Deductive Logic/Categorical Sentence Schemata

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

Consider these arguments:

And

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

Such categorical propositions can take on one of the following four forms (known as moods), traditionally know by the letters AEIO (modern mnemonics are suggested for each letter):


 * 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 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

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

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

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

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

Now consider:

This is of the general form:

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

Assignment
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. {All men are mortal Socrates is a man  Socrates is mortal + Valid - Invalid
 * type=""}
 * Valid by Modus Barbari, Figure I, AAI
 * Valid by Modus Barbari, Figure I, AAI

{Some Greeks are logicians some logicians are tiresome  some Greeks are tiresome. - Valid + Invalid
 * type=""}
 * Invalid argument: the tiresome logicians might all be Romans (for example).
 * Invalid argument: the tiresome logicians might all be Romans (for example).