2.1 Logic

Reasoning is a process that produces new information given previous data by following certain norms that allow us to describe inference as the unit of measurement of reasoning: inference may be more or less (in) correct depending on the compliance or violation of such norms. The science that studies such norms is logic.

The structural understanding of logic has depended, typically but not exclusively, on equivalent sentential approaches: the semantical, the syntactical, and the abstract. From the semantical standpoint the central concept of such norms is that of interpretation and defines our notions of satisfaction, model, and logical truth as denoted by ⊨ and is usually attributed to ^{Tarski (1956a)}. From the syntactical point of view the main concept of those norms is that of deducibility; it characterizes our intuitions of proof, demonstration, and theorem as denoted by ⊢ and is usually attributed to ^{Carnap (1937)}. Finally, from the abstract standpoint the idea behind those norms is that of a function of consequence that generalizes the previous accounts as typically attributed to ^{Tarski (1956b)}. These approaches are sentential and they define logical consequence as the proprium of logic. As an example, consider classical logic: its logical consequence relation follows the next structural norms where and are sentences and is a set of sentences: reflexivity (if φ∈Γ, then Γ⊢φ), monotonicity (if Γ⊢φ, then Γ∪{ψ}⊢φ ), and cut (if Γ⊢ψ and Γ∪{ψ}⊢φ , then Γ⊢φ).

2.2 Logical systems

Logical systems, the tools used to model and better understand the relation of consequence may be defined by a pair 〈L,B〉 where L is a language, and B is a semantic base (often equivalent to a calculus). Usually, the vocabulary is made up of two sets of signs: variables (nonlogical signs) and constants (logical signs). Syntax is used to determine, uniquely and recursively, the well-formed expressions of the system and semantics is used to provide meaning to such well-formed expressions. A well-defined logical system must have these elements. To illustrate this notion of logical system let us consider classical propositional logic, L ₀ . Its vocabulary has the constants CONS={¬,∨} and the variables VAR={φ_1,φ_2,…}. It has two syntax rules: i) if φ∈VAR, then φ is a wellformed formula (wff) of L ₀ ; and ii) if φ and ψ are wffs of L ₀ , then ¬φ and φ∨ψ are also wffs of L ₀ . The semantics of L ₀ is composed by a domain and a function of interpretation. The domain of L ₀ is the set of truth values, D={1,0}, where 1 stands for the designated value and 0 for the antidesignated value. The function of interpretation f maps the variables to the truth values, f:VAR⟶D. With this function a valuation v is defined in such way that v(φ)=f(φ), v(¬φ)=1-v(φ), and v(φ∨ψ)=max(v(φ),v(ψ)). This valuation provides the rules of correspondence that allow us to build the truth tables of L ₀ along with the remaining connectives and tautologies which, due to soundness and completeness, are equivalent to a mechanichal calculus.

2.3 Diagrams and diagrammatic logical consequence

I shall show that diagrammatic logical systems can be defined in a similar fashion and that we can describe a well-behaved notion of diagrammatic logical consequence between diagrams (and not sentences); but before we do that, I would like to introduce diagrams by paying attention to their expressive power.

Pop culture already somehow recognizes this expressive power: the 19^th century proverb “a picture is worth 1000 words” is quite representative in this sense; but the basic idea is much older. Notable examples of confidence in the expressive power of diagrams can be found in different historical periods. Ramon Llull (1232-1315), for instance, is arguably the most famous case: he developed his Ars Magna, a diagrammatic device used to explain the divine nature to those unable to understand God, under the assumption that diagrammatic methods were more convincing or expressive than sentential representations (Figure 1). Thomas Murner (1475-1537) also used diagrams in his Logica Memorativa in order to teach logic (Figure 2). Dutch mathematician and philosopher of science Simon Stevin (1548-1620) developed another remarkable diagram in his demonstration that the efficiency of the inclined plane is a logical consequence of the impossibility of perpetual motion (Figure 3). And, of course, besides these examples we have Descartes’ (1596-1650), who is perhaps the most famous case of a modern philosopher who made good use of diagrams in order to model hypotheses, such as the mechanics of the pineal gland (Figure 4).

