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1. Mhb03.htm 03B25, Decidability of theories and sets of sentences See also 11U05, 12L05, 20F10. 03B30, Foundations of classical theories (including reverse http://www.mi.imati.cnr.it/~alberto/mhb03.htm

03-XX Mathematical logic and foundations General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. General logic Classical propositional logic Classical first-order logic Higher-order logic and type theory Subsystems of classical logic (including intuitionistic logic) Abstract deductive systems Decidability of theories and sets of sentences [See also Foundations of classical theories (including reverse mathematics) [See also Mechanization of proofs and logical operations [See also Combinatory logic and lambda-calculus [See also Logic of knowledge and belief Temporal logic ; for temporal logic, see ; for provability logic, see also Probability and inductive logic [See also Many-valued logic Fuzzy logic; logic of vagueness [See also Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.)

2. List For KWIC List Of MSC2000 Phrases sensitivity, stability, wellposedness 49K40 sentences Decidability of theories and sets of 03B25 sentences unDecidability and degrees of sets of 03D35 http://www.math.unipd.it/~biblio/kwic/msc/m-kl_11_48.htm

semigroupoids, semigroups, groups (viewed as categories) # groupoids, semigroups semigroups # $C$- semigroups # analysis on topological semigroups # commutative semigroups # integrated semigroups # inverse semigroups # mappings of semigroups # orthodox semigroups # regular semigroups # representations of general topological groups and semigroups # structure of topological semigroups # transformation groups and semigroups # varieties of semigroups and applications to diffusion processes # Markov semigroups and linear evolution equations # one-parameter semigroups and monoids # ordered semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions # almost periodic functions on groups and semigroups in $C^*$-algebras # derivations, dissipations and positive semigroups in automata theory, linguistics, etc. semigroups of linear operators # groups and semigroups of linear operators, their generalizations and applications # groups and semigroups of nonlinear operators semigroups of nonlinear operators # groups and semigroups of rings # semigroup rings, multiplicative

3. Sachgebiete Der AMS-Klassifikation: 00-09 classical logic (including intuitionistic logic) 03B22 Abstract deductive systems 03B25 Decidability of theories and sets of sentences, See also {11U05, http://www.math.fu-berlin.de/litrech/Class/ams-00-09.html

Sachgebiete der AMS-Klassifikation: 00-09 nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen 01-XX 03-XX 04-XX 05-XX 06-XX 08-XX nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen

4. 03Bxx deductive systems 03B25 Decidability of theories and sets of sentences See also 11U05, 12L05, 20F10 03B30 Foundations of classical theories (including http://www.emis.de/MSC2000/03Bxx.html

General logic 03B05 Classical propositional logic 03B10 Classical first-order logic 03B15 Higher-order logic and type theory 03B20 Subsystems of classical logic (including intuitionistic logic) 03B22 Abstract deductive systems 03B25 Decidability of theories and sets of sentences [See also ] 03B30 Foundations of classical theories (including reverse mathematics) [See also ] 03B35 Mechanization of proofs and logical operations [See also ] 03B40 Combinatory logic and lambda-calculus [See also ; for temporal logic see ; for provability logic see also ] 03B50 Many-valued logic 03B52 Fuzzy logic; logic of vagueness [See also ] 03B53 Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.) 03B55 Intermediate logics 03B60 Other nonclassical logic 03B65 Logic of natural languages [See also ] 03B70 Logic in computer science [See also 68-XX ] 03B80 Other applications of logic 03B99 None of the above, but in this section Version of December 15, 1998

5. Ebbinghaus, Flum, Thomas. Mathematical Logic. A set is decidable iff it and its complement are enumerable. theories (as consistent sets of sentences closed under consequence) may be Renumerable, http://mathgate.info/cebrown/notes/ebbinghaus.php

The Omega Group TPS A higher-order theorem proving system This page was created and is maintained by Chad E Brown Ebbinghaus, Flum, Thomas. Mathematical Logic.Springer-Verlag, 1984. Chapter 1: Introduction. Chapter 2: Syntax of First-Order Languages. Chapter 3: Semantics of First-Order Languages. Chapter 4: A Sequent Calculus. ... Chapter 12: Characterizing First-Order Logic. Part A. Chapter 1: Introductionprovides motivational text (distinguishing between traditional philosophical logic and mathematical logic) and motivational examples (group theory and equivalence relations). The idea behind Godel's Completeness Theorem is explained, with an intuitive idea of "propositions," (semantic) "consequences," and "proofs." Also, an outline of the material after Godel's Completeness Theorem is given. The authors claim to show that first-order logic is a "best possible language." However, this assumes the point of view that first-order semantics are the only appropriate notion of semantics. Chapter 2: Syntax of First-Order LanguagesThe standard material is covered: alphabets, strings, countability of languages, first-order languages (terms/formulas), induction "in the calculus of terms and in the calculus of formulas" (what Andrews calls "induction on the construction of a wff" ), free and bound variables, sentences.

