In this chapter we take up the three simplest connectives: conjunction, disjunction, and negation, corresponding to simple uses of the English and, or, and it is not the case that. To form complex claims, fol provides us with connectives and quantifiers.
While there is much more to logic than we can even hint at in this book, or than any one person could learn in a lifetime, we can at least cover these most basic of issues.The Boolean Connectives So far, we have discussed only atomic claims. The second is to help you learn 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. The first is to help you learn a new language, the language of first-order logic. More particularly, we have two main aims. Convention is crucial in giving meaning to a language, but once the meaning is established, the laws of logic follow inevitably. We want you to understand just how the laws of logic follow inevitably from the meanings of the expressions we use to make claims. Our goal is to provide detailed and systematic answers to the questions raised above. This book is intended to introduce you to some of the most important concepts and tools of logic. 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. More recently, the study of logic has played a major role in the development of modern day computers and programming languages. The Encyclopedia Brittanica lists logic as one of the seven main branches of knowledge. 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. Over the past century the study of logic has undergone rapid and important advances. To study logic is to use the methods of rational inquiry on rationality itself. These are the questions that one takes up when one studies logic itself. it becomes crucial to understand just what the laws of logic are, and even more important, why they are laws of logic. If there is something arbitrary about logic, then the same must hold of all rational inquiry. After all, no science can be any more certain than its weakest link. The importance of logic has been recognized since antiquity. But we can't even imagine a world in which there both are and are not nine planets. We can imagine a country in which a red traffic light means go, and a world on which water flows up hill. 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.
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. Some people have claimed that the laws of logic are simply a matter of convention. 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. Acceptance of these commonly held principles of rationality is what differentiates rational inquiry from other forms of human activity. 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. 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. In other words, these fields all presuppose an underlying acceptance of basic principles of logic. 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. What they do have in common, with each other and with many other fields, is their dependence on a certain standard of rationality.
And not all that much in the way of methodology. 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.
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