IMMEDIATE INFERENCES FOR CATEGORICAL STATEMENTS
 

When any two statements are:

1. CONTRADICTORIES [A & O; E & I], they always have opposite truth values:

(a) when one is TRUE, we validly deduce the other is FALSE;

(b) when one is FALSE, we validly deduce the other is TRUE.

2. CONTRARIES [A & E], they cannot be TRUE at the same time, but they can either be FALSE at the same time or have opposite values:

(a) when one is TRUE, we validly deduce the other is FALSE;

(b) when one is FALSE, we cannot validly deduce the other's truth value.

3. SUBCONTRARIES [I & O], they can never be FALSE at the same time, but they can either be true at the same time or have opposite values: (a) when one is FALSE, we validly deduce the other is TRUE;

(b) when one is TRUE, we cannot validly deduce truth value.

4. SUPERAltern [A of I; E of O] & SUBAltern [I of A; O of E],

(a) when SUPERaltern is TRUE, its SUBaltern must be TRUE;

(b) when SUPERaltern is FALSE, value of the SUBaltern is not validly deducible.

(c) when SUBaltern is TRUE, value of SUPERaltern is not validly deducible;

(d) when SUBaltern is FALSE, we validly deduce the FALSITY of the SUPERaltern.

5. EQUIVALENT STATEMENTS, such as validly deducible obverses, converses, and contrapositives (and combinations), always have the same truth value.

(a) when a statement is TRUE, its validly deduced, equivalent statement is also TRUE.

(b) when a statement is FALSE, its validly deduced, equivalent statement is also FALSE.

6. NONEQUIVALENT STATEMENTS, the truth value of one can be used to determine the value of the other, if one or more of the above relationships (1-4) holds between them; otherwise the value of the other cannot be determined without further evidence or a valid deductive proof with true premises.
 

GIVEN: CONTRAD. CONTRARY SUBALT. CONVERSE OBVERSE CONTRAP.

'Undet.' means 'undeterminable, given this premise'

A--true: O--false E--false I--true undet. ] true true

A--false: O--true E--undet. I--undet. Undet. false false
 

E--true: I--false A--false O--true true true undet.
 

E--false: I--true A--undet. O--undet. false false undet.
 

SUBCONTRARY SUPERALT.

I--true: E--false O--undet. A--undet. true true undet.
 

I--false: E--true O--true A--false false false undet.
 

O--true: A--false I--undet. E--undet. undet. true true
 

O--false: A--true I--true E--false undet. false false

BEGINNING STATEMENT: CONVERSE [valid for E, I]--SWITCH S- & P-terms
 

A: All S are P All P are S --invalid

ALSO: Some P are S --a valid converse by limitation

E: No S are P No P are S

I: Some S are P Some P are S

O: Some S are not P Some P are not S --invalid
 

OBVERSE [valid for all]--CHANGE Quality & REPLACE P-term with complement

A: All S are P No S are nonP [ things that are not P]

E: No S are P All S are nonP [things that are not P]

I: Some S are P Some S are not nonP [ things that are not P]

O: Some S are not P Some S are nonP [ things that are not P]
 

CONTRAPOSITIVE [valid for A, O]--SWITCH S- & P-terms & REPLACE with complements

A: All S are P All nonP are nonS

E: No S are P No nonP are nonS --invalid

ALSO: Some nonP are not nonS --a valid contrapositive by limitation

I: Some S are not P Some nonP are nonS --invalid

O: Some S are not P Some nonP are not nonS
 

CONVERSE OF OBVERSE [valid for A,O]

A: All S are P No nonP are S

E: No S are P All nonP are S

I: Some S are not P Some nonP are not S

O: Some S are not P Some nonP are S
 

OBVERSE OF CONVERSE [valid for E, I]

A: All S are P No P are nonS --- invalid

E: No S are P All P are nonS

I: Some S are not P Some P are not nonS

O: Some S are not P Some P are nonS ---invalid
 

* * * * * * * * * * * *

DENIALS AND CONTRADICTORIES

To deny a statement, we add "It is not the case that" in front.

The denial of a statement is equivalent to its contradictory.
 

For A and O statements:

"It is not the case that all S are P" == "Some S are not P."

"It is not the case that some S are not P" == "All S are P."
 

For E and I statements:

"It is not the case that no S are P" == "Some S are P."

"it is not the case that some S are P" == "No S are P."
 

COMPLEMENT TERMS:

Add or subtract "non-" (or similar prefix like 'un-') to/from the term to get its complement.

OR add or subtract 'not' to/from the clause or phrase modifying the main noun in the term.