*Except for this preface, this study is completely self-contained.*

**Author**: Raymond R. Smullyan

**Publisher:** Springer Science & Business Media

**ISBN:** 9783642867187

**Category:** Mathematics

**Page:** 160

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Except for this preface, this study is completely self-contained. It is intended to serve both as an introduction to Quantification Theory and as an exposition of new results and techniques in "analytic" or "cut-free" methods. We use the term "analytic" to apply to any proof procedure which obeys the subformula principle (we think of such a procedure as "analysing" the formula into its successive components). Gentzen cut-free systems are perhaps the best known example of ana lytic proof procedures. Natural deduction systems, though not usually analytic, can be made so (as we demonstrated in [3]). In this study, we emphasize the tableau point of view, since we are struck by its simplicity and mathematical elegance. Chapter I is completely introductory. We begin with preliminary material on trees (necessary for the tableau method), and then treat the basic syntactic and semantic fundamentals of propositional logic. We use the term "Boolean valuation" to mean any assignment of truth values to all formulas which satisfies the usual truth-table conditions for the logical connectives. Given an assignment of truth-values to all propositional variables, the truth-values of all other formulas under this assignment is usually defined by an inductive procedure. We indicate in Chapter I how this inductive definition can be made explicit-to this end we find useful the notion of a formation tree (which we discuss earlier).
There are many kinds of books on formal logic. Some have philosophers as their intended audience, some mathematicians, some computer scientists. Although there is a common core to all such books they will be very dif ferent in emphasis, methods, and even appearance. This book is intended for computer scientists. But even this is not precise. Within computer sci ence formal logic turns up in a number of areas, from program verification to logic programming to artificial intelligence. This book is intended for computer scientists interested in automated theorem proving in classical logic. To be more precise yet, it is essentially a theoretical treatment, not a how-to book, although how-to issues are not neglected. This does not mean, of course, that the book will be of no interest to philosophers or mathematicians. It does contain a thorough presentation of formal logic and many proof techniques, and as such it contains all the material one would expect to find in a course in formal logic covering completeness but not incompleteness issues. The first item to be addressed is, what are we talking about and why are we interested in it. We are primarily talking about truth as used in mathematical discourse, and our interest in it is, or should be, self-evident. Truth is a semantic concept, so we begin with models and their properties. These are used to define our subject.
This book introduces some extensions of classical first-order logic and applies them to reasoning about computer programs. The extensions considered are: second-order logic, many-sorted logic, w-logic, modal logic type theory and dynamic logic. These have wide applications in various areas of computer science, philosophy, natural language processing and artificial intelligence. Researchers in these areas will find this book a useful introduction and comparative treatment.
This work makes available to readers without specialized training in mathematics complete proofs of the fundamental metatheorems of standard (i.e., basically truth-functional) first order logic. Included is a complete proof, accessible to non-mathematicians, of the undecidability of first order logic, the most important fact about logic to emerge from the work of the last half-century. Hunter explains concepts of mathematics and set theory along the way for the benefit of non-mathematicians. He also provides ample exercises with comprehensive answers.
In this paper, a comparison is made of several proof calculi in terms of the lengths of shortest proofs for some given formula of first order predicate logic with function symbols. In particular, we address the question whether, given two calculi, any derivation in one of them can be simulated in the other in polynomial time. The analogous question for propositional logic has been intensively studied by various authors because of its implications for complexity theory. And it seems there has not been as much endeavour in this field in first order logic as there has been in propositional logic. On the other hand, fOr most of the practical applications of logic, a powerful tool such as the language of first order logic is needed. The main interest of this investigation lies in the calculi most frequently used in automated theorem proving, the resolution calculus, and analytic calculi such as the tableau calculus and the connection method. In automated theorem proving there are two important aspects of complexity. In order to have a good theorem proving system, we must first have some calculus in which we can express our derivations in concise form. And second, there must be an efficient search strategy. This book deals mainly with the first aspect which is a necessary condition for the second since the length of a shortest proof always also gives a lower bound to the complexity of any strategy.
