The formal language of propositional logic

After briefly introducing Aristotle’s syllogistics in the last blog post, I should now actually explain how it were received and elaborated in antiquity, the Middle Ages and into modern times. In particular, the work of Gottfried Wilhelm Leibniz (1646 to 1716), in which important approaches to modern logic can already be found, should be honoured. It would also be his great project to describe a “Scientia generalis” based on the model of mathematics, in order to “bring about in all scients what Cartesius and others did by algebra and analysis in Arithmetica et Geometria”. (All writings II,1. p.160, after Schupp III, p.238). For this purpose Leibniz wanted to develop a “lingua characteristica” with suitable characters (characteristica universalis). But this and other developments in this direction can be better assessed in retrospect. Therefore, I want to skip this and talk about modern logic.

The mathematician and philosopher Gottlob Frege (1848-1925) founded modern logic with a book entitled “Begriffsschrift – Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens” (“Conceptual writing – one of the arithmetic formula languages of pure thought”). The foreword says: In this “small script I have now tried a rapprochement with the Leibniz thought of a lingua characteristicistica”. Leibniz is thus the godfather of the modern formal logic (Wille, 2018, p. 96).

Gottlob Frege was a philosopher and mathematician, his father had written a New High German grammar (Wikipedia: Gottlob Frege). So Frege probably knew the power of a formal language; with such a language he then raised logic to a new level, on which one could lead proofs as in mathematics. He thus discovered a “new science” for logic, as Galileo had done for natural research. And again one can see how formalization leads to an accelerated development of a field of knowledge, then “emigrating” from philosophy.

The final form of a “propositional logic” was then created somewhat later by Alfred North Whitehead (1981 to 1947) and Bertrand Russell (1872 to 1970). The “term logic”, as the philosophers had always practiced until then, was soon replaced by this propositional logic; logic developed into a predominantly mathematical discipline. Its mathematical form made it possible to transfer logical reasoning rules to machines, which has led, among other things, to the now flourishing field of “artificial intelligence” research.

Since one experiences again and again, which shyness most contemporaries have at mathematical formulas, one has to exercise great caution with the representation of a formal language for the logic. The best thing to do is to look at the structure of a formal language in analogy to the structure of our mother tongue and to demonstrate the similarity of a formal language with our colloquial language. It’s just all so much easier, and so simple, that at first you feel yourself as a stranger. The language of mathematics may serve as an intermediate station between the formal language of logic and our colloquial language, because it is already formal and the contemporary has at least some experience with it.

Thus, we now look at three languages: logic, mathematics and our colloquial language. In all three languages there is a certain set of characters with which you can form strings. In colloquial language this set represents the usual alphabet, strings are words and several strings represent sentences. So you learn to write the letters first, then the words and then the sentences.
Here the important difference between syntax and semantics has to be discussed. Syntax is what is called grammar in natural languages. It shows, for example, how words are declined or conjugated in a sentence regardless of their meaning, depending on their function and position, and in which order they can occur in a sentence. The syntax or grammar thus regulates the formal structure of a language. Semantics, on the other hand, deals with the meaning of words and sentences.

In formal languages, the distinction between syntax and semantics becomes even clearer. The syntax specifies exactly what should be allowed with regard to the formation of character strings. At the syntactic level, therefore, we are only dealing with strings of different complexity, without these having already been assigned a meaning.

Even a rudimentary knowledge of mathematics gives us an indication of the usefulness of a formal language. One can “calculate” in it, i.e. formally argue, in such a way that one does not have to know the meaning of the characters and character strings, which one changes, generates or removes there according to certain rules. The whole thing happens on the syntactic level and the rules are made in such a way that true statements are transformed into true statements – completely independent of the meaning of the signs and strings of signs. Only in this way is it understandable that we can transfer the calculating and logical closing to a machine, that we can develop algorithms to solve tasks that would require a certain intelligence from ourselves.

