II. Zależności statystyczne

Marek Jerzy. Minakowski, Prehistoria logiki formalnej

Marek Jerzy Minakowski, The Pre-Aristotelian Formal Logic

Paper read during the 11th International Congress of Logic, Methodology and Philosophy of Science in Cracow (August, 1999)

Four years ago, in Florence, during the previous Congress, I proposed a programme of investigating the pre-Aristotelian logic. The programme was the subject of my doctoral dissertation which I had a pleasure to defend a year ago at the University here, in Cracow. Now, let me present the results.

The predominant question was: since when the history of logic should be measureed, which can be divided into more elementary questions:

1. was anything known before Aristotle;
2. and — if so —
   1. what was it,
   2. since when was it known, and
   3. what was the difference between this and the Aristotle’s logic.

There have never been common agreement concerning any of this points. For over 2,000 years, since the edition by Alexander of Aphrodisias the Arislotle’s logical works, the disagreement about the authorship of logic lasts. Although the disagreement is about facts, nobody has taken upon the burden to collect all the proofs, and the solutions have been made mainly due to subjective impressions and belonging of the people taking floor.

My aim, therefore, could have been achieved only by collecting all the instances of the use of logic in the pre-Aristotelian texts.

Several times I read all the preserved pre-Aristotelians’ texts (both in Greek and in modern translations). It is about 9.000 pages of standard typescript. There is no wzmianka, no świadectwo there, that anybody created a logical system before Aristotle.

What is more, there are no traces of existing of such a system. Well, I have found 159 fragments with the use of formal logic: 397 instances of application of rules of formal logic. But there are no regularities, neither in logical form of this rules, or in the linguistic pattern of the arguments, that could point to the existence of a most loosely understood logical system. Such a minimal system, that could have been found, would be a finite set of rules, existing at least in the consciousness of the user (not necessarily written), which could produce (after application of some rule of substitution or replacement) a sound proof.

Having concluded that there is no such system grants us that:

1. If the preserved texts of the prearistotelian authors are representative to the period, we can say that there was no known logical system before Aristotle, that could have been comared to the syllogistic.

2. The Aristotle’s thesis that it was he, who first formulated a logical system, can not be falsified.

3. The Socratic Philosophers did not know any logical system, comparable to the Aristotle’s syllogistic.

The difference in the force of the statements is due to the fact that the Socratic Philosophers (Socrates’ disciples) we do not need to assume the representativity, because here the próbka is big enough.

Another problem to solve, apart from the question about the first theoreticians of logic (authors of logical systems) was the question about the first users of the formal logic. It could happen that there was no system, but, e.g., all or some people employed logic every day, naturally, and nobody needed to discover it. It could be so, e.g., that — as a widely known viewpoint says — logic is discovering the universal laws of thought.

My results deny it. Apart from philosophical, or — as now we would rather say — theoretical writings, logic is hard to be found. In the philosophical texts (if we do not count marginal cases: Plato’s Republic II-X and Laws together with Xenophon’s Cyropaedia) one instance of a logical rule can be found in 5,7 pages (if we take that a page has 2,000 characters). In the remaining texts (again, not counting marginal cases: a few short theoretical works by Hippocrates and Antiphon) — one instance of a logical rule can be found in 542 pages (to be scrupulous: 11 times in 5962 pages). The relation is 1:95 in favour of philosophical texts: a logical rule can be found about once in five pages in philosophical texts and once in 500 pages in non-philosophical texts.

That grounds our fourth theorem:

4. The use of logic before philosophy emerged was only accidental and very rare.

Was not then, that logic is a twin of philosophy? No.

I have analysed all the preserved fragments of Pre-Socratic philosophers (collected by Diels and Kranz’ Die Fragmente der Vorsokratiker). I have found 51 instances of the use of formal logic there. Out of them, 13 can be found in the preserved fragments of Eleatic philosopers Zeno and Melissos,19 in two works by Gorgias (The Encomium of Helen and The Apology of Palamedes), while 15 in a work by unknown author (presumably a sophist), known as Dissoi logoi. All the remaining cases are only four instances (Hippon, Philolaus, Anaxagoras and Diogenes of Apollonia).