Figure 1 A diagram from Llull’s Ars Magna (^{Llull, 1501}) 

Figure 2 A diagram from Murner’s Logica Memorativa (^{Murner, 1967}) 

Figure 3 A diagram from Stevin’s De weedgaet (^{Stevin, 1586}) 

Figure 4 A diagram from Descartes’ Principles of Philosophy (^{Descartes, Miller & Miller, 1984}) 

Our point is that the expressive power of diagrams for aiding reasoning was not news for modern thinkers and Leibniz was no exception: we can find evidence for this statement in an 18^th century research program for diagrammatic reasoning in the work of Leibniz (Figure 5), Lambert (Figure 6), and Plouquet (Figure 7).

Figure 5 A Barbara syllogism in De formæ logicæ comprobatione per linearum ductus (^{Leibniz & Couturat, 1903}) 

Figure 6 A Barbara syllogism in Neues Organon (^{Lambert, 1764}) 

Figure 7 A Barbara syllogism in Sammlung der Schriften (^{Bök, 1766}) 

This confidence in the expressive power of diagrams is understandable. In order to represent knowledge we use internal and external representations. Internal representations convey mental images, for example; whereas external representations include physical objects on paper, on blackboards, or computer screens. External representations can be divided into two classes: sentential and diagrammatic (^{Larkin & Simon, 1987}).

Sentential representations are sequences of sentences in a particular language. Diagrammatic representations are sequences of diagrams that contain information stored at one particular locus in a configuration, including information about relations with the adjacent loci; and diagrams are information graphics¹ that index information by location on a plane (^{Larkin & Simon, 1987}). The difference between diagrammatic and sentential representations, thus, is that the former preserve explicit information about topological relations, while the latter do not-they may, of course, preserve other kinds of relations.

This confidence in the power of diagrams is comprehensible due to their computational advantages: they group together information avoiding large amounts of text, they automatically support a large number of perceptual inferences-which are extremely easy to interpret-, and they grant the possibility of applying operational constraints (like free rides and overdetermined alternatives (^{Shimojima, 1996})) to allow the automation of perceptual inference (^{Larkin & Simon, 1987}).

However, despite this general confidence, when it comes to reasoning there is a bias (or a tradition) that supports the claim that while proofbased reasoning is essential in logic and mathematics, diagram-based reasoning, no matter how useful (^{Nelsen, 1993}) or elegant (^{Polster, 2004}), is not, for it is not bona fide reasoning. Thus, for example, Tennant has suggested a diagram is only a heuristic to prompt certain trains of inference (^{Tennant, 1986}); Dieudonné has urged a strict adherence to axiomatic methods with no appeal to geometric intuition, at least in formal proofs (^{Dieudonné, 2008}); Lagrange remarked in the Preface to the first edition of his Mécanique Analytique (1788) that no figures were to be found in his work (^{Boissonnade, Lagrange, & Vagliente, 2013}); and even Leibniz himself shared a similar opinion at some point (emphasis mine):

La force de la démonstration est indepéndante de la figure tracée, qui n’est que pour faciliter l’intelligence de ce qu’on veut dire et fixer l’attention; ce sont les propositions universelles, c’est-à-dire les définitions, les axiomes et les théorèmes déjà démontrés qui font le raisonnement et le soutiendraient quand la figure n’y serait pas (^{Leibniz, 1966: 309}).

This bias against diagram-based reasoning is based upon the assumption that diagrams naturally lead to fallacies, mistakes, and are not susceptible of generalization. Nevertheless, in retrospect we can find an argument against this assumption in Newton’s Preface to the First Edition of Principia Mathematica (^{Newton, 1974}) by reducing proof-based reasoning to mechanical reasoning (emphasis mine):

But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry that what is perfectly accurate is called geometrical; what is less so, is called mechanical. However, the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic; and if any could work with perfect accuracy, he would be the most perfect mechanic of all, for the description of right lines and circles, upon which geometry is founded, belongs to mechanics (^{Newton, 1974: 11}).

Along similar lines, ^{Allwein, Barwise, and Etchemendy (1996)} and ^{Shin (1994)} have developed a successful research program around heterogeneous and diagrammatic reasoning that has promoted different studies and model theoretical schemes that help us represent and better understand diagrammatic reasoning in logical terms, thus allowing a well-defined notion of a diagrammatic logical system and a diagrammatic inference.

