A topological description of space is given, based on the relation of connection among regions and the property of being limited. A minimal set of 10 constraints is shown to permit definitions of points and of open and closed sets of points and to be characteristic of locally compact T2 spaces. The effect of adding further constraints is investigated, especially those that characterise continua. Finally, the properties of mappings in region-based topology are studied. Not all such (...) mappings correspond to point functions and those that do correspond to continuous functions. (shrink)

Regions-based theories of space aim—among others—to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed, one should show that they are points of a geometrical or a topological space constructed in a specific way. This paper intends to show how the development of mathematical tools allows showing that Whitehead’s method of extensive abstraction provides a construction of objects that are (...) fundamental building blocks of specific topological spaces. (shrink)

This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roeper's notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local (...) proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roeper's paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormann's notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained. (shrink)

Through contact algebras we study theories of mereotopology in a uniform way that clearly separates mereological from topological concepts. We identify and axiomatize an important subclass of closure mereotopologies called unique closure mereotopologies whose models always have orthocomplemented contact algebras , an algebraic counterpart. The notion of MT-representability, a weak form of spatial representability but stronger than topological representability, suffices to prove that spatially representable complete OCAs are pseudocomplemented and satisfy the Stone identity. Within the resulting class of contact algebras (...) the strength of the algebraic complementation delineates two classes of mereotopology according to the key ontological choice between mereological and topological closure operations. All closure operations are defined mereologically if and only if the corresponding contact algebras are uniquely complemented while topological closure operations highly restrict the contact relation but allow not uniquely complemented and nondistributive contact algebras. Each class contains a single ontologically coherent theory that admits discrete models. (shrink)

While the classical account of the linear continuum takes it to be a totality of points, which are its ultimate parts, Aristotle conceives of it as continuous and infinitely divisible, without ultimate parts. A formal account of this conception can be given employing a theory of quantification for nonatomic domains and a theory of region-based topology.

Due to the advent of deglobalization and regional integration, this article aims to adopt LASSO-based network connectedness to estimate the multiscale tail risk spillover effects of global stock markets. The results show that tail risk varies across frequencies and shocks. In static analysis, the risk is centered mostly on the developed European and North American markets at a low frequency, and regionalization is imposed on the moderate frequency. Moreover, emerging markets could be sources of risk spillover, especially at the (...) highest frequency where there is no absolute risk center. In dynamic analysis, we use rolling window estimation and find that different frequencies identify distinct episodes of shocks, which provides us with the reason for the diverse risk centers at different time scales in static analysis. Our findings provide heterogeneous financial practitioners, regulators, and investors with diverse characteristics of stock markets under multiple time horizons and help them operate their own trading strategies. (shrink)

As one of the most important routing problems in the complex network within a very-large-scale integration circuit, bus routing has become much more challenging when witnessing the advanced technology node enters the deep nanometer era because all bus bits need to be routed with the same routing topology in the context. In particular, the nonuniform routing track configuration and obstacles bring the largest difficulty for maintaining the same topology for all bus bits. In this paper, we first present (...) a track handling technique to unify the nonuniform routing track configuration with obstacles. Then, we formulate the topology-aware single bus routing as an unsplittable flow problem, which is integrated into a negotiation-based global routing to determine the desired routing regions for each bus. A topology-aware track assignment is also presented to allocate the tracks to each segment of buses under the guidance of the global routing result. Finally, a detailed routing scheme is proposed to connect the segments of each bus. We evaluate our routing result with the benchmark suite of the 2018 CAD Contest. Compared with the top-3 state-of-the-art methods, experimental results show that our proposed algorithm achieves the best overall score regarding specified time limitations. (shrink)

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an (...) inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object‐wise, could underlie diagram‐based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions. (shrink)

The system reachability set is calculated by covering all possible behaviours of the system through a finite number of simulation steps to ensure that the system trajectory stays within a set safety region. In this paper, the theory of the game method is applied to the design of the controller, a very small controller is designed, and good control results are obtained by simulation. The system gradually shows a divergent trend and cannot achieve stable control. A multihop channel reservation (...) Medium Access Control protocol based on a parallel mechanism is proposed. The multihop channel reservation mechanism is proposed based on periodic node sleep, and the node uses the reservation frame to make the reservation of the channel and transmits the data according to the reservation information; the parallel mechanism is used to make the reservation information and data transmission simultaneously. (shrink)

It is not only difficult to be a sports expert but also difficult to grow from a sports expert to a sports elite. Professional athletes are often concerned about the differences between an expert and an elite and how to eventually become an elite athlete. To explore the differences in brain neural mechanism between experts and elites in the process of motor behavior and reveal the internal connection between motor performance and brain activity, we collected and analyzed the electroencephalography findings (...) of 14 national archers and 14 provincial archers during aiming and resting states and constructed the EEG brain network of the two archer groups based on weighted phase lag index; the graph theory was used to analyze and compare the network characteristics via local network and global network topologies. The results showed that compared with the expert archers, the elite archers had stronger functional coupling in beta1 and beta2 bands, and the difference was evident in the frontal and central regions; in terms of global characteristics of brain network topology, the average clustering coefficient and global efficiency of elite archers were significantly higher than that of expert archers, and the eigenvector centrality of expert archers was higher; for local characteristics, elite archers had higher local efficient; and the brain network characteristics of expert archers showed a strong correlation with archery performance. This suggests that compared with expert archers, elite archers showed stronger functional coupling, higher integration efficiency of global and local information, and more independent performance in the archery process. These findings reveal the differences in brain electrical network topologies between elite and expert archers in the archery preparation stage, which is expected to provide theoretical reference for further training and promotion of professional athletes. (shrink)