6. 0 Top The TOP Concept In The Hierarchy. 1 Adverbial Modification To be of sufficient strength, the system must (1) have decidable sets of wffs and Most semantic theories would treat all these sentences as having a http://staff.science.uva.nl/~caterina/LoLaLi/soft/ch-data/gloss.txt

7. Baur: Decidability And Undecidability Of Theories Of Abelian Groups With Predica ~0,n e.g. looks as follows In order to prove Theorem 3 it suffices to show that the set of all sentences cp which are consistent with T(p, 3) is recursively http://www.numdam.org/numdam-bin/fitem?id=CM_1975__31_1_23_0

8. JSTOR A Remark Concerning Decidability Of Complete Theories Let S be the set of numbers of sentences of ~, A the set of numbers of axioms of that the set T is general recursive, i.e. that ~ is a decidable theory. http://links.jstor.org/sici?sici=0022-4812(195012)15:4<277:ARCDOC>2.0.CO;2-8

9. Theory Taxonomy A theory is a set of sentences and for a given theory T, M =T denotes that M = for is not computable (decidable), as it is in scientific theories, then http://cs.wwc.edu/~aabyan/Theories/Taxonomy.html

10. Decidability (logic) -- Britannica Online Encyclopedia In one sense, Decidability is a property of sets (of sentences) that of being another desired feature of a formal theory, namely, Decidability that http://www.britannica.com/eb/topic-155081/decidability

Already a member? LOGIN Encyclopædia Britannica - the Online Encyclopedia Home Blog Advocacy Board ... Free Trial Britannica Online Content Related to this Topic Shopping Revised, updated, and still unrivaled. 2008 Britannica Ultimate DVD/CD-ROM The world's premier software reference source. Great Books of the Western World The greatest written works in one magnificent collection. Visit Britannica Store decidability (logic) A selection of articles discussing this topic. analysis in metalogic analysis in metalogic (in metalogic: Discoveries about formal mathematical systems) analysis in metalogic (in metalogic: The undecidability theorem and reduction classes) history of logic ...rather than being strictly in the finitely axiomatizable first-order language that was once preferred. This reply, however, clashes with another desired feature of a formal theory, namely, decidability: that there exists a finite mechanical procedure for determining whether a proposition is, or is not, a theorem of the theory. This property took on added interest after World War II... problem in axiomatic method set theory ...(and there are corresponding results for NBG) assert that, if the system is consistent, then (1) it contains a sentence such that neither it nor its negation is provable (such a sentence is called undecidable), (2) there is no algorithm (or iterative process) for deciding whether a sentence of ZFC is a theorem, and (3) these same statements hold for any consistent theory resulting from ZFC by...

11. Abstracts is definable in the standard model then the Decidability results natural theories of families $\FMA$, such as the set of sentences true in almost http://www.impan.gov.pl/~kz/Abstracts.html

8. Undecidability and concatenation pdf We consider the problem stated by Andrzej Grzegorczyk in ``Undecidability without arithmetization'' (Studia Logica 79(2005)) whether certain weak theory of concatenation is essentially undecidable. We give a positive answer for this problem. 7. The Intended Model of Arithmetic. An Argument from Tennenbaum's Theorem pdf We present an argument that allows to determine the intended model of arithmetic using some cognitive assumptions and the assumptions on the structure of natural numbers. Those assumptions are as follows: the psychological version of the Church thesis, computability of addition and multiplication and first order induction. We justify the thesis that the notion of natural number is determined by 6. Coprimality in finite models pdf We investigate properties of the coprimality relation within the family of finite models being initial segments of the standard of addition and multiplication on indices of prime numbers. of multiplication in the standard model). This result strengthens an analogous theorem of Marcin Mostowski and Anna Wasilewska, 2004

12. Harvey Friedman Countable Models of Set theories, Lecture Notes in Mathematics, Vol. On Decidability of Equational theories, J. of Pure and Applied Algebra, Vol. http://www.math.ohio-state.edu/~friedman/publications.html

