The aim of this paper is to critically review the traditional typology of argument macrostructures, particularly, the dichotomy between linked and convergent structure. We have found an argument structure that does not fall under one of those five traditional categories: basic, serial, divergent, linked and convergent. We show that the new argument structure, which we call the recursive structure, is not rare-earth, but ubiquitous in real argumentation. Then, we propose and justify a new approach to diagramming arguments of the structure. (...) The new argument structure is really new because arguments of the new structure are analyzed and evaluated differently from those of the other structures, especially from hybrid arguments considered in the literature. In light of the new argument structure, we present a hypothesis why assumptions and exceptions of a defeasible argument play different roles in dialectical settings. (shrink)

This work introduces the Attack-Support Argumentation Framework (ASAF), an approach to abstract argumentation that allows for the representation and combination of attack and support relations. This framework extends the argumentation Framework with Recursive Attacks (AFRA) in two ways. Firstly, it adds a support relation enabling to express support for arguments; this support can also be given to attacks, and to the support relation itself. Secondly, it extends AFRA’s attack relation by allowing attacks to the aforementioned support relation. Moreover, since the (...) support relation of the ASAF has a necessity interpretation, the ASAF also extends the Argumentation Framework with Necessities (AFN). Thus, the ASAF provides a unified framework for representing attack and support for arguments, as well as attack and support for the attack and support relations at any level. (shrink)

Over the last decade, several extensions of Dung’s Abstract Argumentation Frameworks (AFs) have been introduced in the literature. Some of these extensions concern the nature of the attack relation, such as the consideration of recursive attacks, whereas others incorporate additional interactions, such as a support relation. Recently, the Attack–Support Argumentation Framework (ASAF) was proposed, which accounts for recursive attacks and supports, attacks to supports and supports to attacks, at any level, where the support relation is interpreted as necessity. Currently, to (...) determine the accepted elements of an ASAF (which may be arguments, attacks, and supports) it is required to translate such an ASAF into a Dung’s AF. In this work, we provide a formal characterization of acceptability semantics directly on the ASAF, without requiring such a translation. We prove that our characterization is sound since it satisfies different results from Dung’s argumentation theory; moreover, we formally show that the approach proposed here for addressing acceptability is equivalent to the preexisting one, in which the ASAF was translated into an AF. Also, we formalize the relationship between the ASAF and other frameworks on which it is inspired: the Argumentation Framework with Recursive Attacks (AFRA) and the Argumentation Framework with Necessities (AFN). (shrink)

This volume focuses on recursion and reveals a host of new theoretical arguments, philosophical perspectives, formal representations, and empirical evidence from parsing, acquisition, and computer models, highlighting its central role in modern science. Noam Chomsky, whose work introduced recursion to linguistics and cognitive science, and other leading researchers in the fields of philosophy, semantics, computer science, and psycholinguistics in showing the profound reach of this concept into modern science. Recursion has been at the heart of generative grammar from the outset. (...) Recent work in minimalism has put it at center-stage with a wide range of consequences across the intellectual landscape. The contributors to this volume both advance the field and provide a cross-sectional view of the place that recursion takes in modern science. (shrink)

On a widely held characterization, triadic joint attention is the capacity to perceptually attend to an object or event together with another subject. In the last four decades, research in developmental psychology has provided increasing evidence of the crucial role that this capacity plays in socio-cognitive development, early language acquisition, and the development of perspective-taking. Yet, there is a striking discrepancy between the general agreement that joint attention is critical in various domains, and the lack of theoretical consensus on how (...) to account for it. This paper pursues three interrelated aims: (i) it examines the contrast between reductive and non-reductive views of (triadic) joint attention, by bringing into focus the notion of recursive mindreading; (ii) it assembles, advances, and discusses a number of arguments against reductive views; (iii) finally, in dialogue with some prominent non-reductive views, it outlines the case for a non-reductive view that gives pride of place to the idea that co-attenders relate to one another as a ‘you’. (shrink)

In this article, we study some new characterizations of primitive recursive functions based on restricted forms of primitive recursion, improving the pioneering work of R. M. Robinson and M. D. Gladstone. We reduce certain recursion schemes (mixed/pure iteration without parameters) and we characterize one-argument primitive recursive functions as the closure under substitution and iteration of certain optimal sets.

