Direct Reduction of Syllogisms with Byzantine Diagrams

Abstract

The paper explores the potential of Byzantine diagrams in syllogistic logic. Byzantine diagrams are originated by Byzantine scholars in the early modern period to use as tools for teaching and studying Aristotelian logic. This paper presents pioneering work on employing Byzantine diagrams for checking syllogistic validity through reduction.

KEYWORDS:

• Byzantine diagrams
• syllogism
• direct proof
• visual reasoning
• early modern logic

1. Introduction

Following Charles Sanders Peirce, a logic diagram is

a diagram composed of dots, lines, etc, in which logical relations are signified by such spatial relations that the necessary consequences of these logical relations are at the same time signified, or can, at least, be made evident by transforming the diagram in certain ways which conventional “rules” permit.¹

Logic diagrams enhance visual cognition of extracting information. Due to its so-called free ride feature, one can extract even implicit information from logic diagrams (Shimojima 1996). Furthermore, it is often said that they are intuitive and easy to use in contrast to other kinds of notations.

Groundbreaking work on logic diagrams was provided by Sun-Joo Shin in the 1990s when she created a logic diagrams system which was sound and complete. She has also shown that her formal diagram system is equivalent to monadic predicate logic (Shin 1994). After Shin, there were much more authors who provided formal logic diagram systems such as Bhattacharjee, Hammer, Jamnik, Howse, Stapelton et al. (Bhattacharjee et al. 2022; Hammer 1996; Jamnik et al. 1999; Howse et al. 2005; Stapleton et al.2017).

Logic diagrams include different kinds of geometric figures. Most famous are the circular ones such as Euler and Venn diagrams (Moktefi and Shin 2012; Moktefi and Lemanski 2022). But there are also other kinds of geometric figures to build logic diagrams such as tabular diagrams (Lemanski and Jansen 2020), linear diagrams (Moktefi 2013; Englebretsen 2019; Moktefi et al. 2014) or tree diagrams (Hacking 2007). Some other diagrams like Byzantine diagrams include crescents and triangles (Panizza 2017), and there exist also diagrams such as the square of opposition (Smessaert and Demey 2014; Schumann 2013).

All the logic diagrams mentioned have in common that they have a long history. Even though they are used today in modern formal systems, they have been used in a heuristic way or for educational purposes since antiquity or since the Middle Ages at the latest (Moktefi and Shin 2012; Lemanski 2021, sect. 2.2). Because of this long tradition, the history of logic diagrams may help us to develop modern formal systems of diagrams, to understand their cognitive features and their interaction. However, there are also types of diagrams that have hardly been studied so far and whose logical function is hardly known. Among these unknown diagrams are Byzantine diagrams.

As the name says Byzantine diagrams have mainly been used by Byzantine scholars for studying and teaching Aristotelian logic. As it was mentioned, these diagrams include crescents and triangles and depict mainly the major, middle and minor terms of a syllogism through their vertices. They also show the relations among these terms by edges. After introducing these diagrams in esp. the Italian and French region in early modern logic, these diagrams have been used and expanded by diverse scholars. Thus, they have been used for a long time, from about the end of the Middle Ages until the Thirty Years' War in Europe. The 16th century was the high period in which these diagrams were used such that nearly all logic textbooks of that time included Byzantine diagrams.

So far Byzantine diagrams have been researched in art history (Safran 2022; Cacouros 2001; Roberts 2022), music history research and regarding the general history and origin of these diagrams (Panizza 2017; Lemanski 2021, pp. 184–6). The most important modern research paper was written by Panizza (Panizza 2017). This paper showed how to represent different syllogistic figures in Byzantine diagrams. However, the paper tells almost nothing about how to find the validity of syllogism with this kind of diagram. In this respect, the paper was more attributed to the history of these diagrams in the course of learning syllogistic reasoning. Thus, we yet only have a simplified idea of how these diagrams work.

The plan of this paper is to present a pioneer work on diagrammatic reasoning with Byzantine diagrams. The main thesis of this paper is that Byzantine diagrams are not only a memory aid and diagrammatic representation of terms and syllogisms but can also be employed for diagrammatic reasoning. Here, with diagrammatic reasoning, it means reduction, not deduction. Thus the focus of the paper is on finding the validity of syllogisms by using a reduction method. As this is the first work in modern times on diagrammatic reasoning with Byzantine diagrams, the paper only focuses on direct proofs here as using Byzantine diagrams in indirect proofs require a lot of discussion which cannot be done in such a constraint space.