Degrees and Employment History Distinctions Publications Others about Friedman ... Back to Home Page Publications Model Theory Beth's Theorem in Cardinality Logics, Israel J. Math., Vol. 14, No. 2, (1973), pp. 205-212. Countable Models of Set Theories, Lecture Notes in Mathematics, Vol. 337, Springer-Verlag, (1973), pp. 539-573. On Existence Proofs of Hanf Numbers, J. of Symbolic Logic, Vol. 39, No. 2, (1974), pp. 318-324. Adding Propositional Connectives to Countable Infinitary Logic, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 77, No. 1, (1975), pp. 1-6. On Decidability of Equational Theories, J. of Pure and Applied Algebra, Vol. 7, (1976), pp. 1-3. The Complexity of Explicit Definitions, Advances in Mathematics, Vol. 20, No. 1, (1976), pp. 18-29. On the Naturalness of Definable Operations, Houston J. Math., Vol. 5, No. 3, (1979), pp. 325-330. (with L. Stanley), A Borel Reducibility Theory for Classes of Countable Structures, J. of Symbolic Logic, Vol. 54, No. 3, September 1989, pp. 894-914. (with Akos Seress), Decidability in Elementary Analysis I, Advances in Math., Vol. 76, No. 1, July 1989, pp. 94-115.

13. Linguistic Theory And Meta-Theory (e) Some scholars elected to set aside prevailing sentence theories in search . as an infinite set of sentences whose wellformedness is decidable with http://www.beaugrande.com/LinguisticTheoryMetaTheory.htm

Original version in Text 113-161 Revised version July 2005 Linguistic Theory and Meta-Theory for a Science of Texts ROBERT DE BEAUGRANDE Abstract This article explores the typical reactions which occur when an established science confronts a new object of inquiry, as when linguistic theory encountered the text. The usual discussions are not productive as long as the old ‘paradigm’ is still accepted as the framework for achievement. The issues are therefore re-examined in terms of the meta-theory of science (e.g. Sneed, Stegmiiller, Lakatos, Feyerabend, Hempel), and some general solutions are expounded for the problems of validating theories on the basis of empirical content. A paradigmatic example is then presented in order to show a possible role for logical linguistics in future theories: a computer grammar that parses text-sentences into a progressive network and back again via theorem-proving, with further capacities for applying schemas, answering’ questions, or generating summaries. This example may serve as an application of general design values and criteria for preferring and comparing alterative theories. 1. Historical background

14. Home Page Of Giangiacomo Gerla (fuzzy Logic, Fuzzy Control, Multi-valued Logic, G. Gerla Sharpness Relation and Decidable Fuzzy sets operator in the class of all fuzzy subsets of the set of sentences of a given language. A theory http://www.dmi.unisa.it/people/gerla/www/

GIANGIACOMO GERLA Department of Mathematica and Information Sciences, University of Salerno, Via Ponte Don Melillo 84084 Fisciano (SA) ITALY gerla@unisa.it Full Professor in Computability Theory and Formal Logic Member of Soft Computing Laboratory Member of AILA (Associazione Italiana Logica, Applicazioni) Interested in: Fuzzy Logic, Vagueness, Fuzzy control, Multi-Valued Logic, Approximate Reasoning, Similarity Logic, Fuzzy Computability, Decidable and Effectively enumerable fuzzy sets, Extension principle, Abstract logic, Closure operators, Point-free geometry, Pedagogical features of Logic. Didattica Scientific interests List of papers Logic sources ... Personale DOWNLOAD AREA My wonderful book I suggest to buy ( ... unfortunately Kluwer is not so magnanimous to give it free) - G. Gerla. Fuzzy Logic: Mathematical Tools for Approximate reasoning (Kluwer Editor). Contents Preface Hajek-comment Gottwald-comment ... Belohlavek-comment Una breve (ed incompleta) introduzione della logica fuzzy in italiano Logica fuzzy e ragionamenti approssimati Paradoxes and vagueness - G. Gerla

15. Disputatio | Craig’s Theorem And The Empirical Underdetermination Thesis Reasse Using the traditional syntactic approach to scientific theories, we define a theory to be a deductively closed set of sentences of a formal language. http://members.tripod.com/Lonego/guestfiles/list.htm

