Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.
Metaphysical graphical structuralism is the view that at some fundamental level the world is a mathematical graph of nodes and edges. Randall Dipert has advanced a graphical structuralist theory of fundamental particulars and Alexander Bird has advanced a graphical structuralist theory of fundamental properties. David Oderberg has posed a powerful challenge to graphical structuralism: that it entails the absurd inexistence of the world or the absurd cessation of all change. In this paper I defend graphical structuralism. A sharper formulation, (...) some theorems about such structures, and careful attention to the interaction of metaphysical and mathematical features, shows that the absurdities depend on assumptions that are not essential to the view and brings to light a surprising fact about the necessary structure of fundamental properties. (shrink)
The semantic paradoxes are often associated with self-reference or referential circularity. Yablo (1993), however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally (...) complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook (2004) and Yablo (2006) with respect to more restricted sentences systems, which we call F-systems. (shrink)
Nicholas Shackel (2011) has proposed a number of arguments to save the Dipert–Bird model of physical reality from the sorts of unpalatable consequence I identified in Oderberg 2011. Some consequences, he thinks, are only apparent; others are real but palatable. In neither case does he seem to me to have deflected the concerns I raised, leaving graph structuralism on Dipert–Bird lines as problematic as ever.
In this paper a logical interpretation of semantic nets and graph grammars is proposed for modelling natural language understanding and creating language understanding computer systems. An example of parsing a Finnish question by graph grammars and inferring the answer to it by a semantic net representation is provided.
This experiment investigated the effect of format (line vs. bar), viewers’ familiarity with variables, and viewers’ graphicacy (graphical literacy) skills on the comprehension of multivariate (three variable) data presented in graphs. Fifty-five undergraduates provided written descriptions of data for a set of 14 line or bar graphs, half of which depicted variables familiar to the population and half of which depicted variables unfamiliar to the population. Participants then took a test of graphicacy skills. As predicted, the format influenced viewers’ interpretations (...) of data. Specifically, viewers were more likely to describe x–y interactions when viewing line graphs than when viewing bar graphs, and they were more likely to describe main effects and “z–y” (the variable in the legend) interactions when viewing bar graphs than when viewing line graphs. Familiarity of data presented and individuals’ graphicacy skills interacted with the influence of graph format. Specifically, viewers were most likely to generate inferences only when they had high graphicacy skills, the data were familiar and thus the information inferred was expected, and the format supported those inferences. Implications for multivariate data display are discussed. (shrink)
It is shown that if ZF is consistent, then so is ZFC + GCH + "There is a graph with cardinality ℵ 2 and chromatic number ℵ 2 such that every subgraph of cardinality ≤ ℵ 1 has chromatic number ≤ ℵ 0 ". This partially answers a question of Erdos and Hajnal.
The study of the autonomic nervous system (ANS) function has shown to provide useful indicators for risk stratification and early detection on a variety of cardiovascular pathologies. However, data gathered during different tests of the ANS are difficult to analyse, mainly due to the complex mechanisms involved in the autonomic regulation of the cardiovascular system (CVS). Although model-based analysis of ANS data has been already proposed as a way to cope with this complexity, only a few models coupling the main (...) elements involved have been presented in the literature. In this paper, a new model of the CVS, representing the ventricles, the circulatory system and the regulation of the CVS activity by the ANS, is presented. The models of the vascular system and the ventricular activity have been developed using the Bond Graph formalism, as it proposes a unified representation for all energetic domains, facilitating the integration of mechanic and hydraulic phenomena. In order to take into account the electro-mechanical behaviour of both ventricles, an electrophysiologic model of the cardiac action potential, represented by a set of ordinary differential equations, has been integrated. The short-term ANS regulation of heart rate, cardiac contractility and peripheral vasoconstriction is represented by means of continuous transfer functions. These models, represented in different continuous formalisms, are coupled by using a multi-formalism simulation library. Results are presented for two different autonomic tests, namely the Tilt Test and the Valsalva Manoeuvre, by comparing real and simulated signals. (shrink)
Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We classify the closed permutation subgroups containing the group $\operatorname {Aut}(\Gamma)^{\ast}$ , where $\operatorname {Aut}(\Gamma)^{\ast}$ is the group of all isomorphisms and anti-isomorphisms of $\Gamma$ preserving the two sides. Our results rely on a combinatorial theorem of Nešetřil and Rödl and (...) a strong finite submodel property for $\Gamma$. (shrink)
This paper presents a Conceptual Graph (CG) framework to the Generation of Referring Expressions (GRE). Employing Conceptual Graphs as the underlying formalism allows a rigorous, semantically rich, approach to GRE. A number of advantages over existing work are discussed. The new framework is also used to revisit existing complexity results in a fully rigorous way, showing that the expressive power of CGs does not increase the theoretical complexity of GRE.
