Deductive Logic/Categorical Sentence Schemata

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Categorical Sentence Schemata[edit]

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[edit]

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[edit]

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
Aai1.png
Eae2.png
Iai3.png
Aee4.png
Eae1.png
Aee2.png
Aii3.png
Iai4.png
Aii1.png
Eio2.png
Oao3.png
Eio4.png
Eio1.png
Aoo2.png
Eio3.png

Examples[edit]

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[edit]

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.

1

All men are mortal
Socrates is a man
Socrates is mortal

Valid
Invalid

2

Some Greeks are logicians
some logicians are tiresome
some Greeks are tiresome.

Valid
Invalid