Disputatio 7 (November 1999) Welcome Editors and board Submissions Subscriptions ... Book reviews Craig’s Theorem and the Empirical Underdetermination Thesis Reassessed Christian List University of Oxford The present paper proposes to revive the twenty-year old debate on the question of whether Craig’s theorem poses a challenge to the empirical underdetermination thesis. It will be demonstrated that Quine’s account of this issue in his paper "Empirically Equivalent Systems of the World" (1975) is mathematically flawed and that Quine makes too strong a concession to the Craigian challenge. It will further be pointed out that Craig’s theorem would threaten the empirical underdetermination thesis only if the set of all relevant observation conditionals could be shown to be recursively enumerable — a condition which Quine seems to overlook —, and it will be argued that, at least within the framework of Quine’s philosophy, it is doubtful whether this condition is satisfiable. 1. Introduction

16. Completeness Vs Decidability Text - Physics Forums Library Archive completeness vs Decidability Set Theory, Logic, Probability, Decidable A set of sentences R is decidable if the set of sentences of its http://physicsforums.com/archive/index.php/t-84586.html

Physics Help and Math Help - Physics Forums Mathematics Set Theory, Logic, Probability, Statistics PDA View Full Version : completeness vs decidability EvLer I don't have a clear idea of distinction between the two, to me the latter seems to be restatement of the former with added "procedure". completeness: every statement in the system can be either proved or disproved in the system; decidability: iff there exists an algorithm such that for every well-formed formula in that system there exists a maximum finite number N of steps such that the algorithm is capable of deciding in less than or equal to N algorithmic steps whether the formula is (semantically) valid or not valid; So a system can be complete and undecidable, right? If system is incomplete, does that necessarily mean that it is also undecidable? Could someone explain on a simple example? Thanks in advance. gravenewworld Completeness: A set of sentences R is complete if every sentence A or ~A is a consequence of R. Decidable: A set of sentences R is decidable if the set of sentences of its language that are consequences of R is recursive. So a system can be complete and undecidable, right?

17. Herbrand Logic -- First-order Syntax And Herbrand Semantics The Proof theory for Herbrand logic is much different than the proof Theorem ( * = * is not semidecidable) Let be a set of sentences in *. http://logic.stanford.edu/~thinrich/herbrand/html/modeltheory-prooftheory.html

Herbrand Logic Overview Syntax and Semantics Proof and Model Theory Goedel ... Applications Proof Theory The Proof theory for Herbrand logic is much different than the proof theory for first-order logic. The Model theory is also quite different. The proof theory is more complicated, and the model theory is much simpler. Entailment It is well known that solving Diophantine equations, i.e. determining whether a polynomial equation has integer roots, written P(x ,...,x n ) = 0, is undecidable. Every such polynomial can be expressed using multiplication and addition. For example, 4x y + 1 is 4*x*x*x*y*y + 1. The proof of the undecidability of Herbrand entailment shows how to encode addition and multiplication. Proof : The proof shows how to encode arithmetic: N is the set of natural numbers, represented in unary. Add(x,y,z) is true when x + y = z. Mult(x,y,z) is true when x * y = z. N(0) Add(0,y,y) Mult(0,y,0) Technically, we need to express addition and multiplication of all the integers, not just the natural numbers. Doing so is straightforward but tedious. Suppose P(x ,...,x

18. List Of First-order Theories - Wikipedia, The Free Encyclopedia List or describe a set of sentences in the language L , called the axioms . The theory of Abelian groups is decidable. The theory of Infinite divisible http://en.wikipedia.org/wiki/List_of_first-order_theories

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; List of first-order theories From Wikipedia, the free encyclopedia Jump to: navigation search In mathematical logic , a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. Contents Preliminaries Pure identity theories Equivalence relations Orders ... edit Preliminaries For every natural mathematical structure there is a signature σ (often implicit) so that the object is naturally a σ-structure . Given a signature σ there is a unique first-order language L that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: List or describe a set of sentences in the language L , called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in L holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. A L theory may: be consistent: no proof of contradiction exists;

19. PlanetMath: First-order Theory A theory $ T$ is a deductively closed set of sentences in $ L$ A theory $ T$ is axiomatizable if and only if $ T$ includes a decidable subset $ \Delta$ http://planetmath.org/encyclopedia/FinitelyAxiomatizableTheory.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About first-order theory (Definition) In what follows, references to sentences and sets of sentences are all relative to some fixed first-order language Definition. A theory is a deductively closed set of sentences in ; that is, a set such that for each sentence only if Remark . Some authors do not require that a theory be deductively closed. Therefore, a theory is simply a set of sentences. This is not a cause for alarm, since every theory . Furthermore, is unique (it is the smallest deductively closed theory including ), and any structure is a model of iff it is a model of Definition. A theory is consistent if and only if for some sentence . Otherwise, is inconsistent . A sentence is consistent with if and only if the theory is consistent.