In this paper the existence or nonexistence of isomorphic mappings between graph models for the untyped lambda calculus is studied. It is shown that Engeler's D A is completely determined, up to isomorphism, by the cardinality of its `atom-set' A. A similar characterization is given for a collection of graph models of the Pω-type; from this some propositions regarding automorphisms are obtained. Also we give an indication of the complexity of the first-order theory of graph models by (...) showing that the second-order theory of first-order definable elements of a graph model is first-order expressable in the model. (shrink)
This article sets up a graph-theoretical framework for argumentation-analysis (dialectical analysis) which expands classical argument-analysis. Within this framework, a main theorem on the existence of inconsistencies in debates is stated and proved: the vicious circle theorem. Subsequently, two corollaries which generalize the main theorem are derived. Finally, a brief outlook is given on further expansions and possible applications of the developed framework.
Machine generated contents note: Foreword Maurice Nivat; Introduction; 1. Overview; 2. Graph algebras and widths of graphs; 3. Equational and recognizable sets in many-sorted algebras; 4. Equational and recognizable sets of graphs; 5. Monadic second-order logic; 6. Algorithmic applications; 7. Monadic second-order transductions; 8. Transductions of terms and words J. Engelfriet; 9. Relational structures; 10. Conclusion and open problems; References; Index.
We present a system for deciding whether a given sentence can be inferred from text. Each sentence is represented as a directed graph (extracted from a dependency parser) in which the nodes represent words or phrases, and the links represent syntactic and semantic relationships. We develop a learned graph matching model to approximate entailment by the amount of the sentence’s semantic content which is contained in the text. We present results on the Recognizing Textual Entailment dataset (Dagan et (...) al., 2005), and show that our approach outperforms Bag- Of-Words and TF-IDF models. (shrink)
This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of vertices; every missing edge corresponds to an independence relation. These features lead to a simple parameterization of the corresponding set of distributions in the Gaussian case.
It is shown that complex adaptations are best modelled as discrete processes represented on directed weighted graphs. Such a representation captures the idea that problems of adaptation in evolutionary biology are problems in a discrete space, something that the conventional representations using continuous adaptive landscapes does not. Further, this representation allows the utilization of well-known algorithms for the computation of several biologically interesting results such as the accessibility of one allele from another by a specified number of point mutations, the (...) accessibility of alleles at a local maximum of fitness, the accessibility of the allele with the globally maximum fitness, etc. A reduction of a model due to Kauffman and Levin to such a representation is explicitly carried out and it is shown how this reduction clarifies the biological questions that are of interest. (shrink)
This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of vertices; every missing edge corresponds to an independence relation. These features lead to a simple parameterization of the corresponding set of distributions in the Gaussian case.
mix of the concrete and the abstract (if we include universals, laws, propositions and the like), but whichever of these is the case, the world is not purely abstract, as a formal structure is. One might claim, however, that the world is a structure1 in the sense that it instantiates a structure and is nothing else. In other words, all there is to the..
The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...) is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)
For digraphs G, we write V(G) for the set of all vertices of G, and E(G) for the set of all edges of G. A digraph on a set E is a digraph G where V(G) = E.
Since then we have been engaged in the development of such results of greater relevance to mathematical practice. In January, 1997 we presented some new results of this kind involving what we call “jump free” classes of finite functions. This Jump Free Theorem is treated in section 2.
This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact (...) mathematical constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures. (shrink)
Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely to individual (...) things but to how those things are related in larger structures, certain aspects of which relations force certain other aspects to be a certain way. (shrink)
Contemporary cognitive neuropsychology attempts to infer unobserved features of normal human cognition, or ?cognitive architecture?, from experiments with normals and with brain-damaged subjects in whom certain normal cognitive capacities are altered, diminished, or absent. Fundamental methodological issues about the enterprise of cognitive neuropsychology concern the characterization of methods by which features of normal cognitive architecture can be identified from such data, the assumptions upon which the reliability of such methods are premised, and the limits of such methods?even granting their assumptions?in (...) resolving uncertainties about that architecture. With some idealization, the question of the capacities of various experimental designs in cognitive neuropsychology to uncover cognitive architecture can be reduced to comparatively simple questions about the prior assumptions investigators are willing to make. This paper presents some of simplest of those reductions. 1Research for this paper was made possible by a fellowship from the John Simon Guggenheim Memorial Foundation and by grant number SBE-9212264 from the National Science Foundation. I thank Martha Farah for teaching me what little I know of cognitive neuropsychology, Jeffrey Bub for stimulating me to think about these issues and for commenting on drafts of this paper, and Peter Slezak for additional comments. Alfonso Caramazza and Michael McCloskey provided very helpful comments on a second draft. (shrink)
Let L ω ∞ω be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. It is shown that if p(n) ≪ n -1 satisfies certain simple conditions on its growth rate, then for every σ∈ L ω ∞ω , the probability that σ holds for (...) the random graph on n vertices converges. In fact, if $p(n) = n^{-\alpha}, \alpha > 1$ , then the probability is either smaller than 2 -n d for some $d > 0$ , or it is asymptotic to cn -d for some $c > 0, d \geq 0$ . Results on the difficulty of computing the asymptotic probability are given. (shrink)