Thus if, as we stated above, reasoning is a process that produces new information given previous data, and information can be represented diagrammatically, it is not unreasonable to suggest that diagrammatic inference is the unit of measure of diagrammatic reasoning: diagrammatic inference would be correct or incorrect depending on the compliance or violation of certain norms. Let us denote the relation of diagrammatic logical consequence or diagrammatic inference by ↦; this relation would define our intuitions around the informal notions of visual inference or visual argument and would follow, in principle, the standard structural norms where δ is a particular diagram and Δ is a set of diagrams: reflexivity (if δ∈Δ , then Δ↦δ), monotonicity (if Δ↦δ, then Δ∪{δ’}↦δ), and cut (if Δ↦δ and ∪{δ}↦δ’, then Δ ↦δ’).

To give a somewhat detailed account of these structural properties we need to describe how the operator ↦ works. Shimojima has developed a logical theory around diagrammatic inference that defines (informally) a free ride as a process in which some reasoner gains information without following any step specifically designed to gain it (^{Shimojima, 1996: 32}), in other words, a free ride is a process that allows us to reach automatically (and sometimes inadvertently) a conclusion from a diagrammatic representation of the premises. Inversely, an overdetermined alternative occurs when a diagram that should not follow from a given diagrammatic configuration of the premises does follow.

Using Shimojima’s approach we could say that reflexivity establishes that a if a diagram is part of a set of diagrams or diagrammatic configuration, then that particular diagram is a visual consequence of such configuration because there is a free ride from the diagrammatic configuration to a particular diagram; monotonicity would say that if a diagram is a free ride from a diagrammatic configuration, and a new diagram is added to such configuration, then the initial diagram is still a free ride from the original configuration.² Finally, cut would establish that if a diagram is a free ride from a diagrammatic configuration and the addition of a new diagram produces a new free ride, there is a free ride from the original diagrammatic configuration to the new diagram.

3.1 Syllogistic

Syllogistic has its origin in Aristotle’s Prior Analytics and is the theory of inference that deals with the consequence relation between two categorical propositions taken as premises and another categorical proposition taken as a conclusion. A categorical proposition is a proposition composed by two terms, a quantity, and a quality. The subject and the predicate of a proposition are called terms: the term-schemadenotes the subject term of the proposition and the term-schema P denotes the predicate. The quantity may be either universal (All) or particular (Some) and the quality may be either affirmative or negative. These categorical propositions are denoted by a label, either A (universal affirmative), E (universal negative), I (particular affirmative), or O (particular negative). A categorical syllogism, then, is a sequence of three categorical propositions ordered in such a way that two propositions are premises and the last one is a conclusion. Within the premises there is a term that appears in both premises but not in the conclusion. This particular term works as a link between the remaining terms and is known as the middle term, which we will denote with the term-schema M. According to this term we can set up four figures that encode and abbreviate all the valid and only the valid syllogisms (Table 2).

Table 2 Valid syllogisms 

3.2 VENN

VENN (^{Venn, 1880}) is a sound and complete diagrammatic logical system (^{Shin, 1994}) that represents syllogistic perspicuously. VENN can be defined as a (diagrammatic) logical system with a well-defined vocabulary, syntax, and semantics (^{Shin, 1994: 48}). Briefly, the vocabulary is determined by the next elements: the closed curve, the rectangle, the shading, the X, and the line (Figure 8).

Figure 8 Vocabulary of VENN 

With this vocabulary a diagram in VENN is defined as any finite combination of diagrammatic elements (^{Shin, 1994}). A region is any enclosed area in a diagram. A basic region is a region enclosed by a rectangle or a closed curve. A minimal region is a region within which no other region is enclosed. An X-sequence is a diagram of alternating X’s and lines with an X in each extremal position. Regions represent sets and the rectangle represents the domain. A shaded region represents an empty region and a region with an X represents a non-empty region. And with this syntax the categorical propositions can be represented as follows (Figure 9):

Figure 9 Categorical propositions in VENN 

Further, the semantics of VENN depends on a natural homomorphism with sets that help define the rules for developing (diagrammatic) proofs in this diagrammatic system (^{Shin, 1994 81}):

The previous rules may be summarized into the rules of erasure, addition, and unification (^{Nakatsu, 2009: 133}). As an example, consider a diagrammatic proof of a syllogism of the form (Figure 10). According to the previous rules we begin with an introduction of areas (step 1) and then a unification is applied (step 2). After that, we apply an erasure of an X-sequence (step 3) and then a spreading of an X-sequence (step 4). Finally, by the erasure of a closed curve rule, we obtain a final diagram (step 5). Since the conclusion got drawn by drawing down the premises, the inference is valid (and corresponds to a Darii syllogism).