Boolean, relation-induced, and other operations for dealing with first-order definability Uniform relations between sequences Diagonal relations Uniform diagonal relations and some kinds of bisections or bisectable relations Presentation of ${\mathbf S}_q$, ${\mathbf S}_p$ and related structures Presentation of ${\mathbf S}_{pq}$, ${\mathbf S}_{pe}$ and related structures Appendix. Presentation of ${\mathbf S}_{pqe}$ and related structures Bibliography Index of symbols Index of phrases and subjects List of relations involved in presentations Synopsis of presentations
The volume includes the proceedings from the conference FOL75 -- 75 Years of First-Order Logic held at Humboldt University, Berlin, September 18 - 21, 2003 on the occasion of the anniversary of the publication of Hilbert's and Ackermann's Grundzuge der theoretischen Logik. The papers provide analyses of the historical conditions of the shaping of FOL, discuss several modern rivals to it, and show the importance of FOL for interdisciplinary research. While there is no doubt that the celebrated book marks a most important step in the development of logic, the volume in hand proves the actuality of the question "Which logic is the right logic." The volume contains articles by: H. Andreka, J. X. Madarasz, I. Nemeti, A. Avron, K. Brunnler, A. Guglielmi, G. Englebretsen, W. Ewald, P. Hajek, J. Hintikka, W. Hodges, M. Kracht, R. Lanzet, H. Ben-Yami, C. Toke, S. P. Odintsov, H. Wansing, J. A. Robinson, M. Rossberg, M. Thielscher, D. E. Willard, andJ. Wole 'nski.
We show that the classifications of prefix classes of first-order logic with equality according to the solvability of the finite satisfiability problem and according to the 0-1 law for the corresponding [sigma superscript 1 over subscript 1] fragment are identical."
The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject.
Expressiveness, and more recently, succinctness, are two central concerns of finite model theory and descriptive complexity theory. Succinctness is particularly interesting because it is closely related to the complexity-theoretic trade-off between parallel time and the amount of hardware. We develop new bounds on the expressiveness and succinctness of first-order logic with two variables on finite words, present a related result about the complexity of the satisfiability problem for this logic, and explore a new approach to the generalized star-height problem from the perspective of logical expressiveness. We give a complete characterization of the expressive power of first-order logic with two variables on finite words. Our main tool for this investigation is the classical Ehrenfeucht-Fraısse game. Using our new characterization, we prove that the quantifier alternation hierarchy for this logic is strict, settling the main remaining open question about the expressiveness of this logic. A second important question about first-order logic with two variables on finite words is about the complexity of the satisfiability problem for this logic. Previously it was only known that this problem is NP-hard and in NEXP. We prove a polynomialsize small-model property for this logic, leading to an NP algorithm and thus proving that the satisfiability problem for this logic is NP-complete. Finally, we investigate one of the most baffling open problems in formal language theory: the generalized star-height problem. As of today, we do not even know whether there exists a regular language that has generalized star-height larger than 1. This problem can be phrased as an expressiveness question for first-order logic with a restricted transitive closure operator, and thus allows us to use established tools from finite model theory to attack the generalized star-height problem. Besides our contribution to formalize this problem in a purely logical form, we have developed several example languages as candidates for languages of generalized star-height at least 2. While some of them still stand as promising candidates, for others we present new results that prove that they only have generalized star-height 1.
In standard first order logic there is no limit on the cardinality of a set of axioms. All proofs, however, must be finite. This means that for infinite sets of axioms it isn't possible to use them all in a given proof. Given a set G of first order formulas is there a way to modify our logic so that we can have a set G' in our new logic that has finitely many axioms yet can prove the same statements as G?\\\\The natural idea here is that of compactness. If G is ``nice enough," can we find a finite set of similar axioms that proves the same things that G proves, e.g. a ``cover" of G? Here we consider the special case where all but finitely many axioms in G do not fall into one of finitely many structures. The obvious example would be the induction scheme on the natural numbers. In first order logic we would need infinitely many axioms to express the induction scheme. We will construct a new type of logic with the means of collapsing these infinitely many formulas into one formula. We will check if the analogous structures in our new language have the same nice properties of first order logic, namely soundness and completeness. We will then investigate which first order statements we can prove with our new logic.