Mereotopology is that branch of the theory of regions concerned with topological properties such as connectedness. It is usually developed by considering the parthood relation that characterizes the, perhaps non-classical, mereology of Space (or Spacetime, or a substance filling Space or Spacetime) and then considering an extra primitive relation. My preferred choice of mereotopological primitive is interior parthood . This choice will have the advantage that filters may be defined with respect to it, constructing “points”, as Peter Roeper has done (...) (“Region-based topology”, Journal of Philosophical Logic , 26 (1997), 25–309). This paper generalizes Roeper’s result, relying only on mereotopological axioms, not requiring an underlying classical mereology, and not assuming the Axiom of Choice. I call the resulting mathematical system an approximate lattice , because although meets and joins are not assumed they are approximated. Theorems are proven establishing the existence and uniqueness of representations of approximate lattices, in which their members, the regions, are represented by sets of “points” in a topological “space”. (shrink)

A region-based model of physical space is one in which the primitive spatial entities are regions, rather than points, and in which the primitive spatial relations take regions, rather than points, as their relata. Historically, the most intensively investigated region-based models are those whose primitive relations are topological in character; and the study of the topology of physical space from a region-based perspective has come to be called mereotopology. This paper concentrates on a (...) mereotopological formalism originally introduced by Whitehead, which employs a single primitive binary relation C(x, y) (read: "x is in contact with y"). Thus, in this formalism, all topological facts supervene on facts about contact. Because of its potential application to theories of qualitative spatial reasoning, Whitehead's primitive has recently been the subject of scrutiny from within the Artificial Intelligence community. Various results regarding the mereotopology of the Euclidean plane have been obtained, settling such issues as expressive power, axiomatization and the existence of alternative models. The contribution of the present paper is to extend some of these results to the mereotopology of three-dimensional Euclidean space. Specifically, we show that, in a first-order setting where variables range over tame subsets of ℝ³, Whitehead's primitive is maximally expressive for topological relations; and we deduce a corollary constraining the possible region-based models of the space we inhabit. (shrink)

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk :228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation. Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze separation structures (...) and Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces. (shrink)

In the first section we briefly describe methodological assumptions of point-free geometry and topology. We also outline history of geometrical theories based on the notion of emph{region}. The second section is devoted to concise presentation of the content of the LLP special issue on point-free theories of space.

The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain from (...) a philosophical interpretation of the results from the paper. (shrink)

Mereotopology is an extension of mereology with some relations of topological nature like contact. An algebraic counterpart of mereotopology is the notion of contact algebra which is a Boolean algebra whose elements are considered to denote spatial regions, extended with a binary relation of contact between regions. Although the language of contact algebra is quite expressive to define many useful mereological relations and mereotopological relations, there are, however, some interesting mereotopological relations which are not definable in it. Such are, for (...) instance, the relation of n-ary contact, internal connectedness and some others. To overcome this disadvantage, we introduce a generalisation of contact algebra, replacing the contact with a binary relation between finite sets of regions and a region, satisfying some formal properties of Tarski consequence relation. The obtained system is called sequent algebra, considered as an algebraic counterpart of a new mereotopology. We develop the topological representation theory for sequent algebras showing in this way certain correspondence between point-free and point-based models of space. As a by-product, we show how one logical relation in nature notion, Tarski consequence relation, may have also certain spatial meaning. (shrink)

In this paper, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson’s nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius and Russell, which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views these (...) authors suggested. Unlike Arntzenius’s proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell’s proposal and does not retreat to mere finite additivity. (shrink)

Undoubtedly the Penrose-Hameroff Orch OR model may be considered as a good theory for describing information processing mechanisms and holistic phenomena in the human brain, but it doesn’t give us satisfactory explanation of human perception. In this work a new approach explaining our perception is introduced, which is in good agreement with Orch OR model and other mainstream science theories such as string theory, loop quantum gravity and holographic principle. It is shown that human perception cannot be explained in the (...) terms of elementary particles and we should introduce new indivisible holistic objects with geometry based on smooth infinitesimal analysis - elastic membranes. The example of such a membrane is our Universe which is an indivisible whole. It is shown that our perception may be considered as the result of elastic oscillations of two dimensional (2D) elastic membranes with closed topology embedded in our bodies. Only one elastic membrane responsible for its perceptions will correspond to the selected organism, but there may be other membranes, even at the cell level. In other words, reality may be considered as the process of time evolution of holistic energetically very weak macro objects - elastic membranes with the geometry based on smooth infinitesimal analysis. An embedded membrane in this multidimensional world will look different for the external and internal observers: from the outside it will look like a material object with smooth infinitesimal geometry, while from the inside our Universe-like space-time fabric. When interacting with elementary particles and other membranes, a membrane will transform their energy into its elastic energy (a new form of energy) - the energy of stretching of the infinitesimal segments. The theory postulates that these elastic deformations will not be observable from the point of view of the internal observer. Heisenberg’s uncertainty principle will work in this physics only from the point of view of the internal observer. For the external observer each embedded elastic membrane may be stretched and even a very small region will become observable. For example, living organisms play the role of internal observers of the Universe, and at the same time they serve as external observers for 2D membranes embedded into our Universe. We can observe our 2D self-membranes through our perceptions, which are encoded in elastic oscillations of the elastic membrane. According to the theory, elastic membranes occupy energetically favorable positions around microtubules involved into Orch OR. The theory not only gives us a really multidimensional holistic picture of reality, but it also provides us with a new method for understanding such phenomena as perception, self-awareness and will. (shrink)