In Section 2, traditional syllogistic reasoning is introduced. The third section represents Byzantine diagrams in general and it also mentions the required improvisation. Here, Panizza's approach concerning diagrammatic representation is extended. The fourth section shows how to use Byzantine diagrams in direct proofs. Finally, the section offers a method of checking the validity of syllogisms. Readers who are familiar with traditional syllogistic reasoning can thus directly jump to Section 3.

2. Traditional Syllogistic Reasoning

A syllogism is a deductive inference that contains three categorical judgements. A categorical judgement is either universal or particular and either affirmative or negative. The relations between the pairs of categorical judgements are described by the square of opposition (see Figure 1) (Demey 2020; Chatti and Schang 2013).

In Figure 1, the letter ‘a’ denotes the universal affirmative judgement of the form ‘All A is B’, denoted as aAB (A and B are any two terms). The letter ‘e’ denotes the universal negative judgement, ‘No A is B’ (eAB). The letters, ‘i’ and ‘o’ denote the particular propositions. ‘i’ is a particular affirmative judgement of the form ‘Some A is B’ (iAB) and ‘o’ is a particular negative judgement of the form ‘Some A is not B’ (oAB). The logical relations between these judgments are named as contradiction, contrariety, subcontrariety and subalternation. Where,

• a and o as well as i and e form a contradiction, i.e. if one of the pairs is true, the other is false.

• The contrarities a and e cannot be true but false at the same time.

• i and o are subcontrarity, i.e. both can be true together but not false together.

• The subalternation states that if a and e is true, then i and o are also true respectively, but not the converse.

Figure 1. Square of opposition.

Going back to syllogism, the first two categorical judgements are called premises. The last categorical judgement, which is the logical consequence of the premises, is called the conclusion. There are exactly three terms in each syllogism and each of them occurs exactly twice in different categorical judgements. The term that occurs only in the premises and not in the conclusion is called the middle term ‘M’. The term that appears in the subject position of the conclusion is called the minor term and it is denoted by ‘S’. The term in the predicate position of the conclusion is called the major term and is denoted by ‘P’. Major and minor premises are the categorical judgements that contain P and S respectively. Depending on the position of the middle term in the two premises, there are the following four figures in syllogisms.

• 1st figure: MS, PM

• 2nd figure: MS, MP

• 3rd figure: SM, PM

• 4th figure: SM, MP

With four figures and four categorical sentence types each, which can occur in the two premises and in the conclusion, there are $4^4=256$ moods. However, only 24 are considered to be valid (5 of them are conditionally valid). Among these 24 valid moods, the four 1st figure moods are called the ‘perfect’ moods. The names of these four moods are Barbara, Celarent, Darii and Ferio. All the other moods are considered imperfect, and their validity is proven by reducing them to the perfect moods.

There are two methods of reduction, the direct method and the indirect method. This paper only deals with the direct proof method, which can be applied to almost all syllogisms except for the moods Baroco and Bocardo. In the direct proof method, an imperfect mood is transformed into a perfect one by using one of the following transformation rules.

• conversio simplex: reversal of the subject-predicate relations in either e or i judgements by means of pure conversion.

• conversio per accidens: transformation of the a or e judgements into i or o judgements respectively by means of impure conversion.

• mutatio: reversal of the order of the premises.