The syntax of the language for a propositional logic

In general, a construction of a formal language begins by first providing a character set, a certain set of symbols, i.e. defining an alphabet.
In propositional logic, the alphabet consists of a stock of

– Variables A, B, …, i.e. capital letters of our everyday language.
– the characters ‘¬’ and ‘→’, ‘∧’ and ‘∨’.
– the sign “≡”,
– the brackets “(“” and “)”; these should always be used, if you want to see a couple of strings as a unit, to keep an overview. Also, sometimes a space is used to increase the clarity, and finally
– the constants “0” and “1”.

With the help of these characters or “letters” you can form character strings. We will see that in the formal language of propositional logic there is no need to distinguish between letters, words and sentences. You only know characters and strings, and these are uniformly called expressions.
How character strings, i.e. more complex expressions, can be formed from the characters of the alphabet is regulated by certain laws, namely:

– If A is an expression, then ¬A is also an expression.
– If A and B are expressions, so are A ∧ B and A ∨ B.

You can always create new expressions in this way, e.g. ¬A ∨ B or A ∧ (¬A ∨ B). The expression ¬A ∨ B will later play a major role, so we will introduce the abbreviation A → B for him. This is another sign at our disposal. Any expression of the form A → B will always be understood as ¬A ∨ B.
With A and A → B and A ∧ (A → B) we can e.g. get also (A ∧ (A → B)) → B. We will deal with this expression later.

One calls expressions, which are formed according to these rules, well formed. So A → B and A ∧ B ∧ C C are well-formed, but ABA or AB→ are not.

Comments:
Our natural languages also have an alphabet. We are most familiar e.g. with the alphabet of the German language. Other alphabets deviate more or less from it. These characters are also used to form strings, which we call words. These words are formed by the way people deal with the things of the environment. This is a historical process, and through the experience of always new things in this world, new words are always invented. Here there are hardly any rules, at most the condition that the word must be pronounceable. For example, the string mjk?a will not be regarded as a meaningful candidate for a word. So the word formation happens on the semantic level: “The thing must get a name”. A rule is then formed for how the accepted words must be written. Orthography is the teaching of “correct writing”, the correct representation of “accepted character strings”. Words are used to form sentences in colloquial language, which can then each represent a statement. The words are then modified, declined or conjugated depending on their position in the sentences.

In contrast, in a formal language “words” are formed at the syntactic level, and according to only a few rules. “Orthography” here means respecting the rules for the formation of strings. There’s no such thing as sentences.

– Mathematics is also a formal language. Here the character set usually consists of the set of symbols {0,1, … ,9}, i.e. the numbers from 0 to 9, and the characters {+, -,  ∙, :, =}. Depending on the mathematical theory, further characters are added. Furthermore, you still need symbols as placeholders for numbers that are not further specified at first. Often one takes letters of the alphabet of the colloquial language. There are also clear rules here for the formation of “words” and “sentences”. “Words” are then such strings as x + 0 or 5-4, which are also called “terms”. “Sentences” are equations like x + 2 = 0, and they are also formulated according to certain rules.

Herewith we have described the syntax of three languages. It regulates which character strings may be formed on the basis of a character set.

The semantics respectively the evaluation

In semantics we must now say what we want to understand by the characters we have introduced in syntax, what meaning we want to give them.

In propositional logic, the variables A, B, … should stand for statements in our colloquial language that are either true or false. Note that this is not about the content of the statement and its meaning, but only about an evaluation: true or false. In this respect one should actually only talk about evaluation instead of semantics. But we don’t have to become that exact in our colloquial language, in which we construct the formal language and in which we later talk about it. The colloquial language we call the “meta language”, the language of the propositional logic is then here the “object language”. We have to be very specific in this one. But in our colloquial language, however, we may have enough vague terms.

This condition that we only want to consider statements that are either true or false is constitutive for the entire propositional logic. So it’s not about interrogative sentences or exclamations. We also disregard the fact that there may be statements where you can never decide whether they are true or false. The fact that this limitation is not self-evident for some mathematicians does not need to be discussed here.

Let’s look at some simple statements. Be, for example.
A:= “It’s raining.”
B:= “The road’s wet.”
So e.g. A is true exactly when it rains. Truth is given when the statement is consistent with the facts.

