Prior Analytics
The Prior Analytics is a work by Aristotle on deductive reasoning, known as his syllogistic, composed around 350 BCE. Being one of the six extant Aristotelian writings on logic and scientific method, it is part of what later Peripatetics called the Organon. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Łukasiewicz of a revolutionary paradigm. His approach was replaced in the early 1970s in a series of papers by John Corcoran and Timothy Smiley—which inform modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.
The term analytics comes from the Greek words analytos and analyo. However, in Aristotle's corpus, there are distinguishable differences in the meaning of ἀναλύω and its cognates. There is also the possibility that Aristotle may have borrowed his use of the word "analysis" from his teacher Plato. On the other hand, the meaning that best fits the Analytics is one derived from the study of Geometry and this meaning is very close to what Aristotle calls episteme, knowing the reasoned facts. Therefore, Analysis is the process of finding the reasoned facts.
Aristotle's Prior Analytics represents the first time in history when Logic is scientifically investigated. On those grounds alone, Aristotle could be considered the Father of Logic for as he himself says in Sophistical Refutations, "When it comes to this subject, it is not the case that part had been worked out before in advance and part had not; instead, nothing existed at all."
A problem in meaning arises in the study of Prior Analytics for the word syllogism as used by Aristotle in general does not carry the same narrow connotation as it does at present; Aristotle defines this term in a way that would apply to a wide range of valid arguments. Some scholars prefer to use the word "deduction" instead as the meaning given by Aristotle to the Greek word syllogismos. At present, syllogism is used exclusively as the method used to reach a conclusion which is really the narrow sense in which it is used in the Prior Analytics dealing as it does with a much narrower class of arguments closely resembling the "syllogisms" of traditional logic texts: two premises followed by a conclusion each of which is a categorical sentence containing all together three terms, two extremes which appear in the conclusion and one middle term which appears in both premises but not in the conclusion. In the Analytics then, Prior Analytics is the first theoretical part dealing with the science of deduction and the Posterior Analytics is the second demonstratively practical part. Prior Analytics gives an account of deductions in general narrowed down to three basic syllogisms while Posterior Analytics deals with demonstration.
In the Prior Analytics, Aristotle defines syllogism as "a deduction in a discourse in which, certain things being supposed, something different from the things supposed results of necessity because these things are so." In modern times, this definition has led to a debate as to how the word "syllogism" should be interpreted. Scholars Jan Lukasiewicz, Józef Maria Bocheński and Günther Patzig have sided with the Protasis-Apodosis dichotomy while John Corcoran prefers to consider a syllogism as simply a deduction.
In the third century AD, Alexander of Aphrodisias's commentary on the Prior Analytics is the oldest extant and one of the best of the ancient tradition and is available in the English language.
In the sixth century, Boethius composed the first known Latin translation of the Prior Analytics. No Westerner between Boethius and Bernard of Utrecht is known to have read the Prior Analytics. The so-called Anonymus Aurelianensis III from the second half of the twelfth century is the first extant Latin commentary, or rather fragment of a commentary.
The syllogism
The Prior Analytics represents the first formal study of logic, where logic is understood as the study of arguments. An argument is a series of true or false statements which lead to a true or false conclusion. In the Prior Analytics, Aristotle identifies valid and invalid forms of arguments called syllogisms. A syllogism is an argument that consists of at least three sentences: at least two premises and a conclusion. Although Aristotle does not call them "categorical sentences," tradition does; he deals with them briefly in the Analytics and more extensively in On Interpretation. Each proposition P belongs to S, 2) P is predicated of S and 3) P is said of S. Aristotle does not explain why he introduces these innovative expressions but scholars conjecture that the reason may have been that it facilitates the use of letters instead of terms avoiding the ambiguity that results in Greek when letters are used with the linking verb. In his formulation of syllogistic propositions, instead of the copula, Aristotle uses the expression, "... belongs to/does not belong to all/some..." or "... is said/is not said of all/some..." There are four different types of categorical sentences: universal affirmative, particular affirmative, universal negative and particular negative.- A - A belongs to every B
- E - A belongs to no B
- I - A belongs to some B
- O - A does not belong to some B
Following this tradition then, let:
a = belongs to every
e = belongs to no
i = belongs to some
o = does not belong to some
Categorical sentences may then be abbreviated as follows:
AaB = A belongs to every B
AeB = A belongs to no B
AiB = A belongs to some B
AoB = A does not belong to some B
From the viewpoint of modern logic, only a few types of sentences can be represented in this way.