To find to which perfect mood an imperfect mood should be reduced or which transformation rules should be used for reduction, coded letters have been used for ages. The first letter of the imperfect moods gives a hint as to which perfect mood it is to be reduced. For example, the fourth figure mood ‘Bamalip’ started with ‘B’, hence it is reduced to the perfect mood Barbara. Similarly, any imperfect moods having first letters C, D or F shall be reduced to perfect moods Celarent, Darii or Ferio respectively. The letter codes ‘s’, ‘p’ and ‘m’ denote the transformation rules conversio simplex, conversio per accidens and mutatio respectively. As previously mentioned a, e, i and o denote the four categorical judgements. The vowels, in the name for the imperfect moods, through the order of their position indicate the two premises and the conclusion. For the 2nd figure mood Celarent, this indicates that this mood contains e and a judgements in the first and second premise respectively. It also contains, e judgement as the conclusion. The presence of the consonants ‘s’ or ‘p’ means that the preceding vowel should undergo the respective transformation rules. For example, the e judgement in the 2nd figure mood ‘Cesare’ undergo the conversio simplex rule as it is preceding ‘s’. Similarly, the a judgement preceding ‘p’ undergoes conversio per accidens in the 3rd figure mood Darapti. The consonant ‘m’ between any two premises represents the reversal of order for those premises. For example, the a judgement in the first premise and the e judgement in the second premise reverse their order and becomes the second and first premise respectively in Camestres. If the consonant ‘c’ is present in the name of any moods, like Baroco and Bocardo, then the indirect method should be used for its reduction. If an imperfect mood can be reduced into a perfect mood with the help of at least one transformation rule, then the imperfect is valid. If the mood cannot be reduced, then it is invalid. In Sections 4 and 5, it will be shown that to prove whether an imperfect mood is valid or not, using Byzantine diagrams, coded letters are not needed.

3. Byzantine Diagrams

The purpose of this paper is not to investigate the history of Byzantine diagrams as this was done by Panizza before. In Panizza 2017, Panizza explores the history, in which syllogisms are illustrated by Byzantine diagrams. These diagrams were mainly used by Byzantine Greeks and later imported to Italy. Panizza specifically focuses on the ‘Latinization of Byzantine diagrams’ (Panizza 2017, p. 23). The Byzantine diagram was brought to Italy along with Greek manuscripts of Aristotle and Byzantine commentators on Aristotle and was spread by Greek scholars and teachers through their Latin translations of Aristotle, and later popularised by some of the Italian scholars during the early modern period esp. in central Europe. Panizza also examines how this tradition was received in ‘Latin translations, commentaries, and manuals of logic’ (Panizza 2017, p. 23) and how these diagrams changed their appearance over the course of time.

When writing about how these diagrams work in Italian texts, Panizza mainly discussed Johannes Argyropoulos (c. 1415–87), Ermolao Barbaro (1454–93) and some others. In order to see how many Byzantine diagrams there are in history, the following four books are added for the first decade of the 16th century in which Byzantine diagrams are included primarily in Aphrodisiensis and Philoponus 1501; Aristotle and d'Étaples 1503; Zimara and Aristotle 1508; Trapezuntius 1509. A lot more books with byzantine diagrams, along with the aforementioned ones, can be found via the digital repository in Section E (tag: ‘Byzantine diagrams’): History of Euler-Venn-Diagrams 2015.

But this paper is not going to delve into that historical discussion as it is not the main objective of this paper. Rather it will focus on how these diagrams work. The first, second, and third figures of traditional syllogistic are denoted by the Byzantine diagrams in Figures 2, 3 and 4, respectively. The three corners of each diagram represent the major (P), minor (S) and middle (M) terms.

Figure 2.  

Figure 3.  

Figure 4.  

Panizza improvised these diagrams to enhance the visual perception of this method: ‘I have added arrows and numbers to the diagrams found in Renaissance books to show not only the order of the premises but also the position of Subject, Predicate and Middle Term within each’ (Panizza 2017, p. 36).

This paper is going to incorporate Panizza's improvisation in the current approach but it will extend it much further. With the help of this extension, Byzantine diagrams can not only represent syllogisms, but one can also reason with them. Following Panizza, as shown in Figures 5, 6 and 7, the vertices of these three diagrams denote the major, middle and minor terms. In all of these diagrams, 1, 2 and 3 denote the major premise, minor premise and the conclusion respectively. The relation among the major, middle, and minor terms are shown by the arrows. In these diagrams, the arrowhead denotes the position of the predicate and the tail denotes the position of the subject. Since the middle term in the 1st premise of the 1st Figure 5 is in the subject position and the major term is in the predicate position, the arrow goes from M to P. Similarly, for the 2nd premise of the 1st Figure 5, the arrow goes from S to M, as S is the subject and M is the predicate of the 2nd premise. The same convention goes for other figures.