The Presocratic Philosophers’ fragments occupy 895 pages in Diels’ work. Fragments by Zeno and Melissos, Gorgias and Dissoi logoi take only 77 pages: one twelfth only. Such a big disproportion entitles us to say:

5. If the Presocratic Philosophers’ fragments are representative to them, we can state that, excluding Eleatic philosophers and some sophists, they did not use logic.

What does the Eleatic and sophist logic look like, then?

The preserved Eleatic logic is mainly propositional logic, especially indirect proving. 7 out of 13 rules they employed is indirect deduction and three is the rule of detachment. The same case can be seen in a passage from Plato’s Parmenides, which is a copy of Melissos’ proof (so-called first hypothesis: 137c-142a).

The logic of the two Gorgias’ orations is mainly a pure propositional calculus (15 out of 19 cases).

The author of Dissoi logoi 15 times uses rules of logic. These are only rules of the logic of names; the author tries to disprove theorems that good and evil, beauty and ugliness, true and false etc. are the same; he reduces it to absurd conclusions. While the arguments are almost identical, we cannot say anything about the author’s invention and his general logical knowledge.

The question about the logic of Eleatic philosophers, Gorgias and the author of Dissoi logoi, should be answered then:

6. If the preserved texts of Eleatic philosophers, Gorgias and the author of Dissoi logoi are representative to them, it should be stated that none of them employ the logic of names on a larger scale than the non-philosophers.

7. If the preserved texts by Zeno, Melissos and Gorgias are representative to them, we should state that they were quite proficient in the use of rudimentary propositional calculus and it can not be excluded that they knew some sets of valid logical rules.

The next subject of our analysis are the texts of the Socratic philosophers - Plato and Xenophon. Socrates himself did not wrote anything (at least the tradition claims so). However, we have a vast collection of Plato’s (his most famous pupil) writings and a large set of socratic writings by Xenophon. Aristotle can also be treated as a Socratic philosopher.

Could have Aristotle employ Socrates’ achievements? Were they any in the field of logic?

Both in the young Plato’s writings and in Xenophon’s memoirs, Socrates is deducing very numerous syllogistic proofs. In the corpus under examination, rules of names (i.e., rules of the logic of names) are 84% out of all instances of logical rules (12% are pure propositional rules and 4% — mixed ones). Out of the propositional rules, half are indirect deductions, a quarter — pure implicational rules, while other propositional and mixed propositional-nominal rules are only 23 out of 322 all instances of logical rules in the Socratic (Pre-Aristotelian) texts.

We should state, then:

8. Socrates was competent in the use of syllogistic (the logic of names).

9. Socrates almost never employed propositional logic.

We don’t know, to what extent the fact of appearance of logic in Socrates was his own virtue, and to what extent it was:

a) a product of his teachers or — more generally — the intellectual milieu of Athens; b) a result of the subject treated by Socrates (looking for the essence of virtues); c) a result of the question-and-answer method.

Was there any progress in logic between Socrates and Aristotle? Having examined the frequency of the use of logic in different periods of Plato’s life, we should firmly state:

10. Plato did not put anything to the formal logic.

Plato most frequently used logic in the beginning, when he remembered Socrates well enough. The frequency of the use of logic in his writings is less and less, and in his late writings there is almost no use of logic.On the other hand, the logic of Plato is indiscernible from the logic of Xenophon. It is highly unlikely that other Socratic philosophers (older than Aristotle) knew in logic anything not known to Socrates and present in Aristotle. I did not try to explain why the frequency of the use of logic in Plato is diminishing; it is enough to say that it is and therefore Socrates needs not to share his primacy with Plato.

