Logic Reference - J. Barry Krasner, MD, FACP

Classical Logic

Categorical Proposition Types
Type	Quantity	Quality	Exposition	Distribution	Diagram Truth table
A	Universal	Affirmative	All S are P	Subject
A	Universal	Affirmative	All S are P	Subject	T-F-?-?
E	Universal	Negative	No S are P	Subject Predicate
E	Universal	Negative	No S are P	Subject Predicate	F-T-T-?
I	Particular	Affirmative	Some S are P	-
I	Particular	Affirmative	Some S are P	-	T-?-?-?
O	Particular	Negative	Some S are not P	Predicate (No P is in the class referred to by "some S")
O	Particular	Negative	Some S are not P	Predicate (No P is in the class referred to by "some S")	?-T-?-?

Diagram legend
+	Exists by proposition definition
	Does not exist by proposition definition
?	Existence cannot be determined by proposition definition

Classic categorical propositions have the structure: [subject quantifier] [subject] [copula] [predicate]
In the classic categorical proposition, "some" means "at least one, possibly all"
By existential import, there are no non-empty sets (members of subject and predicate classes must exist)

Immediate Inferences: Eduction
Inference	Process	Validity
		A	E	I	O
		All S is P	No S is P	Some S is P	Some S is not P
Simple ConvErsIon	Interchange subject and predicate	No All P is S	Yes No P is S	Yes Some P is S	No (Some P is not S)
per AccidEns Conversion	Interchange subject and predicate Invert quantity	Yes Some P is S	Yes Some P is not S	No	No
Obversion	Change quality, invert predicate	Yes No S is non-P	Yes All S is non-P	Yes Some S is not non-P	Yes Some S is non-P
ContrApOsition	Invert subject and predicate, Exchange subject and predicate	Yes All non-P is non-S	By limitation Some non-P is non-S	No (Some non-P is non-S)	Yes Some non-P is not non-S

Immediate Inferences: Opposition

	Oppositional relationships
Relationship	Definition	Proposition pairs
Contradictories	One must be true and the other false	A, O I, E
Contraries	Both cannot be true, but both can be false	A, E
Subcontraries	Both cannot be false, but both can be true	I, O
Subalternates	The superalternate implies the subalternate	A, I E, O

If	is	Then
		A	E	I	O
		is
A	True	True	False	True	False
A	False	False	???	???	True
E	True	False	True	False	True
E	False	???	False	True	???
I	True	???	False	True	???
I	False	False	True	False	True
O	True	False	???	???	True
O	False	True	False	True	False

Syllogism Structure
Figure	1	2	3	4
Major Premise	M - P	P - M	M - P	P - M
Minor Premise	S - M	S - M	M - S	M - S
Conclusion	S - P	S - P	S - P	S - P

Syllogism Validity Rules

Relating to structure
1. There are only three terms in a syllogism (by definition).
2. The middle term is not in the conclusion (by definition).
3. The major premise contains the predicate (major term) of the conclusion (by definition).
4. The minor premise containes the subject (minor term) of the conclusion (by definition).
Relating to premises irrespective of conclusion or figure
1. No inference can be made from two negative premises, i.e., at least one premise must be affirmative. (Fallacy of Exclusive Premises)
2. No inference can be made from two particular premises, i.e., at least one premise must be universal.
Relating to propositions irrespective of figure
1. If one premise is negative, the conclusion must be negative. (Fallacy of Drawing an Affirmative Conclusion from a Negative Premise)
2. If both premises are affirmative, the conclusion must be affirmative -
  A negative conclusion requires a negative premise (Fallacy of Drawing a Negative Conclusion from Affirmative Premises)
3. In extensional logic, if both premises are universal, the conclusion must be universal. (Existential Fallacy)
  This is not a necessary condition for validity if empty classes are disallowed
4. If one premise is particular, the conclusion must be particular.
Relating to the distribution of terms
1. The middle term must be distributed in at least one premise (Fallacy of the Undistributed Middle)
2. A predicate distributed in the conclusion must be distributed in the major premise. (Fallacy of the Illicit Major)
3. A subject distributed in the conclusion must be distributed in the minor premise. (Fallacy of the Illicit Minor)
4. The quantity of a term cannot become greater in the conclusion.
5. The number of terms distributed in the premises must be greater than the number distributed in the conclusion

Notes on Validity Rules

The rules in (1) determine whether a syllogism has valid form. If so, Rules (2a), (3ab) and (4abc) define all logically valid syllogisms. Rule (3c) is only necessary when existential import is not assumed.
Enforcement of rules (2a), (3abc) and (4abc) yields 15 valid syllogisms. If rule (3c) is not enforced, there are an additional 9 valid syllogisms. These include the so-called "weak" syllogisms, each of which can also be derived from an otherwise valid syllogism with a universal conclusion by subalternation of that conclusion (A becomes I or E becomes O). Graphic derivation
Rules (2b) and (3d) are corollaries of the above rules which do not identify additional invalid syllogisms. Graphic illustration
Rule (4e) follows from rules (4abc) - each term distributed in the conclusion must be distributed in the corresponding premise and the middle term must also be distributed in at least one premise.

