Ebook: Closure Spaces and Logic

This book examines an abstract mathematical theory, placing special emphasis on results applicable to formal logic. If a theory is especially abstract, it may find a natural home within several of the more familiar branches of mathematics. This is the case with the theory of closure spaces. It might be considered part of topology, lattice theory, universal algebra or, no doubt, one of several other branches of mathematics as well. In our development we have treated it, conceptually and methodologically, as part of topology, partly because we first thought ofthe basic structure involved (closure space), as a generalization of Frechet's concept V-space. V-spaces have been used in some developments of general topology as a generalization of topological space. Indeed, when in the early '50s, one of us started thinking about closure spaces, we thought ofit as the generalization of Frechet V­ space which comes from not requiring the null set to be CLOSURE SPACES ANDLOGIC XlI closed(as it is in V-spaces). This generalization has an extreme advantage in connection with application to logic, since the most important closure notion in logic, deductive closure, in most cases does not generate a V-space, since the closure of the null set typically consists of the "logical truths" of the logic being examined.

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The book exmaines closure spaces, an abstract mathematical theory, with special emphasis on results applicable to formal logic. The theory is developed, conceptually and methodologically, as part of topology.
At the least, the book shows how techniques and results from topology can be usefully employed in the theory of deductive systems. At most, since it shows that much of logical theory can be represented within closure space theory, the abstract theory of derivability and consequence can be considered a branch of applied topology. One upshot of this appears to be that the concepts of logic need not be overtly linguistic nor do logical systems need to have the syntax they are usually assumed to have.
Audience: The book presupposes very little technical knowledge, but can probably be read most easily by someone with a background in symbolic logic or, even better, upper division or graduate mathematics. It should be of interest to logicians and, to a lesser degree, computer scientists and other mathematicians.

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The book exmaines closure spaces, an abstract mathematical theory, with special emphasis on results applicable to formal logic. The theory is developed, conceptually and methodologically, as part of topology.
At the least, the book shows how techniques and results from topology can be usefully employed in the theory of deductive systems. At most, since it shows that much of logical theory can be represented within closure space theory, the abstract theory of derivability and consequence can be considered a branch of applied topology. One upshot of this appears to be that the concepts of logic need not be overtly linguistic nor do logical systems need to have the syntax they are usually assumed to have.
Audience: The book presupposes very little technical knowledge, but can probably be read most easily by someone with a background in symbolic logic or, even better, upper division or graduate mathematics. It should be of interest to logicians and, to a lesser degree, computer scientists and other mathematicians.
Content:
Front Matter....Pages i-xvii
Logic and Topology....Pages 1-18
Basic Topological Properties....Pages 19-45
Some Theorems of Tarski....Pages 47-57
Continuous Functions....Pages 59-83
Homeomorphisms....Pages 85-107
Closed Bases and Closure Semantics I....Pages 109-142
Theory of Complete Lattices....Pages 143-169
Closed Bases and Closure Semantics II....Pages 171-204
Truth Functions....Pages 205-224
Back Matter....Pages 225-230

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The book exmaines closure spaces, an abstract mathematical theory, with special emphasis on results applicable to formal logic. The theory is developed, conceptually and methodologically, as part of topology.
At the least, the book shows how techniques and results from topology can be usefully employed in the theory of deductive systems. At most, since it shows that much of logical theory can be represented within closure space theory, the abstract theory of derivability and consequence can be considered a branch of applied topology. One upshot of this appears to be that the concepts of logic need not be overtly linguistic nor do logical systems need to have the syntax they are usually assumed to have.
Audience: The book presupposes very little technical knowledge, but can probably be read most easily by someone with a background in symbolic logic or, even better, upper division or graduate mathematics. It should be of interest to logicians and, to a lesser degree, computer scientists and other mathematicians.
Content:
Front Matter....Pages i-xvii
Logic and Topology....Pages 1-18
Basic Topological Properties....Pages 19-45
Some Theorems of Tarski....Pages 47-57
Continuous Functions....Pages 59-83
Homeomorphisms....Pages 85-107
Closed Bases and Closure Semantics I....Pages 109-142
Theory of Complete Lattices....Pages 143-169
Closed Bases and Closure Semantics II....Pages 171-204
Truth Functions....Pages 205-224
Back Matter....Pages 225-230
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