20. Re: Quantifier Free Sentences In Set Theory (redirected From Franco Parl Re Quantifier free sentences in set theory (redirected from Franco Parl That can be inferred from the Decidability of truth in V for existential http://osdir.com/ml/science.mathematics.fom/2005-05/msg00049.html

var addthis_pub = 'comforteagle'; science.mathematics.fom Top All Lists Date Thread Re: Quantifier free sentences in set theory (redirected from Franco Parl Subject Re: Quantifier free sentences in set theory (redirected from Franco Parlamento) List-id Robinson's Arithmetic is complete with respect to quantifier free sentences. I am wondering whether anyone can tell me if an analog of this holds in set theory. Suppose, for instance, that the language contains two constants one for the empty set and one for the set of finite ordinals as well as function symbols for the basic set theoretic operations like set union, set other theory complete with respect to all the quantifier free sentences in the language? More with this subject... Current Thread Previous by Date: primer on vagueness Stewart Shapiro Next by Date: Re: primer on vagueness Charles Silver Previous by Thread: primer on vagueness Stewart Shapiro Next by Thread: 248:Relational System Theory 2/restated Harvey Friedman Indexes: Date Thread Top All Lists Recently Viewed: qnx.openqnx.dev...

21. On Decidability Properties Of Local Sentences On Decidability properties of local sentences. Source, Theoretical Computer Science 12 12 T. Jech, Set Theory, third ed., Springer, Berlin, 2002. http://portal.acm.org/citation.cfm?id=1226597

22. Who Are Boole, Fitch, And Tarski? | Richard Zach | Philosophy | University Of Ca He also worked on ealry set theory, and suggested the following definition . proved the Decidability of firstorder logic with only one-place predicate http://www.ucalgary.ca/rzach/279/logicians.html

@import "/rzach/misc/drupal.css"; @import "http://www.ucalgary.ca/templates2/styles/uofc-level-c.css"; @import "/rzach/files/rzach/custom_colors/uofc_c_v7.css"; Skip to Headline Skip to Navigation Skip to Content Skip to Sidebar UofC Navigation Prospective Students Current Students Alumni Community Search UofC: IT MY U OF C Contacts Richard Zach Site Navigation Primary links Home Research Teaching CV ... LogBlog Courses and Teaching Evidence (Phil 409.02) Logic I (Phil 279) Frequently Asked Questions about Logic I (Phil 279) How to write logical symbols in email or on BlackBoard ... Philosophy of Mathematics (Phil 567.03/667.16) Search Navigation search Who are Boole, Fitch, and Tarski? Language, Proof, and Logic Abelard, Peter (Pierre Ab©lard) (1079-1142) French theologian and philosopher best known for his work on the problem of universals. He is also known for his poetry and for his celebrated love affair with Heloise Abelard and Heloise were the original celebrity couple. Ackermann, Wilhelm German logician and student of Hilbert . Gave the first direct consistency proof of a non-trivial mathematical theory, and contributed to research on the decision problem. Co-author (with Hilbert) of

23. FOM: Urbana Thoughts The set of true sentences in RCF is decidable, but is very very rich from and generalized quantifiers is descriptive set theory, according to Pillay! http://cs.nyu.edu/pipermail/fom/2000-June/004049.html

FOM: Urbana Thoughts Harvey Friedman friedman at math.ohio-state.edu Sat Jun 10 00:08:46 EDT 2000 Previous message: FOM: conference on diagrammatic reasoning Next message: FOM: Urbana Thoughts Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: conference on diagrammatic reasoning Next message: FOM: Urbana Thoughts Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the FOM mailing list

24. [CL] Clarifications On CL & 2 Possible Problems (dialect Expressiveness; Definit It will of course not be decidable, as CL with seq vars is expressively equivalent to . Underlying set theory and Russell(like) sentences The general http://philebus.tamu.edu/pipermail/cl/2006-August/000939.html

