Categorical Syllogism

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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.[1] 
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 sentences. Each of the three sentences will be called a proposition.[define 1]

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[edit]

A categorical proposition is a sentence. And like all sentences, 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" 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

Examples include the following:
"Is" in "Socrates is mortal"
"Throw" in "All throwers throw something"
"Can" in "Virtually all sparrows can fly"

The Quantifier
The extend 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"
"Virtually all" in "Virtually all sparrows can fly"

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

Quantifier     Subject     Copula     Predicate

All

   

S

   

is

   

P

All

   

throwers

   

throw

   

something

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


Minor/Major Premise and Term[edit]

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. [2]

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. [3]

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.



Five rules of Categorical Syllogisms[edit]

On top of the five rules on Minor, Major Terms, the following are additional rules that must be met for Categorical Syllogism:

1) There must be only three terms in a syllogism
2) Conclusion will follow the weaker premise
3) No conclusion follows two negative premises
4) No conclusion follows from two simple particular premises
5) No negative conclusions follows from two affirmative premises

1) There must be only three terms in a syllogism
Namely the minor term, major term and middle term.

2) Conclusion will follow the weaker premise
Consider the following example:

All wicked persons will face judgment
Some humans are wicked
Some humans will face judgment

3) No conclusion follows two negative premises
Consider the following example:

No mammals are fish
No fish can fly
Therefore ???
  • No conclusion can be derived from the example. Reason: There is no relationship between the two premises.

4) No conclusion follows from two simple particular premises

  • A simple particular means 'some' not more than half

5) No negative conclusions follows from two affirmative premises

  • You can't say "Yes,Yes" in your premises and then say "No" in your conclusion.
  • Reference to Rule 4: Two positive premises will yield a positive conclusion as the weakest link will still be positive.


Types of Proposition[edit]

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

Definitions[edit]

  1. A proposition in this case will be a statement/sentence that affirms or deny something.

References[edit]

  1. Howard Kahane, Logic and Philosophy: A Modern Introduction [Belmont: Wadsworth Publishing Co., 1990], p.270. cited in Jordana Wiener, Aristotle's Syllogism: Logic Takes Form, accessed 25 Oct 2008, http://www.perseus.tufts.edu/GreekScience/Students/Jordana/LOGIC.html
  2. Stock, St. George William Joseph (1850 - ?), Sep 2004, Deductive Logic, accessed 26 Oct 2008, http://www.gutenberg.org/etext/6560
  3. History of Logic (Origins of logic in the West > Aristotle > Syllogisms), accessed 27 Oct 2008, Britannica Online Encyclopedia

Further reading[edit]

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)