Figure 5.  

Figure 6.  

Figure 7.  

But instead of marking the premises and conclusion with numbers here the premises are marked depending on what kind of judgements they are i.e. with a, e, i and o. The following diagrams in Figures 8, 9 and 10 are examples of Barbara, Cesare and Datisi respectively.

Figure 8.  

Figure 9.  

Figure 10.  

As far as we know, there were no different diagrams proposed by the Byzantine Greeks for the 4th figure. This paper proposes to use the inverted crescent Figure 11 as the diagram for the 4th figure.

Figure 11.  

This method is analogous to how the 3rd figure is shown by inverting the diagram used for the 2nd figure. If one looks at the structure of the premises for the 2nd and 3rd figures in the following, it can be seen that they are like mirror images to one another. Similarly, the diagrams for these two figures, not including the direction of the arrows, are mirror images of each other.

Likewise, the premise structure of the 1st and 4th figures are mirror images of each other so they should have similar inverted position diagrams like the 2nd and 3rd figures. However, the inverted crescent has been used in this paper for showing the reduction in Byzantine diagrams (it will be explained much more clearly in Sections 4 and 5). Furthermore, to check the validity of the 1st figure, an inverted crescent is also going to be used. Thus, from here onward the perfect mood will have a dotted line crescent or inverted crescent as a diagram to avoid any ambiguity (see Figure 12). Any other 1st figure will have the diagram as shown in Figure 5.

Figure 12.  

4. Direct Proofs with Byzantine Diagrams

This section explains how these diagrams can represent syllogistic reduction through direct proofs. Each reduction method will be discussed individually and only the rules for application will be stated. Later, in Section 5, it will be shown how to use these rules to check the syllogism's validity.

4.1. Conversio Simplex

Conversio simplex denotes which premise or conclusion has to undergo pure conversion by observing the position of the consonant ‘s’ in the code of the moods of the syllogisms. This rule can be easily represented using Byzantine diagrams without knowing the position of ‘s’. Firstly, the diagram for the perfect mood will be placed over the 2nd figure or placed below the 1st (imperfect moods), 3rd or 4th figures. Secondly, by comparing the arrows from the subject to the predicate in the 1st, 2nd, 3rd or 4th figures with those in the perfect mood, the premises or the conclusion, where the arrows do not have the same direction as the arrows in the perfect mood, are converted. For example, let us consider the moods Cesare (Figure 13) and Festino (Figure 14) of the 2nd figure. When comparing the diagrams of both figures, it can be observed that the arrows of the 1st premises of the 2nd figure are in the opposite direction to those of the perfect mood. The wavy lines mark these arrows. Therefore, if the subject and predicate of the first premises of these two imperfect moods undergo pure conversion, then these figures can be reduced to the perfect moods Celarent and Ferio, respectively.

Figure 13.  

Figure 14.  

It should be noted that one should not convert the subject-predicate relation of random propositions in the imperfect moods based solely on the direction of the arrows. The conversions mentioned above are valid only for judgements that are either e or i. Therefore, the reduction rule for simple conversion in Byzantine diagrams is outlined below:

1. Place the diagrams for the perfect mood either over the 2nd figure or below the 1st, 3rd or the 4th figures.

2. Compare the direction of the arrows in the imperfect moods with those in the perfect mood.

3. The subject-predicate relation in the premise or conclusion of the imperfect mood that has an arrow in the opposite direction to the perfect mood may be altered provided the judgement for that premise or conclusion is either e or i.

By applying the conversio simplex rule to the previous examples as shown in Figures 13 and 14, the subject-predicate relation in the first premises of the imperfect moods now have been reversed (see Figures 15 and 16, respectively).

Figure 15.  

Figure 16.  

4.2. Conversio per Accidens

According to the square of opposition, the universal judgements a and e entails the particular judgements i and o respectively. Thus, the traditional conversio per accidens rule permits these universal judgements to undergo a partial conversion and convert into their respective particular consequences. This reduction method can also be used in Byzantine diagrams to partially convert the premise or the conclusion of a particular mood under validity checking. Similar to conversio simplex, in the conversio per accidens rules, the diagram for the perfect mood is placed over or below the moods going under the reduction. For example, the following two 3rd figure moods Darapti (Figure 17) and Felapton (Figure 18) can be reduced into the perfect moods Darii and Ferio, respectively by converting a to i judgement (as marked by the wavy arrow).

