Some examples of singular terms are ‘Socrates’, ‘Plato’, ‘Xenocrates’. Singular terms cannot have a universal quantity, we can not for example say “All Plato” (here we consider ‘Plato’ to be a unique individual). Aristotle presented the following dichotomy “Some things are universal, others individual.”([AI] 1:9). Elaborating: “By the term ‘universal’ I mean that which is of such a nature as to be predicated of many subjects”([AI] 1:9) then defining ‘individual’ as “that which is not thus predicated”([AI] 1:9). The inability to deal with individual subjects is considered to be one of the greatest flaws of Aristotelian logic. Although many have argued that we could universally predicate an individual subject such as ‘Socrates’, Aristotle himself disagreed: “ ‘Socrates’ is not predicable of more than one subject, and therefore we do not say ‘every Socrates’ as we say ’every man’.”([AM] 5:9). The following example illustrates the awkwardness of propositions containing singular terms.

All men are mortals,
all Socrates are men,
therefore all Socrates are mortals.

Although one would be tempted to convert an ungrammatical sentence like “All Socrates are men” into the singular proposition “Socrates is a man” this type of sentence would never have been considered by Aristotle. All singular terms and consequently singular propositions are ignored in Aristotelian logic.

The square of opposition is used to analyze the relation between propositions in Aristotelian logic. Two propositions with the same subject and same predicate are said to be opposites if one of or both their quality and quantity are different (i.e. All men are mortal and Some men are mortal, is an example of opposite propositions). There are different types of opposite propositions: two propositions are said to be alterns if they only differ in quantity (i.e. All S are P and Some S are P). Two universal propositions are said to be contraries if they only differ in quality (i.e. All S are P and No S are P). Two particular propositions are said to be subcontraries if they differ in quality (i.e. Some S are P and Some S are not P). Two propositions are said to be contradictories if they differ in quality and quantity (i.e. No S are P and Some S are P, also All S are P and Some S are not P). The following diagram illustrates the relation between the opposites:

The square itself was not thought up by Aristotle but the information to devise the square comes directly from the writings of Aristotle. Below is Aristotle’s definition of contraries:

… a man states a positive and a negative proposition of universal character with regard to a universal, these two propositions are ‘contrary’. ([AI] 1:7)

and his definition of contradictory:

An affirmation is opposed to a denial in the sense which I denote by the term ‘contradictory’, when, while the subject remains the same, the affirmation is of universal character and the denial is not. ([AI] 1:7)

The diagram itself is at least as old as the second century C.E. where it is present in the writings of Boethius.[PT]

In Aristotelian logic, as noted earlier, there are no singular propositions. Therefore propositions like ‘Socrates is a man’ would never be present in his logic. This is why the original example

All men are mortal,
Socrates is a man,
therefore Socrates is mortal.

is not an Aristotelian syllogism.

A conversion is the process of changing a proposition by reversing the subject and predicate while maintaining the same quality. There are two types of conversions: simple conversions where the quantity of the proposition is kept unchanged and conversion per accidens where the quantity of the proposition is changed from being universal to being particular. It is important to note that not all conversions are valid and that some conversions do not exist. Below is a table of all conversions of Aristotelian propositions which have a conversion.[PH]

We see that Aristotle accepts the conversion per accidens of $Asp$ to $Ips$ but not the simple conversion of $Asp$ to $Aps$ stating “the terms of the affirmative must be convertible, not however, universally, but in part”([AP] 1:2) and gives the following example of this accepted conversion: “if every pleasure,is good, some good must be pleasure”([AP] 1:2). He also accepts the conversion of $Esp$ to $Eps$ and also the per accidens conversion $Esp$ to $Osp$. Below is Aristotle’s proof of the simple conversion of $Esp$ to $Eps$:

If no B is A, neither can any A be B. For if some A (say C) were B, it would not be true that no B is A; for C is a B. But if every B is A then some A is B. For if no A were B, then no B could be A. But we assumed that every B is A. ([AP] 1:2)

He also proves the simple conversion $Isp$ to $Ips$ and also states various counter examples for non valid conversions. He presents a counter example for the simple conversion of $Osp$ to $Ops$: “but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal.”([AP] 1:2)

It is important to note that for the conversion per accidens to be valid there must be a presupposition. The presupposition being that there exists at least one subject in the universe of discourse. This presupposition is the simplest one that keeps the Aristotelian logic valid. Many controversies and attacks on Aristotelian logic have been based on this problem of having a possible empty universe of discourse.

The position of the middle term (the term which appears in both the major and minor premise) gives rise to what are called figures. The figures represent the possible placement of the middle term in a syllogism. The first figure is based on what Aristotle thought was the deduction which was closest to natural reasoning. So the first figure will only generate perfect syllogisms. The figures below are illustrated using $M$ as the middle term, $P$ as the major term, and as $S$ the minor term. It is clear that there are only 4 such figures.