Figure 10 Example of a proof in VENN 

As we can see, with these rules VENN provides an essential feature of a well-defined diagrammatic logic: a linear time diagrammatic method of decision for syllogistic that consists in drawing the diagrams for the premises and then checking (by mere observation) whether it is possible to “read off” the conclusion from the drawing of the premises; in case it does, the syllogism is valid; otherwise it is invalid.

4.1 LEIB as a logical system

My first goal is to present LEIB by reconstructing its vocabulary, syntax, and semantics. In order to do that I pay special attention to LEIB’s diagrams for categorical propositions (Figure 11):

Figure 11 Categorical propositions in LEIB (^{Leibniz & Couturat, 1903, 292-293}) 

From these diagrams we can infer that the vocabulary of LEIB has the following basic diagrammatic elements: the solid horizontal line and the dotted vertical line (Figure 12).

Figure 12 Vocabulary of LEIB 

With this vocabulary we can define the syntax of the wfds for LEIB. The solid horizontal lines stand for terms and the vertical lines stand for a relation between terms. Given two horizontal lines representing terms, one could be completely included in another; they could be completely disjoint; they could partially intersect each other; or one could be partially not included in another (Figure 13).

Figure 13 Syntax of LEIB 

The semantics of these wfds is straightforward: a diagram of a proposition A shows that all that is S is indexed in P, but not inversely. Proposition E shows that no S is indexed in P and vice versa. Proposition I represents the fact that some S is indexed in some P, and vice versa. Proposition O states that some S is not indexed in all O (Figure 14).

Figure 14 Reconstruction of categorical propositions in LEIB 

To exemplify how LEIB works let us represent a syllogism in a diagrammatic fashion. Figure 15 shows what a Barbara syllogism looks like:

Figure 15 A Barbara syllogism as it originally appears in ^{Leibniz & Couturat (1903: 294)}  

The construction of the previous syllogism is relatively simple. We produce the diagrams that stand for the premises of a Barbara syllogism, namely, MAPSAM. In order to do that we draw the proposition MAP (Figure 16). Then, since SAM shares M with MAP, SAM must be drawn with respect to M (Figure 17):

Figure 16 MAP 

Figure 17 SAM 

Hence, following the reconstruction, a Barbara syllogism would look like the following diagram:

Figure 18 A Barbara syllogism reconstructed in LEIB 

I shall show what canonical syllogisms look like in LEIB, but before I do that I would like to focus on the most interesting aspect about LEIB: a linear time diagram-based algorithm of decision, call it A, that takes any syllogism as an input and provides a decision about the (in)validity of such syllogism by checking whether the wfd of the conclusion is automatically represented by representing the wfds of the premises (otherwise, the syllogism is invalid) (Table 3).

Table 3 LEIB’s algorithm of decision 

Prem takes a syllogism σ and produces a diagram for the premises; ↦ stands for a free ride; and Conc checks the diagram of the conclusion of σ. In Figure 19 it is easy to see how the conclusion was automatically obtained by representing the premises, i.e., the conclusion “got drawn” automatically by “drawing down” the premises: this is a fair example of a free ride.

Figure 19 Free ride in a Barbara syllogism in LEIB 

Using the previous decision algorithm and our reconstruction of LEIB we can show diagrammatic proof of the valid syllogisms in LEIB (Figures 20, 21, 22 & 23):

Figure 20 Valid syllogisms from figure 1  

Figure 21 Valid syllogisms from figure 2  

Figure 22 Valid syllogisms from figure 3  

Figure 23 Valid syllogisms from figure 4  

Finally, to provide a more comprehensive explanation of how LEIB works I would like to show a couple of examples of invalid syllogistic forms. First consider a form with premises MEPSEM, which should be invalid. LEIB shows that such syllogism is actually invalid because the conclusion is not a free ride, but an overdetermined alternative (because the diagram of the conclusion is not even a wfd) (Figure 24).