Figure 17.  

Figure 18.  

The reduction rule is as follows.

1. Place the diagrams for the perfect mood either over the 2nd figure or below the 1st, 3rd or the 4th figures.

2. Compare the direction of the arrows in the 1st, 2nd, 3rd or 4th figures with those in the perfect mood.

3. The a or e judgement of the premises of the 1st, 2nd or 3rd or 4th figures that have an arrow in the opposite direction to that of the perfect mood may be converted to i or o judgement respectively.

4. The a or e judgement of the perfect mood that has an arrow in the opposite or same direction to that of the 1st, 2nd or 3rd or 4th figures may be converted to i or o judgement respectively provided only the conclusion is under conversion.

The following diagrams represent the moods Darapti (Figure 19) and Felapton (Figure 20) where the a judgements in the second premises have now converted to i judgements using the conversio per accidence rule.

Figure 19.  

Figure 20.  

4.3. Mutatio

In the mutatio rule, the order of the premises of the imperfect moods is reversed to match the order of the premises in the perfect moods. As an example, consider the 2nd figure mood Camestres. Camestres is reduced to Celarent to prove its validity. By placing the diagram for Celarent over Camestres, as shown in Figure 21, the judgements of the premises of these two moods can be compared. The major premise of Celarent has the same judgement as that of the minor premise of Camestres. A similar situation can be seen for the remaining two premises of these two moods. Thus the order of the premises for the mood Camestres (see Figure 22) has been reversed.

Figure 21.  

Figure 22.  

After the reversion, the previously minor premise for Camestres is now its major premise. So, this minor premise is now denoted as $P^{\mathrm{\prime }}$ (see Figure 23) and it is the new major premise of Camestres after the reversion. Similarly, the previously major premise, which has now become the minor premise is denoted by $S^{\mathrm{\prime }}$ (Figure 23). Now choose the perfect mood Celarent that has $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ as the major and minor terms respectively (unlike the mood Celarent in Figure 21 which had P and S as the major and minor term respectively). Finally, compare the mood Camestres to Celarent (Figure 24). Now, in Figure 24, it can been seen that the order of the premises in Camestres is the same as the order of the premises in the perfect mood Celarent.

Figure 23.  

Figure 24.  

Mutatio rule can also be used when both the imperfect and perfect moods have the same judgements for both major and minor premises but the subject-predicate relation is altered. For example, consider the 4th figure mood Bramantip. To reduce Bramantip into the perfect mood Barbara, the diagram for Barbara is placed below Bramantip as shown in Figure 25. Comparing the direction of arrows, it can be found that the subject-predicate relation in both the premises of the mood Bramantip is exactly opposite to that of the mood Barbara (Figure 26).

Figure 25.  

Figure 26.  

As previously mentioned in Section 4.1, the direction of the arrow for the a judgement cannot be altered. However, the direction of the arrow in the major premise of Bramantip is the same as that of the minor premise of Barbara i.e. in both cases M is in the position of predicate. Similarly, the direction of the arrow in the minor premise of Bramantip is the same as the arrow in the major premise of Barbara i.e. M is in the position of the subject. Therefore, by applying mutatio rule, these premises in Bramantip can be reversed like the above example. After reversion, the newly obtained terms are named $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ like before (Figure 27). Now Bramantip can be compared to the perfect mood Barbara which has the terms $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ (Figure 28).

Figure 27.  

Figure 28.  

Thus there are two cases of the reduction rule as mentioned below.

Case-1: If the premise set for imperfect mood contained two different judgements, then the reduction rule is as follows.

1. Place the diagrams for the perfect mood either over the 2nd figure or below the 1st, 3rd or the 4th figures.

2. Compare the judgements of the premises of the 1st, 2nd, 3rd or 4th figures to that of the perfect mood.

3. Reverse the order of the premises in the 1st, 2nd, 3rd or 4th figure to match the order of the premises in the perfect mood.

