# The Figures

The Figures

| First Figure | Second Figure | Third Figure |
| Predicate Subject | Predicate Subject | Predicate Subject |

| Premise | a b | a b | a c |
| Premise | b c | a c | b c |
| Conclusion | a c | b c | a b |

Aristotle calls the term which is the predicate of the conclusion the major term and the term which is the subject of the conclusion the minor term.

The premise containing the major term is the major premise, and the premise containing the minor term is the minor premise.

Aristotle then systematically investigates all possible combinations of two premises in each of the three figures.

For each combination, he either demonstrates that some conclusion necessarily follows or demonstrates that no conclusion follows.

The results he states are correct.

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Form Mnemonic Proof
Aab, Abc ⊢ Aac Barbara Perfect
every b is an a, every c is a b ⊢

Eab, Abc ⊢ Eac Celarent Perfect
no b is an a, every c is a b. ⊢no c is an a

Aab, Ibc ⊢ Iac Darii Perfect; also by impossibility, from Camestres
every b is an a, some c is b. ⊢ some c is a.

Eab, Ibc ⊢ Oac Ferio Perfect; also by impossibility, from Cesare
no b is a., some c is b. ⊢ some c is not a.

SECOND FIGURE
Eab, Aac ⊢ Ebc Cesare (Eab, Aac)→(Eba, Aac)⊢CelEbc
no b is a., every c is an a. / no a is b., every c is an a. ⊢ no c is a b.

Aab, Eac ⊢ Ebc Camestres (Aab, Eac)→(Aab, Eca)=(Eca, Aab)⊢CelEcb→Ebc
every b is an a., no c is an a. / every b is an a., no a is c. ⊢ no b is c.

Eab, Iac ⊢ Obc Festino (Eab, Iac)→(Eba, Iac)⊢FerObc
no b is a., some c is a. / no a is b., some c is a. ⊢ some c is not b.

Aab, Oac ⊢ Obc Baroco (Aab, Oac +Abc)⊢Bar(Aac, Oac)⊢ImpObc
every b is a., some c is not a. /

if a belongs to every b / every b is a.
but does not belong to some c, / some c are not a
it is necessary for b not to belong to some c. / some c is not b.

Baroco (27a36-b1)

“Next, if M belongs to every N but does not belong to some X, it is ne
cessary for N not to belong to some X. For if it belongs to every X and M is also predicated of every N, then it is necessary for M to belong to every X: but it was assumed not to belong to some. And if M belongs to every N but not to every X, then there will be a deduction that N does not belong to every X. (The demonstration is the same.)”

if every N is M
and some X is not M.
some X is not N.

if every X is M
and every N is M
then every X is M.

For if it belongs to every X and M is also predicated of every N, then it is necessary for M to belong to every X: but it was assumed not to belong to some.

every N is M.
not every X is M.
not every X is N.

And if M belongs to every N but not to every X, then there will be a deduction that N does not belong to every X.

if a belongs to every b / every b is a.
but does not belong to some c, / some c are not a
it is necessary for b not to belong to some c. / some c is not b.

every b is an a. -> some a are b.
some c are not a
some c are not b

THIRD FIGURE
Aac, Abc ⊢Iab Darapti (Aac, Abc)→(Aac, Icb)⊢DarIab
all c is a., all c is b. / all c is a., some b is c. ⊢ some b is a.

Eac, Abc ⊢ Oab Felapton (Eac, Abc)→(Eac, Icb)⊢FerOab
no c is a., all c is b. / no a is c., all c is b. ⊢ some b are not a.

Iac, Abc ⊢ Iab Disamis (Iac, Abc)→(Ica, Abc)=(Abc, Ica)⊢DarIba→Iab
some c is a., every c is b. / some a is c., every c is b. ⊢ some a is b.

Aac, Ibc ⊢ Iab Datisi (Aac, Ibc)→(Aac, Icb)⊢DarIab
every c is a., some c is b. / every c is a., some b is c. ⊢ some b is a.

Oac, Abc ⊢ Oab Bocardo (Oac, +Aab, Abc)⊢Bar(Aac, Oac)⊢ImpOab
some c is not a. all c is b. / some a is not c. , all c is b.

Eac, Ibc ⊢ Oab Ferison (Eac, Ibc)→(Eac, Icb)⊢FerOab
no c is a., some c is b. ⊢ some b are not a.