In order to make scientific results relevant to practical decision making, it is often necessary to transfer a result obtained in one set of circumstances—an animal model, a computer simulation, an economic experiment—to another that may differ in relevant respects—for example, to humans, the global climate, or an auction. Such inferences, which we can call extrapolations, are a type of argument by analogy. This essay sketches a new approach to analogical inference that utilizes chain graphs, which resemble directed acyclic graphs (...) (DAGs) except in allowing that nodes may be connected by lines as well as arrows. This chain graph approach generalizes the account of extrapolation I provided in my (2008) book and leads to new insights that integrate the contributions of the other participants of this symposium. More specifically, this approach explicates the role of “fingerprints,” or distinctive markers, as a strategy for avoiding an underdetermination problem having to do with spurious analogies. Moreover, it shows how the extrapolator’s circle, one of the central challenges for extrapolation highlighted in my book, is closely tied to distinctive markers and the Markov condition as it applies to chain graphs. Finally, the approach suggests additional ways in which investigations of a model can provide information about a target that are illustrated by examples concerning nanomaterials in sunscreens and Wendy Parker’s discussion of fingerprints in climate science. (shrink)
A sequence generator is a finite graph, more general than, but akin to, the usual state diagram associated with a finite automaton. The nodes of a sequence generator represent complete states, and each node is labeled with an input and an output state. An element of the behavior of a sequence generator is obtained by taking the input and output states along an infinite path of the graph.Sequence generators may be associated with formulas of the monadic predicate calculus, (...) in which the individual variables range over the times 0, 1, 2, 3, [middle dot][middle dot][middle dot], and the predicate variables represent complete states, input states, and output states. An unrestricted singulary recursion is a formula in which the complete state at time [tau] + 1 is expressed as a truth-function of the complete state at time [tau] and the input states from times [tau] + 1 to [tau] + h. Necessary and sufficient conditions are given for a formula derived from a sequence generator being equivalent to an unrestricted singulary recursion. (shrink)
Given any finite graph, which transitive graphs approximate it most closely and how fast can we find them? The answer to this question depends on the concept of “closest approximation” involved. In [8,9] a qualitative concept of best approximation is formulated. Roughly, a qualitatively best transitive approximation of a graph is a transitive graph which cannot be “improved” without also going against the original graph. A quantitative concept of best approximation goes back at least to [10]. (...) A quantitatively best transitive approximation is a transitive graph that makes the minimal number of mistakes against the original graph. In other words, the sum of the edges that are removed from and are added to the original graph is minimal. (shrink)
Previous asymptotically correct algorithms for recovering causal structure from sample probabilities have been limited even in sparse graphs to a few variables. We describe an asymptotically correct algorithm whose complexity for fixed graph connectivity increases polynomially in the number of vertices, and may in practice recover sparse graphs with several hundred variables. From..
In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph’s nodes. Such theories, which we call annotation theories , can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, (...) including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPT ime - or co-NPT ime -complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21]. (shrink)
We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$ , of cofinality $\omega$ , such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ many colours has a g-colour class on which c takes every value. The paper also contains some generalisations of the above statement in which $\mu^{+}$ is (...) replaced by other cardinals < $\mu$. (shrink)
We observe a number of connections between recent developments in the study of constraint satisfaction problems, irredundant axiomatisation and the study of topological quasivarieties. Several restricted forms of a conjecture of Clark, Davey, Jackson and Pitkethly are solved: for example we show that if, for a finite relational structure M, the class of M-colourable structures has no finite axiomatisation in first order logic, then there is no set (even infinite) of first order sentences characterising the continuously M-colourable structures amongst compact (...) totally disconnected relational structures. We also refute a rather old conjecture of Gorbunov by presenting a finite structure with an infinite irredundant quasi-identity basis. (shrink)
We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal µ, of cofinality ω, such that every µ+-chromatic graph X on µ+ has an edge colouring c of X into µ colours for which every vertex colouring g of X into at most µ many colours has a g-colour class on which c takes every value.
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at (...) the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method. (shrink)
This paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite: (a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent, (b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them. A graph $G = \langle\eta, \eta\rangle$ with η as (...) set of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs $G = \langle\nu, \eta\rangle$ with $n = \operatorname{card}(\nu), e = \operatorname{card}(\eta)$ , do generalize to isolic ω-graphs, e.g., n - 1 ≤ e ≤ n(n - 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree $G = \langle\nu, \eta\rangle: N = E + 1$ iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs. The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph. An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum. (shrink)
Let CKDT be the assertion that for every countably infinite bipartite graph G, there exist a vertex covering C of G and a matching M in G such that C consists of exactly one vertex from each edge in M. (This is a theorem of Podewski and Steffens [12].) Let ATR0 be the subsystem of second-order arithmetic with arithmetical transfinite recursion and restricted induction. Let RCA0 be the subsystem of second-order arithmetic with recursive comprehension and restricted induction. We show (...) that CKDT is provable in ART0. Combining this with a result of Aharoni, Magidor, and Shore [2], we see that CKDT is logically equivalent to the axioms of ATR0, the equivalence being provable in RCA0. (shrink)
The aim of this paper is to elucidate the mereological structure of complex states of affairs without relying on the problematic notion of structural universals. For this task tools from graph theory, lattice theory, and the theory of relational systems are employed. Our starting point is the mereology of similarity structures. Since similarity structures are structured sets, their mereology can be considered as a generalization of the mereology of sets ...