Christopher Menzel cmenzel at tamu.edu Tue Aug 8 16:15:37 CDT 2006 Previous message: Next message: Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] we have an interest in using CL instead of some specific FOL notation (only), but we would like to have a rather clear understanding of some of the less common aspects of the foundations of CL. Having invested some time on the (draft) standard documents and the lists archives, we have a couple of open questions and some need for confirmation of our current understanding. 1.) Proof Theory We have read that the specification of a proof theory for CL will not be part of the standard. But does a proof theory exist already, or even corresponding inference engines? If so, for which dialects apart from "standard" FOL? We have checked some of the "standard" tautologies of FOL against the CL semantics. Those we checked seem to be maintained, as indicated in [1] (sometimes adjusting them, e.g. replacement of a name x by a term t in (all dialects). Is there an axiomatization of CL (i.e., of the set of its

25. British Logic Colloquium 2002 The model leads to notions of computability and Decidability on the reals, . The corresponding question in set theory with the axiom of choice would be http://www.cs.bham.ac.uk/~blc02/abstracts.html

Home Search About Us Research ... Resources British Logic Colloquium Birmingham, September 12-14, 2002 British Logic Colloquium HOME Index Abramsky Berger Engström Hamkins ... BLC 2002 home page Abstracts This page contains all abstracts received so far, as well as links to the home-pages of the speakers where known. If any information here is missing or incorrect, please email us at blc02@cs.bham.ac.uk with your corrections. A finitary, interactive semantics for polymorphism Samson Abramsky (University of Oxford) A Kripke-interpretation for a fragment of classical second-order logic Ulrich Berger (University of Wales at Swansea) We introduce a Kripke-style interpretation of a fragment of classical second-order logic into its intuitionistic counterpart that seems to be useful for extracting programs from non-constructive proofs. The main advantages of this interpretation are: (a) extracted programs are Goedel primitive recursive, although the logic is genuinely second-order, (b) program extraction is not restricted to proofs of Pi^0_2-sentences, but applies to more general situations, (c) the interpretation can be used to generalize Parson's result of the Pi^_2-conservativity of classical Sigma^0_1-arithmetic over its intuitionistic counterpart, (d) in many situations the usual cost of achieving (c), namely excessive case distinctions in extracted programs, can be avoided. The Kripke-aspect of this interpretation allowing for point (c) is a generalization of a technique introduced by Coquand and Hofmann for establishing (among other things) Parson's result.

26. Proposal Of A Workshop On : We study the problem of Decidability of MSO theories on various (restricted) Given a sentence of MMSNP, can we decide whether it captures a finite or an http://www.labri.fr/perso/courcell/Logiquecombinatoire/AbstractsSzeged.html

Logic and Combinatorics rd and 24 th September Szeged Hungary Satellite Workshop of the conference : Computer Science Logic th th September 2006 Main page Abstracts of talks ; Slides For all emails, replace DOT by a dot. Isolde Adler Berlin Germany ), Adler@informatik.hu-berlinDOTde : Hyper-tree-width and related invariants (for a longer abstract). Slides http://www2.informatik.hu-berlin.de/~adler/publications.html The hypertree-width of a hypergraph measures how close a hypergraph is to being acyclic. Similar to tree-width, many problems that are NP-complete in general, become tractable when restricted to instances whose underlying hypergraph has bounded hypertree-width. In analogy to tree-width, the hypertree-width of a hypergraph H can be characterised by the number of cops necessary to catch a robber on the hypergraph In this game the robber's escape space must decrease in a monotone way. In contrast to the robber and cops game characterising tree-width, the number of cops necessary to win the game may increase due to this restriction, but at most by a factor of three. Achim Blumensath Darmstadt Germany ), blumensath@mathematik.tu-darmstadtDOTde :

27. Question About Indicator Function Of Recursive And R.e. Sets - Comp.theory | Goo relation, because I think that relation is effectively decidable iff it is recursive (ie. set s indicator function is effectively http://groups.google.co.zm/group/comp.theory/browse_thread/thread/a290689d64eb59

28. Logic A proof theory is said to be sound when every sentence s that can be derived from a set of sentences S is also a valid consequence of S. http://mainline.brynmawr.edu/~dkumar/UGAI/logic.html

Logic What is a logic? First-Order Predicate Calculus Building an Inference Engine Decidability ... To Lecture Notes on Knowledge Representation What is a logic? A study of correct inference. Correct inference typically implies that it is truth preserving. IOW, logic is the study of truth preserving inferences. For example: Truth preserving inference: If there is a potato in the tailpipe, the car will not start. There is a potato in the tailpipe. Therefore, the car will not start. Non-Truth preserving inference: If there is a potato in the tailpipe, the car will not start. My car will not start. Therefore, there is a potato in the tailpipe. The goal is to formalize the notion of a correct inference. This requires three things: The syntax of a formal language. Inference is a relationship between sentences. Thus in order to formalize the notion of a correct inference, one first needs to define what constitutes well-formed sentences. The semantics of the formal language. A specification of the meanings of the well-formed sentences of the formal language. I.e., under what conditions is a sentence true. Meaning is defined in terms of some interpretation . The meaning of a sentence in an interpretation is the truth value of the sentence. A proof theory A formal specification of what constitutes correct inference. A proof theory consists of