This article offers a preliminary critical-historical reconstruction of the concept of dispossession. Part I examines its role in eighteenth- and nineteenth- century struggles against European feudal land tenure. Drawing upon Marx’s critique of French anarchism in particular, I identify a persistent limitation at the heart of the concept. Since dispossession presupposes prior possession, recourse to it appears conservative and tends to reinforce the very proprietary and commoditized models of social relations that radical critics generally seek to undermine. Part II turns (...) to use of the term in Indigenous struggles against colonization, both in order to better grasp the stakes of the above problematic and suggest a way beyond it. Through a reconstruction of arguments by Indigenous scholars and activists, I seek to show the coherence and novelty of their formulation by suggesting that dispossession has come to name a unique historical process, one in which property is generated under conditions that require divestment and alienation from those who appear, only retroactively, as its original owners. In this way, theft and property are related in a recursive, rather than strictly unilinear, manner. Part III provides a specific historical example in the form of nineteenth-century US property law concerning squatters and homesteaders. (shrink)

One of the most striking characteristics of human beings is their ability to function successfully in complex environments about which they know very little. In light of our pervasive ignorance, we cannot get around in the world just reasoning deductively from our prior beliefs together with new perceptual input. As our conclusions are not guaranteed to be true, we must countenance the possibility that new information will lead us to change our minds, withdrawing previously adopted beliefs. In this sense, our (...) reasoning is “defeasible”. The question arises how defeasible reasoning works, or ought to work. In particular we need rules governing what a cognizer ought to believe given a set of interacting arguments some of which defeat others. That is what is called a “semantics” for defeasible reasoning, and this chapter will propose a new semantics that avoids certain clear counter-examples to all existing semantics. (shrink)

The theorem of the title is proven, solving an old question of Remmel. The method of proof uses an algebraic technique of Remmel-Vaught combined with a complex tree of strategies argument where the true path is needed to figure out the final isomorphism.

This little gem is stated unbilled and proved in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows:Second Recursion Theorem. Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of arguments with values in V so that the standard (...) assumptions and hold with. Every n-ary recursive partial function with values in V is for some e. For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let.We will abuse notation and write ž; rather than ž() when m = 0, so that takes the simpler formin this case ). (shrink)

A familiar argument goes as follows: natural languages have infinitely many sentences, finite representation of infinite sets requires recursion; therefore any adequate account of linguistic competence will require some kind of recursive device. The first part of this paper argues that this argument is not convincing. The second part argues that it was not the original reason recursive devices were introduced into generative linguistics. The real basis for the use of recursive devices stems from a deeper philosophical concern; a grammar (...) that functions merely as a metalanguage would not be explanatorily adequate as it would merely push the problem of explaining linguistic competence back to another level. The paper traces this concern from Zellig Harris and Chomsky’s early work in generative linguistics and presents some implications. (shrink)

Let T be Gödel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let $T^{\star}$ be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schütte-style argument the $T^{\star}$ -derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for $T^{\star}$ is obtained. Another consequence is that every $T^{\star}$ -representable number-theoretic function is elementary (...) recursive. Furthermore, it is shown that, conversely, every elementary recursive function is representable in $T^{\star}$ .The expressive weakness of $T^{\star}$ compared to the full system T can be explained as follows: In contrast to $T$ , computation steps in $T^{\star}$ never increase the nesting-depth of ${\mathcal I}_\rho$ and ${\mathcal R}_\rho$ at recursion positions. (shrink)

The complexity of aII 4 set of natural numbers is encoded into a linear order to show that the finite condensation of a recursive linear order can beII 2–II 1. A priority argument establishes the same result, and is extended to a complete classification of finite condensations iterated finitely many times.

