We give recursion-theoretic characterizations of the counting class \(\textsf {\#P} \), the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels \(\{\textsf {\#P} _k\}_{k\in {\mathbb {N}}}\) of the counting hierarchy of functions \(\textsf {FCH} \), which result from allowing queries to functions of the previous level, and \(\textsf {FCH} \) itself as a whole. This is done in the (...) style of Bellantoni and Cook’s safe recursion, and it places \(\textsf {\#P} \) in the context of implicit computational complexity. Namely, it relates \(\textsf {\#P} \) with the implicit characterizations of \(\textsf {FPTIME} \) (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and \(\textsf {FPSPACE} \) (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of \(\textsf {FPSPACE} \). (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 reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as (...) the given totality. (shrink)

When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic (...) into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu. (shrink)

We define, in the spirit of Fenstad [2], a higher type computation theory, and show that countable recursion over the continuous functionals forms such a theory. We also discuss Hyland's proposal from [4] for a scheme with which to supplement S1-S9, and show that this augmented set of schemes fails to generate countable recursion. We make another proposal to which the methods of this section do not apply.

This paper is concerned with the event-based state and fault estimation problem for a class of linear discrete systems with randomly occurring faults and missing measurements. Different from the static event-based transmission mechanism with a constant threshold, a dynamic event-based mechanism is exploited here to regulate the threshold parameter, thus further reducing the amount of data transmission. Some mutually independent Bernoulli random variables are used to characterize the phenomena of ROFs and missing measurements. In order to simultaneously estimate the system (...) state and the fault signals, the main attention of this paper is paid to the design of recursive filter; for example, for all DETM, ROFs, and missing measurements, an upper bound for the estimation error covariance is ensured and the relevant filter gain matrix is designed by minimizing the obtained upper bound. Moreover, the rigorous mathematical analysis is carried out for the exponential boundedness of the estimation error. It is clear that the developed algorithms are dependent on the threshold parameters and the upper bound together with the probabilities of missing measurements and ROFs. Finally, a numerical example is provided to indicate the effectiveness of the presented estimation schemes. (shrink)

An implicit characterization of the class NP is given, without using any minimization scheme. This is the first purely recursion-theoretic formulation of NP.

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 (...) sets. 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the ▵1 1 sets. 5. The closure ordinal for Kripke's construction of the minimal fixed point is ωCK 1. In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen: 1. There may or may not exist nonparadoxical, ungrounded sentences. 2. The number of fixed points may be any positive finite number, ℵ0, or 2ℵ0 . 3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the Π1 1 sets, including intermediate cases. 4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the ▵1 1 sets. 5. The closure ordinal for the construction of the minimal fixed point may be ω, ωCK 1, or any successor limit ordinal in between. In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Godel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language. (shrink)

I survey the syntactic technique of tiering which can be used to restrict the power of a recursion scheme. I show how various results can be obtained entirely proof theoretically without the use of a model of computation.

This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_1}$$\end{document} provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I\Delta_0 + \neg \mathit{exp}}$$\end{document} implies BΣ1\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B\Sigma_1}$$\end{document} to a purely recursion-theoretic question. (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)

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 (...) has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles. (shrink)

A new proof is given of the celebrated theorem of M. Davis, H. Putnam and J. Robinson concerning exponential Diophantine representation of recursively enumerable predicates. The proof goes by induction on the defining scheme of a partial recursive function.

In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

In computational mathematics, it is a matter of deep concern to recognize which of the given iteration schemes converges quickly with lesser error to the desired solution. Fixed point iterative schemes are constructed to be used for solving equations emerging in many fields of science and engineering. These schemes reformulate a nonlinear equation f s = 0 into a fixed point equation of the form s = g s ; such application determines the solution of the original equation via the (...) support of fixed point iterative method and is subject to existence and uniqueness. In this manuscript, we introduce a new modified family of fixed point iterative schemes for solving nonlinear equations which generalize further recursive methods as particular cases. We also prove the convergence of our suggested schemes. We also consider some of the mathematical models which are categorically nonlinear in essence to authenticate the performance and effectiveness of these schemes which can be seen as an expansion and rationalization of some existing techniques. (shrink)

We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result is obtained by (...) constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of $\exists \sum {\begin{array}{*{20}{c}} h \\ 0 \\ \end{array} } $ sentences. We also present an old but previously unpublished proof of the consistency of I∆₀ + ¬ exp + ¬BΣ₁ under the assumption that the size parameter in Lessan's ∆₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of I∆₀ + ¬ exp + ¬BΣ₁ unconditionally has turned out to be so difficult. (shrink)

We define a fragment of Primitive Recursive Arithmetic by replacing the defining axioms for primitive recursive functions by those for functions in some specific complexity class. In this note we consider such theory for AC0. We present a model-theoretical property of this theory, by means of which we are able to characterize its provably total functions. Next we consider the problem of how strong the induction scheme can be in this theory.