29. Knowability: Second-order Logic And Gödel’s First Incompleteness Theore Now a theory T in language L is decidable when there is an algorithm for deciding whether any given sentence of L is a theorem (member) of the theory. http://knowability.blogspot.com/2007/07/second-order-logic-and-gdels-first.html

skip to main skip to sidebar Knowability devoted to issues modal epistemic Author Joe Salerno Guest Authors Bryan Frances John Greco Recent Posts var numposts = 15;var showpostdate = true;var showpostsummary = false;var numchars = 100;var standardstyling = true; Blog Archive December "God bless us, every one!" ANU offers to Schellenberg and Southwood November ... July Feed Atom July 06, 2007 enough Let L be the formal first-order language with a name for zero and function symbols for the successor function, addition, and multiplication (and no other non-logical symbols). So L is a pretty simple formal language. Let N Arithmetic is the set of sentences of L that are true under N. So, arithmetic is a set of sentences of a formal language, where each sentence is most naturally interpreted as an arithmetic truth. This will be an infinite set. Arithmetic counts as a theory Now a theory T in language L is decidable when there is an algorithm for deciding whether any given sentence of L is a theorem (member) of the theory. So a theory T in L is decidable when there is some algorithm that when fed a sentence S of L will tell you in a finite number of steps whether or not S is in T. Roughly put, a theory T is

30. Notes On - Which Undecidable Mathematical Sentences Have Determinate Truth Value WPA is true arithmetic , the set of sentences of PA which are true in the standard model. Corollary. There is only one consistent decidable extension of http://rbjones.com/rbjpub/philos/bibliog/field97.htm

by on Which Undecidable Mathematical Sentences Have Determinate Truth Values? by Hartry Field Section 1. Metaphysical Preamble Section 2. The Objectivity Issue Section 3. Putnam's "Models and Reality" and the concepts of finiteness and natural number Section 4. Extreme Anti-objectivism 's Conclusions conjecture The Final Word? Metaphysical Preamble Field begins by identifying three kinds of ontological stance in relation to mathematics: fictionalism "standard" platonism number theory and set theory and the theory of real numbers are each about a determinate mathematical structure plenitudinous platonism a theory is about its models [Clearly it is possible to have something between standard and plenitudinous platonism, in which the notion of standard model includes more than one but not all models of the theory. One would also expect this to vary with the theory, e.g. we expect a single model of the natural numbers, multiple (but not all) models of set theory (perhaps the well-founded ones, to eliminate non-standard arithmetic), but all the models of group theory.] The Objectivity Issue Now Field puts aside fictionalism and looks at the difference between the two forms of platonism. This he says comes down to asking: "Which undecidable mathematical sentences have determinate truth values?".

31. 03: Mathematical Logic And Foundations The implicit dependence on set theory and the inability to determine a decidable set of firstorder axioms for set theory have caused considerable http://www.math.niu.edu/~rusin/known-math/index/03-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 03: Mathematical logic and foundations Introduction Mathematical Logic is the study of the processes used in mathematical deduction. The subject has origins in philosophy, and indeed it is only by nonmathematical argument that one can show the usual rules for inference and deduction (law of excluded middle; cut rule; etc.) are valid. It is also a legacy from philosophy that we can distinguish semantic reasoning ("what is true?") from syntactic reasoning ("what can be shown?"). The first leads to Model Theory, the second, to Proof Theory. Students encounter elementary (sentential) logic early in their mathematical training. This includes techniques using truth tables, symbolic logic with only "and", "or", and "not" in the language, and various equivalences among methods of proof (e.g. proof by contradiction is a proof of the contrapositive). This material includes somewhat deeper results such as the existence of disjunctive normal forms for statements. Also fairly straightforward is elementary first-order logic, which adds quantifiers ("for all" and "there exists") to the language. The corresponding normal form is prenex normal form. In second-order logic, the quantifiers are allowed to apply to relations and functions to subsets as well as elements of a set. (For example, the well-ordering axiom of the integers is a second-order statement). So how can we characterize the set of theorems for the theory? The theorems are defined in a purely procedural way, yet they should be related to those statements which are (semantically) "true", that is, statements which are valid in every model of those axioms. With a suitable (and reasonably natural) set of rules of inference, the two notions coincide for any theory in first-order logic: the Soundness Theorem assures that what is provable is true, and the Completeness Theorem assures that what is true is provable. It follows that the set of true first-order statements is effectively enumerable, and decidable: one can deduce in a finite number of steps whether or not such a statement follows from the axioms. So, for example, one could make a countable list of all statements which are true for all groups.

