A form of deduction logic that connects two ideas through a third one: "All men are mortal; Socrates is a man; Socrates is mortal" (this form is called Darii), "All coal is black: some rocks are not black; some rocks are not coal" (this form is called Baroko)
The syllogism is composed of major premise; minor premise: conclusion.
The premises and the conclusion are traditionally affirmations or negations of some property of a set of objects. If the set of object is qualified by "all" ("all dogs", "mammals", "all the black things"), the premise is called universal.
If the premise is existential ("some grey things", "there exist some animals that...") it is called particular.
Valid syllogisms are classified according to four figures:
- first figure: M P; S M: S P All men are mortal; Socrates is a man; Socrates is mortal (in this Darii M="all men", P="mortal entities" and S="Socrates")
- second figure: P M; S M; S P All cats are mammals; some animals are not mammals: some animals are not cats
- third figure: M P; M S; S P No snake lives in Sardinia; All snakes are reptiles; some reptiles do not live in Sardinia (this is a Felapton)
- fourth figure: P M; M S; S P Some bows are wood objects; all wood objects are organic: some bows are organic (this is a Dimaris)
What changes here is the position of
M the middle term.
Syllogisms can be constructed in the four figures according to these rules:
- There are only three terms in a syllogism (by definition).
- The middle term is not in the conclusion (by definition).
- The quantity of a term cannot become greater in the conclusion
- The middle term must be universally quantified in at least one premise - you cannot deduct anything from particular observations.
- At least one premise must be affirmative.
- If one premise is negative, the conclusion is negative.
- If both premises are affirmative, the conclusion is affirmative.
- At least one premise must be universal.
- If one premise is particular, the conclusion is particular.
The syllogism appears in Aristotle's works, but it was formalized in the Middle Ages by the Scholastic philosophers according to an elegant scheme.
The five hexameter verses that follow contain the symbolic names (called moods) of all the different syllogistic figures:
“Barbara, Celarent, Darii, Ferioque, prioris.
Cesare, Camestres, Festino, Baroko, secundae.
Tertia, Darapti, Disamis, Datisi, Felapton,
Bokardo, Ferison, habet. Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
The parts in italics are there just to make the verses work; they are not part of the names of the figures.
The vowels mean the following:
- A universal affirmative.
- E universal negative.
- I particular affirmative.
- O particular negative.
For example, Darii is DARII, that's to say
universal affirmative + particular affermative = particular affirmative
All cats are mammals + Beppo is a cat = Beppo is a mammal
Another example, Bokardo is BOKARDO, that's to say
particular negative + universal affirmative = particular negative
some men are not white + all men are mammals; some mammals are not white
What follows is the detailed listing of all the valid syllogism. Notice that you can very well create invalid figures. For example IEI or IEA would be invalid ("Some foos are bars; no bars are quux: ??? "... no conclusion is allowed).
Notice also that the order of the parts inside each member of the syllogism is important; Darii differs from Datisi precisely in that respect.
Barbara all M is P; all S is M: all S is P
Celarent no M is P; all S is M: no S is P
Darii all M is P; some S is M: some S is P
Ferio no M is P; some S is M: some S is not P
Cesare no P is M; all S is M: no S is P
Camestres all P is m; no S is M: no S is P
Festino no P is M; some S is M: some S is not P
Fakofo all P is M; some s is not M: some S is not P
Baroko all P is M; some s is not M: some S is not P
Darapti all M is P; all M is S: some S is P
Disamis some M is P; all M is S: some S is P
Datisi all M is P; some M is S: some S is P
Felapton no M is P; all M is S: some S is not P
Dokamok some M is not P; all M is S: some S is not P
Bokardo some M is not P; all M is S: some S is not P
Ferison no M is P: some M is S: some S is not P
Bramantip all P is M; all M is S: some S is P
Camenes all P is M; no M is S: no S is P
Dimaris some P is M; all M is S: some S is P
Fesapo no P is M; all M is S: some S is not P
Fresison no P is M; some M is S: some S is not P
In greater and painful detail, notice that the initial of all the words shows to which of the first forms (Barbara, Celarent, Darii, Ferio) the syllogism should be reduced. For example, Bokardo (also written Bocardo) should be reduced to Barbara.
How does the reduction take place? This is where the other letters come into play.
- S stands for simpliciter, "simply". This means "invert the subject and the predicate and don't touch the quantifier". Only E (universal negative) and I (particular positive) can be converted simpliciter.
For example "No cats are reptiles" converts to "No reptiles are cats", and "Some cats are female" converts to "Some females are cats". Datisi converts to Darii simpliciter.
- P means per accidens, "special case". Only A (universal positive) and E (universal negative) can be converted this way. For example "All cats are mammals" converts to "Some mammals are cats" (notice that we are going from "For all X that are cats, X is a mammal" to "There exists one mammal M, for which M is a cat") via a quantifier change.
For example Felapton converts to Ferio per accidens (refer to the above table), by converting the universal affirmative to a particular affirmative in the second member.
- M means "exchange the minor and major premise"
- C is to per absurdum
There is an issue here with modern logic, about deriving particular truths from two universal truths. In particular consider Bramantip: "all P is M; all M is S: some S is P". For example, "All Romans are Italians; all Italians are Europeans; some Europeans are Romans". This is all good until nonexistent things enter the scene... for example All unicorns are furry; all furry things are cute; some cute things are unicorns appears to prove the existence of unicorns (since we know that cute things do exist).
Read about this in the excellent article "In defense of Bramantip" at http://www.friesian.com/syllog.htm. The idea is that you cannot really use empty sets as terms in a syllogism; and the syllogism has been around much longer than the idea of the empty set.
A good resource for playing around is the
JavaScript Syllogistic Machine at
http://home.swipnet.se/~w-33039/Syllog.machine.html.
For bonus points, try to work out which moods do Frater's example belong to.
taken in part from The Dictionary of Phrase and Fable (publ. 1894)