Aristotelian Diagrams as Logic Diagrams

Abstract

Recently, diagrams are more seen as logical systems. Curiously, Aristotelian diagrams, such as the square of opposition, have mostly been excluded from this. In this paper we challenge this prejudice and show that Aristotelian diagrams can be considered as proper logic diagrams that can be used to draw inferences. We do so by describing a logical system of Aristotelian diagrams: we provide a diagrammatic vocabulary and syntax, a set of axioms and (transformation) rules, and a formal semantics. We show how this diagrammatic logic system can be used to make inferences and argue that the crucial step here is to not only consider finished diagrams, but also unfinished diagrams; it is in the process of (re)constructing the diagram that the reasoning steps are made. We finish the paper by commenting on the soundness and completeness of the system.

1 Introduction

Aristotelian diagrams such as the square of opposition have been drawn since at least the second century CE [1] and have been used ever since in a variety of disciplines [4, 12]. However, until now Aristotelian diagrams have not been taken seriously as logic diagrams, and are often easily dismissed as ‘merely illustrative’. In their influential overview article, Moktefi and Shin define a logic diagram (which they also call a diagrammatic logic system) as “a logical system (i) which has a list of transformation rules, (ii) which has a formal semantics, but (iii) whose vocabulary is diagrammatic” [8, p. 612]. They state that Aristotelian diagrams are not logic diagrams in this sense, but merely “illustrative diagrams” used as “visual aids in order to make some immediate inferences” [8, p. 614].

In this paper we show that Aristotelian diagrams are logic diagrams. We do so by constructing a logical system based on Aristotelian diagrams that fulfills all three of Moktefi and Shin’s requirements: it has (i) a list of transformation rules, (ii) a formal semantics, and (iii) a diagrammatic vocabulary. After some preliminaries about logic diagrams and Aristotelian diagrams in Sect. 2, we present the vocabulary and syntax of the logical system in Sect. 3. This vocabulary is diagrammatic, thus showing that the logical system fulfills requirement (iii). In Sect. 4 we give an axiomatization consisting of axioms and transformation rules, thus showing that we satisfy requirement (i). Section 5 presents a formal semantics for the system, showing that it satisfies requirement (ii). In Sect. 6 we show how the system (and thus Aristotelian diagrams in general) can be used to draw inferences. We comment on soundness and completeness in Sect. 7. Finally, Sect. 8 contains some concluding thoughts and a short discussion of possible future research paths.

2 Preliminaries

Aristotelian diagrams graphically represent the four Aristotelian relations of contradiction, contrariety, subcontrariety and subalternation holding among a set of formulas or sentences. These four Aristotelian relations are defined in Definition 1. In what follows we use the abbreviations $CD_{\textbf{S}}(\varphi ,\psi )$, $C_{\textbf{S}}(\varphi ,\psi )$, $SC_{\textbf{S}}(\varphi ,\psi )$ and $SA_{\textbf{S}}(\varphi ,\psi )$. These days the contradiction relation is typically represented by a solid line, contrariety by a dashed line, subcontrariety by a dotted line and subalternation by an arrow. Some examples of Aristotelian diagrams following this convention are given in Fig. 1.

Definition 1

Let $\textbf{S}$ be a logical system based on a language $\mathcal {L}_{\textbf{S}}$, which is assumed to have Boolean operators and a model-theoretic semantics $\models _{\textbf{S}}$. The formulas $\varphi , \psi \in \mathcal {L}_\textbf{S}$ are

In this paper we are only concerned with diagrams that contain no equivalent formulas, tautologies or contradictions. Furthermore, we demand that for every formula $\varphi$, the diagram contains exactly one formula that is equivalent to the negation of $\varphi$. Most of the Aristotelian diagrams found in the literature adhere to these constraints, and there are also some theoretical considerations motivating these constraints [4, 12].