32. Logic Colloquium 2003 James Ax studied in 1968 the asymptotic theory of these fields, i.e., the set of sentences in the language of rings which hold in all but finitely many of http://www.helsinki.fi/lc2003/titles.html

Main Awards Registration Accommodation ... ASL Speakers and Titles Tutorial speakers Michael Benedikt Model Theory and Complexity Theory Bell Labs, Lisle, USA. E-mail: benedikt@research.bell-labs.com ABSTRACT: This tutorial concentrates on links between traditional (infinitary) model theory and complexity theory. We begin with an overview of the `classical' connection between complexity theory and finite model theory, giving quickly the basic results of descriptive complexity theory. /We then discuss several ways of generalizing this to take account a fixed infinite background structure. We will start by giving the basics of complexity theory parameterized by a model (algebraic complexity over an arbitrary structure). We then cover results characterizing first-order theories of models via the complexity of query problems (embedded finite model theory). Finally, time permitting, we will look at abstractions of descriptive complexity theory to take into account a background structure. Stevo Todorcevic Set-Theoretic Methods in Ramsey Theory C.N.R.S. - UMR 7056, Paris, France.

33. FTP 2003 - Abstracts Of Accepted Papers MLSS is a decidable sublanguage of set theory involving the constructs sentences to the problem of determining the satisfiability of sentences of MLSS. http://www.dsic.upv.es/~rdp03/ftp/abstracts.html

RDP Home page Organization Sponsors MEETINGS RTA TLCA FTP RULE ... WST VENUE Valencia Registration Accomodation Travelling ... Internet FTP'03 4th International Workshop on First order Theorem Proving Valencia, Spain, June 12-14, 2003 ABSTRACTS OF REGULAR PAPERS ACCEPTED FOR PRESENTATION Quantifier Elimination and Provers Integration S. Ghilardi We exploit quantifier elimination in the global design of combined decision and semi-decision procedures for theories over non-disjoint signatures, thus providing in particular extensions of Nelson-Oppen combination schema. A Decision Procedure for a Sublanguage of Set Theory Involving Monotone, Additive, and Multiplicative Functions D. Cantone (University of Catania, Italy) J.T. Schwartz (New York University, USA) C.G. Zarba (Stanford University, USA) MLSS is a decidable sublanguage of set theory involving the constructs membership, set equality, set inclusion, union, intersection, set difference, and singleton. In this paper we extend MLSS with constructs for expressing monotonicity, additivity, and multiplicativity properties of set-to-set functions. We prove that the resulting language is decidable by reducing the problem of determining the satisfiability of its sentences to the problem of determining the satisfiability of sentences of MLSS. Canonicity N. Dershowitz

34. Formal Systems - Definitions Decidable A formal theory is decidable if there exists an effective procedure that will determine, for any sentence of the theory, whether or not that http://www.rci.rutgers.edu/~cfs/305_html/Deduction/FormalSystemDefs.html

Formal Systems - Definitions (from Ruth E. Davis, Truth, Deduction, and Computation. New York: Computer Science press, 1989.) Listed below are the technical definitions of many of the terms that are used in the investigation of deductive reasoning. Denumerable - A set is denumerable if it can be put into a one-to-one correspondence with the positive integers. Countable - A set is countable if it is either finite or denumerable. A formal theory T consists of: (1) A countable set of symbols . (A finite sequence of symbols of T is called an expression of T.) (2) A subset of the expressions, called the well-formed formulas (abbreviated wffs) of T. The wffs are the legal sentences of the theory. (3) A subset of the wffs called the axioms of T. (4) A finite set of relations R1, ..., Rn on wffs, called rules of inference . For each Ri there is a unique positive integer j such that for every j wffs and each wff A one can effectively decide whether the given j wffs are in the relation Ri to A; if so, A is called a direct consequence Since the set of axioms is often infinite, this set is often specified by providing a finite set of