# 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:
   | 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
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
   | 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. All those valid forms are listed here, combining the four moods and figures. All other forms are invalid and therefore fallacies.

### Summary Table

| Figure I | Figure II | Figure III | Figure IV |
| A: All M are P A: All S are M A: All S are P For Example: All animals are mortal All men are animals All men are mortal | E: No P is M A: All S is M E: No S is P For Example: No reptiles have fur All snakes are reptiles No snakes have fur | I: Some M are P A: All M are S I: Some S are P For Example: Some mugs are beautiful All mugs are useful things Some useful things are beautiful | A: All P are M E: No M are S E: No S are P For Example: All horses have hooves No humans have hooves No humans are horses |
| E: No M are P A: All S are M E: No S are P | A: All P is M E: No S is M E: No S is P | A: All M are P I: Some M are S I: Some S are P | I: Some P are M A: All M are S I: Some S are P |
| A: All M are P I: Some S are M I: Some S are P | E: No P is M I: Some S are M O: Some S are not P | O: Some M are not P A: All M are S O: Some S are not P | E: No P are M I: Some M are S O: Some S are not P |
| E: No M are P I: Some S are M O: Some S are not P | A: All P are M O: Some S are not M O: Some S are not P | E: No M are P I: Some M are S O: Some S are not P | |

### Diagrams

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

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.

### 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.