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*Except for this preface, this study is completely self-contained.*

**Author**: Raymond R. Smullyan

**Publisher:** Springer Science & Business Media

**ISBN:** 9783642867187

**Category:** Mathematics

**Page:** 160

**View:** 611

*Although there is a common core to all such books they will be very dif ferent in emphasis, methods, and even appearance. This book is intended for computer scientists. But even this is not precise.*

**Author**: Melvin Fitting

**Publisher:** Springer Science & Business Media

**ISBN:** 9781468403572

**Category:** Mathematics

**Page:** 242

**View:** 181

**Author**: R. I. G. Hughes

**Publisher:** Hackett Publishing

**ISBN:** 0872201813

**Category:** First-order logic

**Page:** 309

**View:** 440

*This book introduces some extensions of classical first-order logic and applies them to reasoning about computer programs.*

**Author**: Maria Manzano

**Publisher:** Cambridge University Press

**ISBN:** 0521354358

**Category:** Computers

**Page:** 388

**View:** 469

*This work makes available to readers without specialized training in mathematics complete proofs of the fundamental metatheorems of standard (i.e., basically truth-functional) first order logic.*

**Author**: Geoffrey Hunter

**Publisher:** Univ of California Press

**ISBN:** 9780520023567

**Category:** Science

**Page:** 288

**View:** 902

*This book deals mainly with the first aspect which is a necessary condition for the second since the length of a shortest proof always also gives a lower bound to the complexity of any strategy.*

**Author**: Elmar Eder

**Publisher:** Vieweg+Teubner Verlag

**ISBN:** 3528051221

**Category:** Mathematics

**Page:** 173

**View:** 497

*Boolean, relation-induced, and other operations for dealing with first-order definability Uniform relations between sequences Diagonal relations Uniform diagonal relations and some kinds of bisections or bisectable relations Presentation of ...*

**Author**: William Craig

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821841495

**Category:** Mathematics

**Page:** 263

**View:** 401

**Author**: M. Makkai

**Publisher:** Springer Verlag

**ISBN:** UOM:39015017284681

**Category:** Mathematics

**Page:** 301

**View:** 326

**Author**: Stephen Read

**Publisher:**

**ISBN:** OCLC:471620996

**Category:** First-order logic

**Page:** 261

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**Author**: John Heil

**Publisher:** Wadsworth Publishing Company

**ISBN:** 0867209577

**Category:** Philosophy

**Page:** 309

**View:** 542

**Author**: Raymond M. Smullyan

**Publisher:**

**ISBN:** UOM:39015068296204

**Category:** First-order logic

**Page:** 158

**View:** 883

*"Proceedings from the conference FOL75 - 75 Years of First-Order Logic held at Humboldt University, Berlin, Germany, September 18 - 21 2003"--Pref.*

**Author**: Vincent F. Hendricks

**Publisher:** Logos Verlag Berlin

**ISBN:** 3832504753

**Category:** Mathematics

**Page:** 398

**View:** 542

*More specifically, they showed that if C is the class of all finite structures over some relational vocabulary and if P is any property expressible in first-order logic (with equality), then p(P) exists and is either 0 or 1.*

**Author**: P. G. Kolaitis

**Publisher:**

**ISBN:** UCSC:32106020215700

**Category:** Logic, Symbolic and mathematical

**Page:** 20

**View:** 121

*The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics.*

**Author**: Stewart Shapiro

**Publisher:** Clarendon Press

**ISBN:** 9780191524011

**Category:** Mathematics

**Page:** 300

**View:** 451

*We develop new bounds on the expressiveness and succinctness of first-order logic with two variables on finite words, present a related result about the complexity of the satisfiability problem for this logic, and explore a new approach to ...*

**Author**: Philipp Weis

**Publisher:**

**ISBN:** OCLC:756916054

**Category:** Finite model theory

**Page:** 130

**View:** 562

*Given a set G of first order formulas is there a way to modify our logic so that we can have a set G' in our new logic that has finitely many axioms yet can prove the same statements as G?\\\\The natural idea here is that of compactness.*

**Author**: Justin Miller

**Publisher:**

**ISBN:** OCLC:951475978

**Category:**

**Page:**

**View:** 151

**Author**: Albert Atserias

**Publisher:**

**ISBN:** UCAL:X67434

**Category:** Computational complexity

**Page:** 222

**View:** 500

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