Objective: Cognitive impairment is a common neurological complication in patients with end-stage renal disease undergoing maintenance hemodialysis. Brain network analysis based on graph theory is a promising tool for studying CI. Therefore, the purpose of this study was to analyze the changes of functional brain networks in patients on MHD with and without CI by using graph theory and further explore the underlying neuropathological mechanism of CI in these patients.Methods: A total of 39 patients on MHD and 25 healthy (...) controls matched for age, sex, and years of education were enrolled in the study. Resting-state functional magnetic resonance imaging and T1-weighted high-resolution anatomical data were obtained, and functional brain networks for each subject were constructed. The brain network parameters at the global and regional levels were calculated, and a one-way analysis of covariance was used to compare the differences across the three groups. The associations between the changed graph-theory parameters and cognitive function scores in patients on MHD were evaluated using Spearman correlation analysis.Results: Compared with HCs, the global parameters [sigma, gamma, and local efficiency ] in both patient groups decreased significantly. The clustering coefficient in patients with CI was significantly lower than that in the other two groups. The regional parameters were significantly lower in the right superior frontal gyrus, dorsolateral and gyrus rectus of patients with CI than those of patients without CI; however the nodal local efficiency in the left amygdala was significantly increased. The global Cp and regional parameters in the three brain regions were significantly correlated with the cognitive function scores.Conclusion: This study confirmed that the topology of the functional brain network was disrupted in patients on MHD with and without CI and the disruption of brain network was more severe in patients with CI. The abnormal brain network parameters are closely related to cognitive function in patients on MHD. (shrink)

Contact algebra is one of the main tools in region-based theory of space. In it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. Thus we obtain structures, called contact join-semilattices and structures, called distributive contact join-semilattices. We obtain a set-theoretical representation theorem for CJS and a relational representation theorem for DCJS. As corollaries we get also topological representation theorems. We prove that the universal theory of CJS (...) and of DCJS is the same and is decidable. (shrink)

Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in two-dimensional space. Our strategy is to define a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences (...) in our language as this familiar interpretation. This proposal has the merit of transforming a vague, open-ended question about ontologies for practical mereotopological reasoning into a precise question in model theory. We show that (a version of) the familiar interpretation is countable and atomic, and therefore prime. We conclude that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace. (shrink)

The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C, called contact. Standard models of contact algebras are topological and are the contact algebras of regular closed sets in a given topological space. In such a contact algebra we add the predicate of internal connectedness with the following meaning—a regular closed set is internally connected if and only if its (...) interior is a connected topological space in the subspace topology. We add also a ternary relation \ meaning that the intersection of the first two arguments is included in the third. In this paper the extension of a Boolean algebra with \, contact and internal connectedness, satisfying certain axioms, is called an extended contact algebra. We prove a representation theorem for extended contact algebras and thus obtain an axiomatization of the theory, consisting of the universal formulas, true in all topological contact algebras with added relations of internal connectedness and \. (shrink)

On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary relation (‘qualitative (...) probability’) that is tied to _measure_. It expresses that one region is _smaller than or equal in size_ to another. Algebraic models of our theory are _separation_ _σ_-_algebras with qualitative probability_: \((\mathbb {B}, \ll, \preceq )\), where \(\mathbb {B}\) is a Boolean _σ_-algebra, ≪ is a separation relation on \(\mathbb {B}\), and ≼ is a qualitative probability on \(\mathbb {B}\). We show that from algebraic models of this kind we can, in an interesting class of cases, recover a compact Hausdorff topology _X_, together with a countably additive measure _μ_ on a _σ_-field of Borel subsets of that topology, and that \((\mathbb {B}, \ll, \preceq )\) is isomorphic to a ‘standard model’ arising out of the pair (_X_, _μ_). It follows from one of our main results that any closed ball in Euclidean space, \(\mathbb {R}^{n}\), together with Lebesgue measure arises in this way from a separation _σ_-algebra with qualitative probability. (shrink)

The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. There are some problems related to the motivation of the operation of Boolean complementation. Because of this operation is dropped and the language of distributive lattices is extended by considering as non-definable primitives the relations of contact, nontangential inclusion and dual contact. It is obtained an axiomatization of (...) the theory consisting of the universal formulas in the language true in all contact algebras. The structures in, satisfying the axioms in question, are called extended distributive contact lattices. In this paper we consider several logics, corresponding to EDCL. We give completeness theorems with respect to both algebraic and topological semantics for these logics. It turns out that they are decidable. (shrink)

Topological indices are quantitative measurements that describe a molecule’s topology and are quantified from the molecule’s graphical representation. The significance of topological indices is linked to their use in QSPR/QSAR modelling as descriptors. Mathematical associations between a particular molecular or biological activity and one or several biochemical and/or molecular structural features are QSPRs and QSARs. In this paper, we give explicit expressions of two recently defined novel ev-degree- and ve-degree-based topological indices of two classes of benzenoid, namely, linear (...) hexagonal chain and hammer-like benzenoid. (shrink)

With the expansion of the global air cargo transport system, the operating structure of air cargo has become increasingly separate from passenger counterpart, forming an independent organization model. Despite the Chinese air cargo capacity has grown exponentially in the past, its network is still in its infancy. FedEx and UPS have well-established air cargo networks and have operated effectively on both international and domestic scale; thus, understanding the structure and evolution of their air cargo networks is of a high reference (...) value. In conjunction with the division of US regions from the United States Geological Survey, this paper refers to FedEx and UPS as FEPS and analyzes its topological structure, complexity properties, and air cargo accessibility by using social network analysis and accessibility evaluation methods. The results suggest that the structure of the FEPS air cargo network is in the highly developed states and has the typical “small-world” and “scale-free” network characteristics; the degree centrality values for the nodes in the FEPS network suggest that the network complexity has increased; airports in Memphis, Louisville, Indianapolis, and Ontario are the major hubs with both high centrality values and air cargo accessibility; and the FEPS network presents a unique hub-and-spoke structure compared with the passenger counterpart. (shrink)

