Deductive Logic/Categorical Sentence Schemata

Students may wish to supplement this lesson by studying this lecture on Categorical Syllogism.

Consider these arguments:

Therefore:

All men are mortal Socrates is a man Socrates is mortal

And

Therefore:

No reptiles have fur All snakes are reptiles No snakes have fur

Each of these is an example of a syllogism—a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.

In these sentence structures each argument is formed by the major premise, the minor premise, and the conclusion such as:

Major premise: All men are mortal

Minor premise: Socrates is a man

Conclusion: Socrates is mortal

Four letters, traditionally know by the letters AEIO (modern mnemonics are suggested for each letter), describe the type of each premise or conclusion. These letters are:

A – All S are P (All – Universal Affirmative, "All women are mortal")

E – No S are P (Exclusion – Universal negative, "No women are immortal")

I – Some S are P (Inclusion – Particular Affirmation, "Some women are philosophers")

O – Some S are not P (Other – Particular Negative, "Some women are not philosophers")

The mood of every categorical proposition is represented by three of these letters, in a specific order: the first letter names the type of the syllogism’s major premise, the second letter names the type of minor premise, and the third letter names the type of its conclusion.^[1]

The resulting argument forms can also be arranged into any of the following four figures, using these abbreviations:

M – Middle term

S – Subject – Minor Term Variable

P – Predicate of the Conclusion

Figure I	Figure II	Figure III	Figure IV
MP SM SP	PM SM SP	MP MS SP	PM MS SP

Although there are 256 possible propositions, only 15 of those are valid categorical propositions.^[2] All those valid forms are listed here, combining the four moods and figures. All other forms are invalid and therefore fallacies.

Alternatively, these are described in the Wikiversity course on Syllogisms.

The 15 valid forms can be illustrated using diagrams. They are show in this table along with their classical names:

Figure I	Figure II	Figure III	Figure IV
Examples	Examples	Examples	Examples
Examples	Examples	Examples	Examples
Examples	Examples	Examples	Examples
Examples	Examples	Examples

We can use this table to determine if the following argument is valid or invalid.

Therefore:

No reptiles have fur Some reptiles are snakes Some snakes do not have fur

This argument matches the last row in the Figure III column (Modus Ferison), so it is valid.

Now consider:

Therefore:

Some dogs are mammals Some cats are mammals Some dogs are cats

This is of the general form:

Therefore:

Some S are M Some P are M Some S are P

This argument is invalid because it does not match any of the valid forms listed above.

Complete the following quiz. Please select answers to each of the following questions by determining if each syllogism is valid or invalid. Press the "Submit" button after you have made your selections.