Derivation of valid classical syllogisms

		Major premise
		A	E	I	O
Minor premise	A
	E		2a		2a
	I
	O		2a		2a

Existential Fallacy		Conclusion
Existential Fallacy		A	E	I	O
Premises	AA		3b		3b
	AE	3a		3a
	AI		3b		3b
	AO	3a		3a
	EA	3a		3a
	EI	3a		3a
	IA		3b		3b
	IE	3a		3a
	II		3b		3b
	IO	3a		3a
	OA	3a		3a
	OI	3a		3a

Existential Fallacy		Figure
Existential Fallacy		1	2	3	4
Mood	AAA	Barbara	4a	4c	4c
	AAI	Barbari	4a	Darapti	Bramantip
	AEE	4b	Camestres	4b	Camenes
	AEO	4b	Camestrop	4b	Camenop
	AIA	4c	4a,4c	4c	4a,4c
	AII	Darii	4a	Datisi	4a
	AOE	4b,4c	4c	4b	4a
	AOO	4b	Baroco	4b	4a
	EAE	Celarent	Cesare	4c	4c
	EAO	Celaront	Cesaro	Felapton	Fesapo
	EIE	4c	4c	4c	4c
	EIO	Ferio	Festino	Ferison	Fresison
	IAA	4a	4a	4c	4c
	IAI	4a	4a	Disamis	Dimaris
	IEE	4b	4b	4b	4b
	IEO	4b	4b	4b	4b
	IIA	4a,4c	4a,4c	4a,4c	4a,4c
	III	4a	4a	4a	4a
	IOE	4b,4c	4b,4c	4a,4b	4a,4b
	IOO	4b	4b	4a,4b	4a,4b
	OAE	4a	4b	4c	4b,4c
	OAO	4a	4b	Bocardo	4b
	OIE	4a,4c	4b,4c	4a,4c	4b,4c
	OIO	4a	4b	4a	4b

Corollary rules

Rule 2b		Major premise
Rule 2b		A	E	I	O
Minor premise	A
	E		2a		2a
	I
	O		2a		2a

Rule 3d		Conclusion
Rule 3d		A	E	I	O
Premises	AA		3c		3c
	AE	3a		3a
	AI		3c		3c
	AO	3a		3a
	EA	3a		3a
	EI	3a		3a
	IA		3c		3c
	IE	3a		3a
	II		3c		3c
	IO	3a		3a
	OA	3a		3a
	OI	3a		3a

Rule 2b		Figure
Rule 3d		1	2	3	4
Mood	AAA		4a	3c	3c
	AAI		4a
	AEE	4b		4b
	AEO	4b		4b
	AIA	4c	4a,4c	4c	4a,4c
	AII		4a		4a
	AOE	4b,4c	4c	4b	4a
	AOO	4b		4b	4a
	EAE			4c	4c
	EAO
	EIE	4c	4c	4c	4c
	EIO
	IAA	4a	4a	4c	4c
	IAI	4a	4a
	IEE	4b	4b	4b	4b
	IEO	4b	4b	4b	4b
	IIA	4a,4c	4a,4c	4a,4c	4a,4c
	III	4a	4a	4a	4a
	IOE	4b,4c	4b,4c	4a,4b	4a,4b
	IOO	4b	4b	4a,4b	4a,4b
	OAE	4a	4b	4c	4b,4c
	OAO	4a	4b		4b
	OIE	4a,4c	4b,4c	4a,4c	4b,4c
	OIO	4a	4b	4a	4b

Syllogism Reduction

Cesare		Celarent
No P is M	s	No M is P
All S is M		All S is M
No S is P		No S is P

Camestres				Celarent
All P is M		All P is M	m	No M is S*
No S is M	s	No M is S	m	All P* is M
No S is P	s	No P is S		No P* is S*

Festino		Ferio
No P is M	s	No M is P
Some S is M		Some S is M
Some S is not P		Some S is not P