Figure 24 An invalid syllogistic form in LEIB 

In Leibniz’s own account, a form with premises would also be invalid because the conclusion would also be an overdetermined alternative (Figure 25).

Figure 25 An invalid syllogistic form in ^{Leibniz & Couturat (1903:299)}  

4.2 LEIB’s logical attributes

After this exploration of LEIB as a logical system I would like to pursue my second goal by suggesting a proof for a set of propositions that cover attributes of soundness, completeness, decidability, autarchy, and some sort of equivalence with VENN as well as a subset of BOOL (^{Boole, 1951}). But before we look at those properties we need a preliminary result that I like to call Aristotle’s Lemma.

In Prior Analytics A.1, 25b1 Aristotle argued that it is possible to reduce all valid syllogisms to the universal syllogisms from figure 1, that is to say, to Barbara and Celarent. The idea in this proposition is that all valid syllogisms can be reduced, by following some precise instructions (like the ones depicted in Table 4), to other valid syllogisms. This is precisely the phenomenon captured by the medieval tradition of using names as abbreviations as we saw in Table 2 and is also Leibniz’s desideratum.

Table 4 Instructions for the reduction of syllogisms 

Hence, for example, a Cesare syllogism can be reduced to a Celarent syllogism because “Cesare” starts with letter C, and this is possible by a conversion of the proposition E, which is indicated by the letter s right after the letter e that stands for a categorical proposition E. We can see that this idea holds in LEIB.

Proposition 1. (Aristotle’s lemma for LEIB) Every valid syllogism is reducible to some syllogism from figure 1.

Proof. We prove this diagrammatically with the aid of some rigid motions (Figures 25, 26 & 27). In Figure 25 we can see Cesare and Celarent preserve the same diagram by reflection w.r.t. a Y axis; Camestres and Celarent preserve the same diagram by applying a 180° rotation and then a reflection w.r.t. a Y axis. Festino and Ferio preserve the same diagram by reflection w.r.t. the Y axis. Baroco is reduced to Barbara by applying a contradiction of the conclusion using it as a premise.

Figure 25 Reduction of syllogisms from figure 2 to syllogisms from figure 1  

Figure 26 Reduction of syllogisms from figure 3 to syllogisms from figure 1  

Figure 27 Reduction of syllogisms from figure 4 to syllogisms from figure 1  

In Figure 26 we can see Disamis and Darii preserve the same diagram by applying a 180° rotation; Datisi and Darii, and Ferison and Ferio preserve the same diagram. Bocardo is reduced to Barbara by contradiction.

Finally, in Figure 27 we can see Calemes and Celarent preserve the same diagram by applying a 180° rotation and then a reflection w.r.t. a Y axis. Dimaris and Darii preserve the same diagram by applying a 180°rotation. Fresison is reduced to Ferio by reflection w.r.t. a Y axis.

With this result we can proceed to show that LEIB’s algorithm is sound and complete. Let us denote the application of A to a given syllogism σi,j from figure i∈{1,2,3,4} and row j∈{1,2,3,4} in Table 2 by A (thus, for instance, the application of A to a Dimaris syllogism is A (σ_4,2) and, for sake of exposition, A (σ_4,4) is a placeholder).

Proposition 2. (Soundness) If A (σ)=valid, then σ is valid. Proof. We prove this proposition by cases. Since there are four figures, we need to cover each valid syllogism from each figure, which is precisely what we have done in the previous section (Figures 20, 21, 22 y 23). Thus, we have that for every σ_i,j, when A(σ_i,j)=valid, σ_i,j is valid.

Proposition 3. (Completeness) If σ is valid, then A (σ)=valid.

Proof. We prove this by contradiction. Suppose that for all i,j, the syllogism σ_i,j is valid but for some valid syllogism σ_k,j,A(σ_k,j)=invalid. Now, we know σ_1,j is valid and if we apply A(σ_1,j) we obtain A(σ_1,j)=invalid, as we can see from Proposition 2. Now, since all valid syllogisms σ_n>1,j can be reduced to the valid syllogisms from figure 1 by Proposition 1 (Aristotle’s lemma), it follows that A(σ_n>1,j)=valid, and thus, for all valid syllogisms k,A (σ_k,j)=valid, which contradicts our initial assumption.

