If one examines the insights of the ancient thinkers as to whether they can still be regarded as generally valid today, one encounters Aristotelian logic, apart from many mathematical and some physical statements. Although at the end of the 19th century something “better” was found with modern logic, Aristotle’s statements on the laws of thought are still valid and will always remain so. It is highly admirable how clearly Aristotle saw the structure of an argumentation and how he worked out the decisive criteria for reliability.
The first step is a detailed analysis of a conclusion. In a sentence from Topics, Book 1, Chapter 1 of Aristotle:
The conclusion is now a discourse in which some things are presupposed and then something different from them results from it with necessity mediated by those propositions.
The point is here: “Some” is assumed and “some of it different” results. The extent to which this “with necessity” arises will still have to be discussed.
Let’s have a look at a classic example:
The “some” that we presuppose are the two sentences, also called antecedents:
One: “All human beings are mortal.”
Second: “Socrates is a human being.”
The “different” of it, the conclusion is the sentence:
“Socrates is mortal.”
So here we have three terms: “Socrates”, ” human being” and “mortal”. These are put into relation, “human being” is the generic term to “Socrates”, “mortal” to “human being”. If the relations of terms are correct, the antecedents are true. The conclusion then follows from this Dihairesis, i.e. a classification of terms.
The conclusion seems evident to us; to Aristotle it indeed follows “with necessity”. No one would deny that, everything else would be “unreasonable”. However, we still use our “common sense”, which is what we call reason.
We will leave it as it is and deal with this point again later. It should be noted, however, that the protagonists of this conclusion, i.e. those involved in the conclusion, are three terms, or to be more precise: two relationships for a total of three terms, in each case between two terms. For this reason, one speaks here of a term logic.
Logical inference, dialectical inference and false conclusion
However, something is also said about the sentences in which the terms appear, here e.g. that the antecedents are true. With regard to these, Aristotle now makes a decisive case discrimination (Aristotle, no date):
The conclusion provides a proof or a logical inference if it is derived from true and general superordinate sentences, or from such, which are based on true and superordinate sentences of the science concerned.
Dialectical, on the other hand, is the conclusion derived from credible sentences.
So, it depends on whether the first sentences are true or only “credible”. Often the Greek word translated here as “credible” is translated as “probable”. This suggests the idea that one could indicate here a degree of probability for whether the statement is true. However, only in the last century was it possible to develop a theory of probability in which one can calculate with different degrees of probabilities. These can be applied just as well if one assumes degrees of credibility. We will use that in later blog posts.
But in the case of a proof where one can speak of a logical inference, we are dealing with “certain knowledge”, whereas in the case of a dialectical inference we are dealing only with “uncertain knowledge”. In later blog posts we will deal in detail with what kind of knowledge can be gained from certain or uncertain knowledge respectively.
First it is important to distinguish between a “proof”, in which one must assume that the first sentences, the antecedents, are true, and a dialectical conclusion, in which one can only proceed from “credible” antecedents.
About the “true and general superordinate sentences” he says:
True and superordinate propositions are those which are not mediated by others but are certain by themselves. Because for the most fundamental principles of the sciences one must not demand a reason for them, but each of these principles must be certain by itself.
These are principles which were later called “axioms”. They play a major role in an axiomatic-deductive system by Euclid of Alexandria. However, we may assume that there are also sentences “certain by themselves” outside of special sciences, such as: “All human beings are mortal”.
He determines the “credibility” of sentences as follows (ibid.):
Sentences are credible if they are accepted by all, or by most, or by wise men, the latter by all, or by most, or by the most experienced and credible.
Here we are now “in the center of life”. It is almost always the case that we are dealing with sentences that we can only believe. Even as a scientist you will have to believe almost everything, e.g. the statements of the scientists of another subject – even those of your own subject, if you have not checked the statements yourself or cannot check them directly. “Credibility” is therefore a high good for a society. Today in particular, it is often not easy to decide who to believe.
Finally, Aristotle also deals with the fallacy:
A false conclusion is one which is derived from apparently credible sentences without them really being, or which is derived only apparently from credible sentences or from sentences that only seem so.
The error can therefore lie with the antecedents, that’s trivial. More interesting is the case that the rule of inference is not valid, that a inference is only “apparently”, not truly present.
One such false conclusion, which often remains hidden, is the “fallacy of four terms”. This is demonstrated particularly clearly in the following example (Wikipedia: Fehlschluss):
One: What’s got a beard can be shaved.
Second: Keys have a beard.
Conclusion: Keys can be shaved.
A shift in meaning has occurred here in the transition from the first to the second antecedent. A “beard” in the first sentence means something different than a “beard” in the second sentence. One should better speak of two terms “beard1” and “beard2”, and there would not be three, but four terms in the game – hence the name.
Since we constantly argue with unclear terms in our considerations and discussions, we are often undermined by such false conclusions.
The first major step in the analysis of an argumentation is thus taken: a discrimination between a rule of inference and the “presupposed”, a case discrimination between true and only credible antecedents as well as an investigation of the possibilities of a false conclusion. Here again the great systematist shines through. In the next blog post we will, again systematically, distinguish between different types of antecedents and rules of inference.