Note that the fourth figure was not explicitly stated in Aristotle’s work. Although he accepted the reasoning which can be derived from that figure. The fourth figure is sometimes called the Galenian figure as it was possibly first used or discovered by Galen(131-201 C.E.) although some contest its discovery.[LJ] Carl Prantl rejects the fourth figure since it does not appear directly in Aristotle’s work, however fails to recognize that Aristotle accepted its derived results.[LJ]

The mood of a syllogism is a sequence of propositions and conclusions. The figures associated with a mood make a syllogism. Below is a table of the valid syllogisms with their associated moods and figures.

The table below show the valid syllogistic moods derived from the first figure. As stated above these syllogisms are perfect.

Although all the above are perfect syllogisms, for Aristotle, the most clear syllogisms are Barbara and Celarent. In his later work he deduced the Darii, and Ferio syllogisms as they were less natural to him.[LJ] The mnemonics are used to remember the valid moods. Take the classical perfect syllogism ‘Barbara’, the important information in the mnemonic are the vowels. ‘Barbara’ represents the mood $AAA$. In addition to observing the vowels one must know ahead of time that ‘Barbara’ is a mood of the first figure. We now have all the information to deduce that the syllogism has the following form: $Amp\;\&\;Asm\;\therefore Asp$. This syllogism can be written as

All M are P
and all S are M
then all S are P.

Where the placement of the $M$’s, $P$’s, and $S$’s correspond to the first figure (shown below).

To remember in which figure each mood is derived from, medieval logicians invented verses to remember them. For example this verse:

Barbara, Celarent, Darii, Ferio-que prioris.
Cesare, Camestres, Festino, Baroko, secundae.
Tertia Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison habet. Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The key words in the above verse are prioris which refers to moods of the first figure, secundae to those of second figure, tertia to those of the third figure and finally quarta to those of the fourth figure.

The moods of the second figure are listed below, they are all imperfect syllogisms.

Since the above syllogisms are imperfect, their validity must be proven. For example Aristotle proves the validity of the syllogism Cesare:

Let M be predicated of no N, but of all O. Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O. This has already been proved.([AP] 1:5)

Therefore Cesare is a valid mood, here is a decomposition of Aristotle’s proof:

1.	Let M be predicated of no N,	No N are M	$Enm$	premise
2.	but of all O.	All O are M	$Aom$	premise
3.	N will belong to no M	No M are N	$Emn$	since the simple conversion from $Esp$ to $Eps$ is valid
4.	consequently N will belong to no O.	No O are N	$Eon$	using line 2 and 3 we reduce to the syllogism Celarent which is an axiom

The process illustrated above is called reduction where premises and conclusions of a certain argument are converted to a first figure syllogism to conclude that the argument is valid.

The moods of the third figure are listed below, noting that they are also all imperfect

The mnemonics contain more information than simply the mood. 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 (for example Felapton will be reduced to Ferio). In addition some of the letters that succeed vowels in the mnemonic have a particular meaning. If a ‘c’ follows an ‘o’ it indicates that the proof must be done reductio ad impossibile.(There are two such syllogisms, Bocardo and Baroco which are not correctly proven by Aristotle. To see the reason why consult Aristotle’s Syllogistic, From the Standpoint of Modern Formal Logic by Jan Lukasiewicz, section 17.) If an ‘s’(or ‘p’) follows the first or second vowel, this indicates that the proposition corresponding to this vowel is simply(or 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. Finally an ‘m’ indicates a rearrangement of the premises to satisfy the order: major premise, minor premise and conclusion.[PH]

The moods of the fourth figure are listed below, these are all imperfect

The proof of the validity of all imperfect syllogisms can now be given by using the rules associated to the mnemonics. This will be illustrated using the syllogism associated to the mnemonic ‘Dimaris’. It is known to begin with, that ‘Dimaris’ is a fourth figure syllogism. Its vowels are $IAI$ thus the following syllogism will be proven: $Ipm\;\&\;Ams\;\therefore Isp$. Since ‘Dimaris’ begins with the letter ‘D’ a reduction to the first figure syllogism ‘Darii’ will be used. The ‘s’ following the last vowel indicates that a simple conversion in the conclusion of the syllogism ‘Darii’ will be necessary. In addition the ‘m’ indicates a reordering of the premises. The details of the proof are given below

1.	Some P are M	$Ipm$	premise
2.	All M are S	$Ams$	premise
3.	Some P are M	$Ipm$	repetition of line 1 to reorder the premises
4.	Some P are S	$Ips$	by line 2 and 3 and the axiom ‘Darii’
5.	Some S are P	$Isp$	by simple conversion of line 4