Even though Aristotelian diagrams represent logical relations, they have generally not been considered logic diagrams. Shin [11, p. 1] complains of a “general prejudice against diagrams”, namely the view “that diagrams can be only heuristic tools but not valid proofs”. She argues against this prejudice by showing that Venn diagrams can be viewed as a logical system in its own right, with a syntax and a semantics. This shows that Venn diagrams are real logic diagrams, and not merely illustrative diagrams. Since then, this approach has been used to show that other diagrammatic methods also need to be taken seriously as logical systems (see e.g. [6] for a recent example).

However, as was already indicated in the introduction, even if one is sympathetic to the general idea of viewing diagrams as logical systems, remainders of the “prejudice against diagrams” reported by Shin still continue to surround particular cases, in this case Aristotelian diagrams. For example, Moktefi and Shin claim that, in contrast to e.g. Venn diagrams, Aristotelian diagrams are not logic diagrams, but merely “illustrative diagrams” and “visual aids” [8, p. 614]. And they are by no means alone in this prejudice. For example, already in 1967 MacQueen claimed that the square of opposition “is not a true logic diagram” [7, p. 105]. More recently, Lemanski mentions in passing that three texts “do not contain analytical logic diagrams in the strict sense, but rather some squares of oppositions” [5, p. 9], thus confirming that he does not consider Aristotelian diagrams to be logic diagrams. In what follows, we challenge this remaining prejudice by presenting a diagrammatic logic system for Aristotelian diagrams that satisfies the three requirements set out by Moktefi and Shin in [8]: it has (i) a list of transformation rules, (ii) a formal semantics, and (iii) a diagrammatic vocabulary.

Fig. 1.

(a) the traditional square of opposition (with formula variables), (b) one instantiation for the modal logic $\textbf{S5}$, (c) a degenerate square for the modal logic $\textbf{K}$, and (d) a Sherwood-Czeżowski hexagon (with formula variables).

3 Vocabulary and Syntax

In this section we focus on fulfilling requirement (iii). We define a diagrammatic vocabulary, as well as a syntax. For now, we forget the meaning of the diagrams set out in Sect. 2, and look at the diagrams as purely syntactical objects. In Definition 2 we give the vocabulary and in Definition 3 we give the syntactical rules that determine whether a diagram is well-formed. By Definition 2 the wffs of the system are diagrams, thus the vocabulary is diagrammatic. Hence, requirement (iii) has been fulfilled.

Definition 2 (Vocabulary)

The vocabulary consists of (1) a set of formula-constants $\mathcal {C}$, (2) solid, dotted and dashed lines, and (3) arrows.

We use the formula-variables $\alpha , \beta , \gamma , \ldots$ as meta-variables ranging over $\mathcal {C}$, i.e. to refer to the formula-constants in the vocabulary.

Definition 3 (Syntax)

A well-formed Aristotelian diagram, WFAD, is defined as follows:

• 1. Every non-zero, but finite number of formula-constants written on a page is a WFAD.

• 2. If D is a WFAD containing the formula-constants $\alpha$ and $\beta$, then the diagram obtained by connecting $\alpha$ and $\beta$ with a line (solid, dotted, dashed) or an arrow is also a WFAD.

• 3. Only diagrams obtained in this way are WFAD’s.

Fig. 2.

Four well-formed Aristotelian diagrams contradicting the axioms

Note that according to these definitions, the diagrams in Figs. 1, 2, and 3 are all well-formed Aristotelian diagrams. A diagram that is not a WFAD would for example be a Venn diagram, a diagram with a dashed arrow, or a diagram where one of the lines does not end at a formula-constant. It might strike some readers as odd that the diagrams in Figs. 2 and 3 are well-formed. We will comment on this at the end of the next section.