In order to evaluate all the expressions that we can form with such statements according to the rules of syntax, we must now say what we intend to do with the other characters from the character set:

We interpret the “1” as true, the “0” as false.

The sign “¬” before a statement should negate it:
¬A = It is not the case that A, i.e. in our example: “It does not rain”.
It’s the same:
¬1 = 0, “not true” means “false”,
¬0 = 1, “not wrong” means “true”.

The expression A ∧ B represents a combination of two statements, reminiscent of 5 + 4 or 5 – 4. We determine the meaning of the sign “∧ ” by saying when this expression should be true depending on the truth values A and B and when it should be false. In principle, we have 16 possibilities there. The default choice is shown in the following truth table:

A B A B
1 1 1
1 0 0
0 1 0
0 0 0

This assignment is done in such a way that the symbol “∧” corresponds to the colloquial “and”. We’ll test this against our example:
A ∧ B = “It’s raining” and “the road is wet”.
We obviously perceive this statement as true exactly when both statements A and B are true, as it is also stated in the table.

We define the meaning of the sign “∨” in such a way that in the expression A ∨ B it resembles the colloquial “or”:

A B A B
1 1 1
1 0 1
0 1 1
0 0 0

The expression A ∨ B only has the rating 0, i.e. is only false, if both statements A and B are false. So with this “or” the statement “A or B” is true, if either A or B are true, but also if both, A like B, are true. This corresponds to the inclusive “or”. When we use the “or” in colloquial language, we sometimes mean, however, the exclusive “or”, i.e. the “either-or”. This is a typical example of the ambiguity of our colloquial language.

These three rules are actually sufficient to construct arbitrary strings.
The expression ¬A ∨ B with the truth tablet is particularly important and useful:

A ¬A B ¬A B
1 0 1 1
1 0 0 0
0 1 1 1
0 1 0 1

The link ¬A ∨ B, also written as A → B, is called implication. This applies to this:
If A, then B. The statement A is therefore a sufficient condition for B: Always, if A is true, B is also true.
The implication is only false if B is false.
If A is false, then the implication A → B according to the truth table is always true, no matter if B is true or false. That means: If the statement A is not sufficient, then one cannot say anything about the truth value of B in the case of a true implication.

In this way, the truth value can always be determined for longer character strings. We would like to give an example here of what will still concern us in the future. We’re looking at the string:

(A ∧ (A → B)) → B.

For these one receives the following truth table:

A B A B A (A B) (A (A B))B
1 1 1 1 1
1 0 0 0 1
0 1 1 0 1
0 0 1 0 1

So there are expressions in propositional logic that are true in every case, regardless of the truth value of the individual statements. Such expressions are called tautologies. We will see in the next blog post what important role such expressions play in the formation of conclusions.

The Syllogisms of Aristotle

In the last blog post, Aristotle presented the three types of conclusions: the logical conclusion, the dialectical conclusion and the false conclusion. Now we will have to deal with the logical conclusion, which derives a true conclusion from two true premises.

This is, of course, the most important conclusion, which ensures a safe “transport” of the truth of statements. That such a possibility exists is the good news, and you can’t overestimate it. The “bad news” would then be that truth is not gained in any way, it is only passed on. The premises must be true. The question as to how and where one can begin with true statements in practice will be of great concern to us later on.

With the logical conclusion one can distinguish now again several kinds. The individual proofs, as he also called them, differ in the nature of the premises and the conclusion, which then results from a “summation” of the premises. In the Greek “summation” means “συν-λογισμός” (syllogismos); thus, one also speaks of a syllogism, the doctrine of syllogisms is called syllogistics. 

Let us first look at an example of syllogism:

All humans are mortal.
All Greeks are humans.
Therefore: All Greeks are mortal.

So, there are three statements, two propositions from which one proceeds, and a conclusion in which the propositions are “added together”.

The form of sentences and their representation

In this example, all sentences are of the form “All A are B”.  The sentence “All A are B” can also be formulated as follows: “B applies to all A.”  This formulation suggests that B is a predicate of all A: B is attributed to all A (predicated), e.g.: To be mortal applies to all human beings.

