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Deductive Logic by Warren Goldfarb (Hackett Publishing Company) provides a straightforward, lively but rigorous, introduction to truth-functional and predicate logic, complete with lucid examples and incisive exercises, for which Warren Goldfarb is renowned.

This is one of the kindest, most avuncular logic books I have ever read on logic. Every page is evidence of the author's warmth toward his students and his dedication to conveying logic to them in a way that respects them as mature persons. His thorough mastery of the subject and its philosophy is another feature that distinguishes this book from the mountain of logic texts written by inexperienced assistant professors and by persons for whom logic is a mere sideline, not a professional specialty as it is with Goldfarb, an accomplished and respected logician who has been teaching this material for over twenty years. No logic book I know of conveys kind warmth toward the readers or deeply modest non-dogmatic competence in the field more than Goldfarb's DEDUCTIVE LOGIC. The usual scientistic hocus-pocus, formalistic pedantry and breezy dogmatism are nowhere to be found in this book. Its examples are chosen to appeal to the intelligent humanities student, not merely to the mathematical science or computer engineering student. They are carefully and tastefully crafted to avoid irrelevant linguistic complexities, both logical and sociological.

The author took over responsibility for Harvard's legendary introductory logic course Philosophy 140 in 1979 when W. V. Quine retired. A form of Quine's distinctive, if not idiosyncratic, philosophy and organization of logic has been meticulously and creatively implemented. Accordingly, but perhaps to the surprise of readers not familiar with the Quinean approach, deduction in the sense of step-by-step inferring of conclusions implied by given premises is substantially deferred until Section 33 of the books 44 Sections. The 44 sections averaging six pages in length are unequally divided into four Parts titled respectively: Truth-functional Logic, Monadic Quantification Theory, Polyadic Quantification Theory, and Identity and Names. The material in this book has been thoroughly classroom-tested. Most first-edition logic texts are loaded with errors that are exasperating to students and instructors alike. My reading has turned only one (non-exasperating) error: on pages 18, 69, and 289 the space in Augustus De Morgan's last name is omitted. Despite an honest effort to detect further errors typographical and otherwise the reviewer, to his amazement, has found none.
If a college instructor wants to present a Quinean form of modern first-order logic with identity and names but without functions in a competent, accurate and thoughtful way while avoiding patronizing spoon-feeding, this might be the best text. No other book I know comes close. On the other hand, if an instructor wants to convey the sometimes agonizing rough-and-tumble of contemporary or historical philosophy of logic, or the astounding struggles, dead-ends, missed opportunities, lapses in objectivity and embarrassing errors, even inconsistencies, involved in the historical development of currently accepted versions of the science painfully born in Ancient Athens, this book is not even a candidate. The words `Aristotle', `Boole', `contradiction', `epistemic', `ontic', `paradox', proof', `speech-act', `Tarski' and `tautology' do not occur in the index. There is no bibliography of readings in history and philosophy of logic and no list of current journals in the field. From the study of this excellent text, some students might infer that logic is a fascinating, rewarding and useful science that is virtually complete and uncontroversial. But they might also get the impression that it has no past and no future, that it will persist eternally in its present perfect form, and that it is an island of peaceful rationality. Logic may seem to lift the mind's eye toward the Platonic Form of Reason.

Excerpt: Logic is the study of principles of reasoning. It is concerned not with how people actually reason, but rather with how people ought to reason if they wish to ensure the truth of their results. That is, by "principles of logic" we mean those that yield correct reasoning. Moreover, the principles of logic are general: they do not govern reasoning in one specific subject matter or another, but with reasoning as it applies to any and all areas of study.

Reasoning is a matter of drawing conclusions, or inferring. Hence in logic we are often concerned with arguments, that is, inferences from premises to conclusions. An example familiar since antiquity is this:

All persons are mortal.
Socrates is a person.
Therefore, Socrates is mortal.