The Stoics’ logic
About 100 years later, a different approach to logic emerged in the philosophical school of the so-called Stoa. The philosopher Chrysippus (-276 to -204) of Soli (Cilicia) was probably the representative of this school who most successfully dealt with logic. According to Diogenes Laertius his extraordinarily numerous books were very famous at that time (Laertius, 2015, p. 415ff).
In the long run, however, Aristotle’s approach was far more effective. In all centuries up to the time of Gottlob Frege (1848 to 1925), who founded modern mathematical logic, logic was associated with the name Aristotle; stoic logic was almost forgotten in the Middle Ages, its significance was rediscovered only in 1950 by the American science theorist Benson Mattes. I think it is still underestimated.
The stoic logic was based on the findings that had already been gained by the Megarian school who referred to Euclid of Megara. Stiplon, Diodoros Kronos and Philon von Megara were the most prominent representatives of this group. Stoic logic was already a propositional logic in its approach, while Aristotelian logic, as already mentioned, was a term logic. Aristotle had grown up in the Platonic Academy and had therefore probably incorporated the Platonic terminology (Dihairesis). His logic thus became term logic. The Megarian school was free of such influence, and probably saw dialectics more directly as the problem of checking an argumentation for its conclusiveness. For them, the statements were thus in the foreground.
Aristotle had already seen that it depends on whether the antecedents were true, only credible or not. The rules of inference, however, had to be concerned with the relationships between terms. But now the rules of inference are focused on the “transport of truth” – from the antecedents to the conclusion. Then why shouldn’t the protagonists who are supposed to accomplish this be the sentences themselves? In a propositional logic then it must be only a matter of whether the antecedents are true or not. Terms no longer appear explicitly.
It is then also no longer of interest to “categorical judgements”, such as “All human beings are mortal”, in which a judgement is made about the categories, i.e. in which the category “human being” is set in relation to the category “mortal”, for example. Such a judgement, which corresponds to a division of terms, is to be distinguished from the “synthetic judgement”, which in today’s language corresponds to the connection “A and B” or “A or B” of two statements A and B respectively. So, here sentences A and B are connected in various ways.
A particularly important connection is the “implication”: If A, then B, e.g. “If it rains, then the road is wet”. Where A = “it rains”, B = “the road is wet”. So, if A is the case, then B is the case. An implication can also be true, credible or false. The Stoics were already familiar with this connection: “An implication is true, if the after-sentence is contained in the preceding sentence in the sense”. (Sextus Empiricus: Pyrronic Skepticism II,112. p.181, after Schupp, I, p. 346). With this statement they excluded the case that the implication makes no sense, as in the example: “If 2 + 2 = 4, then my friend has birthday today”. Such a case led to difficulties in formulating certain rules of inference.
The Stoics formulated five “unprovable rules of inference”, and there are also said to have been rules on how more general final formulas can be traced back to these fundamental “unprovable conclusions”. One of these conclusions is identical with the “modus ponens”, a rule of inference in which implication plays an important role. That conclusion is:
Be the statement A true, and be also true:
If A is true, then B is true.
Then follows: Statement B is true.
This can be seen immediately by everyone, and so it is not surprising that this rule already belongs to the “unprovable conclusions” of the Stoics.
But there were always great discussions about the “modus ponens”. One always discussed all logical conclusions only by given meaning of sentences or terms. However, the “modus ponens” contains the implication as an antecedent. Because this could now also be senseless as in the above example, “If 2 + 2 = 4, then my friend has birthday today”, the logicians always doubted the general correctness of this rule of inference.
The “unprovable conclusions” long belonged to the school knowledge of late antiquity; writings by Cicero (-106 to -43) or Isodor of Seville (560 to 630) bear witness to this.
The difference between Aristotelian and Stoic logic was not seen by many even in late antiquity, as can be seen from works by Cicero or Galen. However, the neo-Platonist Porphyrios (234 to 305) still compared the stoic with the Aristotelian logic in terms of terminology and objective. Boethius (477 to 524) could then only report on the stoic logic. When one spoke later of logic, one always meant the Aristotelian logic (Schupp, I, p.349) and one referred thereby predominantly to the writings of Boethius. What remained of stoic logic was the distinction between categorical and synthetic judgments and the knowledge of “modus ponens” as a rule of inference, without knowing stoic logic.
Stoic logic, by being an early form of propositional logic, was much closer to modern logic as formulated by Gottlob Frege at the end of the 19th century. What it lacked was a decisive step: the discovery that a formal language of its own is necessary for logic, just as it is for mathematics, so that the laws of logical thought can be formulated independently of the meaning of the statements. Then you can “compute” like in mathematics. The correctness of conclusions can then be defined and checked at this level. With such a strict separation of syntax and semantics, i.e. of grammar and meaning, the “meaning problem” of the implication described above becomes irrelevant.
Thus, we will have to deal first with formal languages, before the modern propositional logic can be introduced. But first we should study explicitly the rules of Aristotelian logic for the sake of completeness.