A two-sorted term system characterizing NC implicitly is described. The term system is defined over the tree algebra [MATHEMATICAL DOUBLE-STRUCK CAPITAL T], the free algebra generated by 0, 1 and ∗, and the recursion scheme uses pointers over tier 0. This differs from previous characterizations of NC, where tier 1 pointers were used or full parameter substitution over tier 0 was allowed.

One theoretical charge (of Optimality Theory in its early conception) must have been to retain that sense of qualitative particularity as affecting as constraining theory relevant to a proscribed field when clearly a motivation was to divine in circumscriptions operational consequences conceived on a deferred abstractive level. An attraction of the theory's embodying results of constraint interactions as responsive to theory-internal qualitative implementation, as being in fact supplementarily transparent to co-ordinations of variously language specific implementations, qualitative identifications, was apparent naturalistic (...) coordination of defined features taken here—in OT—at one fell-swoop. Differently put, the uses of analyses of the relevant types of spaces (‘primitive from the viewpoint of linguistic theories’) will (latterly) involve a period of time-stretch or compression over a previously normal (I take it) process of embeddings in a new theorised and recursively enacted invocation/re-invocation of base as digestively present in results of corrected theories. Or as Prince and Smolensky have it -/- 'What we have done, in essence, is to replace the iterative procedure (act/evaluate, act/evaluate, . . . ) with a recursive scheme: collect the results of all possible actions, then sort recursively.' OT, 16. (shrink)

The notions of health, illness, and disease are fuzzy-theoretically analyzed. They present themselves as non-Aristotelian concepts violating basic principles of classical logic. A recursive scheme for defining the controversial notion of disease is proposed that also supports a concept of fuzzy disease. A sketch is given of the prototype resemblance theory of disease.

In the first part of this paper it is argued that Vendler's classification of verbs into aspectual categories, called activities, accomplishments, achievements, and states, is better seen as classifying the meanings of sentences, and a recursive scheme for describing the aspectual character of sentences is presented.In the second part, this scheme is applied to the discussion of the epistemic and deontic meanings of the modal verbs must, will, and may. In particular, the relation between the “future” and “nonfuture” (...) senses of epistemic will is examined. (shrink)

This paper gives an implicit characterization of the class of functions computable in polynomial space by deterministic Turing machines – PSPACE. It gives an inductive characterization of PSPACE with no ad-hoc initial functions and with only one recursion scheme. The main novelty of this characterization is the use of pointers to reach PSPACE. The presence of the pointers in the recursion on notation scheme is the main difference between this characterization of PSPACE and the well-known Bellantoni-Cook (...) characterization of the polytime functions – PTIME. (shrink)

In his recent article “Verbs, Time and Modality,” M. J. Steedman (1977) proposes a recursive scheme for identifying the aspectual character of a proposition. The purpose of this squib is to point out some faults and gaps in that account that may suggest some interesting directions for future research.

We give a term rewriting characterization of the polyspace functions. Our work follows investigations on term rewriting characterizations of some classes of (sub-) recursive functions as initiated by Cichon and Weiermann [4] and continued by Beckmann and Weiermann [1].The main novelty of this paper is a technique for reformulating recursion schemes. The aim of this technique is to provide rewriting rules which give rise to rewriting chains whose terms are suitably bounded. This bounding is crucial when dealing with computational (...) classes with space constraints. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...) n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at (...) more than one website. Paste the whole URL. http://archive.org/stream/RobertiAudi_The.Cambridge.Dictionary.of.Philosophy/Robert.Audi_The.Cambrid ge.Dictionary.of.Philosophy -/- The 2015 third edition will be available soon. Before you think of buying it read some reviews on Amazon and read reviews of its competition: For example, my review of the 2008 Oxford Companion to Philosophy, History and Philosophy of Logic,29:3,291-292. URL: http://dx.doi.org/10.1080/01445340701300429 -/- Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. The discussion provides an opportunity to bring more flaws to light. -/- Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition. Besides the help received from people in Buffalo, help from elsewhere was received by email. We gratefully acknowledge the following: José Miguel Sagüillo, John Zeis, Stewart Shapiro, Davis Plache, Joseph Ernst, Richard Hull, Concha Martinez, Laura Arcila, James Gasser, Barry Smith, Randall Dipert, Stanley Ziewacz, Gerald Rising, Leonard Jacuzzo, George Boger, William Demopolous, David Hitchcock, John Dawson, Daniel Halpern, William Lawvere, John Kearns, Ky Herreid, Nicolas Goodman, William Parry, Charles Lambros, Harvey Friedman, George Weaver, Hughes Leblanc, James Munz, Herbert Bohnert, Robert Tragesser, David Levin, Sriram Nambiar, and others. -/- . (shrink)

