AristotelianSyllogisms

Aristotle's Complete Syllogisms

Terms

The A's, B's, C's, etc. in the following arguments are called the "terms"
A term is a universal, not an individual.

"Horse" or "Human" or "dog" or "desk" can be a term
Socrates cannot

In an Aristotelian syllogism, there are two premises and one conclusion
Each premise has two terms
There is one term that the two premises share in common.

That term is called the "middle" term, usually B
The other two are called the "extremes", usually A and C
The subject term of the conclusion is called the "minor" term, usually A
The predicate term of the conclusion is called the "major" term, usually C

First Figure Syllogisms:

A is the predicate and B (the middle term) is the subject of the first premise: AxB
B (the middle term) is the predicate and C is the subject of the second premise. BxC

AaB, BaC, and therefore AaC
- Nickname: "Barbara"
AeB, BaC, and therefore Aec

Nickname: "Celarent"

AaB, BiC, and therefore AiC

Nickname: "Darii"

AeB, BiC, and therefore AoC

Nickname: "Ferio"

also rans...

AaB, BaC, and therefore AiC

Nickname: "Barbari"

simply a case of Barbara: it is obvious that if AaC, then AiC

AeB, BaC, and therefore AoC

Nickname: "Celaront"

simply a case of Celarent: it is obvious that if AeC, then AoC

The nicknames are codes to help one remember what the pattern is: they were invented in the Middle Ages.

Note that in the nicknames, the vowels are the important parts: the first two vowels identify the premises, and the third identifies the conclusion. Note that the first letters are unique within the 4 chief first-figure syllogisms: there is only one B, only one C, only one D, and only one F. Also Note that s, p, m, and c do not occur after the first letters of the nicknames.

Second and Third Figure Syllogisms:

Second Figure

A is the predicate, B is the subject of the first premise: AxB
A is the predicate, C is the subject of the second premise: AxC

AaB, AeC, and therefore BeC

Nickname: "Camestres"

AeB, AaC, and therefore BeC

Nickname: "Cesare"

AeB, AiC, and therefore BoC

Nickname: "Festino"

AaB, AoC, and therefore BoC

Nickname: "Baroco"
completion thru impossibility

AeB, AaC, and therefore BoC

Nickname: "Cesaro"
simply a case of Cesare: it is obvious that if BeC, then BoC

AaB, AeC, and therefore Boc

Nickname: "Camestros"
simply a case of "Camestres": it is obvious that if BeC, then BoC

Third Figure

A is the predicate, B is the subject of the first premise: AxB
C is the predicate, B is the subject of the second premise: CxB

AiB, CaB and therefore AiC

Nickname: "Disamis"

AaB, CiB, and therefore AiC

Nickname: "Datisi"

AoB, CaB, and therefore AoC

Nickname: "Bocardo"
completion thru impossibility

AeB, CiB, and therefore AoC

Nickname: "Ferison"

AaB, CaB, and therefore AiC

Nickname: "Darapti"

AeB, CaB, and therefore AoC

Nickname: "Felapton"

Aristotle manipulates these syllogisms into first order syllogisms as follows: this is called 'conversion.'
The first letter identifies the first-figure syllogism that will result from the conversions and switches indicated by 's,' 'p,' and 'm.'
Note that in the nicknames, the vowels identify the premises and conclusions. An 's' tells the user to apply a simple conversion to the 'i' or 'e' that precedes it.

Simple Conversions:

AeB implies that BeA

example: "No dog is human" implies that "no human is a dog."

AiB implies that BiA

example: "Some pets are fish" implies that "some fish are pets"

A 'p' tells the user to apply an accidental conversion to the 'a' or 'i' that precedes it.

Accidental conversion:

AaB implies that AiB

example: "all pigs are mammals" implies that "some pigs are mammals"

IS THE FOLLOWING LEGIT???
AeC implies AoC?

"A belongs to no C" implies that "A does not belong to some C."
"no pigs are fish" implies that "some pigs are not fish"

An 'm' tells the user to switch the order of the premises (switching does not affect the result).
The letter 'c' after the first or second vowel (i.e. after the first or second premise) indicates that the syllogism must be proven by 'completion through impossibility' (i.e. by 1. assuming the denial of the conclusion, 2. combining it with the first or second premise and then 3. discovering that that implies the denial of the other premise (which is assumed)).

So, take "Datisi" for example:

AaB, CiB, and therefore AiC

Nickname: "Datisi"
Being a mammal belongs to every dog.
Being called 'Rover' belongs to some dogs.
Therefore Being a mammal belongs to some things called 'Rover.'

"Datisi" starts with a D and therefore it will be manipulated into a "Darii" (AaB, BiC, and therefore AiC)
There's an s after the second premise (Datisi), and therefore, we are to change CiB into BiC

after that change, we have the premises of a first figure syllogism: AaB, BiC

So, we have:

AaB, BiC, and therefore AiC
which is the same form as a Darii

Take a BaRoCo for another example (i.e. we are following what A. does in 27a37-b1)

AaB: Having fins belongs to every thing that is a trout.
AoC: Having fins does not belong to some dogs.
BoC: Therefore being a trout does not belong to some dogs.