The first two statements are the premises; the third is the conclusion. (Of course, in everyday life, arguments are seldom laid out quite so neatly. That is a rhetorical matter, and not our concern here.) The argument is a deductive argument: the conclusion follows logically from the premises. This feature is often characterized in intuitive terms, in several different ways: if the premises are true then the conclusion must be true; it is impossible that the premises be true and the conclusion false; the truth of the premises assures the truth of the conclusion; to commit oneself to the truth of the premises is ipso facto to commit oneself to the truth of the conclusion. Much of this book is devoted to the project of assessing arguments which claim to be deductive, but to do this we also have to analyze what it means to say that a conclusion logically follows from premises. The task is to formulate a precise and rigorous definition to replace the intuitive characterizations.

Clearly, whether or not the conclusion of an argument logically follows from the premises is not simply a matter of the truth or falsity of the premises and conclusion. Rather, as we shall see in detail, the correctness of the argument depends on the form of the statements that make up the argument: the way those statements are constructed from smaller parts, some of which will occur multiply in those statements. Thus, we are led to investigate structural features of statements, in particular, how the truth or falsity of a statement depends on the parts from which it is constructed and the way they are put together. As W. V. Quine memorably put it, "Logic chases truth up the tree of grammar."

This book is divided into four parts. In the first, we treat truth functional logic, which concerns those structures signaled in ordinary language by "and", "or", "not", and "if ... then". The second takes up simple quantificational logic, which treats "all" and "some". The third extends quantificational logic to cover cases that result when nested structures of "all" and "some" are allowed, as in statements like "Everybody loves somebody sometime". Finally, Part IV discusses the logic of identity ("is equal to", "is the same as") and of complex names.

Each of parts IIII is divided into three chapters, representing three stages of our enterprise.

A) Analysis of Discourse. We seek to discern in ordinary statements their structural features, and to characterize those features. We aim at displaying their logical construction: how language is used to express logical forms. Now, ordinary language is extraordinarily variegated. To make the assessment of logical relations possible, we seek to paraphrase ordinary statements into a more uniform symbolic notation. Paraphrased statements display transparently how they are constructed from simpler parts.

B) Logical Assessment. Having put statements into a symbolic form, we can now investigate the formal relations of statements that yield deductive arguments. We show how to manipulate the forms, and we give procedures for ascertaining whether the conclusion of an argument does indeed follow logically from the premises.

C) Reflection. Here we reflect on the logical concepts and the methods developed in the previous chapter, and we investigate their general properties. For example, we might seek general earmarks of any correct deductive argument or inquire about the adequacy and comprehensiveness of the techniques for logical assessment. In this stage, we reason about reasoning, thereby enacting Frege's dictum, "Reason's proper study is itself".

Exercises appear at the end of the volume, arranged by the part and chapter to which they pertain.

A Companion to Philosophical Logic by Dale Jacquette (Blackwell Companions to Philosophy: Blackwell) This collection of newly commissioned essays by international contributors offers a representative overview of the most important developments in contemporary philosophical logic. Written by experts from a variety of different logical and philosophical perspectives, the volume presents controversies in philosophical implications and applications of formal symbolic logic.

A Companion to Philosophical Logic is likely to become the standard reference to the introductory scope of philosophical logics. Its accessible articles allow a general reader to grasp the defining characteristics of much exciting work in logic. Highly recommended.

Each section features contributors currently active in research who explain the central ideas of their special field and take a philosophical stand on recent issues in the intersection of logic and analytic philosophy. Taken together the essays survey major trends and offer original insights to advance research and philosophical discussion. A Companion to Philosophical Logic provides a comprehensive state-of-the-art handbook for students and professional researchers in philosophical logic.