4. Denote the new major and minor terms as $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ respectively.

5. Choose a perfect mood that has the same premises as the perfect mood in step 1, but has $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ as the major and minor premises respectively. Draw the diagram for this perfect mood over or below the imperfect mood that is being compared.

Case-2: If the premise set for imperfect mood contained two same judgements, then the reduction rule is as follows.

1. Place the diagrams for the perfect mood either over the 2nd figure or below the 1st, 3rd or the 4th figures.

2. Compare the direction of the arrows in the 1st, 2nd, 3rd or 4th figures with those in the perfect mood.

3. If any of the arrows are of different directions to that of the perfect mood, then reverse the order of the premises in the 1st, 2nd or 3rd or 4th figure to match the direction of the premises in the perfect mood.

4. Denote the new major and minor terms as $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ respectively.

5. Choose a perfect mood that has the same premises as the perfect mood in step 1, but has $P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ as the major and minor premises respectively. Draw the diagram for this perfect mood over or below the imperfect mood that is being compared.

5. Checking Validity of the Syllogisms

In the previous section, it has been shown how Byzantine diagrams can represent the three ‘direct proofs’ of reduction. However, it is not mentioned how these representation methods can be used to check validity. For the validity checking of an imperfect mood with Byzantine diagrams, one just needs to compare the diagram for that imperfect mood with the diagrams for one of the perfect moods (Figure 29).

Figure 29.  

The main aim of this comparison is to make needful changes to the diagram for the imperfect mood, according to the rules mentioned in Section 4., so that the diagram has the exact features (the direction of arrows and judgements of the premises/conclusions) of the diagrams for one of the perfect mood. If these changes can be done, then it can be said that the imperfect mood under comparison has been reduced to the perfect mood, hence it is valid. If the imperfect mood cannot be reduced to a perfect mood, then it is an invalid syllogism. So, basically the diagrams in Figure 29 acts here like a set of axioms. Rearrangements are done to the structure of every other mood to illustrate them in a similar structure to that of the perfect moods.

But before proceeding towards the reduction method, to make the validity checking easier, the following types of Byzantine diagrams (Figures 30 and 31) are introduced.

Figure 30.  

Figure 31.  

These types of diagrams are also found throughout some early modern books containing Byzantine diagrams, for example Pacius and Aristotle 1598. However, these diagrams have been used differently here than in their original usage. In this paper, an extended curve over the curve of the conclusion denotes the judgments that follow from the conclusion of the perfect moods. The solid curve over the conclusion in Figures 30 and 31 denotes the judgements i and o which follow from the judgements a and e respectively.

The process of reduction using Byzantine diagrams is very intuitive. By placing the diagrams for the perfect mood over or below the imperfect moods, it becomes evident what kind of structural changes need to be done to the imperfect mood. A user only needs to be careful about the restrictions mentioned in the rules explained in the previous section. Nevertheless, to have an idea of the step-by-step process of the reduction method a reduction algorithm is presented below.

Reduction Algorithm

1. Draw the respective diagram for the mood to be compared and then denote the premises with a, e, i and o.

2. For the imperfect mood with conclusion a and e, follow the following steps.

   1. Choose the perfect mood which has the same set of premises as that of the mood to be compared and go to step (iii) (for example, if the previous mood has the premises, say $\{i,a\}$, then choose the perfect mood Darii which has the same set of premises, i.e. $\{a,i\}$). Otherwise, go to step (ii).

   2. If a perfect mood with the same set of premises cannot be found, then go to step (xvii).

   3. Draw the diagram for the perfect mood over or below the diagram for the imperfect mood and go to the next step.

   4. If the order of the premises is the same for both the moods (perfect and imperfect), then go to step (v). Otherwise, go to step (vi).

   5. If the order of the premises is the same for both the moods (perfect and imperfect) due to having the same judgements in both premises but the order of the subject-predicate relation in the major and minor premises are different, then go to step (vi). If there is no difference, then go to step (vii)

   6. For both steps (iv) and (v) apply the mutatio rule on the imperfect mood. Next, go to step (iv).

   7. If the direction of all the arrows in the imperfect mood is the same as that of the perfect mood, then go to step (x). Otherwise, go to step (viii).

   8. If possible, apply the conversio simplex rule on the imperfect mood. Then go to step (ix). If conversio simplex cannot be applied, then go to step (ix).