Argumentation is an important research area in the field of AI. There is a substantial amount of work on different aspects of Dung’s abstract Argumentation Framework (AF). Two relevant aspects considered separately so far are: i) extending the framework to account for recursive attacks and supports, and ii) considering dynamics, i.e., AFs evolving over time. In this paper, we jointly deal with these two aspects. We focus on High-Order Argumentation Frameworks with Necessities (HOAFNs) which allow for attack and support relations (...) (interpreted as necessity) not only between arguments but also targeting attacks and supports at any level. We propose an approach for the incremental evaluation of the credulous acceptance problem in HOAFNs, by “incrementally” computing an extension (a set of accepted arguments, attacks and supports), if it exists, containing a given goal element in an updated HOAFN. In particular, we are interested in monitoring the credulous acceptance of a given argument, attack or support (goal) in an evolving HOAFN. Thus, our approach assumes to have a HOAFN Δ, a goal \rho occurring in Δ, an extension E for Δ containing \rho, and an update u establishing some changes in the original HOAFN, and uses the extension for first checking whether the update is relevant; for relevant updates, an extension of the updated HOAFN containing the goal is computed by translating the problem to the AF domain and leveraging on AF solvers. We provide formal results for our incremental approach and empirically show that it outperforms the evaluation from scratch of the credulous acceptance problem for an updated HOAFN. (shrink)

Lange offers an argument that, according to him, “does not show merely that some proofs by mathematical induction are not explanatory. It shows that none are […]”. The aim here is to present a counterexample to his argument.

The present paper focuses on Graham Priest’s claim that even primitive recursive relations may be inconsistent. Although he carefully presented his claim using the expression “may be,” Priest made a definite claim that even numerical equations can be inconsistent. His argument relies heavily on the fact that there is an inconsistent model for arithmetic. After summarizing Priest’s argument for the inconsistent primitive recursive relation, I first discuss the fact that his argument has a weak foundation to explain that the existence (...) of a model for some relations does not guarantee that they are primitive recursive. Moreover, since his identity relation is a combination of a standard identity and a congruence relation, it does not simply represent the standard identity function. Then, I argue that his identity relation cannot be both inconsistent and primitive recursive. Furthermore, I extend the argument to the general case that there is no inconsistent primitive recursive relation. These arguments show that the standard notions of “function” and “primitive recursive function” do not fit Priest’s inconsistent model. New definitions are necessary to show the existence of a dialetheic primitive recursive relation. (shrink)

This book brings together in one place David Hitchcock’s most significant published articles on reasoning and argument. In seven new chapters he updates his thinking in the light of subsequent scholarship. Collectively, the papers articulate a distinctive position in the philosophy of argumentation. Among other things, the author:• develops an account of “material consequence” that permits evaluation of inferences without problematic postulation of unstated premises.• updates his recursive definition of argument that accommodates chaining and embedding of arguments and allows any (...) type of illocutionary act to be a conclusion. • advances a general theory of relevance.• provides comprehensive frameworks for evaluating inferences in reasoning by analogy, means-end reasoning, and appeals to considerations or criteria.• argues that none of the forms of arguing ad hominem is a fallacy.• describes proven methods of teaching critical thinking effectively. (shrink)

I propose a characterization of the dialectical dimension of argumentation by considering the activity of arguing as involving a “second order intersubjectivity”. I argue that argumentative communication enables this kind of intersubjectivity as a matter of the recursive nature of acts of arguing—both as justificatory and as persuasive devices. Calling attention to this feature is a way to underline that argumentative discourses represent the explicit part of a dynamic activity, “a mechanism of rational validation”, as Rescher showed, which (...) is a practice that presupposes the possibility of attaining objectivity. (shrink)