Contents: Preface. Acknowledgments. List of Contributors. Introduction: Logic, Philosophy, and Philosophical Logic: Dale Jacquette (Pennsylvania State University). Part I: Historical Development of Logic: 1. Ancient Greek Philosophical Logic: Robin Smith (Texas A&M University). 2. History of Logic: Medieval: B.G. Sundholm (Leiden University) and E.P. Bos (Leiden University). 3. The Rise of Modern Logic: Rolf George (University of Waterloo) and James Van Evra (University of Waterloo). Part II: Symbolic Logic and Ordinary Language: 4. Language, Logic, and Form: Kent Bach (San Francisco State University). 5. Puzzles About Intensionality: Nathan Salmon (University of California, Santa Barbara). 6. Symbolic Logic and Natural Language: Emma Borg (University of Reading) and Ernest Lepore (Rutgers University). Part III: Philosophical Dimensions of Logical Paradoxes: 7. Logical Paradoxes: James Cargile (University of Virginia). 8. Semantical and Logical Paradox: Keith Simmons (University of North Carolina at Chapel Hill). 9. Philosophical Implications of Logical Paradoxes: Roy A. Sorensen (Dartmouth College). Part IV: Truth and Definite Description in Semantic Analysis: 10. Truth, the Liar, and Tarski's Semantics: Gila Sher (University of California, San Diego). 11. Truth, the Liar, and Tarskian Truth Definition: Greg Ray (University of Florida). 12. Descriptions and Logical Form: Gary Ostertag (New York University). 13. Russell's Theory of Definite Descriptions as a Paradigm for Philosophy: Gregory Landini (University of Iowa). Part V: Concepts of Logical Consequence: 14. Necessity, Meaning, and Rationality: The Notion of Logical Consequence: Stewart Shapiro (Ohio State University). 15. Varieties of Consequence : B.G. Sundholm (Leiden University). 16. Modality of Deductively Valid Inference : Dale Jacquette (Pennsylvania State University). Part VI Logic, Existence, and Ontology: 17. Quantifiers, Being and Canonical Notation: Paul Gochet (University of Lige). 18. From Logic to Ontology: Some Problems of Predication, Negation and Possibility: Herbert Hochberg (University of Texas). 19. Putting Language First: The "Liberation" of Logic from Ontology: Ermanno Bencivenga (University of California, Irvine). Part VII: Metatheory and the Scope and Limits of Logic: 20. Metatheory: Alasdair Urquhart (University of Toronto). 21. Metatheory of Logics and the Characterization Problem: Jan Wolenski (Jagiellonian University). 22. Logic in Finite Structures: Definability, Complexity, and Randomness: Scott Weinstein (University of Pennsylvania). Part VIII: Logical Foundations of Set Theory and Mathematics: 23. Logic and Ontology: Numbers and Sets: Jos Benardete (Syracuse University). 24. Logical Foundations of Set Theory and Mathematics: Mary Tiles (University of Hawaii). 25. Property-Theoretic Foundations of Mathematics: Michael Jubien (University of California, Davis). Part IX: Modal Logics and Semantics: 26. Modal Logic: Johan van Benthem (University of Amsterdam). 27. First Order Alethic Modal Logic: Melvin Fitting (City University of New York). 28. Proofs and Expressiveness in Alethic Modal Logic: Maarten de Rijke (University of Amsterdam) and Heinrich Wansing (Dresden University of Technology). 29. Alethic Modal Logics and Semantics: Gerhard Schurz (University of Erfurt). 30. Epistemic Logic: Nicholas Rescher (University of Pittsburgh). 31. Deontic, Epistemic, and Temporal Modal Logics: Risto Hilpinen (University of Miami). Part X: Intuitionistic, Free, and Many-Valued Logics: 32. Intuitionism: Dirk van Dalen (University of Utrecht) and Mark van Atten (University of Utrecht). 33. Many-Valued, Free, and Intuitionistic Logics: Richard Grandy (Rice University). 34. Many-Valued Logic: Grzegorz Malinowski (University of Lodz). Part XI: Inductive, Fuzzy, and Quantum Probability Logics: 35. Inductive Logic : Stephen Glaister (University of Washington). 36. Heterodox Probability Theory: Peter Forrest (University of New England). 37. Why Fuzzy Logic?: Petr Hjek (Academy of Sciences of the Czech Republic). Part XII: Relevance and Paraconsistent Logics: 38. Relevance Logic: Edwin Mares (Victoria University of Wellington). 39. Paraconsistency: Bryson Brown (University of Lethbridge). 40. Logicians Setting Together Contradictories: A Perspective on Relevance, Paraconsistency, and Dialetheism: Graham Priest (University of Melbourne). Part XIII: Logic, Machine Theory, and Cognitive Science: 41. The Logical and the Physical: Andrew W. Hodges (Wadham College, Oxford University). 42. Modern Logic and its Role in the Study of Knowledge: Peter A. Flach (University of Bristol). 43. Actions and Normative Positions: A Modal-Logical Approach : Robert Demolombe (Toulouse Center) and Andrew J.I. Jones (University of Oslo). Part XIV: Mechanization of Logical Inference and Proof Discovery: 44. The Automation of Sound Reasoning and Successful Proof Finding: Larry Wos (Argonne National Laboratory) and Branden Fitelson (Yale University). 45. A Computational Logic for Applicative Common LISP: J. Strother Moore (University of Texas) and Matt Kaufmann (Advanced Micro Devices, Inc). 46. Sampling Labelled Deductive Systems: D.M. Gabbay (King's College). Resources for Further Study. Index.