The formulation is also the one that comes closest to the Greek text, so it is the more original form. In scholasticism however, this was rewritten then as “All A are B”. All A has the predicate B: All humans are mortal. 

The above syllogism is then in its original form:

To be mortal belongs to all human beings,
To be a human being belongs to all Greeks.
Therefore: To be mortal belongs to all Greeks.

Moreover, this formulation also suggests a method that was discovered much later to illustrate the relationship between two terms. One takes advantage of the fact that each term applies to a set of entities. “Human beings” can be Greeks but also Egyptians, Thracians or Asians or Europeans. If one represents the set of human beings by a circle or any closed curve (B), the set of Greeks as well (A), then the circle for the Greeks lies within the circle for the human beings.

Fig.1: The set of Greeks (A) is a subset of the set human beings (B). To be a human being (B) applies to all Greeks (A).

Such pictures are called Venn diagrams, after the mathematician John Venn (1834 to 1923), who introduced them following Leonard Euler (1707 to 1783). Actually, the philosopher Gottfried Wilhelm Leibniz (1646 to 1716) already used them. These Venn diagrams thus illustrate relationships between two sets in general, no matter of what kind the elements are. Within the framework of set theory, we would also write: A ⊂ B, i.e. A is a subset of B.

But this form of a sentence is not the only one at syllogisms. Aristotle obtained an overview of all possible forms of sentences and statements. This resulted in: (Prior Analytics, Book I, 1,  see
https://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html):

A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite.
By universal I mean the statement that something belongs to all or none of something else;
by particular that it belongs to some or not to some or not to all;

by indefinite that it does or does not belong.

Thus, beside the general sentences like “All Greeks are human beings” or “To be human belongs to all Greeks” there is also the negation (no-compliance) and the particular statement. Altogether you get the following types of sentences:

B belongs to all A, (all A are B),
B doesn’t belong to any A, (no A is B),
B belongs to some A, (Some A are B),
B doesn’t belong to some A, (some A are not B).

We can illustrate the other types of statements as follows:

 Fig. 2: Left: B doesn’t belong to any A. Middle: B belongs to some A. Right: B doesn’t belong to some A.

In scholastic circles these forms are abbreviated as (A a B), (A e B), (A i B) and (A o B). The letters “a” and “i” should remind us of “affirmo”, “e” and “o” of “nego”.

The structure of syllogisms

Two such forms of statement then form the premises, one such the conclusion. Each conclusion can thus be characterized by three letters from the set {a,e,i,o} and by the position of the three letters, which stand for the concepts in the respective sentence.

One of these terms, the so-called middle term, must occur in both conditions, it can be in first or second place, or in one condition in first, in the other in second. This results in four different shapes or figures. The first figure is the following (see example above)

A – B

B – C

A – C.

Now Aristotle selects in all forms that combination of two forms of statement that necessarily lead to a conclusion. He simply sorts out the conclusions for which he finds a counterexample.

The valid conclusions can then each be represented by the form and a specific combination of the letters a,e,i,o. And in order to be able to remember such combinations better, one has integrated them into corresponding peculiar words, e.g. one remembers the combination a a a with the word “Barbara”, and knows in addition that here the 1st figure is present. This conclusion corresponds exactly to the above example.

Fig.3: The Barbara syllogism

Another important conclusion, also of the 1st figure, is called “Celarent” in this way:

Fig.4: The syllogism Celarent

Aristotle demonstrated in this way that one could systematically formulate a system of conclusions in which from the truth of the premises necessarily follows the truth of the conclusion.

With the systematics of Aristotle, one now has a complete overview of all possible conclusions. Previously however, in syllogistics, “the whole art of syllogistics had consisted in searching around with great effort of time and effort”. as Aristotle wrote in his work On Sophistical Refutations (after Schupp, I 275).

But here one already has a system of statements that is reminiscent of the formal predicate logic that will be introduced later. Like there, you can “quantify” using the predicates, i.e. you can operate with quantities such as “all”, “none” and “some” for the predicates.