The above ten theses or theorems makes up my doctoral thesis The Pre-Aristotelian Formal Logic. On this ground, we can outline the history of logic before the oldest known tractatus about logic was written, the Prior Analytics.

To sum up, I will present my proposal of the pre-history of logic, which can be build on the above theses.

Everything suggests that our ancestors did not think logically. It needn’t be any harm; the problem they had to solve did not need logic to be solved. I do not want to say that they did not need knowledge of logic, like Mr Jourdain did not need to know that he speaks in prose. They did not employ logic: like Mr Jourdain who did not speak Greek. He, like them did not need logic at all. The prevailing majority of people even now does not need logic at all. The teaching in schools can have only transitional effect, because it cannot create the need.

Just like the knowledge of grammar of your mother-tongue is needed only in rare doubtful situations, when your native speaker’s intuition fails, in the same way logic become important only after such kinds of reasoning emerged, where common sense failed. Such a situation was the appearance of theoretical science, called then ‘philosophy’.

It didn’t happen at once. Well — even now many people calling themselves ‘philosophers’ do not employ logic in their work. First, such a situation should have emerged, that philosophers started to discuss, persuade one another and other people. The existence of thoretical discussions was necessary for the need of the use of logic.

The Greeks liked to discuss very much; but their debates were mainly orations, speeches made on a market-square, public gathering, in a court. We cannot find any instances of the use of logic in the written records of such discussions: to persuade to mob, something different was needed: rhetoric. Speeches, were logic can be found, are directed only towards a closed group of people: disciples and friends; they are only ‘handbook’ or ‘pretended’ speeches.

Theoretical discussions were quite popular among the Greeks; as far as they had a form of sayings of Oracle or Wise Men, there had been no place for logic there.

And it happened that both ways crossed one another. People started to discuss general questions. The discussions based upon propositional calculus; Melissos was proving that the Being cannot have dimensions, because if it was limited, it would not be; while it is. Gorgias was defending Palamedes in such a way that Palamedes could not have been plotting against the Greeks, because to betray them he should have contact them, to contact them he should have spoken with them, while it is impossible, because he does not speak Troyan; therefore Palamedes could not have betrayed the Greeks, q.e.d. To make Helen praiseworthy, Gorgias employed a ramified alternative of four parts; because each of them leads to the same conclusion, the conclusion has been proved.

These argument were convincing, but they did not touch the essence. Logic was unable to get into the simple proposition and, e.g., deduce properties from a definition. And then Socrates appeared. Socrates started to define; he was walking around a market-square talking with peaople and asking them, what virtue is, what prowess, what love and what justice is. Thereupon they were concluding: virtue is good and the good is beautiful; therefore virtue is beautiful. However, shrewdness is not beautiful; therefore it is no virtue.

At this very moment logic started to rule; everywhere, where Socrates appears, such reasonings appear. In his disciples’ writings (he didn’t wrote) almost such proofs appear. Somebody can doubt, whether Socrates was a creative philosopher, but nobody can doubt that in logic he deserved great honours.

Two Socrates’ disciples wrote much (at least their production we know): Plato and Xenophon. Whilst Xenophon was rather a memoirist than a philosopher, Plato is still regarded by some people as the greatest philosopher of all times. No one of them did not have — presumably — much purport in logic; they are both quoting Socrates, maybe with high fidelity, but nothing more. Plato, in older age, started to promote his own doctrines and he ceased to need logic. In one of his late dialogues (Parmenides) he put a long proof in a propositional convention, but we can doubt, whether it was Plato, who was it’s author.

And then Aristotle entered history; educated in cult of Socrates, he was in touch with his logic for many years. Because it was almost unique way of using logic in his milieu, it is nothing strange, that it become the subject of his analysis, when he decided to treat it from the theoretical point of view. First he collected and catalogued examples of deductions (in Topica), and then he generalized it and put in a system(in Analytics).

The rest you know from handbooks.