The following tutorials will be part of the program of Diagrams 2022:

- Interpreting and understanding relational database queries using diagrams
- Representational Systems Theory: What, Why and How?
- Square of Opposition: Theory and Applications
- Marlo diagrams: design and manipulation

The tutorials will take place on **September 13 2022**. Please click on the links to find the abstracts of the tutorials.

## Interpreting and understanding relational database queries using diagrams

*Wolfgang Gatterbauer*

The goal of this tutorial is to survey the most important visual query representations common across various Visual Query Languages (VQLs) for relational databases. Our focus will be on the core relational fragment equivalent in expressiveness to relational algebra. Since that fragment is basically first-order logic, this concrete real-world problem of finding intuitive diagrammatic representations for relational queries may be of great interest to and future motivation for the diagrammatic reasoning community.

While VQLs have been heavily studied since the early days of the relational model, and their basic premise is that they can help users formulate their queries faster and more easily, they have never been able to replace the more widely used textual specification of queries. We believe the main reason for this is a mismatch between the intended purpose of VQLs and what they in effect afford. It has been well observed that diagrammatic depictions of queries are better used for understanding existing queries instead of composing new queries. We assert that helping users understand existing relational queries expressed in text will be increasingly important for future applications, such as collaborative data platforms, collaborative query management systems, and even spoken dialogue systems with visual responses.

With that in mind, this tutorial will focus on relational VQLs from the perspective of query interpretation instead of query composition. What are the most important metaphors developed for visualizing relational queries? How do these metaphors compare to visual formalisms developed by the diagrammatic reasoning community, both in their structure and their expressiveness? What are the basic principles that can guide the future development of visual query representations?

## Representational Systems Theory: What, Why and How

*Daniel Raggi and Aaron Stockdill*

Representational Systems Theory (RST) provides a formal framework for modelling representational systems, the structure of individual representations, their properties, and transformations between them. In this example-driven tutorial we want to introduce participants to the theory and applications of RST, with the help of an interactive web application for specifying representations and manipulating their structure.

One of the main innovations of RST is the concept of a construction space, where many

structures of interest for the study of representations can be defined in graph-theoretic terms. Notably, the concept of a construction generalises that of a syntax tree, but its weaker restrictions admit directed and indirected cycles. This enables construction spaces to model the structure of representations that are often considered informal, such as geometric and topological figures, plots, and other kinds of diagrams. Moreover, RST allows us to model complex relations between the objects of different representational systems, which opens up the door for using transformations that convert representations from one system into another.

Representational systems are encoded in RST using three construction spaces. The first

space encodes grammatical relations between representations (how they are built), the second space encodes entailment relations (how they can be meaningfully manipulated), and the third space encodes meta-properties of representations (e.g., whether one representation uses more ink than another, or whether numeral 2 represents the number two). Moreover, in RST we can encode links between construction spaces, which can be exploited to produce transformations across representational systems.

There are many computational frameworks available for formalising mathematical theories

(Isabelle, HOL-Light, Lean, Coq, Metamath). These frameworks typically that rely on some

meta-logic that restrict us, from the outset, to encoding statements as strings with an underlying tree structure. RST differs from this frameworks in that it provides the foundations for a meta-language that takes diagrams and ‘atypical’ representations as seriously as the aforementioned frameworks treat formal theories.

In this tutorial we want participants to get a sense of the variety of representations whose

structure can be modelled using construction spaces, by using an interactive web application that allows users to build and manipulate graphs for construction spaces. This web application connects to a back-end that allows the use of algorithms for transforming across representational systems.

## Square of Opposition: Theory and Applications

*Jean-Yves Beziau*

The square of opposition is one of the most famous logic diagrams, having

a 2.500-year story. It is based on a theory, the theory of opposition, initiated

by Aristotle. This theory leads to many other diagrams and has a great

variety of applications.

The name of the theory is “Theory of Opposition”, but it also sometimes

called the “Square of Opposition Theory”, or simply “Square of Opposition”,

referring then both to the theory and its offspring, i.e. the square of

opposition diagram. And in fact, most of the time when one is talking at the

same time about the square of opposition s/he is talking of the diagram and

the theory surrounding it.

This diagram was suggested by the Stagyrite himself but explicitly

designed only several centuries later, by Apuleius and Boethius. And then

other diagrams of opposition were designed: triangles, hexagons, octagons,

decagons, cubes and other polyhedra. These diagrams do not limit to

geometrical figures and objects, because they incorporate additional data

and features.