Most importantly, Aristotle must select the valid conclusions “by hand” from the set of all possible combinations, simply by discarding those that he recognizes as invalid with the help of an example. This “recognition” is an intuitive one, one with “common sense”. One does not doubt the correctness of the conclusion, but for a strict science in today’s sense this kind of knowledge is not sufficient.

Even if one describes the relations of terms mentioned above with the help of set theory, and thus justifies the conclusions within the framework of set theory, the insight is a mathematical one. Thus, the conclusions would be only indirectly logically justified, because mathematics only uses, as we know today, the rules of inference that are ultimately gained in modern predicate logic. Only in this predicate logic can the conclusions be strictly justified by deriving them from tautologies. We’ll see about that in a later blog post. Only in this logic is the bottom reached on which one can incontestably win a true sentence again from true sentences.

We still would have to comment on Aristotle’s remark on the case that the conditions are only likely to be true or credible. Then the conclusion could also only probably be true. Saying more here was not possible at that time.  It was not until the beginning of the 20th century that a theory of probability was developed with which one can become more precise in this case. This will be explained in more detail in a later blog post on the topic “How to deal with insecure knowledge”.

Syllogisms in an axiomatic-deductive system

But Aristotle’s logic has not only shown us how safe knowledge can be passed on safely. He has also shown that “It is possible also to reduce all syllogisms to the universal syllogisms in the first figure.”, as he writes in the Prior Analytics, 1st book, 7th chapter (see https://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html).

Thus, we already have a concrete axiomatic-deductive system. This logical order of the statements in a field of knowledge represented for him the ideal of a science. In the Posterior Analytics, 1st book, 3rd chapter he writes:

On the other hand, I maintain that any science must be based on proofs, but that the knowledge of the unmediated principles is not provable. And it is clear that this must be necessary. For since a knowledge of the earlier propositions from which the proof is made is necessary, but one stops once at unmediated propositions, these must necessarily be unprovable. This is my view and I maintain that there are not only sciences, but also supreme principles of them through which we learn the concepts of conclusion. (Here I prefer the translation of a german version in http://www.zeno.org/Philosophie/M/Aristoteles/Organon/Zweite+Analytiken+oder+Lehre+vom+Erkennen/1.+Buch/3.+Kapitel).

The mathematician Euclid of Alexandria logically arranged the geometric knowledge of that time in this way in the time around -300. This organization of a scientific theory as an axiomatic-deductive system is still a model for any rigorous science today. One tried in all centuries to imitate this organization of a thought building, thus “more geometrico”, after kind of the Euclidean geometry, to arrange the knowledge of his science. In “Die Idee der Wissenschaft – Ihr Schicksal in Physik, Rechtswissenschaft und Theologie” I described how successful it has been to this day to realize this idea (Honerkamp, 2017).

Who first had this idea is not clear. The mathematical proof was already known to the Pythagoreans. If there are enough statements in an area, secured by proofs, one will probably at some point consider which statements could be regarded as axioms. This poses the question: Which statements do I need as a basis in order to be able to deduce all the others from them? Or also: How do I create a logical order?

Such specifications will then always be special statements, have special properties. They can be immediately insightful, i.e. “certain by themselves”, as in theories of mathematics, but also highly abstract and far from our idea, as in theories of physics.

Aristotle and the Stoa

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.