Philosophy of Logic: An Anthology by Dale Jacquette (Blackwell Philosophy Anthologies: Blackwell) The essays in this anthology include some of the most important recent scholarship in philosophy of logic. I have deliberately avoided republishing papers that are readily available in other anthologies, or that are more closely related to philosophy of language or philosophy of mathematics, regardless of their influence in contemporary work in logic. My intention has been to make this volume a more unique distinctive resource that will complement rather than duplicate other selections of readings currently available. Although some of the papers are more technical than others, all are intended for and can be read with good understanding by beginning students in philosophy who have completed a first course in symbolic logic.

My choice of papers has been guided by a sense of major issues in philosophy of logic that have shaped recent discussion and contributed to ongoing research programs in theoretical and applied philosophical logic. To this end, I have organized the papers thematically rather than chronologically, to give the best overview of philosophical issues connected with logical analysis and the development of formal systems of symbolic logic. The papers range from general topics in classical logic to specialized investigations of the concept of meaning and truth, the interpretation of quantifiers in predicate logic, the theory of valid inference and logical entailment, and problems of alethic modality, intensionality, and propositional attitude. These are undoubtedly among the central problems of philosophical logic reflecting some of the most intriguing new directions in the field, but they by no means exhaust the possibilities.

Additional writings related to the philosophy of logic can be found in my Blackwell collection Philosophy of Mathematics: An Anthology. Newly commissioned papers on additional topics, concerning the metatheory of logic, logical and semantic paradoxes, nonstandard logics of many different sorts, fuzzy logic, relevance logics, paraconsistent logics, free logics, monotonic versus nonmonotic systems, applied logics in mathematics, science, probability theory, formal semantics, linguistic modeling, computer and cognitive applications, ethics, epistemology, and time, are collected in my Blackwell Companion to Philosophical Logic. The present book will serve its purpose if it helps provide readers at all levels with the necessary background and a sufficient sense of interest in its subject to continue philosophical inquiry and pursue advanced study of the methods, uses and longstanding problems in the philosophy of logic.

Understanding Symbolic Logic by Virginia Klenk (Prentice Hall) This book is intended as a comprehensive introduction to symbolic logic. It presupposes no prior acquaintance with either logic or mathematics, and it includes all the standard topics through relational predicate logic with identity. The book was written in the conviction that any student can master symbolic logic, and it is designed to give the student as much help as possible in attaining that mastery.