Mood Figure	Explication	Distributed terms Premises : Conclusion	Rules violated	Validity
AAA1	All M are P All S are M All S are P	2 : 1	-	Valid Barbara
AAE1	All M are P All S are M No S are P	2 : 2	Affirmative premises, Negative conclusion Illicit Major	Invalid
AAI1	All M are P All S are M Some S are P	2 : 0	Existential Fallacy	Conditionally Valid (S exists) Barbari
AAO1	All M are P All S are M Some S are not P	2 : 1	Affirmative premises, Negative conclusion Illicit Major Existential Fallacy	Invalid
AEA1	All M are P No S are M All S are P	3 : 1	Negative premise, affirmative conclusion	Invalid
AEE1	All M are P No S are M No S are P	3 : 2	Illicit Major	Invalid
AEI1	All M are P No S are M Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
AEO1	All M are P No S are M Some S are not P	3 : 1	Illicit Major Existential Fallacy	Invalid
AIA1	All M are P Some S are M All S are P	1 : 1	Illicit Minor	Invalid
AIE1	All M are P Some S are M No S are P	1 : 2	Affirmative premises, Negative conclusion Illicit Major Illicit Minor	Invalid
AII1	All M are P Some S are M Some S are P	1 : 0	-	Valid Darii
AIO1	All M are P Some S are M Some S are not P	1 : 1	Affirmative premises, Negative conclusion Illicit Major	Invalid
AOA1	All M are P Some S are not M All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
AOE1	All M are P Some S are not M No S are P	2 : 2	Illicit Major Illicit Minor	Invalid
AOI1	All M are P Some S are not M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
AOO1	All M are P Some S are not M Some S are not P	2 : 1	Illicit Major	Invalid
EAA1	No M are P All S are M All S are P	3 : 1	Negative premise, affirmative conclusion	Invalid
EAE1	No M are P All S are M No S are P	3 : 2	-	Valid Celarent
EAI1	No M are P All S are M Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
EAO1	No M are P All S are M Some S are not P	3 : 1	Existential Fallacy	Conditionally Valid (S exists) Celaront
EEA1	No M are P No S are M All S are P	4 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
EEE1	No M are P No S are M No S are P	4 : 2	2 negative premises	Invalid
EEI1	No M are P No S are M Some S are P	4 : 0	2 negative premises Negative premise, affirmative conclusion Existential Fallacy	Invalid
EEO1	No M are P No S are M Some S are not P	4 : 1	2 negative premises Existential Fallacy	Invalid
EIA1	No M are P Some S are M All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
EIE1	No M are P Some S are M No S are P	2 : 2	Illicit Minor	Invalid
EII1	No M are P Some S are M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
EIO1	No M are P Some S are M Some S are not P	2 : 1	-	Valid Ferio
EOA1	No M are P Some S are not M All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion Illicit Minor	Invalid
EOE1	No M are P Some S are not M No S are P	3 : 2	2 negative premises Illicit Minor	Invalid
EOI1	No M are P Some S are not M Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
EOO1	No M are P Some S are not M Some S are not P	3 : 1	2 negative premises	Invalid
IAA1	Some M are P All S are M All S are P	1 : 1	Undistributed Middle	Invalid
IAE1	Some M are P All S are M No S are P	1 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IAI1	Some M are P All S are M Some S are P	1 : 0	Undistributed Middle	Invalid
IAO1	Some M are P All S are M Some S are not P	1 : 1	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IEA1	Some M are P No S are M All S are P	2 : 1	Negative premise, affirmative conclusion	Invalid
IEE1	Some M are P No S are M No S are P	2 : 2	Illicit Major	Invalid
IEI1	Some M are P No S are M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
IEO1	Some M are P No S are M Some S are not P	2 : 1	Illicit Major	Invalid
IIA1	Some M are P Some S are M All S are P	0 : 1	Undistributed Middle Illicit Minor	Invalid
IIE1	Some M are P Some S are M No S are P	0 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major Illicit Minor	Invalid
III1	Some M are P Some S are M Some S are P	0 : 0	Undistributed Middle	Invalid
IIO1	Some M are P Some S are M Some S are not P	0 : 1	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IOA1	Some M are P Some S are not M All S are P	1 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
IOE1	Some M are P Some S are not M No S are P	1 : 2	Illicit Major Illicit Minor	Invalid
IOI1	Some M are P Some S are not M Some S are P	1 : 0	Negative premise, affirmative conclusion	Invalid
IOO1	Some M are P Some S are not M Some S are not P	1 : 1	Illicit Major	Invalid
OAA1	Some M are not P All S are M All S are P	2 : 1	Negative premise, affirmative conclusion Undistributed Middle	Invalid
OAE1	Some M are not P All S are M No S are P	2 : 2	Undistributed Middle	Invalid
OAI1	Some M are not P All S are M Some S are P	2 : 0	Negative premise, affirmative conclusion Undistributed Middle	Invalid
OAO1	Some M are not P All S are M Some S are not P	2 : 1	Undistributed Middle	Invalid
OEA1	Some M are not P No S are M All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
OEE1	Some M are not P No S are M No S are P	3 : 2	2 negative premises	Invalid
OEI1	Some M are not P No S are M Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OEO1	Some M are not P No S are M Some S are not P	3 : 1	2 negative premises	Invalid
OIA1	Some M are not P Some S are M All S are P	1 : 1	Negative premise, affirmative conclusion Undistributed Middle Illicit Minor	Invalid
OIE1	Some M are not P Some S are M No S are P	1 : 2	Undistributed Middle Illicit Minor	Invalid
OII1	Some M are not P Some S are M Some S are P	1 : 0	Negative premise, affirmative conclusion Undistributed Middle	Invalid
OIO1	Some M are not P Some S are M Some S are not P	1 : 1	Undistributed Middle	Invalid
OOA1	Some M are not P Some S are not M All S are P	2 : 1	2 negative premises Negative premise, affirmative conclusion Illicit Minor	Invalid
OOE1	Some M are not P Some S are not M No S are P	2 : 2	2 negative premises Illicit Minor	Invalid
OOI1	Some M are not P Some S are not M Some S are P	2 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OOO1	Some M are not P Some S are not M Some S are not P	2 : 1	2 negative premises	Invalid