Topological indices are numeric parameters which portray the topology of a subatomic structure. In QSAR/QSPR analysis, topological descriptors play a vital role to examine the topology of a network. An interconnection network is a structure whose components are connected physically according to some pattern. In this paper, an interconnection network, ternary hypertree, which is a structural combination of complete ternary tree and hypertree, is introduced. We have evaluated the topological descriptors grounded on the distances for the ternary hypertree. (...) The analytical expressions for Wiener, different types of Szeged, and Mostar indices are determined. (shrink)

A topological index can be focused on uprising of a chemical structure into a real number. The degree-based topological indices have an active place among all topological indices. These topological descriptors intentionally associate certain physicochemical assets of the corresponding chemical compounds. Graph theory plays a very useful role in such type of research directions. The hex-derived networks have vast applications in computer science, physical sciences, and medical science, and these networks are constructed by hexagonal mesh networks. In this paper, (...) we determined the exact values of vertex-edge-degree-based topological descriptors for hex-derived networks HDN 1 p and HDN 2 p, which are generated by the hexagonal network of dimension p. (shrink)

Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α, β -Zagreb index and several other (...) general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks. (shrink)

A topological index is a numeric quantity related with the chemical composition claiming to correlate the chemical structure with different chemical properties. Topological indices serve to predict physicochemical properties of chemical substance. Among different topological indices, degree-based topological indices would be helpful in investigating the anti-inflammatory activities of certain chemical networks. In the current study, we determine the neighborhood second Zagreb index and the first extended first-order connectivity index for oxide network O X n, silicate network S L n, (...) chain silicate network C S n, and hexagonal network H X n. Also, we determine the neighborhood second Zagreb index and the first extended first-order connectivity index for honeycomb network H C n. (shrink)

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. (...) [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras. (shrink)

The aim of this paper is to give new kinds of modal logics suitable for reasoning about regions in discrete spaces. We call them dynamic logics of the region-based theory of discrete spaces. These modal logics are linguistic restrictions of propositional dynamic logic with the global diamond E. Their formulas are equivalent to Boolean combinations of modal formulas like E(A ∧ ⟨α⟩ B) where A and B are Boolean terms and α is a relational term. Examining what we (...) can say about dynamic models when we use formulas to describe them, we successively address the axiomatization/completeness issue and the decidability/complexity issue of our dynamic logics of the region-based theory of discrete spaces. (shrink)

There are many network topology designs that have emerged to fulfill the growing need for networks to provide a robust platform for a wide range of applications like running businesses and managing emergencies. Amongst the most famous network topology designs are star network, mesh network, hexagonal network, honeycomb network, etc. In a star network, a central computer is linked with various terminals and other computers over point-to-point lines. The other computers and terminals are directly connected to the central (...) computer but not to one another. As such, any failure in the central computer will result in a failure of the entire network and computers in star network will not be able to communicate. The star topology design can be represented by a graph where vertices represent the computer nodes and edges represent the links between the computer nodes. In this paper, we study the vertex-edge-based topological descriptor for a newly designed hexagon star network. (shrink)

최근에 전국 시대의 초 나라 지역에서 발굴된 텍스트는 ‘고대 중국 다시 쓰기’를 추동할 정도로 이 시기 지식인들을 새로운 시각으로 조명할 것을 요구한다. 이 글은 이제까지 전래문헌을 중심으로 추출한 고대 중국 사상사의 서술에서 벗어나 종교적 의례를 위한 서주 시대 청동기 명문에서 전국 시대 초 나라 죽간에 이르기까지 유가 사상과 관계된 고고학적 텍스트를 분석한다. 이러한 분석은 초나라 지역의 지식인들이 과거의 종교적 전통을 계승하면서 지역의 특수성에 기반한, 북쪽 중국과는 다른 유가 사상 전개의 방향성을 제시한다. 과거 주 나라의 조상숭배를 위한 의례에서 樂舞를 통하여 ‘威儀’로 (...) 드러난 덕은 초 나라 지역의 유가적 지식인들에 의하여 내면적 수양의 대상으로 전화하였다. 초 지역의 의례 전문가들(儒)은 이전 시대에 天과 조상에서 확인하여 전할 수 있었던 덕의 개념을 누구라도 수양할 수 있는 대상으로서 보편적 덕성의 개념으로 해석하였다. 주 나라의 이념과 통치가 붕괴된 시기에 과거 종교적 의례의 전통과는 별개로, 철저히 철학적 담론의 주제로서 인간 본성의 회복과 실현에 논의의 초점을 맞추는 맹자와는 달리, 초 나라 지역의 의례 전문가들은 의례 행위 그 자체를 도덕적 수양의 토대로 삼는 유가적 교육을 전개하였다. (shrink)

