Издательство Seven Bridges Press, 1999, -597 pp.
What do the fields of astronomy, economics, finance, law, mathematics, medicine, physics, and sociology have in common? Not much in the way of subject matter, that's for sure. And not all that much in the way of methodology. What they do have in common, with each other and with many other fields, is their dependence on a certain standard of rationality. In each of these fields, it is assumed that the participants can differentiate between rational argumentation based on assumed principles or evidence, and wild speculation or nonsequiturs, claims that in no way follow from the assumptions. In other words, these fields all presuppose an underlying acceptance of basic principles of logic.
For that matter, all rational inquiry depends on logic, on the ability of logic and rational people to reason correctly most of the time, and, when they fail to reason inquiry correctly, on the ability of others to point out the gaps in their reasoning. While people may not all agree on a whole lot, they do seem to be able to agree on what can legitimately be concluded from given information. Acceptance of these commonly held principles of rationality is what differentiates rational inquiry from other forms of human activity.
Just what are the principles of rationality presupposed by these disciplines? And what are the techniques by which we can distinguish correct or valid" reasoning from incorrect or invalid" reasoning? More basically, what is it that makes one claim \follow logically" from some given information, while some other claim does not?
Many answers to these questions have been explored. Some people have claimed that the laws of logic are simply a matter of convention. If this is so, logic and convention we could presumably decide to change the conventions, and so adopt different principles of logic, the way we can decide which side of the road we drive on. But there is an overwhelming intuition that the laws of logic are somehow more fundamental, less subject to repeal, than the laws of the land, or even the laws of physics. We can imagine a country in which a red traffic light means go, and a world on which water flows up hill. But we can't even imagine a world in which there both are and are not nine planets.
The importance of logic has been recognized since antiquity. After all, no science can be any more certain than its weakest link. If there is something arbitrary about logic, then the same must hold of all rational inquiry. it becomes crucial to understand just what the laws of logic are, and even more important, why they are laws of logic. These are the questions that one takes up when one studies logic itself. To study logic is to use the methods of rational inquiry on rationality itself.
Over the past century the study of logic has undergone rapid and important advances. Spurred on by logical problems in that most deductive of disciplines, mathematics, it developed into a discipline in its own right, with its own concepts, methods, techniques, and language. The Encyclopedia Brittanica lists logic as one of the seven main branches of knowledge. More recently, the study of logic has played a major role in the development of mode day computers and programming languages. Logic continues to play an important part in computer science; indeed, it has been said that computer science is just logic implemented in electrical engineering.
This book is intended to introduce you to some of the most important concepts and tools of logic. Our goal is to provide detailed and systematic answers to the questions raised above. We want you to understand just how the laws of logic follow inevitably from the meanings of the expressions we use to make claims. Convention is crucial in giving meaning to a language, but once the meaning is established, the laws of logic follow inevitably.
More particularly, we have two main aims. The first is to help you lea a new language, the language of first-order logic. The second is to help you lea about the notion of logical consequence, and about how one goes about establishing whether some claim is or is not a logical consequence of other accepted claims. While there is much more to logic than we can even hint at in this book, or than any one person could lea in a lifetime, we can at least cover these most basic of issues.
Introduction
I Propositional Logic
Atomic Sentences
The Logic of Atomic Sentences
The Boolean Connectives
The Logic of Boolean Connectives
Methods of Proof for Boolean Logic
Formal Proofs and Boolean Logic
The Logic of Conditionals
Introduction to Quantification
The Logic of Quantifiers
Multiple Quantifiers
Methods of Proof for Quantifiers
Formal Proofs and Quantifiers
More about Quantification (optional)
III Applications and Metatheory
First-order Set Theory
Mathematical Induction
Advanced Topics in Propositional Logic
Advanced Topics in FOL
Completeness and Incompleteness
Summary of Formal Proof Rules
What do the fields of astronomy, economics, finance, law, mathematics, medicine, physics, and sociology have in common? Not much in the way of subject matter, that's for sure. And not all that much in the way of methodology. What they do have in common, with each other and with many other fields, is their dependence on a certain standard of rationality. In each of these fields, it is assumed that the participants can differentiate between rational argumentation based on assumed principles or evidence, and wild speculation or nonsequiturs, claims that in no way follow from the assumptions. In other words, these fields all presuppose an underlying acceptance of basic principles of logic.
For that matter, all rational inquiry depends on logic, on the ability of logic and rational people to reason correctly most of the time, and, when they fail to reason inquiry correctly, on the ability of others to point out the gaps in their reasoning. While people may not all agree on a whole lot, they do seem to be able to agree on what can legitimately be concluded from given information. Acceptance of these commonly held principles of rationality is what differentiates rational inquiry from other forms of human activity.
Just what are the principles of rationality presupposed by these disciplines? And what are the techniques by which we can distinguish correct or valid" reasoning from incorrect or invalid" reasoning? More basically, what is it that makes one claim \follow logically" from some given information, while some other claim does not?
Many answers to these questions have been explored. Some people have claimed that the laws of logic are simply a matter of convention. If this is so, logic and convention we could presumably decide to change the conventions, and so adopt different principles of logic, the way we can decide which side of the road we drive on. But there is an overwhelming intuition that the laws of logic are somehow more fundamental, less subject to repeal, than the laws of the land, or even the laws of physics. We can imagine a country in which a red traffic light means go, and a world on which water flows up hill. But we can't even imagine a world in which there both are and are not nine planets.
The importance of logic has been recognized since antiquity. After all, no science can be any more certain than its weakest link. If there is something arbitrary about logic, then the same must hold of all rational inquiry. it becomes crucial to understand just what the laws of logic are, and even more important, why they are laws of logic. These are the questions that one takes up when one studies logic itself. To study logic is to use the methods of rational inquiry on rationality itself.
Over the past century the study of logic has undergone rapid and important advances. Spurred on by logical problems in that most deductive of disciplines, mathematics, it developed into a discipline in its own right, with its own concepts, methods, techniques, and language. The Encyclopedia Brittanica lists logic as one of the seven main branches of knowledge. More recently, the study of logic has played a major role in the development of mode day computers and programming languages. Logic continues to play an important part in computer science; indeed, it has been said that computer science is just logic implemented in electrical engineering.
This book is intended to introduce you to some of the most important concepts and tools of logic. Our goal is to provide detailed and systematic answers to the questions raised above. We want you to understand just how the laws of logic follow inevitably from the meanings of the expressions we use to make claims. Convention is crucial in giving meaning to a language, but once the meaning is established, the laws of logic follow inevitably.
More particularly, we have two main aims. The first is to help you lea a new language, the language of first-order logic. The second is to help you lea about the notion of logical consequence, and about how one goes about establishing whether some claim is or is not a logical consequence of other accepted claims. While there is much more to logic than we can even hint at in this book, or than any one person could lea in a lifetime, we can at least cover these most basic of issues.
Introduction
I Propositional Logic
Atomic Sentences
The Logic of Atomic Sentences
The Boolean Connectives
The Logic of Boolean Connectives
Methods of Proof for Boolean Logic
Formal Proofs and Boolean Logic
The Logic of Conditionals
Introduction to Quantification
The Logic of Quantifiers
Multiple Quantifiers
Methods of Proof for Quantifiers
Formal Proofs and Quantifiers
More about Quantification (optional)
III Applications and Metatheory
First-order Set Theory
Mathematical Induction
Advanced Topics in Propositional Logic
Advanced Topics in FOL
Completeness and Incompleteness
Summary of Formal Proof Rules