This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ?moving pictures of thought?. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, (...) the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege?Russell thesis, and the failure of the Language of Thought hypothesis. (shrink)
This paper examines Charles Peirce's graphical notation for first-order logic with identity. The notation forms a part of his system of existential graphs, which Peirce considered to be his best work in logic. In this paper a Tarskian semantics is provided for the graphical system.
Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), Sperber (...) and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning. (shrink)
In this paper, a game-theoretical semantics is developed for the so-called alpha part of Charles S. Peirce's System of Existential Graphs of 1896. This alpha part is that portion of Peirce's graphs that corresponds to propositional logic. The paper both expounds a game-theoretical semantics for the graphs that seems close to Peirce's own intentions and proves for the alpha part of the graphs that this semantics is adequate.
Given a finite relational language L is there an algorithm that, given two finite sets A and B of structures in the language, determines how many homogeneous L structures there are omitting every structure in B and embedding every structure in A? For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether QΓ, the class of finite tournaments omitting every tournament in Γ, is well-quasi-order? First, we give a nonconstructive (...) proof of the existence of an algorithm for the case in which Γ consists of one tournament. Then we determine explicitly the set of tournaments each of which does not have an antichain omitting it. Two antichains are exhibited and a summary is given of two structure theorems which allow the application of Kruskal's Tree Theorem. Detailed proofs of these structure theorems will be given elsewhere. The case in which Γ consists of two tournaments is also discussed. (shrink)
We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition result (...) involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”. (shrink)
We present results from numerical studies of supervised learning operations in recurrent networks considered as graphs, leading from a given set of input conditions to predetermined outputs. Graphs that have optimized their output for particular inputs with respect to predetermined outputs are asymptotically stable and can be characterized by attractors which form a representation space for an associative multiplicative structure of input operations. As the mapping from a series of inputs onto a series of such attractors generally depends on the (...) sequence of inputs, this structure is generally noncommutative. Moreover, the size of the set of attractors, indicating the complexity of learning, is found to behave non-monotonically as learning proceeds. A tentative relation between this complexity and the notion of pragmatic information is indicated. (shrink)
One of the claims made for C. S. Peirce's existential graphs has been that they are a deductively complete formulation of first-order logic with identity. As Peirce presented them, this is true only for certain versions of first-order logic :those which do not include terms for individuals. I amend Peirce's rules here, showing, in particular, how they are capable of demonstrating that, for instance, ?Jack is in the kitchen? contradicts ?Jack is not in the kitchen?
This paper raises the question of the prominence and use of statistical graphs in science, and argues that their use in problem solving analysis can best be understood in an ‘interactionist’ frame of analysis, including bio-emotion, culture, social organization, and environment as elements. The frame contrasts both with philosophical realism and with social constructivism, which posit two variables and one way causal flows. We next posit basic differences between visual, verbal, and numerical media of perception and communication. Graphs are thus (...) seen as key interactive sites where different media are transformed into more interpretable forms. Examples are taken from Limnology where numbers are transformed into graphs to find patterns in them, and thus, by implication in the environmental materials from which the numerical measurements were taken. Their revisualization by passes a human cognitive limitation, for the direct analysis — interpretation of lists and tables of numbers, visual imaging being a cognitive strength. Sense of problem, conceptual repertoire, and social relations are seen to direct this pattern search and interpretive process. (shrink)
Logicians have strongly preferred first-order natural deductive systems over Peirce's Beta Graphs even though both are equivalent to each other. One of the main reasons for this preference, I claim, is that inference rules for Beta Graphs are hard to understand, and, therefore, hard to apply for deductions. This paper reformulates the Beta rules to show more fine-grained symmetries built around visual features of the Beta system, which makes the rules more natural and easier to use and understand. Noting that (...) the rules of a natural deductive system are natural in a different sense, this case study shows that the naturalness and the intuitiveness of rules depends on the type of representation system to which they belong. In a diagrammatic system, when visual features are discovered and fully used, we have a more efficacious deductive system. I will also show that this project not only helps us to apply these rules more easily but to understand the validity of the system at a more intuitive level. (shrink)