Socrates, Plato and Aristotle

If I wanted to write a treatise on the pre-Socratics, I would now have to deal in this blog post with the so-called pluralists and atomists such as Anaxagoras, Empedocles, Leukipp and Democritus. But I do not want to give an outline of the philosophy of the pre-Socratics here. This has happened often enough elsewhere, and certainly after a much more detailed study of the literature of the past millennia than I can show. I am essentially interested in the thoughts of the pre-Socratics, in which one can discover the precursors of the way of thinking of modern physics, mathematics and logic.
There it is without doubt the three basic ideas which have been expressed so aptly and so early in the history of ideas by the school of Miletus, by the Pythagoreans and by Xenophanes: In nature there is such a thing as a causal connection, the regularities in nature can be formulated in the language of mathematics, and knowledge can be attained by “finding the better by searching”. But there can be no ultimate knowledge.
The first tentative steps of the pre-Socratics towards basic concepts such as ” motion ” are also remarkable. The long history of the clarification of this concept over the millennia clearly shows how much effort it takes to gradually move from the darkness of the first reflections to a clear idea with which one can argue reliably.
If I now follow the development of the “logos” in the history of philosophy, I will be even more selective. I will concentrate on the highlights, i.e. the works where substantial progress has been made. Of course, one often recognizes these only by looking back from today’s standpoint. The focus will be on such developments in logic, mathematics and physics that have become significant for the present state of the art in terms of both methods and the formation of concepts.
So, if one begins to look at the development in the time after the pre-Socratics, one must first speak of the three greats of Greek philosophy: Socrates (-469 to -399), Plato (-428 to -348) and Aristotle (-384 to -322). As already mentioned in the first blog post, Socrates introduced a new topic into philosophy: Ethics. But it was not only this new topic that led to a new era in philosophy. Anaxagoras from Kleizomenai ( Ionia ) had moved to Athens in the year -462 and had made the thoughts of the Milesians and other pre-Socratics known there.
Where more people can come together, it is also more likely that people with the same interests will meet and talk about them. In addition to teacher-pupil relationships, communities can now also emerge in which discussions can take place on an equal footing. Dialectics came into fashion. Already Zenon of Elea had been her great friend, Aristotle had later even praised him as their inventor (Mansfeld & Primavesi, 2011, pp. 361, No.4). Attic democracy flourished, it was an early type of democracy in which the “people of state” ruled, where the people of state consisted of the full male citizens of the city of Athens who had reached the age of 30. These full citizens could also turn to matters which were not directly necessary to daily life; for such work there were enough slaves, women or immigrants. Especially in political meetings and in court, eloquence and the art of dialectics were in demand, i.e. to find the right reasons for opinions and to spread those that could be applauded by listeners. Sophists taught for money to master this art, even to be able to skilfully involve the opponent in a contradiction in discussions.
About Socrates we know some about Diogenes Laertius (Laertius, 2015, pp. 67-90), in particular he is supposed to have said: ” I know nothing except that I know nothing” (DL 83). And “the good is nothing small, but start with small things” (DL 83). But we don’t know any of his writings. Plato, however, puts a lot into his mouth in his works. These works always represent dialogues that Socrates conducts with a wide variety of dialogue partners and in which it is always a matter of clarifying a concept or a question.

Plato

After the death of Socrates, Plato, in his years of teaching and travel, first stayed with a follower of the philosophy of Parmenides, “travelled to Cyrene to the mathematician Theodoros and from there to Italy to the Pythagoreans Philolaos and Eurytos; from there to Egypt to the prophets” (Laertius, 2015, p. 141). Thus he collected various impressions and combined them to a grandiose mythical thought building, which is still effective today and continues to inspire many people.
He must have been a great storyteller. In his works he described in the form of a dialogue the effort to find concepts and answers to the question of correct behaviour or a good life. Who first chose this literary form is controversial. “It is said that Zenon, the Eleate, was the first to write dialogues, but Aristotle names as such […] Alexamos from Styra […]. I, on the other hand, think that Plato, through his strict treatment and training […], has secured himself the right to first place […]”. Dialogue is “a question-and-answer execution of a philosophical or political theme”, but dialectics “is the art of persuasion, through which we prove something void or right on the basis of the question-and-answer procedure of the undermentioners” (all quotations: (Laertius, 2015, p. 159)).
Diogenes Laertius says of Plato’s works: “TIMAIOS belongs to the field of physics, to the logic belongs POLITIKOS, KRATYLOS, PARMENIDES and SOPHISTES” (Laertius, 2015, p. 160).
When he, as in TIMAIOS, came to talk about the ideas of the pre-Socratics about the cosmos, he did so in mythical form or tied to Pythagoras and saw the world built up from geometric forms. At that time, logic was still predominantly understood to mean the work on terms, i.e. the uncovering of the relationships between related terms, e.g. their classification as superordinate or subordinate terms. A clearer definition was expected from such a division of terms, called “dihairesis”. One could already see, however, that a definition often required definitions of the determinants, thus leading to an infinite regress.
Aristotle grew up in the Academy with this method. Thus, in his logic, which he was soon to develop, the concepts were also at the centre. This thus became a so-called term logic. Modern logic, on the other hand, is a propositional logic. In the next blog post I will work out the difference exactly.
Plato mainly used induction, i.e. the conclusion from special to general, for a demonstration of evidence, “which through some true cases opens up the same truth for other cases in an appropriate way”, as Laertius (Laertius, 2015, p. 161) says. For a long time this conclusion was considered “appropriate”, only David Hume stressed that this conclusion is not mandatory, thus not always “appropriate”. In modern logic there is no place for this conclusion either.
Plato’s central theme, however, was the human soul and how it can express itself in language, ethics, art and politics. He adorns existing myths, redesigns them or invents completely new stories.