AAA2	All P are M All S are M All S are P	2 : 1	Undistributed Middle	Invalid
AAE2	All P are M All S are M No S are P	2 : 2	Affirmative premises, Negative conclusion Undistributed Middle	Invalid
AAI2	All P are M All S are M Some S are P	2 : 0	Undistributed Middle Existential Fallacy	Invalid
AAO2	All P are M All S are M Some S are not P	2 : 1	Affirmative premises, Negative conclusion Undistributed Middle Existential Fallacy	Invalid
AEA2	All P are M No S are M All S are P	3 : 1	Negative premise, affirmative conclusion	Invalid
AEE2	All P are M No S are M No S are P	3 : 2	-	Valid Camestres
AEI2	All P are M No S are M Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
AEO2	All P are M No S are M Some S are not P	3 : 1	Existential Fallacy	Conditionally Valid (S exists) Camestrop
AIA2	All P are M Some S are M All S are P	1 : 1	Undistributed Middle Illicit Minor	Invalid
AIE2	All P are M Some S are M No S are P	1 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Minor	Invalid
AII2	All P are M Some S are M Some S are P	1 : 0	Undistributed Middle	Invalid
AIO2	All P are M Some S are M Some S are not P	1 : 1	Affirmative premises, Negative conclusion Undistributed Middle	Invalid
AOA2	All P are M Some S are not M All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
AOE2	All P are M Some S are not M No S are P	2 : 2	Illicit Minor	Invalid
AOI2	All P are M Some S are not M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
AOO2	All P are M Some S are not M Some S are not P	2 : 1	-	Valid Baroco
EAA2	No P are M All S are M All S are P	3 : 1	Negative premise, affirmative conclusion	Invalid
EAE2	No P are M All S are M No S are P	3 : 2	-	Valid Cesare
EAI2	No P are M All S are M Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
EAO2	No P are M All S are M Some S are not P	3 : 1	Existential Fallacy	Conditionally Valid (S exists) Cesaro
EEA2	No P are M No S are M All S are P	4 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
EEE2	No P are M No S are M No S are P	4 : 2	2 negative premises	Invalid
EEI2	No P are M No S are M Some S are P	4 : 0	2 negative premises Negative premise, affirmative conclusion Existential Fallacy	Invalid
EEO2	No P are M No S are M Some S are not P	4 : 1	2 negative premises Existential Fallacy	Invalid
EIA2	No P are M Some S are M All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
EIE2	No P are M Some S are M No S are P	2 : 2	Illicit Minor	Invalid
EII2	No P are M Some S are M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
EIO2	No P are M Some S are M Some S are not P	2 : 1	-	Valid Festino
EOA2	No P are M Some S are not M All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion Illicit Minor	Invalid
EOE2	No P are M Some S are not M No S are P	3 : 2	2 negative premises Illicit Minor	Invalid
EOI2	No P are M Some S are not M Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
EOO2	No P are M Some S are not M Some S are not P	3 : 1	2 negative premises	Invalid
IAA2	Some P are M All S are M All S are P	1 : 1	Undistributed Middle	Invalid
IAE2	Some P are M All S are M No S are P	1 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IAI2	Some P are M All S are M Some S are P	1 : 0	Undistributed Middle	Invalid
IAO2	Some P are M All S are M Some S are not P	1 : 1	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IEA2	Some P are M No S are M All S are P	2 : 1	Negative premise, affirmative conclusion	Invalid
IEE2	Some P are M No S are M No S are P	2 : 2	Illicit Major	Invalid
IEI2	Some P are M No S are M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
IEO2	Some P are M No S are M Some S are not P	2 : 1	Illicit Major	Invalid
IIA2	Some P are M Some S are M All S are P	0 : 1	Undistributed Middle Illicit Minor	Invalid
IIE2	Some P are M Some S are M No S are P	0 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major Illicit Minor	Invalid
III2	Some P are M Some S are M Some S are P	0 : 0	Undistributed Middle	Invalid
IIO2	Some P are M Some S are M Some S are not P	0 : 1	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IOA2	Some P are M Some S are not M All S are P	1 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
IOE2	Some P are M Some S are not M No S are P	1 : 2	Illicit Major Illicit Minor	Invalid
IOI2	Some P are M Some S are not M Some S are P	1 : 0	Negative premise, affirmative conclusion	Invalid
IOO2	Some P are M Some S are not M Some S are not P	1 : 1	Illicit Major	Invalid
OAA2	Some P are not M All S are M All S are P	2 : 1	Negative premise, affirmative conclusion	Invalid
OAE2	Some P are not M All S are M No S are P	2 : 2	Illicit Major	Invalid
OAI2	Some P are not M All S are M Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
OAO2	Some P are not M All S are M Some S are not P	2 : 1	Illicit Major	Invalid
OEA2	Some P are not M No S are M All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
OEE2	Some P are not M No S are M No S are P	3 : 2	2 negative premises Illicit Major	Invalid
OEI2	Some P are not M No S are M Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OEO2	Some P are not M No S are M Some S are not P	3 : 1	2 negative premises Illicit Major	Invalid
OIA2	Some P are not M Some S are M All S are P	1 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
OIE2	Some P are not M Some S are M No S are P	1 : 2	Illicit Major Illicit Minor	Invalid
OII2	Some P are not M Some S are M Some S are P	1 : 0	Negative premise, affirmative conclusion	Invalid
OIO2	Some P are not M Some S are M Some S are not P	1 : 1	Illicit Major	Invalid
OOA2	Some P are not M Some S are not M All S are P	2 : 1	2 negative premises Negative premise, affirmative conclusion Illicit Minor	Invalid
OOE2	Some P are not M Some S are not M No S are P	2 : 2	2 negative premises Illicit Major Illicit Minor	Invalid
OOI2	Some P are not M Some S are not M Some S are P	2 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OOO2	Some P are not M Some S are not M Some S are not P	2 : 1	2 negative premises Illicit Major	Invalid
AAA3	All M are P All M are S All S are P	2 : 1	Illicit Minor	Invalid