From these results it follows that a syllogism σ is valid if and only if A (σ)=valid, and since A is a diagrammatic decision procedure we can infer, as a corollary, that:

Proposition 4. (Decidability) LEIB is decidable.

Indeed, algorithm A is a mechanical and diagrammatic procedure that is sound and complete; but perhaps the importance of these results lies in their relation to two important properties: autarchy and some sort of equivalence with VENN using three sets or regions, call it VENN ₃ .

The autarchy of a diagrammatic system corresponds to a trade-off between free rides and overdetermined alternatives, that is, a compromise between valid and invalid diagrams (i.e., diagrams that do not follow from the configuration of the premises). An autarchic diagrammatic system is, thus, a system with a set of operational constraints that always give rise to free rides and never to overdetermined alternatives (^{Bellucci, Moktefi & Pietarinen, 2013}). Since LEIB is sound, complete, and decidable, it follows that:

Proposition 5. (Autarchy) LEIB is autarchic.

This is an important result that will have impact on Leibniz’s own account of LEIB, as we shall see in Section 4.3; specially because:

Proposition 6. (Equivalence) LEIB is equivalent to VENN ₃ w.r.t. syllogistic.

Proof. In order to provide proof for this statement we show that every valid syllogism in LEIB (say, theorem of LEIB) is a valid syllogism in VENN ₃ (say, theorem of VENN ₃ ) and vice versa. From left to right: suppose that for any valid syllogisms, is a valid syllogism in LEIB but an invalid one in VENN ₃ . Given the soundness and completeness of LEIB, being σ_i,j a valid syllogism in LEIB implies that σ_i,j is valid simpliciter. But since VENN is sound and complete as well (^{Shin, 1994}), if σ_i,j is invalid in VENN ₃ , then σ_i,j must be invalid, which is a contradiction. From right to left the proof is similar.

What this brief metalogical exploration shows is that LEIB is not only a logical system, but an actual bona fide diagrambased logic because it produces valid (soundness) and only valid inferences (completeness) by providing a time efficient (O(n)) mechanical method of decision (decidability) that helps the automation of perceptual inference (autarchy), while preserving equivalence with VENN ₃ (w.r.t. syllogistic) and, thus, with some subset of BOOL (equivalence).

4.3 Leibniz vs LEIB

After these results we would like to introduce an interesting puzzle that originates from a conflict between the previous formal account of LEIB and Leibniz’s own account. Consider that, by Proposition 6, LEIB is equivalent to VENN ₃ w.r.t. syllogistic and thus, it is equivalent to some subset of BOOL. This fact is important because it means that LEIB does not allow us to infer particular propositions from universal propositions due to issues with empty classes: in fact if, for instance, we try to get a Barbari syllogism in LEIB we will find that a such task is impossible and any attempt to build a Barbari necessarily yields a Barbara syllogism, due to autarchy, and the same happens with other syllogisms that require existential import.

In the opinion of ^{Kneale and Kneale (1962)}, Leibniz was by no means an Aristotelian purist, but he was committed to the assumption of existential import (p. 322). Indeed, Leibniz thought that his diagrammatic system was capable of modeling and representing syllogisms with existential import such as Barbari, Cesaro, Fesapmo, Calemos, and so forth, which implies the possibility of deriving particular propositions from universal ones (Phil., IV, 50; Math., V, 27; Cf. New essays, IV, XVII, 4). Nevertheless, it is not clear how this is possible since LEIB is autarchic and equivalent to VENN ₃ and BOOL.

The previous situation leads to an interesting puzzle between consistency and eligibility: if Leibniz proposed his system only as a representation system, then it would be a trivial system because it would represent any syllogism, both valid and invalid; but it is quite clear that Leibniz himself argued against such a thesis. Therefore, his system was proposed rather as a non-trivial reasoning system, but then, by Propositions 1-6, by being equivalent to VENN ₃ and BOOL, it has a model in a modern interpretation of syllogistic that assumes empty terms. The puzzle is, thus: why would Leibniz argue that his system is capable of modeling such imperfect syllogisms (implying the acceptance of empty terms) when it actually does not (implying the rejection of empty terms)? I believe the answer to this puzzle is beyond the logical scope, and so, we leave the question open.