Conference Agenda

According to Carnap’s AUFBAU, the concept of similarity occupies center stage in the logical constitution of the world. The basic method of constitution for Carnap was the method of quasi-analysis. Quasi-analysis crucially depends on the concept of similarity.

According to Goodman, Carnap’s similarity-based method of constitution was fatally flawed. On the other hand, Goodman praised the AUFBAU as a philosophical work of first rank relevant for future philosophy of science. “The AUFBAU cannot yet be relegated to the status of a monument having purely historical interest. Its lessons have not been fully enough learned.” Goodman claimed: “The function of an AUFBAU constructional system is …to map experience. A map is schematic, selective, conventional and uniform. And these characteristics are virtues rather than defects.” In this talk I want to argue that a representational generalization of quasi-analysis renders plausible Goodman’s reading of the AUFBAU as a theory of mapping scientific knowledge. This mapping account may be an interesting project even for contemporary philosophy of science.

I review the works of Gärdenfors (1990) and Scorzato (2013) and show that their combination provides an elegant solution of Goodman’s new riddle of induction. The solution is based on two main ideas: (1) Clarifying what is expected from a solution: understanding that philosophy of science is a science itself, with the same limitations and strenghts as the other scientific disciplines. (2) Understainding that the concept of complexity of a model’s assumptions and the concept of direct measurements must be characterized together. Direct measurements are the context where the concept of similarity becomes essential. I argue that conceptual spaces are part of the solution, but I emphasise that a concept of epistemic complexity must also be part of it, to remediate to the limitations of conceptual spaces. In particular, the fact that conceptual spaces are more robust for single-domain concepts (Strößner 2022) is not a serious limitation if their application is needed only for direct measurements.

As Sznajder (2016) argued, there are resemblances between Carnap’s attribute spaces and the conceptual spaces introduced by Gärdenfors (2000,2014). This connection keeps leading to interesting results, e.g. (Douven,Verheyen,Elqayam,Gärdenfors,Osta-Vélez,2025). Earlier in his Aufbau (1923,1928), Carnap sketched a theory of constructional systems of concepts based on the method of quasi-analysis, which reconstructs concepts as sets of similar objects, see (Richardson,1998), (Del Sordo; Mormann, 2022). But Carnap’s quasi-analysis faced Goodman’s (1953) companionship and imperfect community objections. These were discussed by Proust (1986), Brockhaus (1963), Mormann (1994,2009) and Leitgeb (2007,2011). Independently, Rumfitt (2015) and Mormann (2020,2021) developed polar spaces as an application of conceptual spaces to vagueness. Then, Belastegui (2022) showed that polar spaces are mathematically equivalent to the similarity structures of Carnap (1923), and that these avoid Goodman’s objections because they require the existence of objects that behave like prototypes (Rosch,1975). Unless one constrains the convex regions in a conceptual space, imperfect communities can arise. Nevertheless, these cannot arise when concepts are represented as Voronoi cells, because Voronoi tessellations are Carnapian similarities. Thus, using Voronoi tessellations avoids Goodman’s objections justifiably because it appeals to the prototype theory of concepts. Providing this justification is, I claim, what Carnap’s Aufbau can do for conceptual spaces.

The literature is abundant with similarity conceptions: qualitative similarity, structural similarity, set similarity, a.s.o. Do these similarity conceptions have anything in common, in virtue of which they are all similarity conceptions? I will propose that they can all be related to the concept of sameness, although in three different ways, corresponding to three different types of similarity conceptions: commonality conceptions, closeness conceptions and hybrid conceptions. These conceptions can be used to answer three different questions, relating similarity to sameness (where “sameness” should be understood as “full property-overlap”, not as “numerical identity”). The commonality conception can be used to answer the question: in what ways are A and B (partially) the same? The similarity of A and B is regarded as their commonality. The closeness conception can be used to answer the question: what changes would make A and B the same? The similarity of A and B is regarded as their closeness, the opposite of their distance. The hybrid conception tries to answer both questions. Examples of commonality conceptions are partial identity and number of shared properties. Examples of closeness conceptions are metric similarity and transformations. Tversky’s contrast similarity is an example of the hybrid type.