Many of our concepts are introduced to us via, and seem only to be constrained by, roughand-ready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (...) (from numbers to numbers) – i.e. one whose value for any given input can be determined by a step-by-step calculating procedure, where each step is fully determined by some antecedently given finite set of calculating rules. We are told that we are to abstract from practical considerations of how many steps will be needed and how much ink will be spilt in the process, so long as everything remains finite. We are also told that each step is to be ‘small’ and the rules governing it must be ‘simple’, available to a cognitively limited calculating agent: for we want an algorithmic procedure, step-by-minimal-step, to be idiot-proof. For a classic elucidation of this kind, see e.g. Rogers (1967, pp. 1–5). Church’s Thesis, in one form, then claims this informally explicated concept in fact has a perfectly precise extension, the set of recursive functions. Church’s Thesis can be supported in a quasi-empirical way, by the failure of our searches for counterexamples. It can be supported too in a more principled way, by the observation that different appealing ways of sharpening up the informal chararacterization of algorithmic computability end up specifying the same set of recursive functions. But such considerations fall short of a demonstration of the Thesis. So is there a different argumentative strategy we could use, one that could lead to a proof? Sometimes it is claimed that there just can’t be, because you can never really prove results involving an informal concept like algorithmic computability.. (shrink)

A notion of strictly primitive recursive realizability is introduced by Damnjanovic in 1994. It is a kind of constructive semantics of the arithmetical sentences using primitive recursive functions. It is of interest to study the corresponding predicate logic. It was argued by Park in 2003 that the predicate logic of strictly primitive recursive realizability is not arithmetical. Park’s argument is essentially based on a claim of Damnjanovic that intuitionistic logic is sound with respect to strictly primitive recursive realizability, but that (...) claim was disproved by the author of this article in 2006. The aim of this paper is to present a correct proof of the result of Park. (shrink)

A common argument in support of a beginning of the universe used by advocates of the kalām cosmological argument (KCA) is the argument against the possibility of an actual infinite, or the “Infinity Argument”. However, it turns out that the Infinity Argument loses some of its force when compared with the achievements of set theory and it brings into question the view that God predetermined an endless future. We therefore defend a new formal argument, based on the nature of time (...) (just as geometrical reasoning is based on the nature of space), which addresses more directly the question of beginningless time. (shrink)

The foundation scheme in set theory asserts that every nonempty class has an ∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\in$$\end{document}-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-recursion theory. We take KP set theory without foundation as the base theory. We show that KP-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^-$$\end{document} + Π1\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _1$$\end{document}-Foundation + V=L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=L$$\end{document} is enough to carry out finite injury arguments in α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP. (shrink)

It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist non-standard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number.” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward L ̈owenheim-Skolem theorem, most theorists agree that the number theoretic version (...) does not have skeptical consequences about the reference of “natural number” analogous to the ‘relativity’ Skolem claimed pertains to notions such as “uncountable” and “cardinal.” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extra-mathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we consider the manner in which the natural numbers figure in our computational practices – e.g. in characterizing how we compute sums and products or determine prime factorizations – it follows that our application of computational terms such as “computable function” and “tractable computational problem” indirectly uniquely determines their structure. (shrink)

Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 3–37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results for any (...) ideal of the r.e. degrees. (shrink)

Cultural evolutionary psychology (CEP) accounts for the cultural evolution of cognition. It is based on evolutionary psychology and cultural evolutionary theory and aims at a synthetic attempt which is achieved by what we want to call ‘generalization by recursion’. We argue in this paper that the evolutionary psychology program as a whole could greatly benefit from CEP’s generalization. As we will show, there is one theoretical element in particular, that CEP relevantly generalizes in contrast to its predecessors. It comes from (...) cultural evolutionary theory and has been referred to as the ‘dual inheritance argument’. CEP’s novelty consists in showing that not only cognitive products (‘grist’) but also cognitive mechanisms (‘mills’) are subject to cultural evolution and provides lots of empirical evidence for this claim. This account is ‘recursive’, because CEP’s generalization of the dual inheritance argument theoretically employs a recursive feedback-loop between cultural learning and cultural evolution. We also argue that this account might be considered to supersede unificatory and reductionistic efforts of its competitors, because it is stronger than purely structural or analogical unification while at the same time it is not too strong in order to fall prey to implausible reductionism. (shrink)

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis.