AAE3	All M are P All M are S No S are P	2 : 2	Affirmative premises, Negative conclusion Illicit Major Illicit Minor	Invalid
AAI3	All M are P All M are S Some S are P	2 : 0	Existential Fallacy	Conditionally Valid (M exists) Darapti
AAO3	All M are P All M are S Some S are not P	2 : 1	Affirmative premises, Negative conclusion Illicit Major Existential Fallacy	Invalid
AEA3	All M are P No M are S All S are P	3 : 1	Negative premise, affirmative conclusion	Invalid
AEE3	All M are P No M are S No S are P	3 : 2	Illicit Major	Invalid
AEI3	All M are P No M are S Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
AEO3	All M are P No M are S Some S are not P	3 : 1	Illicit Major Existential Fallacy	Invalid
AIA3	All M are P Some M are S All S are P	1 : 1	Illicit Minor	Invalid
AIE3	All M are P Some M are S No S are P	1 : 2	Affirmative premises, Negative conclusion Illicit Major Illicit Minor	Invalid
AII3	All M are P Some M are S Some S are P	1 : 0	-	Valid Datisi
AIO3	All M are P Some M are S Some S are not P	1 : 1	Affirmative premises, Negative conclusion Illicit Major	Invalid
AOA3	All M are P Some M are not S All S are P	2 : 1	Negative premise, affirmative conclusion	Invalid
AOE3	All M are P Some M are not S No S are P	2 : 2	Illicit Major	Invalid
AOI3	All M are P Some M are not S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
AOO3	All M are P Some M are not S Some S are not P	2 : 1	Illicit Major	Invalid
EAA3	No M are P All M are S All S are P	3 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
EAE3	No M are P All M are S No S are P	3 : 2	Illicit Minor	Invalid
EAI3	No M are P All M are S Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
EAO3	No M are P All M are S Some S are not P	3 : 1	Existential Fallacy	Conditionally Valid (M exists) Felapton
EEA3	No M are P No M are S All S are P	4 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
EEE3	No M are P No M are S No S are P	4 : 2	2 negative premises	Invalid
EEI3	No M are P No M are S Some S are P	4 : 0	2 negative premises Negative premise, affirmative conclusion Existential Fallacy	Invalid
EEO3	No M are P No M are S Some S are not P	4 : 1	2 negative premises Existential Fallacy	Invalid
EIA3	No M are P Some M are S All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
EIE3	No M are P Some M are S No S are P	2 : 2	Illicit Minor	Invalid
EII3	No M are P Some M are S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
EIO3	No M are P Some M are S Some S are not P	2 : 1	-	Valid Ferison
EOA3	No M are P Some M are not S All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
EOE3	No M are P Some M are not S No S are P	3 : 2	2 negative premises	Invalid
EOI3	No M are P Some M are not S Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
EOO3	No M are P Some M are not S Some S are not P	3 : 1	2 negative premises	Invalid
IAA3	Some M are P All M are S All S are P	1 : 1	Illicit Minor	Invalid
IAE3	Some M are P All M are S No S are P	1 : 2	Affirmative premises, Negative conclusion Illicit Major Illicit Minor	Invalid
IAI3	Some M are P All M are S Some S are P	1 : 0	-	Valid Disamis
IAO3	Some M are P All M are S Some S are not P	1 : 1	Affirmative premises, Negative conclusion Illicit Major	Invalid
IEA3	Some M are P No M are S All S are P	2 : 1	Negative premise, affirmative conclusion	Invalid
IEE3	Some M are P No M are S No S are P	2 : 2	Illicit Major	Invalid
IEI3	Some M are P No M are S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
IEO3	Some M are P No M are S Some S are not P	2 : 1	Illicit Major	Invalid
IIA3	Some M are P Some M are S All S are P	0 : 1	Undistributed Middle Illicit Minor	Invalid
IIE3	Some M are P Some M are S No S are P	0 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major Illicit Minor	Invalid
III3	Some M are P Some M are S Some S are P	0 : 0	Undistributed Middle	Invalid
IIO3	Some M are P Some M are S Some S are not P	0 : 1	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IOA3	Some M are P Some M are not S All S are P	1 : 1	Negative premise, affirmative conclusion Undistributed Middle	Invalid
IOE3	Some M are P Some M are not S No S are P	1 : 2	Undistributed Middle Illicit Major	Invalid
IOI3	Some M are P Some M are not S Some S are P	1 : 0	Negative premise, affirmative conclusion Undistributed Middle	Invalid
IOO3	Some M are P Some M are not S Some S are not P	1 : 1	Undistributed Middle Illicit Major	Invalid
OAA3	Some M are not P All M are S All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
OAE3	Some M are not P All M are S No S are P	2 : 2	Illicit Minor	Invalid
OAI3	Some M are not P All M are S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
OAO3	Some M are not P All M are S Some S are not P	2 : 1	-	Valid Bocardo
OEA3	Some M are not P No M are S All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
OEE3	Some M are not P No M are S No S are P	3 : 2	2 negative premises	Invalid
OEI3	Some M are not P No M are S Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OEO3	Some M are not P No M are S Some S are not P	3 : 1	2 negative premises	Invalid
OIA3	Some M are not P Some M are S All S are P	1 : 1	Negative premise, affirmative conclusion Undistributed Middle Illicit Minor	Invalid
OIE3	Some M are not P Some M are S No S are P	1 : 2	Undistributed Middle Illicit Minor	Invalid
OII3	Some M are not P Some M are S Some S are P	1 : 0	Negative premise, affirmative conclusion Undistributed Middle	Invalid
OIO3	Some M are not P Some M are S Some S are not P	1 : 1	Undistributed Middle	Invalid
OOA3	Some M are not P Some M are not S All S are P	2 : 1	2 negative premises Negative premise, affirmative conclusion Undistributed Middle	Invalid
OOE3	Some M are not P Some M are not S No S are P	2 : 2	2 negative premises Undistributed Middle	Invalid
OOI3	Some M are not P Some M are not S Some S are P	2 : 0	2 negative premises Negative premise, affirmative conclusion Undistributed Middle	Invalid
OOO3	Some M are not P Some M are not S Some S are not P	2 : 1	2 negative premises Undistributed Middle	Invalid
AAA4	All P are M All M are S All S are P	2 : 1	Illicit Minor	Invalid
AAE4	All P are M All M are S No S are P	2 : 2	Affirmative premises, Negative conclusion Illicit Minor	Invalid