The interaction between the theory of opposition and the related

diagrams is strong because it is interactive. New theoretical developments

lead to new diagrams, and vice-versa, new diagrams generate

improvements of the theory.

In the first part of this tutorial, we will explain the basic concepts of this

theory and their relations with other relational frameworks such as

structures, networks or conceptual architectures. We will present many

diagrams illustrating this theory and transforming it.

We will show that there is no cube of opposition of which each face is a

square of opposition, comparing a formal proof of this result with a quicker

and simpler one using a reasoning based on a diagram.

We will examine the exact relation between the square of opposition

and the hexagon of opposition, the second most famous diagram of the

theory, mainly promoted by Robert Blanché in the second half of the

twentieth century. We will explain in particular why Blanché’s hexagon can

be considered as an improvement of the square, not just as an extension of

it.

We will emphasize the difference between extending the theory of

opposition mainly on the basis of geometrical inspiration by contrast to

motivated by logical and philosophical ideas.

We will discuss n-opposition theory generalizing the theory to n-sides

polygons of opposition and promoting many-dimensional objects

represented by different diagrams.

We will also stress the usefulness of the systematic use of colors for

diagrams and discuss the best ways to present diagrams from the point of

view of the theory as well as from a geometrical and design perspectives.

In a second part, we will give examples of many applications, showing that

the square of opposition theory is an interdisciplinary theory not only

internally, mixing geometrical intuitions with philosophical ideas, but also

externally, by its wide range of applications.

We will present applications of this theory, with corresponding

diagrams to:

- Classification of signs in semiotics
- The notion of analogy
- Music theory
- Categorization of paintings
- Fundamental concepts of modern logic (truth, logical truth and consequence)
- Articulation of Kant’s double duality a priority /a posterior, analytic/synthetic
- Chance and determinism

We will also present the reflexive meta-aspect of the square of

opposition theory, showing how it applies to the theory of opposition itself,

presenting diagrams about oppositions.

This tutorial is self-content, not requiring any previous knowledge of the

theory of opposition and may interest a wide audience due to its

interdisciplinary character.

We will promote strong interaction with the participants by

incentivizing them in particular to think about, and propose, some new

diagrams, new configurations of the theory of opposition, giving hints of

how to proceed.

## Marlo diagrams: design and manipulation

*Marcos Bautista López Aznar*

Marlo diagrams are graphical tools for the didactics of logic, complementary to Venn

diagrams, which have already been used successfully with more than 400 students

between 12 and 18 years old. Diagrams can serve as models of logical reasoning

because they analyze their processes and their constituent elements and reflect them on

external memory devices for better execution, revision and translation into natural and

formal language. In this sense, it cannot be accidental that we verify in them the

correspondence of the Aristotelian figures of the demonstrative syllogism with the

formal laws of the syllogism that Boole theorized in 1854 through algebra.

Marlo’s diagrams do not work with a fixed image of compartmentalized terms,

variables, or classes. On the contrary, it generates a representation of the phases of

dynamic and flexible inference, which promotes the synchronization of mathematical,

logical, and linguistic reasoning. That is, greater parallelism is established between the

steps of the graphic and formal demonstration, facilitating, at the same time, the

interpretation of each step in natural language. An experienced student uses the

graphical representation to check the validity of his formal inferences and, at the same

time, relies on these formal inferences to construct his diagrams. This means that the

teacher, in addition to the student’s conclusion, can visualize the processes that have led

to it.

Marlo’s diagrams allow plausible conclusions to be drawn, distinguishing between

possible and probable.

Marlo’s diagrams can intuitively represent propositions with intermediate quantifiers

that can be chained or connected to solve problems with multiple premises.

The tutorial has a purely practical approach in which we will explain the following

topics:

- Formalization of premises and representation of logical connectives from the perspective of quantification of the predicate.
- Formal notation and its correspondence with the diagrams: When to split propositional models and when to add a variable outside the set.
- Semantics of propositional models: Everything that is not prohibited is allowed.
- Conversion and transformation: Looking for a common denominator that acts as a

middle term in propositional calculus. - The laws of inference: Principles of identity, non-contradiction, and precaution

when synthesizing the models. - Inferences through the synthesis of propositional models or superposition of sets.
- Logical exercises: Syllogisms and propositional calculus

https://fersoler.github.io/MarloDiagrams/