Over a decade ago, John Sowa (1984) did the AI community the great service of introducing it to the Existential Graphs of Charles Sanders Peirce. EG is a formalism which lends itself well to the kinds of thing that Conceptual Graphs are aimed at. But it is far more; it is a central element in the mathematical, logical, and philosophical thought of Peirce; this thought is fruitful in ways that are seldom evident when we first encounter it. In one (...) of his major works on Existential Graphs, Peirce remarks that.. (shrink)
Using the programming-language concept of continuations, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an (...) improved account of quantificational binding, weak crossover, wh-questions, superiority, and polarity licensing. (shrink)
This paper presents our work on textual inference and situates it within the context of the larger goals of machine reading. The textual inference task is to determine if the meaning of one text can be inferred from the meaning of another and from background knowledge. Our system generates semantic graphs as a representation of the meaning of a text. This paper presents new results for aligning pairs of semantic graphs, and proposes the application of natural logic to derive inference (...) decisions from those aligned pairs. We consider this work as first steps toward a system able to demonstrate broad-coverage text understanding and learning abilities. (shrink)
We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...) Thomason). The constructions use the result of Erd $\H{o}$ s that there are finite graphs with arbitrarily large chromatic number and girth. (shrink)
The conditional independence relations present in a data set usually admit multiple causal explanations — typically represented by directed graphs — which are Markov equivalent in that they entail the same conditional independence relations among the observed variables. Markov equivalence between directed acyclic graphs (DAGs) has been characterized in various ways, each of which has been found useful for certain purposes. In particular, Chickering’s transformational characterization is useful in deriving properties shared by Markov equivalent DAGs, and, with certain generalization, is (...) needed to justify a search procedure over Markov equivalence classes, known as the GES algorithm. Markov equivalence between DAGs with latent variables has also been characterized, in the spirit of Verma and Pearl (1990), via maximal ancestral graphs (MAGs). The latter can represent the observable conditional independence relations as well as some causal features of DAG models with latent variables. However, no characterization of Markov equivalent MAGs is yet available that is analogous to the transformational characterization for Markov equivalent DAGs. The main contribution of the current paper is to establish such a characterization for directed MAGs, which we expect will have similar uses as Chickering’s characterization does for DAGs. (shrink)
Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain "built-in" relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence.
In discussions about whether the Principle of the Identity of Indiscernibles is compatible with structuralist ontologies of mathematics, it is usually assumed that individual objects are subject to criteria of identity which somehow account for the identity of the individuals. Much of this debate concerns structures that admit of non-trivial automorphisms. We consider cases from graph theory that violate even weak formulations of PII. We argue that (i) the identity or difference of places in a structure is not to (...) be accounted for by anything other than the structure itself and that (ii) mathematical practice provides evidence for this view. We want to thank Leon Horsten, Jeff Ketland, Øystein Linnebo, John Mayberry, Richard Pettigrew, and Philip Welch for valuable comments on drafts of this paper. We are especially grateful to Fraser MacBride for correcting our interpretation of two of his papers and for other helpful comments. CiteULike Connotea Del.icio.us What's this? (shrink)
In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem‘s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure with nodes (...) and links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in this mentally supported space, actively construction possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)
My starting point is some widely accepted and intuitive ideas about justified, well-founded belief. By drawing on John Pollock’s work, I sketch a formal framework for making these ideas precise. Central to this framework is the notion of an inference graph. An inference graph represents everything that is relevant about a subject for determining which of her beliefs are justified, such as what the subject believes based on what. The strengths of the nodes of the graph represent (...) the degrees of justification of the corresponding beliefs. There are two ways in which degrees of justification can be computed within this framework. I argue that there is not any way of doing the calculations in a broadly probabilistic manner. The only alternative looks to be a thoroughly non-probabilistic way of thinking wedded to the thought that justification is closed under competent deduction. However, I argue that such a view is unable to capture the intuitive notion of justification, for it leads to an uncomfortable dilemma: either a widespread scepticism about justification, or drawing epistemically spurious distinctions between different types of lotteries. This should worry anyone interested in well-founded belief. (shrink)
In this article, we reflect on the use of formal methods in the philosophy of science. These are taken to comprise not just methods from logic broadly conceived, but also from other formal disciplines such as probability theory, game theory, and graph theory. We explain how formal modelling in the philosophy of science can shed light on difficult problems in this domain.