Poverty alleviation through tourism is an important way for China to achieve targeted poverty alleviation and win the battle of poverty alleviation. As a region with deep poverty and great difficulty in poverty alleviation, whether tourism development has injected key impetus into ethnic minority areas needs to be tested by both qualitative analysis and quantitative measurement. This paper takes eight ethnic provinces in China as an example to conduct an empirical study. Based on the Data Envelopment Analysis -BCC (...) model and Malmquist index, it evaluates the tourism investment and tourism poverty alleviation efficiency of the ethnic regions in the two stages of tourism poverty alleviation, and analyzes them by classification. The results of the study show: The pure technical efficiency in the first stage is relatively high, but the total factor productivity of each region is declining; The pure technical efficiency in the second stage is also relatively high, but the scale efficiency is low, and the change rate of total factor productivity of the provinces in China has increased significantly; The “double high” type includes Guangxi, Inner Mongolia, and Guizhou, and the “double low” type includes Qinghai, Yunnan, Tibet, Xinjiang, and Ningxia. The results of the study generally show that tourism poverty alleviation has brought about the improvement of the living standards of residents and the development of local economy, but the efficiency of tourism poverty alleviation needs to be improved. On this basis, the article puts forward corresponding improvement measures, in order to further help the ethnic minority areas get rid of poverty in a comprehensive way by promoting the efficient and sustainable development of tourism. (shrink)

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions (...) as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (shrink)

The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space. Most conveniently, a Euclidean vector space is used, representing phenotypic properties by real-valued variables. Computational work on the biophysical genotype-phenotype model of RNA folding, however, suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and can be formalized only in terms of a relatively unfamiliar (...) structure. Patterns of phenotypic evolution—such as punctuation, irreversibility, and modularity—result naturally from the properties of the genotype-phenotype map, which, given the genetic accessibility structure, define accessibility in the phenotype space. The classical framework, however, addresses these patterns exclusively in terms of natural selection on suitably constructed fitness landscapes. Recent work has extended the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. Lewontin’s notion of “quasi-independence” of characters can also be formalized in topological terms: it corresponds to the assumption that a region of the phenotype space is represented by a product space of orthogonal factors. In this picture, each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a “type”, i.e., as a set of phenotypes that share the same set of phenotypic characters. Thus, a theory of character identity can be developed that is based on the correspondence of local factors in different regions of the phenotype space. (shrink)

Topological indices are of incredible significance in the field of graph theory. Convex polytopes play a significant role both in various branches of mathematics and also in applied areas, most notably in linear programming. We have calculated some topological indices such as atom-bond connectivity index, geometric arithmetic index, K-Banhatti indices, and K-hyper-Banhatti indices and modified K-Banhatti indices from some families of convex polytopes through M-polynomials. The M-polynomials of the graphs provide us with a great help to calculate the topological indices (...) of different structures. (shrink)

This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language ℒ with a distinguished unary predicate c(x), function-symbols +, · and - and constants 0 and 1 is defined. An interpretation ℜ for ℒ is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as 'region x is connected' and the function-symbols (...) and constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation ������ base on the real closed plane which turns out to be isomorphic to ℜ. A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation. (shrink)

Farm to school programs are at the vanguard of efforts to create an alternative agrifood system in the United States. Regionally-based, mid-tier food distributors may play an important role in harnessing the potential of farm to school programs to create viable market opportunities for small- and mid-size family farmers, while bringing more locally grown fresh food to school cafeterias. This paper focuses on the perspectives of food distributors. Our findings suggest that the food distributors profiled have the potential to (...) help institutionalize farm to school programs. Notably, their relationships with farmers may be a critical element in expanding the scale and scope of local school food procurement. Their ability to catalyze local school food procurement however, is limited by the structural context in which farm to school programs operate. Specifically, the oppositional school year and agriculture production cycle, and tight food service budget constraints disembed and limit the potential of farm to school programs to decrease the “marketness” of school food procurement and to shift it from a process based largely on price to one that is more territorially embedded. As farm to school programs continue to gain support, regionally-based food distributors that have the meaningful relationships necessary to re-embed the school food service market back into the larger society may be critical to enabling advocates to achieve their goals. (shrink)

Background and PurposeTranscranial direct current stimulation is an emerging non-invasive neuromodulation technique for focal epilepsy. Because epilepsy is a disease affecting the brain network, our study was aimed to evaluate and predict the treatment outcome of cathodal tDCS by analyzing the ctDCS-induced functional network alterations.MethodsEither the active 5-day, −1.0 mA, 20-min ctDCS or sham ctDCS targeting at the most active interictal epileptiform discharge regions was applied to 27 subjects suffering from focal epilepsy. The functional networks before and after ctDCS were (...) compared employing graph theoretical analysis based on the functional magnetic resonance imaging data. A support vector machine prediction model was built to predict the treatment outcome of ctDCS using the graph theoretical measures as markers.ResultsOur results revealed that the mean clustering coefficient and the global efficiency decreased significantly, as well as the characteristic path length and the mean shortest path length at the stimulation sites in the fMRI functional networks increased significantly after ctDCS only for the patients with response to the active ctDCS. Our prediction model achieved the mean prediction accuracy of 68.3% after the nested cross validation. The mean area under the receiver operating curve was 0.75, which showed good prediction performance.ConclusionThe study demonstrated that the response to ctDCS was related to the topological alterations in the functional networks of epilepsy patients detected by fMRI. The graph theoretical measures were promising for clinical prediction of ctDCS treatment outcome. (shrink)