AAI4	All P are M All M are S Some S are P	2 : 0	Existential Fallacy	Conditionally Valid (P exists) Bramantip
AAO4	All P are M All M are S Some S are not P	2 : 1	Affirmative premises, Negative conclusion Existential Fallacy	Invalid
AEA4	All P are M No M are S All S are P	3 : 1	Negative premise, affirmative conclusion	Invalid
AEE4	All P are M No M are S No S are P	3 : 2	-	Valid Camenes
AEI4	All P are M No M are S Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
AEO4	All P are M No M are S Some S are not P	3 : 1	Existential Fallacy	Conditionally Valid (M exists) Camenop
AIA4	All P are M Some M are S All S are P	1 : 1	Undistributed Middle Illicit Minor	Invalid
AIE4	All P are M Some M are S No S are P	1 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Minor	Invalid
AII4	All P are M Some M are S Some S are P	1 : 0	Undistributed Middle	Invalid
AIO4	All P are M Some M are S Some S are not P	1 : 1	Affirmative premises, Negative conclusion Undistributed Middle	Invalid
AOA4	All P are M Some M are not S All S are P	2 : 1	Negative premise, affirmative conclusion Undistributed Middle	Invalid
AOE4	All P are M Some M are not S No S are P	2 : 2	Undistributed Middle	Invalid
AOI4	All P are M Some M are not S Some S are P	2 : 0	Negative premise, affirmative conclusion Undistributed Middle	Invalid
AOO4	All P are M Some M are not S Some S are not P	2 : 1	Undistributed Middle	Invalid
EAA4	No P are M All M are S All S are P	3 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
EAE4	No P are M All M are S No S are P	3 : 2	Illicit Minor	Invalid
EAI4	No P are M All M are S Some S are P	3 : 0	Negative premise, affirmative conclusion Existential Fallacy	Invalid
EAO4	No P are M All M are S Some S are not P	3 : 1	Existential Fallacy	Conditionally Valid (M exists) Fesapo
EEA4	No P are M No M are S All S are P	4 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
EEE4	No P are M No M are S No S are P	4 : 2	2 negative premises	Invalid
EEI4	No P are M No M are S Some S are P	4 : 0	2 negative premises Negative premise, affirmative conclusion Existential Fallacy	Invalid
EEO4	No P are M No M are S Some S are not P	4 : 1	2 negative premises Existential Fallacy	Invalid
EIA4	No P are M Some M are S All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
EIE4	No P are M Some M are S No S are P	2 : 2	Illicit Minor	Invalid
EII4	No P are M Some M are S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
EIO4	No P are M Some M are S Some S are not P	2 : 1	-	Valid Fresison
EOA4	No P are M Some M are not S All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
EOE4	No P are M Some M are not S No S are P	3 : 2	2 negative premises	Invalid
EOI4	No P are M Some M are not S Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
EOO4	No P are M Some M are not S Some S are not P	3 : 1	2 negative premises	Invalid
IAA4	Some P are M All M are S All S are P	1 : 1	Illicit Minor	Invalid
IAE4	Some P are M All M are S No S are P	1 : 2	Affirmative premises, Negative conclusion Illicit Major Illicit Minor	Invalid
IAI4	Some P are M All M are S Some S are P	1 : 0	-	Valid Dimaris
IAO4	Some P are M All M are S Some S are not P	1 : 1	Affirmative premises, Negative conclusion Illicit Major	Invalid
IEA4	Some P are M No M are S All S are P	2 : 1	Negative premise, affirmative conclusion	Invalid
IEE4	Some P are M No M are S No S are P	2 : 2	Illicit Major	Invalid
IEI4	Some P are M No M are S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
IEO4	Some P are M No M are S Some S are not P	2 : 1	Illicit Major	Invalid
IIA4	Some P are M Some M are S All S are P	0 : 1	Undistributed Middle Illicit Minor	Invalid
IIE4	Some P are M Some M are S No S are P	0 : 2	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major Illicit Minor	Invalid
III4	Some P are M Some M are S Some S are P	0 : 0	Undistributed Middle	Invalid
IIO4	Some P are M Some M are S Some S are not P	0 : 1	Affirmative premises, Negative conclusion Undistributed Middle Illicit Major	Invalid
IOA4	Some P are M Some M are not S All S are P	1 : 1	Negative premise, affirmative conclusion Undistributed Middle	Invalid
IOE4	Some P are M Some M are not S No S are P	1 : 2	Undistributed Middle Illicit Major	Invalid
IOI4	Some P are M Some M are not S Some S are P	1 : 0	Negative premise, affirmative conclusion Undistributed Middle	Invalid
IOO4	Some P are M Some M are not S Some S are not P	1 : 1	Undistributed Middle Illicit Major	Invalid
OAA4	Some P are not M All M are S All S are P	2 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
OAE4	Some P are not M All M are S No S are P	2 : 2	Illicit Major Illicit Minor	Invalid
OAI4	Some P are not M All M are S Some S are P	2 : 0	Negative premise, affirmative conclusion	Invalid
OAO4	Some P are not M All M are S Some S are not P	2 : 1	Illicit Major	Invalid
OEA4	Some P are not M No M are S All S are P	3 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
OEE4	Some P are not M No M are S No S are P	3 : 2	2 negative premises Illicit Major	Invalid
OEI4	Some P are not M No M are S Some S are P	3 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OEO4	Some P are not M No M are S Some S are not P	3 : 1	2 negative premises Illicit Major	Invalid
OIA4	Some P are not M Some M are S All S are P	1 : 1	Negative premise, affirmative conclusion Illicit Minor	Invalid
OIE4	Some P are not M Some M are S No S are P	1 : 2	Illicit Major Illicit Minor	Invalid
OII4	Some P are not M Some M are S Some S are P	1 : 0	Negative premise, affirmative conclusion	Invalid
OIO4	Some P are not M Some M are S Some S are not P	1 : 1	Illicit Major	Invalid
OOA4	Some P are not M Some M are not S All S are P	2 : 1	2 negative premises Negative premise, affirmative conclusion	Invalid
OOE4	Some P are not M Some M are not S No S are P	2 : 2	2 negative premises Illicit Major	Invalid
OOI4	Some P are not M Some M are not S Some S are P	2 : 0	2 negative premises Negative premise, affirmative conclusion	Invalid
OOO4	Some P are not M Some M are not S Some S are not P	2 : 1	2 negative premises Illicit Major	Invalid