4 Axiomatization

In this section, we focus on fulfilling requirement (i). We define transformation rules, which are in effect introduction rules, in Definition 5, as well as axioms, that deal with consistency, (Definition 4), structural rules, among which is an elimination rule, (Definition 6) and a consequence relation (Definition 7). The transformation rules are based on earlier work, in particular: the properties of the relations in AD-frames from [4], the theorems in Section. 3.3 of [12], the Aristotelian subdiagrams in [13], and Proposition 19 in [9].

Definition 4 (Axioms)

Within a given WFAD:

• 1. For every $\alpha$, there is no solid, dashed or dotted line between $\alpha$ and $\alpha$, and no arrow from $\alpha$ to $\alpha$.

• 2. For any $\alpha$ and $\beta$ there is at most one line or arrow between them.

• 3. For any $\alpha$, $\beta$ and $\gamma$, if $\alpha \ne \beta$ and there is a solid line between $\alpha$ and $\gamma$, then there is no solid line between $\beta$ and $\gamma$.

• 4. For every $\alpha$, there is at least one $\beta$ such that there is a solid line between $\alpha$ and $\beta$.

Definition 5 (Transformation rules)

Apply the transformation rules only if the consequent hasn’t been fulfilled. For every $\alpha$, $\beta$, $\gamma$ and $\delta$ in a WFAD:

• $-$I If there is a solid line between $\alpha$ and $\beta$, and an arrow from $\gamma$ to $\beta$, then draw a dashed line between $\alpha$ and $\gamma$.

• $\bullet$I If there is a solid line between $\alpha$ and $\beta$, a solid line between $\gamma$ and $\delta$ and a dashed line between $\alpha$ and $\gamma$, then draw a dotted line between $\delta$ and $\beta$.

• $\rightarrow$I If there is a solid line between $\alpha$ and $\beta$, and a dotted line between $\delta$ and $\beta$, then draw an arrow from $\alpha$ to $\delta$.

• Tr. If there is an arrow from $\alpha$ to $\beta$ and an arrow from $\beta$ to $\gamma$, draw an arrow from $\alpha$ to $\gamma$.

Definition 6 (Structural Rules)

• EFSQ If any of the axioms does not hold, then draw any new WFAD.

• Weak Remove any dashed line, dotted line, or arrow.

Fig. 3.

Four well-formed but unfinished diagrams.

Definition 7 (Consequence Relation)

A Diagram D follows from a diagram $D'$, written as $D' \vdash D$, iff D can be obtained by applying a finite number of transformation and/or structural rules to $D'$.

Figure 3 illustrates the transformation rules. We can apply $-$I (dashed line introduction) to diagram (a) to obtain diagram (b), hence Diagram 3(b) follows from Diagram 3(a). Applying a transformation rule to obtain a new diagram is exactly like deriving a new wff from another wff in any other logic. So we can in turn use $\bullet$I (dotted line introduction) to obtain diagram (c) from (b), and use $\rightarrow$I (arrow introduction) to obtain diagram 1(a) from 3(c). Similarly, Diagram 1(d) follows from Diagram 3(d).

As long as it is still possible to apply such transformation rules, we call the diagram unfinished. More precisely: a diagram is unfinished iff it is still possible to apply transformation rules, i.e. there is at least one transformation rule whose antecedent is fulfilled while the consequent has not been fulfilled. A diagram is finished iff it is not unfinished.

The idea of working with unfinished diagrams instead of only finished diagrams is crucial for our approach. If one only looks at finished diagrams, then it is indeed tempting to view Aristotelian diagrams as purely illustrative visual aids. However, it is in the process of constructing a diagram that important reasoning steps are made.¹

At the end of the previous section we mentioned that some readers might feel uneasy about classifying not only the diagrams in Fig. 1, but also those in Figs. 2 and 3 as well-formed diagrams. This unease can now be explained. Firstly, all of the diagrams in Fig. 2 contradict at least one of the axioms, just like in classical propositional logic, CPL, $p \wedge \lnot p$ is well-formed, but contradicts the axioms of CPL. This should explain the unease about these diagrams. Secondly, the diagrams in Fig. 3 are unfinished diagrams, while those in Fig. 1 are finished diagrams. Thus (of the diagrams in this paper) only the diagrams in Fig. 1 are well-formed, finished, and satisfy all of the axioms of Definition 4.