During the COVID-19 epidemic, draconian countermeasures forbidding nonessential human activities have been adopted in several countries worldwide, providing an unprecedented setup for testing and quantifying the current impact of humankind on climate and for driving potential sustainability policies in the postpandemic era from a perspective of complex systems. In this study, we consider heterogeneous sources of environmental and human activity observables, considered as components of a complex socioenvironmental system, and apply information theory, network science, and Bayesian inference to analyze their (...) structural relations and nonlinear dynamics between January 2019 and August 2020 in northern Italy, i.e., before, during, and after the national lockdown. The topological structure of a complex system strongly impacts its collective behavior; therefore, mapping this structure is essential to fully understand the functions of the system as a whole and its fragility to unexpected disruptions or shocks. To this aim, we unravel the causal relationships between the 16 environmental conditions and human activity variables, mapping the backbone of the complex interplay between intervening physical observables—such as NO2 emissions, energy consumption, intervening climate variables, and different flavors of human mobility flows—to a causal network model. To identify a tipping point during the period of observation, denoting the presence of a regime shift between distinct network states, we introduce a novel information-theoretic method based on statistical divergence widely used in statistical physics. We find that despite a measurable decrease in NO2 concentration, due to an overall decrease in human activities, locking down a region as a climate change mitigation is an insufficient remedy to reduce emissions. Our results provide a functional characterization of socioenvironmental interdependent systems, and our analytical framework can be used, more generally, to characterize environmental changes and their interdependencies using statistical physics. (shrink)

ObjectiveTo explore alterations in white matter network topology in de novo Parkinson's disease patients with rapid eye movement sleep behavior disorder.Materials and MethodsThis study included 171 de novo PD patients and 73 healthy controls recruited from the Parkinson's Progression Markers Initiative database. The patients were divided into two groups, PD with probable RBD and PD without probable RBD, according to the RBD screening questionnaire. Individual structural network of brain was constructed based on deterministic fiber tracking and analyses were (...) performed using graph theory. Differences in global and nodal topological properties were analyzed among the three groups. After that, post hoc analyses were performed to explore further differences. Finally, correlations between significant different properties and RBDSQ scores were analyzed in PD-pRBD group.ResultsAll three groups presented small-world organization. PD-pRBD patients exhibited diminished global efficiency and increased shortest path length compared with PD-npRBD patients and HCs. In nodal property analyses, compared with HCs, the brain regions of the PD-pRBD group with changed nodal efficiency were widely distributed mainly in neocortical and paralimbic regions. While compared with PD-npRBD group, only increased Ne in right insula, left middle frontal gyrus, and decreased Ne in left temporal pole were discovered. In addition, significant correlations between Ne in related brain regions and RDBSQ scores were detected in PD-pRBD patients.ConclusionsPD-pRBD patients showed disrupted topological organization of white matter in the whole brain. The altered Ne of right insula, left temporal pole and left middle frontal gyrus may play a key role in the pathogenesis of PD-RBD. (shrink)

According to the identity version of spacetime supersubstantivalism, material objects are numerically identical to spacetime regions. While the view has been commended for its parsimony and concordance with physics, it has not properly been assessed in light of the metaphysics of properties. The present essay fills this void by discerning several subspecies of identity supersubstantivalism, corresponding to various property ontologies. Finding them all wanting, the essay develops and defends a novel brand of supersubstantivalism based on the trope ontology. On (...) this view, here dubbed ‘supertropestantivalism’, regions are pluralities of geometrically and topologically alike tropes, some of which bundle together as material objects. The resulting picture is not itself an identity version of supersubstantivalism, but it retains the originally attractive features of the identity version while avoiding the shortcomings of competing ontologies. (shrink)