Quantified Categorical Propositions (Venn)

Nomenclature	EQU (U*) Equivalence	SUB (A*) Subset	INT (I*) Intersect	SUP (Y*) Superset	DIS(E*) Disjoint	η	O*	ω
Nomenclature	Toto-total affirmative	Toto-partial affirmative	Parti-total affirmative	Parti-partial affirmative	Toto-total negative	Toto-partial negative	Parti-total negative	Parti-partial negative
Verbal form	All A is all B	All A is some B	Some A is some B	Some A is all B	No A is any B	All A is not some B	Some A is not any B	Some A is not some B
Graphic form
Venn diagram
Truth table	T-F-F-?	T-F-T-?	T-T-T-?	T-T-F-?	F-T-T-?	F-F-T-?	F-T-F-?	F-F-F-?
Classical equivalents	A		O			These cannot exist in traditional logic, since they deny the existence of one or both of the classes to which they refer
Classical equivalents	I				E
Conversion	EQU All A is all B	SUP Some A is all B	INT Some A is some B	SUB All A is some B	DIS No A is any B

Determination of Validity of Venn Syllogisms

Existential import is enforced
A valid Venn syllogism has 3 terms - 2 premises and a conclusion. Since propositions are always convertible, premises are not designated major and minor.
There is a middle term, but since propositions are always convertible, terms are not designated major and minor
A middle term must be present in each premise
There are 4 Figures, as per classical syllogisms. Since propositions are always convertible, all syllogisms can be reduced to First Figure.
A valid syllogism must have a compatible pair of donor-recipient premises based upon the middle term (Y to the right). One non-middle term is designated X and the other is designated Z.
Corollary: Nothing can be inferred from an I* premise and a non-U* premise (the 2 never constitute a compatible premise pair)
Corollary: Nothing can be inferred from 2 E* premises (the 2 never constitute a compatible premise pair)
If one premise is U*, the conclusion is of the same type as the remaining premise
Corollary: If one premise is E*, the conclusion must be E*