The main part of the book is divided into twenty units, each of which has an introduction and a statement of study objectives so that the student has an overview of what is to come and knows exactly what is required in order to master the unit. The explanatory material for each unit is divided into several subsections, each of which has a specific function and covers one relatively small, clearly defined topic. The clear separation of topics and the division into easily comprehended small "bites" allow the student to master the material step by step without being overwhelmed by an indigestible mass of information.

One‑variable predicate logic is developed, in detail, independently of relational predicate logic, and identity is presented in two separate units. The semantics of predicate logic is also developed in a separate unit, as is the semantics for sentential logic. In addition to the basic material, there are several "extra credit" units, which provide a glimpse into alternative methods of logic and more advanced topics.

I have tried to give as detailed explanations as possible, both for specific techniques, such as drawing up truth tables or constructing proofs, and for the rationale behind these techniques. It seems to me as important for a student to understand why things are done in a certain way as to learn the techniques themselves, and in this book I have tried to supply the whys" as well as the hows."

The book does, however, supply the hows" in abundance. Aside from the detailed explanations, there are numerous examples worked out in the text: various types of truth tables, a great many detailed, step‑by‑step symbolizations, and over fifty fully worked out proofs. In addition, there are copious exercises, with answers to fully half of these provided at the back of the book. Stars indicate problems for which answers are given.

Because of the detailed explanations, the extensive coverage, and the clear division of topics, the book is extremely flexible. It can be used in either freshman courses or upper‑division courses and is suitable for quarter, semester, or even two-quarter courses. In one quarter, for instance, one might cover just Units 1 through 14; in a semester course, Units 1 through 15,17, and 18; and in a two‑quarter course one might cover the entire book, including the supplementary units. Because of the step‑by‑step approach and the numerous examples and exercises, the book can also be used in self‑paced classes. Suggestions on how to structure such a course are included in the Instructor's Manual.

A new edition has given me the opportunity to make numerous changes that should clarify and streamline the presentation. In addition to updating examples and exercises, I have provided new or expanded explanations for many topics that students might find puzzling and have made scores of relatively minor changes that significantly clarify the material. The most substantial changes are in sections covering logical form and the distinction between form and substitution instance.

Logic, Form, and Grammar by Peter Long (International Library of Philosophy: Routledge) contains Peter Long's important essay, Logic, Form and Grammar, which resolves many difficulties for the logical form of an argument where the reasoning is hypothetical. Also included are two essays on classical problems in philosophical logic, relating to logical form and formal relations. The notion of logical form and its application are at the heart of some of the classical problems in philosophical logic and are the focus of Peter Longs investigations in the three essays that comprise this volume.

Peter Long first examines the notion of logical form as it applies to arguments involving hypothetical statements. In particular, he considers what logicians take to be paradigms of 'formally valid' arguments, such as 'If today is Wednesday then tomorrow is Thursday; today is Wednesday: therefore tomorrow is Thursday'. Long points to an important problem with such arguments. Whilst they are valid under the form If p then q; p: therefore q, this form is not a logical form. But in that case how can logic claim to be the science of formal inference? Long resolves this difficulty by drawing a fundamental distinction within the notion of the form under which an argument is valid. With this distinction it becomes possible for the first time to determine the status of any formally valid argument involving hypotheticals.

The remainder of the book takes up the notion of logical form as it applies to such simple propositions as `This sheet is white' and `London is north of Paris'. When we speak of, say, the first as giving expression to the relation of a thing's having a property, what is in question is a formal relation and we call it such because the relation is expressed through this proposition having the form. Peter Long shows that the confusion of such formal relations with relations proper is common in philosophy and is at the root of the theory that properties and relations are universals, and is responsible for the assimilation of facts to complexes.