5 Semantics

In this section we focus on fulfilling requirement (ii). We provide a formal semantics for our logical system. This is mostly analogous to the formal semantics that one is used to. Definition 8 defines models on which WFAD’s are interpreted, Definition 9 provides the semantic clauses, and Definition 10 gives the semantic consequence relation. Just as in any other logic, a diagram can be satisfied by multiple models, and one model can satisfy multiple diagrams.

Note however that one of the elements of a model is an underlying logic $\textbf{S}$. Such an underlying logic can for example be CPL, first-order logic, or a normal modal logic. This underlying logic should not be confused with the diagrammatic logic system developed in this paper. In most practical applications of Aristotelian diagrams, the diagram is interpreted on a model $\langle \mathcal {F}, \textbf{S}, I \rangle$ such that the formula-constants used in the diagram will be the elements of $\mathcal {F} \subseteq \mathcal {L}^S$. In that case $\mathcal {C}$ is a subset of the wff’s of $\textbf{S}$ and $I(\varphi ) = \varphi$. However, to avoid confusion we have clearly distinguished the two languages in this paper (cf. also [4], where a similar distinction is made).

Definition 8 (Models)

A model is a triple $\langle \mathcal {F}, \textbf{S}, I \rangle$ such that $\textbf{S}$ is a logical system that has a language $\mathcal {L}_{\textbf{S}}$, Boolean operators and a model-theoretic semantics $\models _{\textbf{S}}$, $I: \mathcal {C} \rightarrow \mathcal {L}_\textbf{S}$ is a bijection, and $\mathcal {F} \subseteq \mathcal {L}_{\textbf{S}}$ is such that:

• 1. for every $\varphi \in \mathcal {F}$ there is a $\psi \in \mathcal {F} \setminus \{\varphi \}$ such that $\models _\textbf{S} \psi \leftrightarrow \lnot \varphi$,

• 2. for every $\varphi \in \mathcal {F}$ there is no $\psi \in \mathcal {F} \setminus \{\varphi \}$ such that $\models _\textbf{S} \varphi \leftrightarrow \psi$,

• 3. there is no $\varphi \in \mathcal {F}$ such that $\models _\textbf{S} \varphi$ or $\models _\textbf{S} \lnot \varphi$.

Definition 9 (Semantic Clauses)

Let $M = \langle \mathcal {F}, \textbf{S}, I \rangle$ be a model, then M satisfies an Aristotelian diagram D iff all of the following conditions are met:

• 1. For every $\alpha$ occurring in D, $I(\alpha ) \in \mathcal {F}$.

• 2. For every $\alpha$ and $\beta$ occurring in D such that $\alpha$ and $\beta$ are connected by a solid line, $CD_{\textbf{S}}(I(\alpha ),I(\beta ))$.

• 3. For every $\alpha$ and $\beta$ occurring in D such that $\alpha$ and $\beta$ are connected by a dashed line, $C_{\textbf{S}}(I(\alpha ),I(\beta ))$.

• 4. For every $\alpha$ and $\beta$ occurring in D such that $\alpha$ and $\beta$ are connected by a dotted line, $SC_{\textbf{S}}(I(\alpha ),I(\beta ))$.

• 5. For every $\alpha$ and $\beta$ occurring in D such that there is an arrow from $\alpha$ to $\beta$, $SA_{\textbf{S}}(I(\alpha ),I(\beta ))$.

Definition 10 (Semantic Consequence)

D is a semantic consequence of $D'$, written as $D' \models D$, iff every model that satisfies $D'$ also satisfies D.