Chapter 1 First Person Access to Mental States. Mind Science and Subjective Qualities -/- Abstract. The philosophy of mind as we know it today starts with Ryle. What defines and at the same time differentiates it from the previous tradition of study on mind is the persuasion that any rigorous approach to mental phenomena must conform to the criteria of scientificity applied by the natural sciences, i.e. its investigations and results must be intersubjectively and publicly controllable. In Ryle’s view, philosophy (...) of mind needs to adopt an antimentalist stance to achieve this aim. Antimentalism not only definitively rejects the idea that mind is a substance separated from the body, it also denies that mental phenomena radically differ from physical phenomena by virtue of several unique features. Most problematically, mental phenomena have a conscious character (mental states are related to specific qualitative feelings) and are accessible only to the first-person (only the subject knows directly what s/he is experiencing inside his/her mind). Ryle takes a strong stance on antimentalism going so far as to maintain that an approach to mind which aims to meet the criteria of scientificity set by the natural sciences must avoid any reference to internal, unobservable mental states. In his view (which is considered a specifically philosophical version of psychological behaviorism and also addresses questions put forward in psychological research), mental states can be redescribed in terms of behavioral dispositions. In this chapter, we address the historical roots of the antimentalist view and analyze its relation to the later tradition of research on mind. We show that, compared to the antimentalist stance, functionalism and cognitivism take a step back when they maintain that direct reference to mental states is necessary since mental states cause and therefore explain human behavior. This step backwards is often interpreted as a return to mentalism. However, this is only partially true. Indeed, we suggest that these later traditions retain one important element of Ryle’s antimentalism, i.e. the idea that mental states must be uniquely identified using external and publicly observable criteria, while excluding any reference to introspection and those qualitative dimensions of a mental state, which are accessible only to the first-person. According to the perspective we put forward, this epistemological stance has continued to influence contemporary research on mind and current philosophical and psychological theories which both tend to exclude the subjective qualities of human experience from their accounts of how the mind works. The issue we raise here is whether this is legitimate or whether subjective qualities do play a role with respect to the way our mind works. The conclusion of this chapter anticipates the argument the book makes in in favor of this latter position, starting from a particular angle, i.e. the problem of how we categorize concepts related to our internal states. -/- Chapter 2 The Misleading Aspects of the Mind/Computer Analogy. The Grounding Problem and the Thorny Issue of Propriosensitive Information -/- Abstract. After the crisis of behaviorism, cognitivism and functionalism became the predominant models in the field of psychology and of philosophy, respectively. Their success is mainly due to the new key they use for interpreting mental processes: the mind/computer analogy. On the basis of this analogy, mental operations are seen as cognitive processes based on computations, i.e. on manipulations of abstract symbols which are in turn understood as informational unities (representations). This chapter identifies two main problems with this model. The first is how these symbols can relate to and communicate with perception and thus allow us to identify and classify what we perceive through the senses. Here we limit ourselves to presenting this issue in relation to the classical symbol grounding problem originally put forward by Harnad on the basis of Searle’s Chinese room argument. An attempt to address the problem raised here will be made in chap. 3. The second point we discuss in relation to the mind/computer analogy concerns the idea of information it fosters. Indeed, following this analogy, information is something available in the external world which can be captured by the senses and transmitted to the central system without being influenced or modified by the procedures of transmission. This perspective does not take into account that – unlike computers – in living beings information is acquired by means of the body. As Ulric Neisser has already pointed out, the body is itself an informational source that provides us with additional sensory experience that influences (modifies or complements) the information extracted from the external world by the senses. To develop this line of analysis and to determine exactly what information is provided by the body and how this might influence cognition, we examine Sherrington’s and Gibson’s positions. Moving on from their views, we qualify bodily information in terms of ‘proprioception’. We use ‘proprioception’ in a broad sense to describe any kind of experience we have of our internal states (including postural information as well as sensations related to the general state of the body and its parts). Following Damasio’s and Craig’s studies, we further elaborate this position, arguing that living beings are equipped with an internal propriosensitive monitoring system which maps all the changes that constantly occur in our body and that give us perceptual (‘proprioceptive’ or propriosensitive) information about what happens inside us. Moreover, relying on Goldie’s and Ratcliffe’s view, we show that emotional information can also be considered as a form of ‘proprioception’ which contributes to determining everything we perceive. This analysis leads us to the second main thesis of this book: ‘proprioception’ is a form of internal perception and it is an essential component of the sensory information we can access and use for all cognitive purposes. -/- Chapter 3 Semantic Competence from the Inside: Conceptual Architecture and Composition -/- Abstract. Concepts are essential constituents of thought: they are the instruments we use to categorize our experience, i.e. to classify things and group them together in homogeneous sets. Here we define concepts as the internal mental information (representations) that allows us, among other things, to master words in natural language. By analyzing the way in which individuals master word meanings we explore a number of hypotheses regarding the nature of concepts. Following Diego Marconi’s research, we differentiate between two kind of abilities that underpin lexical competence – so-called ‘referential’ and ‘inferential competence’ – and we suggest that, in order to support these abilities, concepts must also include two corresponding kinds of information, i.e. inferential and referential information. We point out that the most widely used and acknowledged theories of concepts do not make this distinction, instead broadly characterizing the information used for categorization in terms of propositionally described feature lists. However, we show that while feature lists can explain inferential competence, they do not account for referential competence. To address the issue of referential competence we examine Ray Jackendoff’s hypothesis that to account for the possibility of linking perceptual and conceptual information we need to assume the existence of a (visual) representation that encodes the geometric and topological properties of objects and bridges the gap between the percept and the concept. Furthermore, we analyze the extension of this work by Jesse Prinz who introduced the notion of a proxytype, a perceptual representation of a class of objects that incorporates structural and parametric information related to their appearance. However, as we point out, proxytypes can only explain the relationship between perception and concepts with respect to instances that can be perceived through the senses and that belong to the same class by virtue of their physical similarity. We suggest that this notion be extended to include larger conceptual classes. To accomplish this, we further develop Mark Johnson, George Lakoff and Jean Mandler’s idea of a schematic image and argue that conceptual representations include a perceptual schema. Perceptual schemata are non-linguistic, structured experiential gestalts (patterns or maps) that make use of information taken from all sensory modalities, including body perception. They accomplish a quasi-conceptual function: they allow us to recognize and to classify different instances. In this work, we hypothesize that perceptual schemata are an essential component of concepts, but not identical to them. Instead, we suggest that concepts include both perceptual and propositional information with perceptual schemata providing the ‘perceptual core concept’ that grounds related propositional information. -/- Chapter 4 In the Beginning There Were Categories. The Bodily Origin of Prelinguistic Categorical Organization: The Example of Folkbiological Taxonomies -/- Abstract. Studies of categorization in psychology and the cognitive sciences have made use of the notions of ‘category’ and ‘concept’ without precisely defining what is meant by either; in fact, often these terms have been used as synonyms, making it difficult to address specific issues related to conceptual development. This chapter begins by discussing the definitions of, as well as the distinctions between, ‘categories’ and ‘concepts’ in the classical philosophical tradition (Aristotle, Kant and Husserl). We introduce our view of categories with reference to Husserl. Categorization is defined as the way in which our experience is originally (pre-linguistically) organized in a passive and fully unconscious manner on the basis of universal structuring principles. Conceptualization is explained as a later process in which the earlier categorical macro-classes are further subdivided into more specific and detailed sets; this later process also relies on linguistic learning and exposure to culture. On the basis of Ray Jackendoff, Jean Mandler, George Lakoff and Mark Johnson’s work, we hypothesize that categorization is a two-step process that begins with the formation of homogeneous sets of entities partitioned into regions that describe the ontological boundaries of the objects that humans perceive (categories) and continues at a later stage with the development of more specific classifications (concepts). In this chapter, we mainly address three related issues: Why should we assume that there is categorical organization which precedes the development of a conceptual system? How do categories and concepts relate to each other? Shall we hypothesize that categories are innate or that they are formed before concepts on the basis of information and organizational structure available at a very early developmental stage? We show that categorical partitions are necessary for categorization and, following Mandler, that the general categories we form at an early age do not match our adult superordinate concepts. As for the third issue, we argue that there might be no need to assume that categories are innately present in the human mind, since their formation can be explained – at least in certain cases – by basic mechanisms that work on body (propriosensitive) information. This hypothesis will not be discussed in general, but in relation to a particularly relevant example of categorical partition, i.e. the folk-biological dichotomy between ANIMATE/INANIMATE. This is compared with other dichotomies that derive from it, but are not directly categorical such as LIVING/NON-LIVING and BIOLOGICAL/NON-BIOLOGICAL. -/- Chapter 5 Internal States: From Headache to Anger. Conceptualization and Semantic Mastery -/- Abstract. Here we ask if we can also apply the distinction between referential and inferential competence we introduced in Chapter 3 to words that do not refer to things that are perceived using the external senses, especially to words/concepts that denote bodily experiences (such as pain, thirst, hunger, etc.) or emotions. We introduce and discuss the hypothesis that – even though such words/concepts do not refer to intersubjectively identifiable entities in the external world – they do have a kind of referent that can be accessed via direct perception, more specifically ‘proprioception’, as we have defined it in terms of all propriosensitive information we can consciously access. In the first part of the chapter, we specifically consider terms denoting bodily experiences such as ‘pain’ or ‘hunger’ and argue that their referents are identified and classified from a first-personal point of view on the basis of four main characteristics: their specific intensity, their localization in the body, their co-occurrence with other signals and above all their specific qualitative sensations. Emotions are addressed in the second part of the chapter. We suggest that there is a continuity between bodily experiences and emotions. In particular, we argue for a perceptual theory of emotions in line with that proposed by James and Lange at the end of the 19th century and developed more recently by authors such as Damasio (see also chap. 2§5, §6). The hypothesis we put forward is that the referential information that supports the categorization of emotions and therefore also our mastery of terms referring to emotions consists in the perception (i.e. the ‘proprioception’) of those bodily states and changes in bodily states which constitute our emotional experience. In the context of this discussion we examine some objections to this line of reasoning that arise from a cognitivist perspective and following authors such as Oatley, Johnson-Laird and Frijda, we distinguish between basic emotions that can be identified and classified solely on the basis on how they feel and complex emotions whose identification and classification additionally depends on cognitive factors. To describe how emotions are identified and classified on the basis of how they feel, we rely on Marcel and Lambie’s distinction between an ‘emotion state’ and an ‘emotion experience’. Both notions indicate kinds of feelings that we consciously experience. However, they describe first-order and second order emotion awareness respectively. The emotion state is the feeling we have of the bodily states and changes that occur when we are experiencing an emotion, while the emotion experience is the fully developed and integrated emotion we both experience and are, with reflection, aware of experiencing. On the basis of this differentiation, we also show that the same characteristics that aid in the identification and classification of bodily experiences (specific qualitative sensations; somatic localization; specific intensity; presence/absence of specific concomitant sensations) can also be used for the identification and classification of emotions – at least basic emotions. In the last two sections of the chapter we present some clinical evidence on the semantic competence of people who suffer from Alexithymia and Autism Spectrum Disorder which supports the conclusions of our preceding analyses. -/- Chapter 6 The ‘Proprioceptive’ Component of Abstract Concepts -/- Abstract. In this chapter, we address the issue of whether the mastery of abstract words requires only inferential knowledge and thus, if the concepts that support the mastery of abstract words include only linguistic information. We start by differentiating the notions of ‘abstract’ and ‘general’ which are often erroneously confused. We then identify a strict definition of abstract, as contrasted with ‘concrete’, that applies to words or concepts whose referent cannot be experienced by the senses. We argue that abstract words/concepts would be better described as theoretical, because they are usually conceived as structured sets of inferential knowledge expressed linguistically; that is, as small theories. Pointing out parallels with Carnap’s analysis of this issue in philosophy of science, we hypothesize that words/concepts denoting non-observable entities are not all ‘equally theoretical’, because their link to sensory experience can be stronger or weaker. We revive the distinction, inspired by Quine, between theoretical and intertheoretical concepts/words. This distinction relies on the fact that the former – unlike the latter – retains a strong, although indirect, connection with perception. In Quine’s discussion, perception is understood uniquely in terms of observability, i.e. of external sensory experience. Here we argue, however, that bodily, ‘proprioceptive’ (i.e. propriosensitive) experience can also serve to referentially ground theoretical (i.e. abstract) concepts/words. We frame this issue using the example of the theoretical concept ‘freedom’ and Lakoff’s hypothesis that this concept is developed on the basis of bodily information. We contrast this with the example of ‘democracy’ which more closely resembles an intertheoretical concept/word. Furthermore, we show that one of the classical views put forward in psycholinguistic research to explain how abstract concepts are mentally represented – i.e. Paivio’s Dual Coding Theory – points in the same direction as our analysis. The same is true of Barsalou’s work suggesting that we use internal information to understand at least some abstract words. To sustain this position, we put forward two lines of evidence: the first comes from psycholinguistic studies while the second examines deficits of semantic competence exhibited by people with Autism Spectrum Disorder. On the basis of our analysis, we put forward a classification that distinguishes between different kinds of concreteness and different degrees of abstraction: concepts/words referring to body experiences and basic emotions are described as analogous to concrete concepts/words because they are grounded in perceptual (i.e. propriosensitive) experience, while abstract concepts/words are considered more or less abstract depending on whether they are intratheoretical (and rely entirely on inferential information) or theoretical (and are partially grounded in perceptual – or more often in propriosensitive perceptual – information). In the last section of the chapter we consider two scales that have been used in psycholinguistic research to measure the degree of concreteness vs. abstractness of words and we show that – used conjointly – they can provide a measure of the internal vs. external grounding of specific words. (shrink)