This paper argues that the new metaphysics of powers, also known as dispositional essentialism or causal structuralism, is an illusory metaphysics. I argue for this in the following way. I begin by distinguishing three fundamental ways of one might see facts of physical modality—facts about physical necessitation and possibility, causation, disposition, and chance—as being grounded in the world. The first way, call it the 1st degree, is that the actual world, or all worlds, in their entirety, are the source of (...) physical modality. Humeanism is the best known such approach, but there are other less well-known approaches. The second way, the 2nd degree, is that the source of physical modality lies in certain 2nd-order facts, involving a relation between universals. Armstrong’s necessitarianism and other views are 2nd-degree. The third way, the 3rd degree, holds that properties themselves are the source of physical modality. This is the powers view. I examine four ways of developing the 3rd degree: relational constitution, graph-theoretic structuralism, dispositional roles, and powerful qualities. All these ways are either incoherent, or just disguised versions of the 1st-degree. The new metaphysics of powers is illusory. With the collapse of the 3rd degree, the 2nd degree, the necessitarian view of law, collapses as well. I end the paper with some reflections on the 1st degree, on the problem of explaining necessary connections between distinct existences, and on the dim prospects of holist ontology. (shrink)
Predicativity requirements of explicit presentability of objects and predicatively acceptable proof are distinguished from predicativist theses of a philosophical character. Familiar among these are expressions of skepticism about the objectivity of full power sets of infinite sets. Articulation of strong, limitative theses, however, turns out to be problematic: impredicative commitments creep into the very formulations, e.g. that “predicative definability'' marks a limit of “intelligibility''. A thought experiment is proposed to undermine the predicativist idea that arbitrary parts of an infinite whole (...) of atoms are “mind- or language-dependent''. On the other hand, weaker claims, e.g., that predicative mathematics is “more secure'' than impredicative, are nearly platitudinous. The interesting philosophical force of predicativism seems to be negative, in its challenge to indispensability arguments, à la Gödel-Friedman, for the transfinite in pure mathematics, and à la Quine-Putnam, for abstract mathematics in the sciences. Evidence is mounting in favor of Gödel-Friedman, e.g. impredicativity of free-variable formulations of theorems such as Kruskal's and Graph Minor, and more far-reaching, recent work in Boolean Relation Theory. This may lead to a realization of Gödel's idea of justifying strong axioms of infinity through their unifying, explanatory role, in analogy with theoretical physics. (shrink)
According to the standard view of definition, all defined terms are mere stipulations, based on a small set of primitive terms. After a brief review of the Hilbert-Frege debate, this paper goes on to challenge the standard view in a number of ways. Examples from graph theory, for example, suggest that some key definitions stem from the way graphs are presented diagramatically and do not fit the standard view. Lakatos's account is also discussed, since he provides further examples that (...) suggest many definitions are much more than mere convenient abbreviations. (shrink)
We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event space. The only models for (...) the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) Another axiom due to Soler forces K to be the field of real, or complex numbers, or the quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's theorem fully characterizes the probability measures on the algebra of events, so that Born's rule is derived. (d) Gleason's theorem is equivalent to the existence of a certain finite set of rays, with a particular orthogonality graph (Wondergraph). Consequently, all aspects of quantum probability can be derived from rational probability assignments to finite "quantum gambles". (e) All experimental aspects of entanglement- the violation of Bell's inequality in particular- are explained as natural outcomes of the probabilistic structure. (f) We hypothesize that even in the absence of decoherence macroscopic entanglement can very rarely be observed, and provide a precise conjecture to that effect .We also discuss the relation of the present approach to quantum logic, realism and truth, and the measurement problem. (shrink)
Possibly the most fundamental scientific problem is the origin of time and causality. The inherent difficulty is that all scientific theories of origins and evolution consider the existence of time and causality as given. We tackle this problem by starting from the concept of self-organization, which is seen as the spontaneous emergence of order out of primordial chaos. Self-organization can be explained by the selective retention of invariant or consistent variations, implying a breaking of the initial symmetry exhibited by randomness. (...) In the case of time, we start from a random graph connecting primitive “events”. Selection on the basis of consistency eliminates cyclic parts of the graph, so that transitive closure can transform it into a partial order relation of precedence. Causality is assumed to be carried by causal “agents” which undergo a more traditional variation and selection, giving rise to causal laws that are partly contingent, partly necessary. (shrink)
Drawing substantive conclusions from linear causal models that perform acceptably on statistical tests is unreasonable if it is not known how alternatives fare on these same tests. We describe a computer program, TETRAD, that helps to search rapidly for plausible alternatives to a given causal structure. The program is based on principles from statistics, graph theory, philosophy of science, and artificial intelligence. We describe these principles, discuss how TETRAD employs them, and argue that these principles make TETRAD an effective (...) tool. Finally, we illustrate TETRAD's effectiveness by applying it to a multiple indicator model of Political and Industrial development. A pilot version of the TETRAD program is described in this paper. The current version is described in our forthcoming Discovering Causal Structure: Artificial Intelligence for Statistical Modeling. (shrink)
A case study of multimodal systems and a new interpretation of Charles S. Peirce's theory of reasoning and signs based on an analysis of his system of ...