We consider questions related to the rigidity of the structure R, the PTIME-Turing degrees of recursive sets of strings together with PTIME-Turing reducibility, pT, and related structures; do these structures have nontrivial automorphisms? We prove that there is a nontrivial automorphism of an ideal of R. This can be rephrased in terms of partial relativizations. We consider the sets which are PTIME-Turing computable from a set A, and call this class PTIMEA. Our result can be stated as follows: There is (...) an oracle, A, relative to which the PTIME-Turing degrees are not rigid . Furthermore, the automorphism can be made to preserve the complexity classes DTIMEA for all k 1, or to move any DTIMEA for n 2. From the existence of such an automorphism we conclude as a corollary that there is an oracle A relative to which the classes DTIME are not definable over R. We carry out the corresponding partial relativization for the many-one degrees to construct an oracle, A, relative to which the PTIMA-many-one degrees relative to A have a nontrivial automorphism, and one relative to which the lattice of sets in PTIMEA under inclusion have a nontrivial automorphism. The proof is phrased as a forcing argument; we construct the set A to meet a particular collection of dense sets in our notion of forcing. Roughly, the dense sets will guarantee that if A meets these sets and we split A into two pieces, A0 and A1, in a simple way, and then switching the roles of A0 and A1 in all computations from A will produce an automorphism of the ideal of PTIMA-degrees below A. We force A0 and A1 to have different PTIME-Turing degree; this will then make the automorphism nontrivial. An appropriately generic set A is constructed using a priority argument. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. We review (...) the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

This paper is a sequel to our [7]. In that paper we constructed a Π1 0 tree of avoidable points. Here we construct a Π1 0 tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2<ω. One interesting fact is that while our Π1 0 tree (...) of sharp filters (a subtree of Δ<ω) is isomorphic to the tree of addresses (a subtree of 2<ω), the tree of addresses is recursively enumerable but not recursive. To achieve this end we use a finite injury priority argument. (shrink)

This article argues that Gilbert Simondon’s philosophy of technology is useful for both science and technology studies and critical theory. The synthesis has political implications. It offers an argument for the rationality of democratic interventions by citizens into decisions concerning technology. The new framework opens a perspective on the radical transformation of technology required by ecological modernization and sustainability. In so doing, it suggests new applications of STS methods to politics as well as a reconstruction of the Frankfurt School’s “rational (...) critique of reason.”. (shrink)

In his famous paper, An Unsolvable Problem of Elementary Number Theory, Alonzo Church identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his (...) general philosophical thoughts on mathematics. (shrink)

Physicalism is the idea that everything either is physical or is nothing over and above the physical. For this formulation of physicalism to have determinate content, it should be identified what the “physical” refers to; i.e. the body problem. Some other closely related theses, especially the ones employed in the causal arguments for different versions of physicalism, and more especially the causal closure thesis, are also subject to the body problem. In this paper, I do two things. First, I explore (...) the structure of causal arguments for physicalism that represents a general argument. To do this, the premises and the conclusion of the general argument are given exact formulations. Second, drawing on those premises, especially the causal closure thesis, I propose a naturalistic formulation of the physical that satisfies the requirements any formulation of the physical is expected to fulfill. Following this proposal, we also have a recursive algorithm to recognize the set of all physical events. (shrink)

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges (such as the relationship between Toulmin warrants and their counterparts in traditional diagrams) represent slightly different ways of looking at old and deep theoretical questions. Others (such as how to allow Toulmin diagrams to be recursive) are diagrammatic versions (...) of questions that have already been addressed in artificial intelligence models of argument. But there are further questions (such as the relationships between refutations, rebuttals and undercutters, and the roles of multiple warrants) that are posed as a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class, are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two. (shrink)

David Hitchcock, in his recent “Informal Logic and the Concept of Argument”, defends a recursive definition of ‘argument.’ I present and discuss several problems that arise for his definition. I argue that refining Hitchcock’s definition in order to resolve these problems reveals a crucial, but minimally explicated, relation that was, at best, playing an obscured role in the original definition or, at worst, completely absent from the original definition.

Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability. I show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I argue that (...) there are such propositions, but that no recognizable example of one can be identified, even in principle. (shrink)

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formulaHaving graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules were (...) replaced by pocket calculators and personal computers, but I stuck to my original decision.My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem, which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functionsf: ℕn→ ℕ are computable?While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:f is recursive,f is computable on an abstract machine,f is definable in the untyped λ-calculus.These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance. (shrink)

In this paper, I propose to understand argumentative decoupling—that is, the structural fact of the argumentative chain self-referring to one of its constituents in subsequent arguments—as part of the way in which cognitive decoupling representation works. In order to support this claim, I make use of part of the discussion developed in cognitive studies and evolutionary theories that describes this phenomenon when explaining intentional communication. By using Toulmin’s model, I exemplify how decoupling representation may be seen as part (...) of a chain of arguments in which a second argumentative move is usually oriented to action. I conclude by reflecting on the relationship between this human cognitive capacity and the problem of recursion to hold that these two concepts are not synonymous but stand in a subordinated and complementary relation to each other. (shrink)

In the paper there is presented the argument for the situational paradigm of theory of language. In comparison to the nominativistic semantics it is argued in the paper that the propositional semantics is a better tool of explaining various speech acts in which the principle of compositionality is collapsed. In the paper there are also described referential mechanisms of metaphorical speech acts. In accordance with these mechanisms the metaphorical status of speech acts is determined by the collapse of Fregean referential (...) compositionality of speech acts. In the paper there are constructed two notions of compositionality. First of them may be called classical Fregean concept of compositionality which establishes a semantic correlate of a complex speech act on a functional basis of semantic correlates of constituents of a complex speech act. According to the second notion of compositionality, correlates of constituents of a complex speech act are recursively deter-mined by a correlate of a complex speech act. Mataphorical speech acts possess such a property that correlates of their constituent expressions are determined in processes of de-coding by recursive compositionality functions. Maximal metaphors are speech acts in which recursive as well as Fregean compositionality collapse. In the paper it is argued that the traditional nominativistic semantics is not able to explain communicational functioning of metaphorical speech acts. That is why the conclusion of the paper resolves itself to the thesis that the situational semantic paradigm which should base on recursive notion of compositionality, is better than the nominativistic paradigm. (shrink)

In this paper the theoretical power of the concept of robustness is discussed in order to characterize the argumentative competence of a speaker. This notion is countered with the extended use of the idea complexity. As a general background some empirical results are used to support the theoretical discussion. The paper mainly relies on the theory of cultural cognition to situate the category of robustness and offers particular criteria to specify the possible operationalization of the notion. These criteria could (...) later be applied, and even tested, in the analysis of other social behaviours that involve collective intentionality, cooperation and recursivity. En este trabajo se discute el alcance teórico del concepto robustez para caracterizar la competencia argumentativa de un hablante. Se contrapone la categoría al uso extendido de la idea de complejidad. Como referencia se utilizan algunos resultados empíricos para apoyar la discusión. El trabajo se apoya principalmente en el anclaje conceptual proveniente de la cognición cultural para situar la categoría robustez y ofrece criterios particulares que especifican la posible operacionalización de la noción, criterios que podrán luego ser aplicados, e incluso testeados, en el análisis de otros comportamientos sociales que involucran intencionalidad colectiva, cooperación y recursividad. (shrink)

We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)

In this note is proved the following:Theorem.Iƒ A × B is universal and one oƒ A, B is r.e. then one of A, B is universal.Letα, τbe 1-argument recursive functions such thatxgoes to, τ) is a map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is calleduniversalif every r.e. set is reducible to A; A × B is calleduniversalif the set.