One of the earliest applications of transfinite numbers is in the construction of derived sequences by Cantor [2]. In [6], the existence of derived sequences for countable closed sets is proved in ATR0. This existence theorem is an intermediate step in a proof that a statement concerning topological comparability is equivalent to ATR0. In actuality, the full strength of ATR0 is used in proving the existence theorem. To show this, we will derive a statement known to be equivalent to ATR0, (...) using only RCA0 and the assertion that every countable closed set has a derived sequence. We will use three of the subsystems of second order arithmetic defined by H. Friedman , which can be roughly characterized by the strength of their set comprehension axioms. RCA0 includes comprehension for Δmath image definable sets, ACA0 includes comprehension for arithmetical sets, and ATR0 appends to ACA0 a comprehension scheme for sets defined by transfinite recursion on arithmetical formulas. MSC: 03F35, 54B99. (shrink)

We examine two connectionist networks—a fractal learning neural network (FLNN) and a Simple Recurrent Network (SRN)—that are trained to process center-embedded symbol sequences. Previous work provides evidence that connectionist networks trained on infinite-state languages tend to form fractal encodings. Most such work focuses on simple counting recursion cases (e.g., anbn), which are not comparable to the complex recursive patterns seen in natural language syntax. Here, we consider exponential state growth cases (including mirror recursion), describe a new training (...) class='Hi'>scheme that seems to facilitate learning, and note that the connectionist learning of these cases has a continuous metamorphosis property that looks very different from what is achievable with symbolic encodings. We identify a property—ragged progressive generalization—which helps make this difference clearer. We suggest two conclusions. First, the fractal analysis of these more complex learning cases reveals the possibility of comparing connectionist networks and symbolic models of grammatical structure in a principled way—this helps remove the black box character of connectionist networks and indicates how the theory they support is different from symbolic approaches. Second, the findings indicate the value of future, linked mathematical and empirical work on these models—something that is more possible now than it was 10 years ago. (shrink)

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ (...) over $\mathsf {RCA}_0$. For each fixed $k \geq 3$, the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$. The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$. Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of our knowledge, these versions of the Rival–Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$, by $\mathsf {ADS}$, and by $\mathsf {SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival–Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf {RCA}_0$. (shrink)

We prove that the Kleene schemes for primitive recursion relative to the μ-operator, relativized to some nondeterministic objects, have the same power to express total functionals when interpreted over the partial continuous functionals and over the Kleene-Kreisel continuous functionals. Relating the former interpretation to Niggl's ℳω we prove Nigg's conjecture that ℳω is strictly weaker than Plotkin's PCF + PA.

We show that, for every partition F of the pairs of natural numbers and for every set C, if C is not recursive in F then there is an infinite set H, such that H is homogeneous for F and C is not recursive in H. We conclude that the formal statement of Ramsey's Theorem for Pairs is not strong enough to prove , the comprehension scheme for arithmetical formulas, within the base theory , the comprehension scheme for (...) recursive formulas. We also show that Ramsey's Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within . In particular, Ramsey's Theorem for Pairs is not conservative over for -sentences. (shrink)

Organization in a tangled world -- Process views of organization -- Alfred North Whitehead on process -- Bruno Latour on relativizing the social, and the becoming of networks -- Niklas Luhmann on autopoiesis and recursiveness in social systems -- James March on decision processes and organization : a logic of streams -- Karl Weick on organizing and sensemaking -- A scheme for process based organizational analysis -- Some implications for organizational analysis.

LetTbe a recursive theory in the language of first order Arithmetic. We prove that ifTextends: (a) the scheme of parameter free Δ1-minimization (plusexp). or (b) the scheme of parameter free Π1-induction, then there are no Σ1-maximal models with respect toT. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ1-maximal models with respect toIΔ0+expdo not satisfy the scheme of Σ1-collectionBΣ1.