6 Reasoning with Aristotelian Diagrams

One of the canonical uses of logic is to facilitate and evaluate reasoning. The logical system presented in this paper can be used for this as well. Suppose that we know that for some logical system S, $SA_{\textbf{S}}(\alpha ,\delta )$, $CD_{\textbf{S}}(\alpha ,\beta )$, and $CD_{\textbf{S}}(\delta ,\gamma )$. We formalise this symbolic information as the well-formed Aristotelian diagram in Fig. 3(a), which will act as a premise in our diagrammatic reasoning system. By applying transformation rules $-$I, $\bullet$I and $\rightarrow$I, we obtain as a conclusion the diagram in Fig. 1(a). Finally, we can return from the diagrammatic to the symbolic realm, for example by reading off from Fig. 1(a) that $SA_{\textbf{S}}(\gamma ,\beta )$. (And thus one can also conclude e.g. that $\models _{\textbf{S}} \gamma \rightarrow \beta$.)

One big advantage of this approach is that one does not need to understand all the ins and outs of the underlying logic $\textbf{S}$ in order to make the derivation. Thus, Aristotelian diagrams are especially useful for very complex underlying logics, or for people not yet familiar with the underlying logic. These include students in an introductory logic course, but also researchers who encounter a logical system for the first time – because the logic is new altogether, or because it belongs to another research tradition, e.g. consider a philosophical logician reading for the first time about the AI formalism of Sugeno integrals [3].

7 Soundness and Completeness

For reasons of space, we cannot provide detailed soundness and completeness proofs in this paper. However, we comment on both in this section. To prove soundness, it suffices to check for every axiom and rule that they hold in every model. As an example, consider part of Axiom 1: there is no solid line between $\alpha$ and $\alpha$. Towards a contradiction, suppose there is such a line in a diagram D satisfied by a model $M = \langle \mathcal {F}, \textbf{S}, I \rangle$. By the semantic clauses, $CD_{\textbf{S}}(I(\alpha ), I(\alpha ))$, which contradicts that the operators of $\textbf{S}$ are Boolean (Definitions 1 and 8).

Proving (strong) completeness would take up even more space than proving soundness. However, results in [4] (especially Theorem 2) suggest a promising proof strategy. In [4] a number of structural properties of Aristotelian diagrams are given, and it is proven that all other structural properties follow from those. Furthermore, the properties in [4] can all be derived from the axioms and transformation rules in this paper. Thus, it is reasonable to suspect that the axiomatization provided in this paper is complete, but this still remains to be proven.

8 Concluding Thoughts

We have shown that there is a persistent prejudice against Aristotelian diagrams even among those who are generally favorable about diagrammatic reasoning. To argue against this prejudice, we constructed a diagrammatic logic system for Aristotelian diagrams that fulfills all three requirements set out by Moktefi and Shin: it has (i) a list of transformation rules (Sect. 4), (ii) a formal semantics (Sect. 5), and (iii) a diagrammatic vocabulary (Sect. 3). In addition, we illustrated that this system can be used to draw inferences (Sect. 6), and we commented on the soundness and completeness (Sect. 7).

This study opens up multiple possibilities for future research. We have only looked at diagrammatic arguments with a single premise. We can also look at deriving a diagram from combining the information in two or more diagrams, or we might investigate the rules for adding new formulas to a diagram.

There are also connections with other diagrams that are worth exploring. For example, one can investigate whether the Diagrammatic system introduced here falls within the general framework of Single Feature Indicator Systems introduced in [10]. One could also look at the relation with Euler diagrams. In [2] it is shown that there is a close relation between Euler diagrams and Aristotelian diagrams. Is there e.g. also a relation between diagrammatic logic systems for Euler diagrams and the system in this paper?

Finally, it is worth pointing out that we have focussed on the description of logic diagrams given by Moktefi and Shin in [8]. However, the definition of logic diagrams differs slightly amongst different authors. For example, in [11] Shin seems to require that the diagrammatic logic system is complete as well. Exploring these other definitions for Aristotelian diagrams is left for future work.