Unit 2: Kinds of Proposition

Kinds of Proposition
Traditional Classification (Categorical Propositions)
- The Four-fold Scheme (A, E, I, O)
- Distribution of Terms
Modern Classification of Propositions
Square of Opposition (Traditional / Aristotelian)
- The Relationships
- Table of Inferences

Kinds of Proposition

A proposition is the "building block" of an argument. It is a statement that asserts or denies something and can be either true or false. In logic, we classify propositions to understand their structure.

Traditional Classification (Categorical Propositions)

Aristotelian logic is built on Categorical Propositions. These are propositions that relate two classes (or categories) of things.
A "class" is a collection of all objects that have some common property (e.g., the class of "dogs," the class of "mammals").

A categorical proposition asserts that a class (the Subject class, S) is either included in or excluded from another class (the Predicate class, P).

The standard form of a categorical proposition has four parts:

Quantifier: "All," "No," or "Some." (Tells us "how many" of the subject class).
Subject Term (S): The class being talked about.
Copula: "are" or "are not." (The linking verb).
Predicate Term (P): The class being related to the subject.

Example: "All [Quantifier] dogs [Subject] are [Copula] mammals [Predicate]."

The Four-fold Scheme (A, E, I, O)

Propositions are classified in two ways:

By Quality: Is it Affirmative (includes) or Negative (excludes)?
By Quantity: Is it Universal (talks about the *whole* subject class) or Particular (talks about *part* of the subject class)?

This gives us the four standard forms:

Type	Vowel	Form	Example	Quantity	Quality
Universal Affirmative	A	All S are P	All cats are mammals.	Universal	Affirmative
Universal Negative	E	No S are P	No cats are reptiles.	Universal	Negative
Particular Affirmative	I	Some S are P	Some cats are black.	Particular	Affirmative
Particular Negative	O	Some S are not P	Some cats are not black.	Particular	Negative

Mnemonic: The vowels come from the Latin words "AffIrmo" (I affirm) and "nEgO" (I deny).

A and I are Affirmative.
E and O are Negative.

Distribution of Terms

This is a crucial concept for understanding inference and fallacies.

A term is Distributed (D) if the proposition makes a claim about ALL members of the class referred to by that term.
A term is Undistributed (U) if the proposition makes a claim about only SOME members of that class.

Proposition	Form	Subject (S)	Predicate (P)
A	All S are P	Distributed (D)	Undistributed (U)
(e.g., "All dogs are mammals." This tells us something about all* dogs, but not about all mammals. The class of "mammals" is larger and we are not talking about all of them.)*
E	No S are P	Distributed (D)	Distributed (D)
(e.g., "No dogs are reptiles." This tells us something about all* dogs (they are all excluded) and all reptiles (they are all excluded).)*
I	Some S are P	Undistributed (U)	Undistributed (U)
(e.g., "Some dogs are brown." This tells us about some* dogs and some brown things. It does not make a claim about all dogs or all brown things.)*
O	Some S are not P	Undistributed (U)	Distributed (D)
(e.g., "Some dogs are not brown." This doesn't talk about all dogs, but it does talk about the entire class of brown things, stating that the "some dogs" in question are excluded from that whole class.)

Mnemonic for Distribution: "AsEbInOp" (A-S, E-Both, I-None, O-P).
A simpler one: Universals (A, E) distribute their Subjects. Negatives (E, O) distribute their Predicates.

Modern Classification of Propositions

Modern logic (post-Aristotle) classifies propositions differently, focusing on how they are combined rather than just S-P relationships.

Simple Proposition: A proposition that contains no other proposition as a component. (e.g., "The cat is on the mat.") This is similar to a categorical proposition.
Compound Proposition: A proposition that contains two or more simple propositions, joined by "logical connectives."
- Conjunctive: "P and Q" (e.g., "The cat is on the mat and the dog is in the garden.")
- Disjunctive: "P or Q" (e.g., "I will have tea or I will have coffee.")
- Hypothetical (or Conditional): "If P, then Q" (e.g., "If it rains, then the ground will be wet.")
General Proposition: This is the modern version of universal propositions, using quantifiers. (e.g., "For all x, if x is a human, then x is mortal.")

Square of Opposition (Traditional / Aristotelian)

The Square of Opposition is a diagram that shows the logical relationships between the four standard forms of categorical propositions (A, E, I, O) when they have the *same subject and predicate terms*.

Crucial Assumption (Boolean vs. Aristotelian): The Traditional/Aristotelian square assumes that the classes we are talking about *actually exist*. (e.g., "All unicorns have horns" implies that unicorns exist). Modern (Boolean) logic does not make this "existential import" assumption, which invalidates some of these relationships.

[Diagram Placeholder: The Square of Opposition]

A square with A, E, I, O at the corners.
A (top left) --- Contraries --- E (top right)
| (Subalternation) | (Subalternation)
I (bottom left) -- Subcontraries -- O (bottom right)
Diagonals (A to O, E to I) are Contradictories.

The Relationships

Contradictories (A and O; E and I):
- They have the opposite truth value. They can never both be true, and they can never both be false.
- If A is True, O *must* be False.
- If A is False, O *must* be True. (Same for E and I).
Contraries (A and E):
- They can never both be True.
- They *can* both be False.
- If A is True, E *must* be False.
- If A is False, E is Undetermined (it could be true or false).
Subcontraries (I and O):
- They can never both be False.
- They *can* both be True.
- If I is False, O *must* be True.
- If I is True, O is Undetermined.
Subalternation (A and I; E and O):
- This is the relationship between a universal and its corresponding particular.
- Truth flows downwards: If A is True, I *must* be True. (If "All cats" are mammals, then "Some cats" are mammals).
- Falsity flows upwards: If I is False, A *must* be False. (If "Some cats are reptiles" is false, then "All cats are reptiles" must also be false).

Table of Inferences (Assuming Aristotelian View)

If this is TRUE...	A is...	E is...	I is...	O is...
A ("All S are P") is True	True	False (Contrary)	True (Subaltern)	False (Contradictory)
E ("No S are P") is True	False (Contrary)	True	False (Contradictory)	True (Subaltern)
I ("Some S are P") is True	Undetermined	False (Contradictory)	True	Undetermined
O ("Some S are not P") is True	False (Contradictory)	Undetermined	Undetermined	True

If this is FALSE...	A is...	E is...	I is...	O is...
A ("All S are P") is False	False	Undetermined	Undetermined	True (Contradictory)
E ("No S are P") is False	Undetermined	False	True (Contradictory)	Undetermined
I ("Some S are P") is False	False (Subaltern)	True (Contradictory)	False	True (Subcontrary)
O ("Some S are not P") is False	True (Contradictory)	False (Subaltern)	True (Subcontrary)	False