Atsushi Shimojima, Doshisha Univeristy (co-author Dave Barker-Plummer, Stanford University): An Account of Some Cases of Free Rides in Diagrams
A free ride property of a diagrammatic system is the system’s ability to “automatically" express a consequential piece of information upon the expression of a certain set of premises that implies it. The general property has been pointed out several times in the literature and analyzed, to a certain extent, by Shimojima (2015) and Stapleton et al. (2017). This talk will delve deeper into the property. I will show that (at least) some cases of free rides are based on the way syntactic conventions regulate the spatial arrangement of “atomic indicators” in a diagram to make some (otherwise very complicated) disjunctive properties of the diagram perceptually accessible.
Gem Stapleton, University of Brighton: Observational Advantages in Euler Diagram Systems
The ability of diagrams to convey information effectively comes, in part, from their ability to make facts explicit, thus avoiding the need to derive them. Such facts are referred to as observational advantages. This talk will provide an overview of the theory of observation, which builds on the idea that representations of information require meaning-carrying relationships to be present. Meaning-carriers are fundamental in that they are precisely the syntactic properties of the representation of information from which its meaning is derived. Using the theory of observation, a formal characterisation of observational advantages can be explored. The talk will proceed to demonstrate the theory of observation and observational advantages in the context of Euler diagram systems, as compared to symbolic set-theoretic statements. The implications for different syntactic and semantic choices that impact the scale of the observational advantages possessed by these diagrams will be explored, considering both theoretical and practical aspects.
Ahti-Veikko Pietarinen, Tallinn University of Technology/Nazarbayev University: Abduction and Diagrams
Peirce took abductive conclusions to be drawn in a special, co-hortative mood (“the investigand”). Such conclusions are representative interpretants that represent abduction (or retroduction) as a form of reasoning that can convey a general conception of the truth. The truth is not asserted; abduction merely delivers the idea of a matter of course, rendering that idea comparatively simple and natural and hence assuring us of its justified assertibility. Hence abductive reasoning is at home in addressing “How Possible” –questions in science. It concerns how things might, could or would conceivably be such that they can under its auspices be plausibly asserted. Now Peirce took all reasoning to be diagrammatic and represented by graphs. Yet there are almost no examples in his corpus of what non-deductive graphs might look like. I propose an interpretation of the sole exception, a sketch of an existential graph from a rejected page from 1903, which might be a preliminary design of what Peirce wanted to result in a representative interpretant of that peculiar inverse type of inference.
Frederik Stjernfelt, Aalborg University Copenhagen: Operating with Dicisigns in the Wild
Given Peirce’s final developments of his analysis of propositions – under technical headlines such as “Dicisigns” and “Phemes” – in the years after 1900, I find it an important task to investigate which empirical sign types are grasped by the new theoretical descriptions. In a certain sense, this forms a continuation of what Bellucci recently called Peirce’s apriori-aposteriori method where the a priori deduction of sign types is followed by a subsequent empirical test in the sense that the priori categories are further interpreted in terms of empirical generalizations (e.g., in the ten-sign taxonomy of the Syllabus). Pursuing such an investigation regarding Peircean propositions, I shall discuss the essentially iconic device of co-localizing Subjects and Predicates in empirical propositions, including issues such as what may be called "graphical case grammar" and empirical generalizations of Peirce’s “Sheet of Assertion” concept to cover the iconic syntheses of rhemes into propositions and propositions into arguments.
Amirouche Moktefi, Tallinn University of Technology: Playing by the Rules: The Philosophical Significance of Carroll’s Diagrams
The role of diagrams in mathematical proofs is disputed. Yet, diagrams have long been used in various mathematical disciplines. In logic, they knew a golden age after their popularisation by Leonhard Euler in his Letters to a German Princess (1768). By the end of the nineteenth century, several schemes were in existence, and to some extent, in competition. In particular, diagrams were invented to handle logic problems known as elimination problems which consist in finding what information regarding any combination of terms follows from a set of premises. For the purpose, John Venn published in 1880 a scheme offered as an improvement over Euler’s well-known circles. The method consisted in representing the complete information contained in the premises on a single diagram, then to see ‘at a glance’ the conclusion regarding specific terms. An inconvenience of this scheme, as pointed out by Louis Couturat (1914), is that it does not really tell how the conclusion is to be ‘extracted’ from the diagram. A rival scheme, published in 1886 by Lewis Carroll, demands that information is transferred from the premises-diagram to another diagram that would depict the conclusion. This transfer is achieved by following rules which are explicitly defined and strictly applied. Although both Venn and Carroll introduced diagrammatic methods for the problem of elimination, they differ in their practices and demands on how a diagram ought to be manipulated. Venn appealed to imagination to work out the conclusion with a single diagram while Carroll applied rules on a diagram to derive other diagrams. The former method was said to lack rigor, but the latter may be accused of lacking naturalness and economy. This difference of practices, and the philosophical views that they embody, will be shown to resurface in the recent debates on the role of diagrams in mathematical practice.