   9. If possible, apply the conversio per accidens rule on the imperfect mood. Then go to step (vii). If conversio per accidens cannot be applied then go to step (xiii).

   10. Check whether the conclusion of the imperfect mood is the same as that of the perfect mood. If it is the same, then go to step (xvi). Otherwise, go to step (xi).

   11. If possible, apply conversio per accidents on the perfect mood. Then go to step (vii).

   12. Check whether or not the conclusion of the imperfect mood is the same as that of the perfect mood. If it is the same, then go to step (xvi). Otherwise, go to step (xii).

   13. Choose the perfect mood which has a subalternation of one of the premises of the mood to be compared and go to step (xiv) (e.g. for Fapesmo choose the perfect mood Ferio where i is the subalternate of a).

   14. Go to step (iv) and repeat the previous steps until step (xii) but do not repeat step (xiii). Then go to step (xv).

   15. Check whether or not the conclusion of the imperfect mood is the same as that of the perfect mood. If it is the same, then go to step (xvi). Otherwise, go to step (xvii).

   16. The mood is valid.

   17. The mood is invalid.

3. For the imperfect mood with conclusion i and o follow the following steps.

   1. If possible, choose the perfect mood which has the same set of premises as that of the mood to be compared and go to step (c). Otherwise, go to step (b).

   2. If a perfect mood with same set of premises cannot be found, then go to step (xvii) of point 2.

   3. Extend the conclusion of the perfect mood with the judgement (as shown in Figures 30 and 31). Then go to step (d).

   4. Repeat steps (iii) to (xvii) as mentioned in point 2 with the condition that the conclusion of the imperfect mood can be the same with the extension of the conclusion of the perfect moods even if its not same with the conclusion of the perfect moods.

A few examples are now shown to clarify the above approach. Start with the 1st figure mood Barbari (Figure 32). Now Barbari is to be compared with the perfect mood Barbara as both of them have the same set of premises $\{a,a\}$ [step (a)]. Also, since the conclusion of Barbari is ‘i’, it is compared with the extended Barbara as shown in Figure 33 [step (c)]. According to step (d), steps (iii) to (xvii) of point 2 will now be repeated. The direction of all the arrows of Barbari is the same as that of Barbara [step (vii)]. The conclusion of Barbari is the same as the extended conclusion of Barbara [step (x)]. Hence, according to step (xvi), Barbari is a valid mood.

Figure 32.  

Figure 33.  

Take Camestres as the next example. Since the set of premises of the 2nd figure mood Camestres i.e. $\{a,e\}$ is the same as the set of premises of the perfect mood Celarent ($\{e,a\}$), Camestres is to be compared with Celarent according to step (i). By step (iii), put the diagram for Celarent over Camestres (Figure 34). As the order of the premises of Camestres is opposite to that of Celarent, change the order of the premises by the mutatio rule (Figure 35) [step (vi)].

Figure 34.  

Figure 35.  

Next, check the direction of all the arrows and denote the arrows having opposite directions (Figure 36) [step (vii)]. Change the direction of the arrow through the conversio simplex rule (Figure 37) [step (viii)]. In Figure 37, it can be seen that all the arrows of Camestres are in the same direction as that of Celarent. By checking the conclusion of Camestres to that of Celarent, according to step (x), it can be found that they are the same. Thus, Camestres is reduced to Celarent and so Camestres is valid according to step (xvi).

Figure 36.  

Figure 37.  

For the third example choose the 3rd figure mood Darapti. According to steps (a) and (c), Darapti should be compared with the extended Barbara as they have the same set of premises (Figure 38). The direction of the arrow in the second premise of Darapti is of the opposite direction to that of the second premise of Barbara (Figure 39). However, conversio simplex rule cannot be applied to an a judgement. Also, conversio per accidens rule is not applicable here. So by step (xiii), Darapti is compared with Darii which has the subalternation of the second premise of Darapti (Figure 40).

Figure 38.  

Figure 39.  

Figure 40.  

Now by step (xiv), all the steps from (xii) are repeated. Since the arrow in the second premise of Darapti is opposite to that of Darii (Figure 41), by step (ix), the a judgement is converted to the i judgement (Figure 42). Finally, the direction of the arrow is reversed (Figure 43) by step (viii) and Darapti is a valid syllogism according to steps (x) and (xvi).