It has a c in it: therefore it is proven by a Proof thru impossibility:

the contradictory opposite of BoC (some C is not B) is BaC (every C is B):
first, assume that contradictory opposite BaC (every C is B)

BaC: Being a trout belongs to every dog.
AaB: having fins belongs to every trout.
that's the start of a Barbara (the order of the premises does not matter), so:
AaC: Having fins belongs to every dog.
But that is incompatible with AoC, which was assumed.
We know that either BoC or BaC is right: both cannot be right, and one must be wrong: either being a trout belongs to every dog or being a trout does not belong to some dogs. They are contradictory opposites.
Therefore, since we are assuming that the premises are correct, and we know that the contradictory of the conclusion is incompatible with a premise, we know that the conclusion is right.

Aristotle did not explicitly classify, but nonetheless noticed, a further set of valid syllogisms that involve AxB and BxC as premises and CxA as the conclusion. Medieval logicians called them the 'Fourth Figure.' Aristotle's follower Theophrastus systematically proved their validity and provided counterexamples to reject the other 'Fourth Figure' possibilities.

Fourth Figure

'Bramantip,'

Example: All Romans are Italians, All Italians are Europeans, so some Europeans are Romans.

'Camenes,'
'Dimaris,'
'Fesapo,' and
'Fresison.'

Conversions of some of the syllogisms above:

Nickname: "Camestres" (2nd figure)

AaB, AeC, and therefore BeC
which changes to
AaB, CeA, and therefore CeB

because s tells us to change the e's

now we can change that into
CeA, AaB, and therefore CeB

we do that, because m tells us to switch order of premises

Which is the same as AeB, BaC, and therefore AeC, a Celarent

we can switch labels A, B, and C any time as long as we switch them in both premises and the conclusion consistently
here we call the C's A's, the A's B's, and the B's C's.
we are NOT thereby switching terms: we are simply calling what used to be called C A, and what used to be called A B, and what used to be called B C. It does not at all affect the logic: we could call them anything at all, as long as we do so consistently to preserve the pattern.

Now it's a Celarent, as the code tells us

Nickname: "Cesare" (2nd figure)

AeB, AaC, and therefore BeC

apply s to get

BeA, AaC, and therefore BeC
Which is the same as AeB, BaC, and therefore AeC
Now it's a Celarent, as the code tells us.

Nickname: "Festino" (2nd figure)

AeB, AiC, and therefore BoC
BeA, AiC, and therefore BoC

AeB, BiC, and therefore BoC

switch labels in all steps

Now it's a Ferio, as the code tells us.

Nickname: "Disamis"

AiB, CaB and therefore AiC
BiA, CaB and therefore CiA

CaB, BiA and therefore CiA

AaB, BiC and therefore AiC

switch labels

Now it's a Darii, as the code tells us.

Nickname: "Datisi"

AaB, CiB, and therefore AiC
AaB, BiC, and therefore AiC

Now it's a Darii, as the code tells us.

Nickname: "Ferison"

AeB, CiB, and therefore AoC
AeB, BiC, and therefore AoC
Which is a Ferio, as the code tells us.

Nickname: "Darapti"

AaB, CaB, and therefore AiC
AaB, CiB, and therefore AiC

AaB, BiC, and therefore AiC

this is legit, but I don't see how the code tells us to convert the second premise again, this time with a simple conversion

And now it's a Darii, as the code tells us.

Nickname: "Felapton"

AeB, CaB, and therefore AoC
AeB, CiB, and therefore AoC

AeB, BiC, and therefore AoC

again, a simple conversion of CiB into BiC: I guess the code included that and I didn't realize it.

A Ferio, as the code tells us

Fourth Figure:
'Bramantip,'

AaB, CaB, and therefore CiA
CaB, AaB, and therefore CiA

AaB, CaB, and therefore AiC

switch consistently

That's a Darapti, and we know that's valid.

'Dimaris,'

AiB, CaB, and therefore CiA
AiB, CaB, and therefore AiC

This is a Disamis, and we know that's valid

'Camenes,'

AaB, CeB, and therefore CeA
CeB, AaB, and therefore CeA

AeB, CaB, and therefore AeC

switch labels

AeB, CaB, and therefore AoC

AeC implies AoC IS THIS LEGIT?

and AeB, CaB, and therefore AoC is a Felapton, which we know is valid.

'Fesapo,'

AeB, CaB, and therefore CoA
BeA, CaB, and therefore CoA

BeA, CiB, and therefore CoA

CiB, BeA, and therefore CoA

AiB, BeC, and therefore AoC

label switch

BiA, BeC, and therefore AoC

AiB, AeC, and therefore BoC

label switch

AiB, CeA, and therefore BoC

simple

AiB, CeA, and therefore CoB

CeA, AiB, and therefore CoB

AeB, BiC, and therefore AoC

which is a Ferio

'Fresison'

AeB, CiB, and therefore CoA

BeA, BiC, and therefore CoA

AeB, AiC, and therefore CoB

label switch

AeB, CiA, and therefore CoB