A graphical model is a graph that represents a set of conditional independence relations among the vertices (random variables). The graph is often given a causal interpretation as well. I describe how graphical causal models can be used in an algorithm for constructing partial information about causal graphs from observational data that is reliable in the large sample limit, even when some of the variables in the causal graph are unmeasured. I also describe an algorithm for estimating (...) from observational data (in some cases) the total effect of a given variable on a second variable, and theoretical insights into fundamental limitations on the possibility of certain causal inferences by any algorithm whatsoever, and regardless of sample size. (shrink)
This paper describes Peirce's systems of logic diagrams, focusing on the so-called ''existential'' graphs, which are equivalent to the first-order predicate calculus. It analyses their implications for the nature of mental representations, particularly mental models with which they have many characteristics in common. The graphs are intended to be iconic, i.e., to have a structure analogous to the structure of what they represent. They have emergent logical consequences and a single graph can capture all the different ways in which (...) a possibility can occur. Mental models share these properties. But, as the graphs show, certain aspects of propositions cannot be represented in an iconic or visualisable way. They include negation, and the representation of possibilities qua possibilities, which both require representations that do not depend on a perceptual modality. Peirce took his graphs to reveal the fundamental operations of reasoning, and the paper concludes with an analysis of different hypotheses about these operations. (shrink)
Contemporary versions of Bell’s argument against local hidden variable (LHV) theories are based on the Clauser Horne Shimony and Holt (CHSH) inequality, and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell, and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper we derive two-particle Bell inequalities for (...) traceless two-outcome observables, whose violation in the singlet spin state go beyond the Grothendieck constant both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations, and applying a recent result of Alon and his associates (Invent. Math. 163 499 (2006)) we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increases to in…nity as the dimension and number of measurements grow. Technically these result are possible because perfect correlations (or anti-correlations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%. (shrink)
1 INTRODUCTION Above the other titles he might justly have claimed, Charles S. Peirce prized the title 'logician'. He expressed in several places his ...
A personal account is presented for the present status of mathematical chemistry, with emphasis on non-numerical applications. These use mainly graph-theoretical concepts. Most computational chemical applications involve quantum chemistry and are therefore largely reducible to physics, while discrete mathematical applications often do not. A survey is provided for opinions and definitions of mathematical chemistry, and then for journals, books and book series, as well as symposia of mathematical chemistry.
1. A particle moves back and forth along a line, increasing in speed. Graph. 2. How many equivalence classes in Galilean spacetime are there for a particle that is at rest? A particle that is moving at a constant speed? Why are the previous two questions trick questions? 3. In Galilean spacetime, there is no such thing as absolute velocity. Is there such a thing as absolute acceleration? If not, why not? If so, describe a spacetime in which there (...) is no notion of absolute acceleration. Hint: to move from Aristotelian spacetime to Galilean spacetime, we got rid of the notion of absolute velocity by counting two graphs as equivalent (picturing the same spacetime) if they differed by a shear transformation. Perhaps we can get rid of absolute acceleration with an analogous move? 4. Draw a two-dimensional Cartesian grid. Label the axes x and t, and mark a scale on these axes. Make the x axis the horizontal axis, and the t axis the vertical one. Pick two points that are not on the same vertical line. Name them Ann and Bob. Label each point with its x and t coordinates. (shrink)
We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most (...) k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle. Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result. (shrink)
No one realized that the book and the labyrinth were one and the same.道可道[也]，非常[恆]道名可名[也]，非常[恆]名无名，天地[萬物]之始有名，萬物之母 故常[恆]無欲，以觀其眇常[恆]有欲，以觀其徼[噭]此兩者同出而異名同謂之玄，玄之又玄，眾眇之門。The dao that can be spoken of is not the constant DaoThe name that can be named is not the constant name;Nameless, it is the beginning of heaven and earth [the myriad things]Named, it is the mother of the myriad things. Therefore,Constantly without desire, observe its marvels;Constantly with desire, observe its manifestationsThese two are the same, when emerged they are named differently.When merged, this is called mystery, (...) mystery upon mystery, the gateway to the numerous marvels. (Daodejing 1)1The paradoxical opening lines of the Daodejing have .. (shrink)
We give in this paper indications about the dynamical impact (as phenotypic changes) coming from the main sources of perturbation in biological regulatory networks. First, we define the boundary of the interaction graph expressing the regulations between the main elements of the network (genes, proteins, metabolites, ...). Then, we search what changes in the state values on the boundary could cause some changes of states in the core of the system (robustness to boundary conditions). After, we analyse the role (...) of the mode of updating (sequential, block sequential or parallel) on the asymptotics of the network, essentially on the occurrence of limit cycles (robustness to updating methods). Finally, we show the influence of some topological changes (e.g. suppression or addition of interactions) on the dynamical behaviour of the system (robustness to topology perturbations). (shrink)
The difficulty of conducting relevant experiments has long been regarded as the central challenge to learning about the economy from data. The standard solution, going back to Haavelmo's famous “The Probability Approach in Econometrics” (1944), involved two elements: first, it placed substantial weight on a priori theory as a source of structural information, reducing econometric estimates to measurements of causally articulated systems; second, it emphasized the need for an appropriate statistical model of the data. These elements are usually seen as (...) tightly linked. I argue that they are, to a large extent, separable. Careful attention to the role of an empirically justified statistical model in underwriting probability explains puzzles not only in economics, but more generally with respect to recent criticisms of Reichenbach's principle of the common cause, which lies behind graph-theoretic causal search algorithms. And it provides an antidote to the pessimistic understanding of the possibilities for passive observation of causal structure in econometrics and related areas of Nancy Cartwright and others. (shrink)