THE IS-OUGHT PROBLEM: An Investigation in Philosophical Logic by Gerhard Schurz ($120.00, hardcover, Kluwer Academic Pubishers; ISBN: 0792344103) Can OUGHT be derived from IS? This book presents a systematic investigation of this time-honored philosophical problem by means of modern alethicdeontic predicate logic. Two comprehensive introductory chapters into the philosophical and logical foundations make the text understandable also for nonlogicians, ethicists, social scientists and students of philosophy. New in this study are two leitmotifs: relevance and metalogical generality. It turns out that is-ought inferences indeed exist, but they are all irrelevant in a precise logical sense. New proof techniques allow to establish this result for very broad classes of logics.

This book investigates a traditional problem of philosophy by means of modern logic. It is addressed to logicians as well as to philosophers or scientists, in particular to ethicists.

The book is a study in philosophical logic. This means that it approaches the is-ought problem mainly from the side of modern logic, but it has not only a logical but also a genuine philosophical ambition, and so it contains several purely philosophical considerations. These are condensed in first on the philosophical background of the is-ought problem, later on ethical applications of the logical results, concluding on the philosophical investigation of is-ought bridge principles, and in several interlude paragraphs of the book.

A profound philosophical investigation of the question of analytical or strongly intersubjective is-ought bridge principles supplements the logical study. The final results imply incisive limitations for the justifiability of ethics as opposed to empirical science.

The most far-reaching logical results of the study are situated as "theorems"; further results, which are neither lemmas nor corollaries, are called "propositions".Examples of is-ought-inferences violating Hume’s thesis are reflected in "facts" The proofs of theorems about logical foundations which do not directly touch the is-ought problem are collected in a separate Appendix.

EXCERPT:

In the famous passage of his Treatise, David Hume put forward a basic argument against the argumentative praxis of ethicists of his time. He stated that from what is (or is not), nothing about what ought to be (or ought not to be) can logically be concluded. This is Hume’s is-ought thesis. Consequently, the is-ought problem is the question whether, and under which conditions, this thesis is true. Let us repeat the frequently quoted passage from Hume once more:

"In every system of morality, which I have hitherto met with, I have always remark’d, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surpriz’d to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is, however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, ‘tis necessary that it shou’d be observ’d and explain’d; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers; and am persuaded, that this small attention wou’d subvert all the vulgar systems of morality..."

The notion of "ought", that is, of an ethical obligation or norm, respectively, is obviously related to the notion of an ethical value. All ethical systems assume some analytically true connection between these two notions to the effect that, roughly speaking, what is ethically good (in itself as well as in its consequences) ought to be done, and vice versa. So the is-ought problem has an obvious twin brother in the question whether ethical value statements can be logically inferred from fact statements. To clarify the terminology, we call a statement about what is i.e. a statement about the facts, may they be singular or general, accidental or necessary a descriptive statement; a statement about what ought to be a normative statement, a statement about what is valuable a valuative statement, and finally a statement which is either normative or valuative an ethical statement. Although we will focus in the following on the is-ought problem, it is clear that all our results simultaneously apply to Hume’ problem in its more general formulation, namely, whether any ethical statements can logically be inferred from descriptive statements.

In our interpretation of the quoted passage from Hume, we have understood "deduction" in the sense of logically valid inference. This is the standard interpretation of Hume’s thesis. If we speak in the following of Hume’s is-ought thesis (or of the is-ought problem) simpliciter we always understand it in this standard sense: as the thesis, that n ethical statement can be logically inferred from any descriptive statement. However, the is-ought problem may also be understood in the extended sense, as the question whether ethical statements may be inferred from descriptive statements in a broader sense of inference which includes also some "nonlogical" kinds of inference (whatever that may be). Indeed, some authors like MacIntyre think that Hume understands "deduction" in this broader sense of "inference" .It is clear that the philosophically extended is-ought thesis entails the standard is-ought thesis, but not vice versa. At a certain stage of our investigation, the standard is-ought problem will automatically turn into the extended is-ought problem.