The Platonic Academy

For the development of the “logos” all this does not give much. It is interesting in this context, however, that he also promoted training in analytical subjects, probably motivated by the impression that Pythagorean mathematicians made on him. He was not only a great narrator, but also a good organizer and designer. He founded a school where young people were taught philosophy and science. For this he acquired, probably in -387, a plot of land near a grove called Akademeia, because it had been dedicated to the Heros Akademos. In addition to metaphysics, ethics, dialectics and the doctrine of the soul, physics and especially mathematics were studied there, which soon became part of the basic education of every student of a philosopher. The academy remained, with interruptions, over many centuries, it was closed only around 530.
The Platonic Academy became a model: since Augustine (354 to 430) and Martianus Capella, a Roman encyclopedist from the 5th or 6th century, a canon of seven subjects had become established for the schools of late antiquity. This was divided into a trivium (grammar, rhetoric and dialectic or logic) and a quadrivium (arithmetic, geometry, astronomy, music theory). These subjects were called the seven liberal arts because they were “worthy of a free man,” and “free” was a man when he was free from the necessity of earning a living.
In the Middle Ages this study of the liberal arts was regarded as preparation for scientific studies in theology, jurisprudence and medicine. In the universities, the liberal arts were soon taught within the framework of a separate faculty, the Facultas Artium. The “Liberal Arts”, which we know from the USA e.g. as “undergraduate studies” and which are introduced nowadays also e.g. in Germany, try to continue this tradition. However, dialectics is more in the foreground here than any of the mathematical disciplines.

Aristotle

Aristotle, a pupil of Plato’s, had particularly fertile ground for his rigorous training at the Platonic Academy. Born into an educated and wealthy family in Stageira (Chaldidike) in the year -384, he was sent to Plato’s Academy in Athens in the year -367 at the age of 17. The mathematician Eudoxos von Knidos (ca. -395 to ca. -350) played an important role there.
Aristotle, on the one hand, loved to argue and discuss with dissenters. On the other hand, he showed – as with people who have not passed a mathematical education without consequences – a tendency to work systematically and to collect the doctrines of former philosophers such as Pythagoras or Democritus (Schupp, I 256).
Aristotle became such a great systematist, attentive to the methodical in a thought process and vigilant for connections. He was the first of this kind, and he was also to establish a completely new kind of philosophy, which is no longer a poiesis, a “production” of doctrines in a poetic manner. Instead, the focus is on analysis and methods, reasons are required and the results are checked for conclusiveness.
It is probably plausible that one feels the urge for such things when one has a large corpus of doctrines before one’s mind’s eye and at some point wonders why the respective philosophers can so firmly hold their views. But you also have to live in a stimulating environment and, above all, have the talent to make such a new start. So Aristotle soon went his own way in the academy. According to Diogenes Laertius, Plato is said to have said: “Aristotle beat me like young fillings do against their own mother” (Laertius, 2015, p. 225).
In the list of Aristotle’s writings, Laertius lists 146 titles and speaks of a total of 445 270 lines. According to him, the work on logic is “very clearly marked as a tool for all sub-areas.” These writings were later combined into a collection of six books entitled “Organon” (gr. ὄργανον = tool).

