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Tipi: A TPTPbased theory development environment emphasizing proof dependencies
"... In some theory development tasks, a problem is satisfactorily solved once it is shown that a theorem (conjecture) is derivable from the background theory (premises). Depending on one’s motivations, the details of the derivation of the conjecture from the premises may or may not be important. In some ..."
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In some theory development tasks, a problem is satisfactorily solved once it is shown that a theorem (conjecture) is derivable from the background theory (premises). Depending on one’s motivations, the details of the derivation of the conjecture from the premises may or may not be important. In some contexts, though, one wants more from theory development than simply derivability of the target theorems from the background theory. One may want to know which premises of the background theory were used in the course of a proof output by an automated theorem prover (when a proof is available), whether they are all, in suitable senses, necessary (and why), whether alternative proofs can be found, and so forth. The problem, then, is to support proof analysis in theory development; the tool described in this paper, Tipi, aims to provide precisely that. 1
Recent Developments in Computing and Philosophy
"... Because the label "computing and philosophy " can seem like an ad hoc attempt to tie computing to philosophy, it is important to explain why it is not, what it studies (or does) and how it differs from research in, say, "computing and history, " or "computing and biology&quo ..."
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Because the label "computing and philosophy " can seem like an ad hoc attempt to tie computing to philosophy, it is important to explain why it is not, what it studies (or does) and how it differs from research in, say, "computing and history, " or "computing and biology". The American Association for History and Computing is "dedicated to the reasonable and productive marriage of history and computer technology for teaching, researching and representing history through scholarship and public history "
Relations versus functions at the foundations of . . .
 FORTHCOMING IN THE JOURNAL OF LOGIC AND COMPUTATION
"... Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in term ..."
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Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell’s reduction of functions to relations over Frege’s reduction of relations to functions. There is an interesting system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an oversimplification: one can’t assimilate predication to functional application.
On the Distinction between Relational and Functional Type Theory
"... It is commonly believed that it makes no difference whether one starts with relational types or functional types in formulating type theory, since one can either start with relations as primitive and represent functions as relations or start with functions as primitive and represent relations as fun ..."
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It is commonly believed that it makes no difference whether one starts with relational types or functional types in formulating type theory, since one can either start with relations as primitive and represent functions as relations or start with functions as primitive and represent relations as functions. It is also commonly believed that the formulabased logic of relational type theory is equivalent to the termbased logic of functional type theory. However, in this paper, the authors argue that there are systems with logics that can be properly characterized in relational type theory, but not in functional type theory. We investigate an important difference between relational type theories (RTTs) and functional type theories (FTTs). It is often thought that for each RTT, there is an FTT which is a mere variant (or vice versa), since relations and functions are interdefinable. It is often concluded,
and
"... Our computational metaphysics group describes its use of automated reasoning tools to study Leibniz’s theory of concepts. We start with a reconstruction of Leibniz’s theory within the theory of abstract objects (henceforth ‘object theory’). Leibniz’s theory of concepts, under this reconstruction, h ..."
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Our computational metaphysics group describes its use of automated reasoning tools to study Leibniz’s theory of concepts. We start with a reconstruction of Leibniz’s theory within the theory of abstract objects (henceforth ‘object theory’). Leibniz’s theory of concepts, under this reconstruction, has a nonmodal algebra of concepts, a conceptcontainment theory of truth, and a modal metaphysics of complete individual concepts. We show how the objecttheoretic reconstruction of these components of Leibniz’s theory can be represented for investigation by means of automated theorem provers and finite model builders. The fundamental theorem of Leibniz’s theory is derived using these tools. 1