We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...) (second-order number theory with a -comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a operator can be introduced in other settings. (shrink)

We define theories of bounded arithmetic, whose definable functions and relations are exactly those in certain complexity classes. Based on a recursion-theoretic characterization of NC in Clote , the first-order theory TNC, whose principal axiom scheme is a form of short induction on notation for nondeterministic polynomial-time computable relations, has the property that those functions having nondeterministic polynomial-time graph Θ such that TNC x y Θ are exactly the functions in NC, computable on a parallel random-access machine in (...) polylogarithmic parallel time with a polynomial number of processors.0 We then define three theories of weak second-order arithmetic which respectively characterize relations in the classes of alternating logarithmic time, logspace and nondeterministic logspace. (shrink)

We show that for every countable recursively saturated model M of Peano arithmetic and every subset A⊆M, there exists a full satisfaction class SA⊆M2 such that A is definable in (M,SA) without parameters. It follows that in every such model, there exists a full satisfaction class which makes every element definable, and thus the expanded model is minimal and rigid. On the other hand, as observed by Roman Kossak, for every full satisfaction class S there are two elements which have (...) the same arithmetical type, but exactly one of them is in S. In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model. The analogue of the first result proved here for full satisfaction classes was obtained also by Roman Kossak for partial inductive satisfaction classes. However, the proof relied on the induction scheme in a crucial way, so recapturing the result in the setting of full satisfaction classes requires quite different arguments. (shrink)

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae νNϕ of formulae ν of a normal modal theory Tm based (...) on T, such that the counterpart `ξi' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T).The article's basic completeness-result (2.13) then establishes that ϕvarphi; is a theorem of Tm iff [[νN(ϕ) is aconsequenceof νN(Tm) for each extension N of T which is a subtheory of the canonical generic theory (ultrafilter) u]] = 1 – or equivalently, that Tm is consistent iff[[there is anextension N of T such that N is a subtheory of the canonical generic theory u, and νN(ϕ) for all ϕ in Tm]] > 0.The proof of thiscompleteness-result also shows that an N which provides a countermodel for a modally unprovable ϕ – or equivalently, a closed set in the Stone space St(T) in the sense of V(C) – is intertranslatable with an `accessibility'-relation of a closely related Kripke-semantics whose `worlds' are generic extensions of an initial universe V via C.This interrelation providesa fairly precise rationale for the semantics' recourse to C-valued structures, and exhibits a sense in which the boolean-valued context is sharp. (shrink)

Prototypes and fuzziness are regarded in this paper as fundamental phenomena in the inherent logic of concepts whose relationship, however, has not been sufficiently clarified. Therefore, modifications are proposed in the definition of both. Prototypes are defined as the elements possessing maximal degree of membership in the given category such thatthis membership has maximal cognitive efficiency in representing theelement. A modified fuzzy set (m-fuzzy set) is defined on aclass (possibly self-contradictory collection) such that its core (the collection of elements with (...) full membership) is aset (self-consistent collection) comprising the finite set of the prototypes. In this scheme of recursive representation the possibility of contradictions, which has been proven to be inevitable in the logic of concepts, islocalized. Finally a related model of the brain''s pattern recognition mechanism is briefly summarized. (shrink)

We investigate the theory Peano Arithmetic with Indiscernibles ( \(\textrm{PAI}\) ). Models of \(\textrm{PAI}\) are of the form \(({\mathcal {M}},I)\), where \({\mathcal {M}}\) is a model of \(\textrm{PA}\), _I_ is an unbounded set of order indiscernibles over \({\mathcal {M}}\), and \(({\mathcal {M}},I)\) satisfies the extended induction scheme for formulae mentioning _I_. Our main results are Theorems A and B following. _Theorem A._ _Let_ \({\mathcal {M}}\) _be a nonstandard model of_ \(\textrm{PA}\) _ of any cardinality_. \(\mathcal {M }\) _has an (...) expansion to a model of _ \(\textrm{PAI}\) _iff_ \( {\mathcal {M}}\) _has an inductive partial satisfaction class._ Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of \(\textrm{PA}\) : _Corollary._ _A countable model_ \({\mathcal {M}}\) of \(\textrm{PA}\) _is recursively saturated iff _ \({\mathcal {M}}\) _has an expansion to a model of _ \(\textrm{PAI}\). _Theorem B._ _There is a sentence _ \(\alpha \) _ in the language obtained by adding a unary predicate_ _I_(_x_) _to the language of arithmetic such that given any nonstandard model _ \({\mathcal {M}}\) _of_ \(\textrm{PA}\) _ of any cardinality_, \({\mathcal {M}}\) _has an expansion to a model of _ \(\text {PAI}+\alpha \) _iff_ \({\mathcal {M}}\) _has a inductive full satisfaction class._. (shrink)