The Organon

The titles of the six books in the Organon Collection are as follows:
1. the categories,
2. on the interpretation (peri hermeneias),
3. the doctrine of the logical conclusion (analytika protera, first analysis),
4. the study of evidence (analytika hystera, second analysis),
5. the topics and
6. the Sophist Refutations
The books 3,4 and 5 are particularly important for the development of logical thinking, i.e. the doctrines of the logical conclusion and the proof as well as the topics. Here the so-called Aristotelian logic is developed. This is therefore the fruit of a time in which dialectics occupied a high position in society. Even in discussions about more fundamental questions such as those about a good life or “reasonable” morals, it was soon more a question of the power of the arguments than of authoritarian settlements, and finally it could not fail to be thought about directly which forms of argumentation possess such power that they are incontestable, i.e. must be accepted by every sensible person.
The new era of philosophy, with its emphasis on dialectics, has therefore by no means allowed the idea of the logos to recede into the background or hindered its dissemination. On the contrary, dialectics was precisely the field in which this idea could prove to be a particularly useful and valuable tool for thinking. The idea took shape in Aristotelian logic, which was to become a model for a science that today plays a significant role within the framework of artificial intelligence. In the next blog post I will deal explicitly with Aristotelian logic.

Aristotle and the System of Sciences

Aristotle’s talent for analytical thinking and systematic work, as well as his passion for collecting the doctrines of earlier philosophers, showed other fruits. He establishes a systematics of all sciences known at that time. It divides the sciences into practical and ethical sciences, into poetic (manufacturing) sciences such as medicine or craftsmanship, and finally into theoretical sciences. In the latter he distinguished mathematics, natural science and “first philosophy”, to which he added theology, ontology and logic. Theology at that time consisted essentially in a study of the unchangeable “divine” stars.
This division of Aristotle is, so to speak, the first “cashing down” of knowledge of a time and subsumes everything that creates knowledge under science. Probably he also counted the theoretical sciences of mathematics and natural science among philosophy, albeit not among the “first”. About 500 years later Diogenes writes of Laertius:
As for the parts of philosophy, there are three: physics, ethics, and dialectics.(Laertius, 2015, p. 10),
whereby dialectic here also stands for the doctrine of thought principles, i.e. for logic and more generally for epistemology. Ethics had thus immigrated into philosophy and mathematics had been eliminated. There could no longer be a thinker like Pythagoras, who practiced mathematics but also represented a certain world view with religious fervour. Important ancient mathematicians, such as Euclid of Alexandria or Archimedes of Syracuse, do not appear in the work of Diogenes Laertius on the lives and opinions of famous philosophers – except for Eudoxos of Knidos, who was a member of Plato’s Academy for a time.
But physics, too, was to emigrate from philosophy, albeit only about 2,000 years later, when Galileo Galilei overcame the Aristotelian theory of motion. By demonstrating how to describe regularities in nature in the language of mathematics, he discovered a “new science”. Natural science became a modern physics. The consequences of this discovery will be discussed in detail.
Finally, at the end of the 19th century, logic in the form of mathematical logic emigrated from philosophy. It is now a branch of mathematics and computer science. Nowadays it is even observed how epistemology is becoming an area of cognitive science that not only deals with our ability to think, but also with all conscious and unconscious processes in our brain such as perceiving, learning or remembering.
Cosmology, the theme of the pre-Socratics and the first theme of philosophy ever, is today an area of modern physics. A history of cosmology from the pre-Socratics to Hawking would be highly interesting: the “question of the whole” has arisen in mythical thinking and in this form is still present in all religions, and the pre-Socratics already began to look for physical reasons for their ideas. But only for about 100 years has there been a physical cosmology in which the logos rule strictly. This cosmology is a reconstruction of the history of the universe with consistent consideration of physical theories. The development of the cosmos could only be convincingly told after the phenomena of nature had been understood on the grounds of reliable theories. Because this knowledge was just the necessary guidelines for a history of the universe, which no longer wants to be a mythical but a logical one.