The three figures
Depending on the position of the middle term, Aristotle divides the syllogism into three kinds: syllogism in the first, second, and third figure. If the Middle Term is subject of one premise and predicate of the other, the premises are in the First Figure. If the Middle Term is predicate of both premises, the premises are in the Second Figure. If the Middle Term is subject of both premises, the premises are in the Third Figure.Symbolically, the Three Figures may be represented as follows:
| First figure | Second figure | Third figure | |
| Predicate — Subject | Predicate — Subject | Predicate — Subject | |
| Major premise | A ------------ B | B ------------ A | A ------------ B |
| Minor premise | B ------------ C | B ------------ C | C ------------ B |
| Conclusion | A ********** C | A ********** C | A ********** C |
The fourth figure
In Aristotelian syllogistic, syllogisms are divided into three figures according to the position of the middle term in the two premises. The fourth figure, in which the middle term is the predicate in the major premise and the subject in the minor, was added by Aristotle's pupil Theophrastus and does not occur in Aristotle's work, although there is evidence that Aristotle knew of fourth-figure syllogisms.
Syllogism in the first figure
In the Prior Analytics translated by A. J. Jenkins as it appears in volume 8 of the Great Books of the Western World, Aristotle says of the First Figure: "... If A is predicated of all B, and B of all C, A must be predicated of all C." In the Prior Analytics translated by Robin Smith, Aristotle says of the first figure: "... For if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C."Taking a = is predicated of all = is predicated of every, and using the symbolical method used in the Middle Ages, then the first figure is simplified to:
If AaB
and BaC
then AaC.
Or what amounts to the same thing:
AaB, BaC; therefore AaC
When the four syllogistic propositions, a, e, i, o are placed in the first figure, Aristotle comes up with the following valid forms of deduction for the first figure:
AaB, BaC; therefore, AaC
AeB, BaC; therefore, AeC
AaB, BiC; therefore, AiC
AeB, BiC; therefore, AoC
In the Middle Ages, for mnemonic reasons they were called respectively "Barbara", "Celarent", "Darii" and "Ferio".
The difference between the first figure and the other two figures is that the syllogism of the first figure is complete while that of the second and fourth is not. This is important in Aristotle's theory of the syllogism for the first figure is axiomatic while the second and third require proof. The proof of the second and third figure always leads back to the first figure.
Syllogism in the second figure
This is what Robin Smith says in English that Aristotle said in Ancient Greek: "... If M belongs to every N but to no X, then neither will N belong to any X. For if M belongs to no X, neither does X belong to any M; but M belonged to every N; therefore, X will belong to no N."The above statement can be simplified by using the symbolical method used in the Middle Ages:
If MaN
but MeX
then NeX.
For if MeX
then XeM
but MaN
therefore XeN.
When the four syllogistic propositions, a, e, i, o are placed in the second figure, Aristotle comes up with the following valid forms of deduction for the second figure:
MaN, MeX; therefore NeX
MeN, MaX; therefore NeX
MeN, MiX; therefore NoX
MaN, MoX; therefore NoX
In the Middle Ages, for mnemonic reasons they were called respectively "Camestres", "Cesare", "Festino" and "Baroco".
Syllogism in the third figure
Aristotle says in the Prior Analytics, "... If one term belongs to all and another to none of the same thing, or if they both belong to all or none of it, I call such figure the third." Referring to universal terms, "... then when both P and R belongs to every S, it results of necessity that P will belong to some R."Simplifying:
If PaS
and RaS
then PiR.
When the four syllogistic propositions, a, e, i, o are placed in the third figure, Aristotle develops six more valid forms of deduction:
PaS, RaS; therefore PiR
PeS, RaS; therefore PoR
PiS, RaS; therefore PiR
PaS, RiS; therefore PiR
PoS, RaS; therefore PoR
PeS, RiS; therefore PoR
In the Middle Ages, for mnemonic reasons, these six forms were called respectively: "Darapti", "Felapton", "Disamis", "Datisi", "Bocardo" and "Ferison".
Table of syllogisms
Boole’s acceptance of Aristotle
's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were “to go under, over, and beyond” Aristotle's logic by:- providing it with mathematical foundations involving equations;
- extending the class of problems it could treat—from assessing validity to solving equations; and
- expanding the range of applications it could handle—e.g. from propositions having only two terms to those having arbitrarily many.