A simple inductive argument shows natural languages to have infinitly many sentences, but workers in the field have uncovered clear evidence of a diverse group of ‘exceptional’ languages from Proto-Uralic to Dyirbal and most recently, Pirahã, that appear to lack recursive devices entirely. We argue that in an information-theoretic setting non-recursive natural languages appear neither exceptional nor functionally inferior to the recursive majority.

It's often hypothesized that the structure of mental representation is map-like rather than language-like. The possibility arises as a counterexample to the argument from the best explanation of productivity and systematicity to the language of thought hypothesis—the hypothesis that mental structure is compositional and recursive. In this paper, I argue that the analogy with maps does not undermine the argument, because maps and language have the same kind of compositional and recursive structure.

A theorem published by A. Grzegorczyk in 1955 states a certain kind of effective uniform continuity of computable functionals whose values are natural numbers and whose arguments range over the total functions in the set of the natural numbers and over the natural numbers. Namely, for any such functional a computable functional with one function-argument and the same number-arguments exists such that the values of the first of the functionals at functions dominated by a given one are completely determined by (...) their restrictions to numbers not exceeding the corresponding values of the second functional at the given function. We prove versions of this theorem for the class of the primitive recursive functionals and for the class of the elementary recursive ones. Analogous results can be proved for many other subrecursive classes of functionals. (shrink)

This article addresses three questions about well-being. First, is well-being future-sensitive? I.e., can present well-being depend on future events? Second, is well-being recursively dependent? I.e., can present well-being depend on itself? Third, can present and future well-being be interdependent? The third question combines the first two, in the sense that a yes to it is equivalent to yeses to both the first and second. To do justice to the diverse ways we contemplate well-being, I consider our thought and discourse about (...) well-being in three domains: everyday conversation, social science, and philosophy. This article’s main conclusion is that we must answer the third question with no. Present and future well-being cannot be interdependent. The reason, in short, is that a theory of well-being that countenances both future-sensitivity and recursive dependence would have us understand a person’s well-being at a time as so intricately tied to her well-being at other times that it would not make sense to consider her well-being an aspect of her state at particular times. It follows that we must reject either future-sensitivity or recursive dependence. I ultimately suggest, especially in light of arguments based on assumptions of empirical research on well-being, that the balance of reasons favors rejecting future-sensitivity. (shrink)

This paper develops concepts and procedures for the evaluation of complex debates. They provide means for answering such questions as whether a thesis has to be considered as proven or disproven in a debate or who carries a burden of proof. While being based on classical logic, this framework represents an (argument-based) approach to non-monotonic, or defeasible reasoning. Debates are analysed as dialectical structures, i.e. argumentation systems with an attack- as well as a support-relationship. The recursive status assignment over the (...) arguments is conditionalised on proponents in a debate. The problem of multiple status assignments arising on circular structures is solved by showing that uniqueness can be guaranteed qua reconstruction of a debate. The notion of burden of proof as well as other discursive aims rational proponents pursue in a debate is defined within the framework. (shrink)

This paper starts from the observation that the standard arguments for compositionality are really arguments for the computability of semantics. Since computability does not entail compositionality, the question of what justifies compositionality recurs. The paper then elaborates on the idea of recursive semantics as corresponding to computable semantics. It is then shown by means of time complexity theory and with the use of term rewriting as systems of semantic computation, that syntactically unrestricted, noncompositional recursive semantics leads to computational explosion (factorial (...) complexity). Hence, with combinatorially unrestricted syntax, semantics with tractable time complexity is compositional. (shrink)

The argument that cumulative technological culture originates in technical-reasoning skills is not the only alternative to social accounts; another possibility is that accumulation ofbothtechnical-reasoning skillsandenhanced social skills stemmed from the onset of a more basic cognitive ability such as recursive representational redescription. The paper confuses individual learning of pre-existing information with creative generation of new information.