Proposition type	Corresponding propositions	Donor Status	Recipient Status
U*:	All Y is all X All Z is all Y All X is all Y All Y is all Z	Inclusive Donor Can donate to U, A, I, Y, E*	Inclusive Recipient Can receive from U, A, I, Y, E*
A*:	All Y is some X Some Z is all Y Some X is all Y All Y is some Z	Exclusive Donor Can donate to U*	Standard Recipient Can receive from U, Y
I*:	Some Y is some X Some Z is some Y Some X is some Y Some Y is some Z	Exclusive Donor Can donate to U*	Exclusive Recipient Can receive from U*
Y*:	Some Y is all X All Z is some Y All X is some Y Some Y is all Z	Standard Donor Can donate to U, A, E*	Exclusive Recipient Can receive from U*
E*:	No Y is any X No Z is any Y No X is any Y No Y is any Z	Exclusive Donor Can donate to U*	Standard Recipient Can receive from U, Y

Graphic Derivation of Validity of Venn Syllogisms

		Premise 2
		U*	A*	I*	Y*	E*
Premise 1	U*	U*	A*	I*	Y*	E*
	A*	A*
	I*	I*
	Y*	Y*
	E*	E*
		Without regard to figure or conclusion

	Figure
Premises	1	2	3	4
A-A	A*	Invalid	Invalid	Y*
A-Y	Invalid	Y*	A*	Invalid
A-E	Invalid	E*	Invalid	E*
Y-A	Invalid	A*	Y*	Invalid
Y-Y	Y*	Invalid	Invalid	A*
Y-E	E*	Invalid	E*	Invalid
E-A	E*	E*	Invalid	Invalid
E-Y	Invalid	Invalid	E*	E*

De Morgan: Formal Logic (DMFL)

A, E, I, O are defined as in classical logic
Terms are designated by upper case letters, e.g., X, Y, Z
The corresponding inverse terms (not-X, not-Y, not-Z) are designated by lower case letters (x, y, z)
A', E', I', O' are obtained by inverting both subject and predicate of A, E, I, O
e.g. A' is defined as "All not-X is not-Y"
A and A', etc. are called contranominals
Copula are abbreviated as follows:
) → A, . → E, (no space) → I, : → O
Existential import is explicitly unenforced by the I' proposition, "Some not-X is not-Y".
The text states that this proposition can be read as "there are things in the universe which are neither Xs nor Ys."

Proposition Types	Classical nomenclature	DMFL nomenclature
A ↔ E	Contraries	Subcontraries
I ↔ O	Subcontraries	Supercontraries

A
	X)Y	X.y	y)x
	X)Y	Equivalence implies existential import
A'		x)y
E		X.Y Y.X
E'		x.y y.x
I		XY
I'		xy
O		X:Y
O'		x:y

De Morgan Propositions

De Morgan propositions can be partially but not totally reconciled with truth tables if the empty set (not-X, not-Y) result is ignored. In this scheme, A)B, A.B, AB and A:B refer to A, E, I, O propositions respectively. a is defined as not-A. On a 2-set Venn diagram which displays the empty set, a universal proposition (A, A', E, E') will have exactly 1 positive and 1 negative area, for a total of 12 possible diagrams.

Proposition type	Associated truth-equivalent propositions	Truth table	De Morgan equivalence assertions	Empty set indifference
A	X)Y - X.y	T-F-?-?	y)x	X)Y - X.y - y)x - y.X
A'	x)y - x.Y	T-?-F-T	Y)X	x)y - x.Y - Y)X - Y.x
E	X)y - X.Y - Y)x - Y.X	F-T-T-?	(same)	X)y - X.Y - Y)x - Y.X
E'	x)Y - x.y	?-?-T-F	y)X	(same)
??	Y)X - Y.x (A')	T-?-F-?
**	y)X - y.x	?-T-?-F	Unique truth table
??	y)x - y.X (A)	T-F-?-T

Truth values AB-AB-A'B-A'B'	Name	Symbolization	Verbal equivalents	Corresponding classical propositions			Corresponding Venn propositions
T-T-T-T	Tautology	⊤ 1			I	O	I*
T-T-T-F	Inclusive disjunction OR	A ∨ B A + B ¬A ⊃ B	Either A or B, possibly both				I*
T-T-F-T	Converse implication	A ⊂ B A ← B ¬A ⊃ ¬B A ∨ ¬B	A is a necessary condition for B				Y*
T-T-F-F	Subject	A	-				Y*
T-F-T-T	Material implication	A → B A ⇒ B A ⊃ B ¬A ∨ B	If A then B A is a sufficient condition for B	A			A*
T-F-T-F	Predicate	B	Not B				A*
T-F-F-T	Biconditional XNOR IFF	A ↔ B A ⇔ B A ≡ B	A if and only if B A is a necessary and sufficient condition for B				U*
T-F-F-F	Conjunction AND	A · B A ∧ B A & B AxB AB	Both A and B				U*
F-T-T-T	Alternative denial NAND	A∣B A↑B A ⊃ ¬B	Not both A and B	E		O	E*
F-T-T-F	Exclusive disjunction XOR	A⊕B A⊻B	Either A or B but not both	E			E*
F-T-F-T	Inverse of predicate	¬B ˜B !B	Not B
F-T-F-F	Material nonimplication	A ⊅ B A ↛ B A ∧ ¬B	A but not B
F-F-T-T	Inverse of subject	¬A ˜A !A
F-F-T-F	Converse nonimplication	A ⊄ B A ↚ B ¬A ∧ B	B but not A
F-F-F-T	Joint denial NOR	A ↓B ¬A ∧ ¬B	Neither A nor B
F-F-F-F	Contradiction	⊥ F

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