition

Type E - Universal Negative proposition
None of the subject will be distributed in the class defined by the predicate.

Example:

No birds have gills

Type I and O proposition

Type I - Particular Affirmative proposition
Some of the subject will be distributed in the class defined by the predicate.

Example:

Some birds can fly

Type O - Particular Negative proposition
Some of the subject will not be distributed in the class defined by the predicate.

Example:

Some birds cannot fly

Further reading

Norman L. Geisler, 1990, Come, Let Us Reason: An Introduction to Logical Thinking, Baker Academic

Context above is heavily referenced from this book --Jestermeister 11:34, 27 October 2008 (UTC)