Figure 41.  

Figure 42.  

Figure 43.  

For the fourth example, check the validity of the 4th figure mood Bamalip, it needs to be compared with extended Barbara (Figure 44). According to Figure 45, both the premises of Bamalip are in the opposite subject-predicate relation to that of Barbara. But since these premises have a judgements, the conversio simplex rule cannot be used. However, mutatio rule can be applied. Thus the same subject-predicate relationship for both moods becomes the same after the application of mutatio (Figure 46).

Figure 44.  

Figure 45.  

Figure 46.  

Finally, the arrow for the conclusion of Bamalip is in the opposite direction to that of both the conclusions of Barbara (Figure 47). The arrow in Bamalip is reversed using the conversio simplex rule (Figure 48). As the conclusion of Bamalip is the same as that of the extended conclusion of Barbara, hence Bamalip is valid.

Figure 47.  

Figure 48.  

Using the reduction algorithm, invalidity of an mood can also be shown. For example, take the 3rd figure mood aaa. The mood aaa in this figure is not valid. It cannot be reduced to Barbara as none of the rules convert the subject-predicate relation in the second premise of the 3rd figure (Figure 49). However, this imperfect mood can be reduced to the premises of Darii but the conclusion is not the same (Figure 50). Thus aaa mood in the 3rd figure is invalid.

Figure 49.  

Figure 50.  

Using the above-mentioned rules it is not possible to make the diagrams for the valid moods of Baroco and Bocardo reduced to the mood of Barbara or any other perfect moods. Hence the method of direct reduction is incapable of showing the validity of both Baroco and Bocardo. This drawback is for the direct proof method only and is not a limitation of Byzantine diagrams itself as they can be used also for indirect proofs for all moods (including Baroco and Bocardo). As was mentioned in Section 1, due to space constrain it is not possible to present the indirect proof methods here. However, since many textbooks of the early modern period first presented direct proof with Byzantine diagrams, this method had to be investigated first.

6. Conclusion

Historically, the combination of Byzantine diagrams for two different figures (i.e. by putting diagrams for the perfect moods over or below the imperfect moods as shown from Figure 13 onward) is presented in Pacius and Aristotle 1598. However, the text never mentioned their purpose in validity checking of the imperfect moods. In Panizza 2017, Panizza also showed Byzantine diagrams as a heuristic tool that is capable of only representing different moods. However, she never talked about further possible usage of Byzantine diagrams in syllogisms. This paper has extended Panizza's method of representation by introducing new diagrammatic elements like wavy arrows or dotted diagrams for the perfect moods. With the help of this small extension, this paper has shown that Byzantine diagrams can be employed for diagrammatic reasoning through reduction by providing a method for checking the validity of syllogistic moods independent of the mnemonic code. The validity checking method has been established by presenting an extended version of Byzantine diagrams in Section 3 and demonstrating how to use Byzantine diagrams in direct proofs in Section 4.

The foundational work of this paper may help to study indirect proofs and also more complicated forms of Byzantine diagrams. For example, there are some historical textbooks that mainly use indirect proofs with byzantine diagrams. However, these presuppose that one already knows the direct proof with the diagrams. Furthermore, there are complex Byzantine diagrams which were combined with a Hexagon of Opposition which is illustrated but not completely explained in Lemanski 2021, p. 191. Lastly, it must be mentioned that there is some evidence that the Byzantine diagrams were also transferred into gestures. This topic, too, needs to be investigated in more detail elsewhere.

In essence, this work bridges the historical origins of Byzantine diagrams with contemporary diagrammatic reasoning techniques by uncovering its full capabilities.

Acknowledgments

The author of this paper is grateful to Jens Lemanski and anonymous reviewers for their comments, and suggestions for improvements.

Additional information

Funding

The paper benefited from ViCom-project Gestures and Diagrams in Visual-Spatial Communication funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – (RE 2929/3-1).

Notes

1 Peirce 1931, pp. III, 293. For more details on ‘rules’ and Peirce's conceptes of rules see Bhattacharjee and Moktefi 2021; Moktefi and Bhattacharjee 2021.