The research described here explores the idea of using Supreme Court oral arguments as pedagogical examples in first year classes to help students learn the role of hypothetical reasoning in law. The article presents examples of patterns of reasoning with hypotheticals in appellate legal argument and in the legal classroom and a process model of hypothetical reasoning that relates them to work in cognitive science and Artificial Intelligence. The process model describes the relationships between an advocate’s proposed test for deciding (...) a case or issue, the facts of the hypothetical and of the case to be decided, and the often conflicting legal principles and policies underlying the issue. The process model of hypothetical reasoning has been partially implemented in a computerized teaching environment, LARGO (“Legal ARgument Graph Observer”) that helps students identify, analyze, and reflect on episodes of hypothetical reasoning in oral argument transcripts. Using LARGO, students reconstruct examples of hypothetical reasoning in the oral arguments by representing them in simple diagrams that focus students on the proposed test, the hypothetical challenge to the test, and the responses to the challenge. The program analyzes the diagrams and provides feedback to help students complete the diagrams and reflect on the significance of the hypothetical reasoning in the argument. The article reports the results of experiments evaluating instruction of first year law students at the University of Pittsburgh using the LARGO program as applied to Supreme Court personal jurisdiction cases. The learning results so far have been mixed. Instruction with LARGO has been shown to help law student volunteers with lower LSAT scores learn skills and knowledge regarding hypothetical reasoning better than a text-based approach, but not when the students were required to participate. On the other hand, the diagrams students produce with LARGO have been shown to have some diagnostic value, distinguishing among law students on the basis of LSAT scores, posttest performance, and years in law school. This lends support to the underlying model of hypothetical argument and suggests using LARGO as a pedagogically diagnostic tool. (shrink)
A reactive graph generalizes the concept of a graph by making it dynamic, in the sense that the arrows coming out from a point depend on how we got there. This idea was first applied to Kripke semantics of modal logic in [2]. In this paper we strengthen that unimodal language by adding a second operator. One operator corresponds to the dynamics relation and the other one relates paths with the same endpoint. We explore the expressivity of this (...) interpretation by axiomatizing some natural subclasses of reactive frames. (shrink)
In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion (...) systems for classical logic and Girard's proofnets for linear logic. (shrink)
I describe in this paper some of my efforts in developing formal theories of social processes. These include work on models of occupational mobility, on models to describe the emergence of expectations out of performance evaluations, and on the graph theory formulation of the Status Characteristics theory. Not all models have been equally significant in developing theory. However, the graph theory formulation has played a central role in the growth of the Expectation States program. It has been involved (...) in the generalization of theories, the integration of theories, and in the construction of highly sensitive tests of theories that would be impossible without the inferential capacities of formalization. (shrink)
Time series of macroscopic quantities that are aggregates of microscopic quantities, with unknown one‐many relations between macroscopic and microscopic states, are common in applied sciences, from economics to climate studies. When such time series of macroscopic quantities are claimed to be causal, the causal relations postulated are representable by a directed acyclic graph and associated probability distribution—sometimes called a dynamical Bayes net. Causal interpretations of such series imply claims that hypothetical manipulations of macroscopic variables have unambiguous effects on variables (...) “downstream” in the graph, and such macroscopic variables may be predictably produced or altered even while particular microstates are not. This paper argues that such causal time series of macroscopic aggregates of microscopic processes are the appropriate model for mental causation. (shrink)
Imagine a very fi ne grid or graph on which dots are placed at various coordinates so that, as a consequence, this or that shape materializes there. Depending on the coordinates of the dots, different shapes will appear, and for every shape there will be a pattern in the coordinates that guarantees its appearance. Take, for example, the diagonal line that slopes rightward and upward at an angle of 45 degrees from the origin. This line is bound to make (...) an appearance so long as the coordinates satisfy the condition or pattern that as they move away from the origin, (0,0), the coordinates are progressively larger pairs of equal numbers: (1,1), (3,3), and so on. In the world of such dots and shapes, it is going to be in principle possible, for any array of dots that realizes a relevant shape, to derive the presence of the shape from the numerical coordinates of the dots. More particularly, it is going to be possible to derive that shape without reliance on anything other than, fi rst, the empirical fact that the given array of coordinates instantiates this or that pattern; and second, the a priori knowable fact that the pattern guarantees the presence of the shape in question. The nature of the shapes on any grid—if indeed there are any relevant shapes present—is going to be a priori derivable from the positions of the dots; it is going to be possible in principle to derive the one from the other. The simplest and most appealing version of physicalism parallels this sort of doctrine about dots and shapes (Pettit 1994, 1995). It holds that just as the positions of the dots determine the nature of the shapes a priori, so the way the natural world is physically organized a priori determines the way it presents itself in psychological and other terms. The way things are physically confi gured entails the presence of psychological and other realities, and it does this without reliance on anything other than what a.. (shrink)