Hume’s argument has lost nothing of its importance in present time, and probably will never do so. For on the ground of our moral attitudes there is a thicket of intuitions about what is good and what is bad, intuitions which stem from what we have learned throughout our childhood and which usually enter our moral reasoning in an unreflective way. We are easily led to allow certain deep but nevertheless basically subjective intuitions to turn a seemingly factual claim into a normative or valuative assertion, without our taking notice of it. Often, this lack of reflection on their own valuations leads people to a dogmatic attitude, because they mistakenly think of their own moral position as based on "facts" and thus as being unrefutable, and so they reject contradicting positions as obviously irrational. This leads, then, to debates about moral affairs which are driven by blind engagement and fanatism, instead of mutual understanding and rational discourse.

An example is the vehement debate about abortion in present day. The crucial point of this debate is the question which factual property of the unborn child is sufficient for attributing to it the same unrestricted right to live as we attribute it to born persons at least in our western civilization. For the one party in this debate, which often appeals to the importance of our moral conscience and instinct, it is obvious that this at this moment a human being has been created and life begins. Going to the other extreme, there factual property is the fertilization are philosophers like Peter Singer or Norbert Hoerster who have argued that this factual property is the beginning of the personality of the baby, which includes elementary self-interests as well as an elementary awareness of them.’ From the latter position it unavoidably follows that not only embryos but even very young babies, which have not developed these marks of personality, do not have the unrestricted right to live which older children or adults have. This consequence is shocking to the former party, which more than once has called the mentioned philosophers inhuman. Vice versa, several of these philosophers have a conspicuous tendency to call the former party irrational and incapable of moral argument. But if Hume’s thesis is true, then there is just no point in trying to ‘prove’ that one of the mentioned ‘ positions is ‘objectively true’ and thus to refute the other because it is not possible to derive the right to live from one of the mentioned empirical facts. There are different possible views on the matter of abortion, as there exist different ethical world views. Of course, this does not mean that there is no possibility of rational argument. But it means that, if Hume’s thesis is true, then all what rational argument can do and this is important enough! is to give a clear exposition of the possible views and their ethical premises; but it cannot lead to the ‘one truth’ among the possible positions. The decision of abortion must be, in the end, a collective social decision based on some form of democratic consensus, but not a matter of refuting or eliminating one or more of the possible moral attitudes. If the parties in the debate would be more aware about this point, much of the hate and demagogy could be avoided.

To summarize: in order to increase tolerance, mutual understanding and, thus, the rationality in our moral discourse, one must constantly be aware of the difficulty, if not impossibility, of justifying moral values as in natural science, by an appeal to the empirical facts, and in connection with that, one must constantly be aware of the multitude of different but equally possible moral attitudes. This is, in my view, the reason why the importance of the is-ought problem will never disappear.

So far we have spoken about the practical importance of the is-ought problem. Clearly, a logical-theoretical study of it, like this study is one, is a quite different thing. Its concepts and results will be necessarily abstract and seemingly distant from practical moral reasoning. But if one goes through them, one will see that in the end they can be applied in various ways to practical moral reasoning...

Nature cannot tell us which ethical concept is the right one: we have to decide. Intersubjective agreement in ethics is not given by a common nature which exists independently from us. In Peirce’s words, it is not the intersubjective result of the empirical research community. It is basically the result of a common culture which enables mutual understanding and a common life practice. But cultures are historically rather divergent. To strive for or against cultural homogeneity is itself an ethical question. However, questions like these are certainly beyond the purpose of this investigation, which was to show that ethical theories cannot be intersubjectively justified like theories of science.

Referee’s comments: "...the most complete and in-depth work written on the logical treatment of Hume’s Law. It has the rare quality of being an authentic work of philosophical logic, in the sense that it uses logical techniques even highly sophisticated and new ones to address a specific theme of philosophical reflection, without restricting itself to pure formal research..."

