Symbolic Logic

Proposition Types
Code	Quantity	Quality	Exposition	Symbolic Notation	Example	Simple ConvErsIon Switch S, P	per AccidEns conversion Replace Quantity	ContrApOsition (1) Switch S, P (2) Replace S, P with ~S, ~P	Obversion (1) Replace Quality (2) Replace P with ~P
A	Universal	Affirmative	All S are P	∀x S(x)→P(x)	All birds have wings	invalid	Some P are S	All non-P are non-S	No S are non-P
E	Universal	Negative	No S are P	∀x S(x)→ ¬P(x)	No birds have fins	No P are S	Some P are not S	Invalid	All S are non-P
I	Particular	Affirmative	Some S are P	∃x (S(x)∧P(x))	Some birds swim	Some P are S	Does not exist	None	Some S are not non-P
O	Particular	Negative	Some S are not P	∃x (S(x)∧¬P(x))	Some birds do not fly	invalid	Does not exist	Some non-P are not non-S	Some S are non-P

Rules of the syllogism

There are only three terms in a syllogism (by definition).
The middle term is not in the conclusion (by definition).
The major premise contains the predicate (major term) of the conclusion (by definition).
The minor premise containes the subject (minor term) of the conclusion (by definition).
At least one premise must be universal.
At least one premise must be affirmative.
If one premise is particular, the conclusion is particular.
If one premise is negative, the conclusion is negative.
If both premises are affirmative, the conclusion is affirmative.
In extensional logic, if both premises are universal, the conclusion is universal.
The quantity of a term cannot become greater in the conclusion.
The middle term must be universally quantified in at least one premise

Syllogistic Moods
Major Premise	A	A	A	A	E	E	E	E	I	I	I	I	O	O	O	O
Minor Premise	A	E	I	O	A	E	I	O	A	E	I	O	A	E	I	O
Valid	Y	Y	Y	Y	Y	N *	Y	N *	Y	Y	N †	N †	Y	N *	N †	N *†
* At least premise must be affirmative † At least one premise must be universal

Syllogistic Figures
	Perfect		Imperfect
Figure	1st		2nd		3 d		4th
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

Figure	1st	2nd	3 d	4th
Valid Syllogisms	AAA Barbara	AEE Camestres	AAI Darapti	AAI Bramantip
	EAE Celarent	EAE Cesare	IAI Disamis	AEE Camenes
	AII Darii	EIO Festimo	AII Datisi	IAI Dimaris
	EIO Ferio	AOO Baroco	EAO Felapton	EAO Fesapo
			OAO Bocardo	EIO Fresison
			EIO Ferison

Mnemonic interpretation

The mnemonic of the second, third and fourth figures have first letter: 'B', 'C', 'D' or 'F.' This indicates to which first figure mood the syllogism is reduced to, to prove its validity

If a 'c' follows an 'o' it indicates that the proof must be done reductio ad impossibile.
s: exchange the subject and predicate of the preceding proposition, without changing the quantity.
p: exchange the subject and predicate of the preceding proposition should be exchanged, while changing the quantity.
If an 's'(or 'p') follows the first or second vowel, this indicates that the proposition corresponding to this vowel is simply (per accidens) converted in the proof.
If an 's'(or 'p') follows the last vowel this implies that the conclusion will be derived by making a simple (or per accidens) conversion of the conclusion of the first figure syllogism that is used in the reduction.
An 'm' indicates a rearrangement of the premises to satisfy the order: major premise, minor premise and conclusion. Exchange the major and minor premises.

Camestres
All P is M
No S is M
No S is P
s - minor
All P is M
No M is S
No S is P
m
No M is P'
All S' is M
No P' is S'
s - conc
Celarent
No M is P'
All S' is M
No S' is P'

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

Festimo
No P is M
Some S is M
Some S is not P
s - major
Ferio
No M is P
Some S is M
Some S is not P

Baroco
All P is M
Some S is not M
Some S is not P
c - Contradict minor Set as conclusion
Barbara
All P is M
All S is P
All S is M