Using multidimensional scaling, Roger Shepard explained accurate generalisation across contexts by appealing to both geometric similarity representations and probabilistic inference. However, such an explanation would be more complex than if one appealed to either principle alone. There is currently no widespread agreement on whether similarity is more fundamental than probability, or vice versa, or how exactly the two relate. A different notion of similarity – structural similarity – highlights the action-relevance (‘exploitability’) of internal representations and requires these to map onto the environment statistics relevant to explain accuracy or adaptive success. I provide a new systematisation according to which the different notions of probability and similarity complement one another in explaining the possibility of accurate generalization based on learned concepts.

Similarity plays a crucial role in Rudolf Carnap's work on inductive logic. Carnap challenged the traditional view that inductive methods contradict empiricism due to their reliance on a synthetic a priori uniformity assumption. Carnap proposed a logical alternative to frequentist probability, advocating for a probabilistic uniformity assumption. He aimed to categorize all probabilistic statements as analytical and part of an inductive logic. In doing so, he faced several challenges. Two of which turned out to be particularly persistent. First, the challenge of an adequate logical probabilistic treatment of universal statements. Second, the challenge of adequately characterising so-called “analogical inferences”. Both, universal statements and analogical inferences, make up for an important core component of scientific theorising. Regarding the former, Carnap saw a solution in transforming the philosophy of science dealing with universal statements towards one of dealing with their inferential role. Regarding the latter, Carnap modified his systems of an inductive logic, adding more and more free parameters and specifying increasingly sophisticated notions and measures of similarity to account for such inferences. In this talk, we will outline the historio-systematic context and cornerstones of Carnap’s programme and detail the development of different notions of similarity involved in it.

This talk explores the connection between the account of qualities set out by Rudolf Carnap’s phenomenalist constitution system, as presented in The Logical Structure of the World and contemporary quality space theory (QST) as advocated by Clark and Rosenthal. Both frameworks assume that qualities can be characterized in terms of similarity relations: Carnap proposes that equivalence classes of elementary experiences, grouped by recollected resemblance, provide the basis for reconstructing qualities. QST, on the other hand, models qualities as locations in similarity-based quality spaces that are determined by perceptual discrimination in the relevant sensory modality.

I argue that, although there is a historical link from Carnap via Goodman to contemporary QST, the differences between the two accounts prevail. This is particularly evident in at least four respects: (1) Carnap considers total experiences, whereas QST considers modalities in isolation; (2) Carnap compares successive experiences, whereas QST compares experiences elicited by simultaneously presented stimuli; (3) Carnap relies on introspection, whereas QST relies on discriminatory behavior that may occur non-consciously; and (4) Carnap treats similarity as fundamental and distinguishability as derived, whereas QST reverses this explanatory direction.

(Joint work with Carl Hodgetts.)

How we judge the similarity between stimuli in the world is connected ultimately to how we represent them. Because of this, decisions about how we model similarity, either in terms of human behavior or patterns of neural activity, can provide key insights into how representations are structured and organized. Despite this, psychology and cognitive neuroscience continue to be dominated by a narrow range of similarity models, particularly those that characterize similarity as distance within “cognitive space.” Despite the appeal of such models, their topological nature places fundamental constraints on their ability to capture relationships between objects and events in the world. To probe this, we created a stimulus set in which the predicted similarity relationships (based on an alternative model of similarity) could not be simply embedded within Euclidean space. This approach revealed that the spatial model distorts these predictions, and the perceived similarities of human observers. These findings indicate that cognitive spaces—that underlie much recent work probing both visual and conceptual representations in cognitive neuroscience—are limited in fundamental ways that restrict their theoretical and practical utility.