Contents:
PREFACE
1. PHILOSOPHICAL BACKGROUND AND PROGRAM OF THE STUDY
1.1 The Is-Ought Problem and its Significance
1.2 Choice of an Adequate Logical Framework
1.3 Distinction between Descriptive and Normative among the Primitive Symbols
1.4 Difficulties in the Explication of Hume’s Thesis: Prior’s Paradox
1.5 In which Logics Shall Hume’s Thesis Be Investigated?
Reflections on the Concept of "Logic"
1.5.1 "Logic" in the Mathematical and Philosophical Sense: Reflections on the Logic Analytic Synthetic Distinction
1.5.2 Varieties of Modal Logics and their Philosophical Importance
1.6 Logics without Bridge Principles: Program and Survey of Results
1.7 The Question of Bridge Principles
1.7.1 On the Relation between the Logical and the Semantic Is Ought Problem
1.7.2 The Open Question Argument and Its Limitations
1.8 The Is-Ought Problem in the Philosophically Extended Sense
1.9 A Short Summary of the Plan of the Book
2. THE LOGICAL BACKGROUND: A.D. I LOGICS
2.1 Language
2.2 The Formalization of Natural Language in :e
2.3 Semantics
2.4 Representation and Axiomatization
2.4.1 The Minimal a.d. I logic Kadj
2.4.2 Normal Extensions of Kado
2.4.3 Uniform Substitution for Predicates
2.4.4 Normal Extensions of Kadj
2.4.5 Deductibility and Consequence
2.5 Correctness and Completeness
2.6 Validity Preserving Operations on Models and Frames
11.10 Consequences for the Scientific Justifiability of Ethical Theories
12. ARE SYNTHETIC BRIDGE PRINCIPLES SCIENTIFICALLY JUSTIFIABLE?
12.1 Ethical Concepts as Theoretical Concepts: Holistic Justification Procedures
12.2 On the Limits of the Justifiability of Synthetic Bridge Principles A Comparison Between Physics and Ethics
APPENDIX
A. I Interchange of substitution for predicates and for individual variables
A.2 Transitivity of predicate substitutions
A.3 Uniform substitution for predicates in Kadl
A.4 Skeletons of Kadl axiom schemata
A.5 Preservation of frame-validity under cosubstitution
A.6 Advancing V, a and d-rule
A.7 Model-completeness for a.d.1 logics
A.8 Singleton frames for a.d.1logics which are not propositionally representable
A.9 Canonical a.0 logics with incomplete I-counterparts
A.10 Canonicity transfer from a.0 to a. I logics
A. 11 Canonicity transfer from monornodal to combined bimodal I logics
A. 12 Halld-6 n-completeness and the Bolzanocriterion
A. 13 Correspondence and canonicity for (NI5)
A.14 Domains of I.l.models
A.15 Characterization of a.d.l Logics
A.16 Characterization of a.d.(G) 2 logics
A. 17 Admissibility of (VGR)
TABLE OF DEFINITIONS, LEMMATA, PROPOSITIONS, THEOREMS, COROLLARIES, FACTS, FIGURES AND PROBLEMS
NOTES
BIBLIOGRAPHY

EFFECTIVE LOGIC COMPUTATION by Klaus Truemper (hardcover, 560 pages, John Wiley & Sons; ISBN: 0471238864) covers the emerging area of logic computation — the use of advanced mathematical methods to solve complex problems in logic. This logic system is useful for the construction of expert systems, such as automated handwriting analysis, traffic control systems, and data mining. A serious book for those interested in algorithms to solve ‘satisfiability’ and ‘minimum satisfiability’ problems. While mostly dealing with the theory behind Truemper’s practical algorithms, the book also provides discussions of applications to major problem classes. The topic of this book is essentially the starting point for solving these problems, that is ways in which a complex problem may be broken down into a number of smaller ones. Required reference for all students of logic interested in practical applications in engineering and computation.

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