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First-Order Logic: Language
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First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. It goes by many names, including: [F]irst-order predicate calculus (FOPC), the lower predicate calculus, the language of first-order logic or predicate logic. Unlike natural languages such as English, FOL uses a wholly unambiguous formal language interpreted by mathematical structures. FOL is a system of deduction extending propositional logic by allowing quantification over individuals of a given domain (universe) of discourse. For example, it can be stated in FOL "Every individual has the property P".
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The Language of First-order Logic [C]omes packaged with the program Tarski's World. Since Hyperproof uses an extension of the natural-deduction-style proof system taught in The Language of First-order Logic, it can be used to accompany this text. The Language of First-order Logic is available in versions for Macintosh and Microsoft Windows. A version of the software for computers running NeXTstep is ... available. To receive it, purchase a copy of either the Macintosh or Windows version and follow the instructions contained in the book.
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Concepts expressed in natural language must be "translated" to first-order logic (FOL) before FOL can be used to address them, and there are a number of potential pitfalls in this translation. In FOL, means "p, or q, or both", that is, it is inclusive. In English, the word "or" is sometimes inclusive (e.g, "cream or sugar?"), but sometimes it is exclusive (e.g., "coffee or tea?" is usually intended to mean one or the other, not both). Similarly, the English word "some" may mean "at least one, possibly all", but other times it may mean "not all, possibly none". The English word "and" should sometimes be translated as "or" (e.g., "men and women may apply"). [2]
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Tarski's World [I]s an innovative and enjoyable way to introduce your students to the language of first-order logic. Using this program students quickly master the meaning of the connectives and quantifiers, and soon become fluent in the symbolic language at the core of modern logic. Tarski's World allows the students to build three-dimensional worlds and to describe them in first-order logic. They evaluate the sentences in the constructed worlds and if their evaluation is incorrect, the program provides them with a game that leads them to understand where they went wrong.
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The use of first-order logic as database logic is shown to be powerful enough for formalizing and implementing not only relational but ... hierarchical and network-type databases. It enables one to treat all the types of databases in a uniform manner. This paper focuses on the database language for heterogeneous databases. The language is shown to be general enough to specify constraints for a particular type of database, so that a specification of database type can be “translated” to the specification given in the database language, creating a “logical environment” for different views that can be defined by users. Owing to the fact that any database schema is seen as a first-order theory expressed by a finite set of sentences, the problems concerned with completeness and compactness of the database logic discussed by Jacobs ("On Database Logic,” J. ACM 29,2 (Apr. 1982), 310-332) are avoided.
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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.
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