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The Berry Paradox
, 1994
"... was Godel's secretary. She said that Godel was very careful about his health and because of the snow he wasn't coming to the Institute that day and therefore my appointment was canceled. And that's how I had two phone conversations with Godel but never met him. I never tried again. I'd like to tell ..."
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was Godel's secretary. She said that Godel was very careful about his health and because of the snow he wasn't coming to the Institute that day and therefore my appointment was canceled. And that's how I had two phone conversations with Godel but never met him. I never tried again. I'd like to tell you what I would have told Godel. What I wanted to tell Godel is the difference between what you get when you study the limits of mathematics the way Godel did using the paradox of the liar, and what I get using the Berry paradox instead. What is the paradox of the liar? Well, the paradox of the liar is "This statement is false!" Why is this a paradox? What does "false" mean? Well, "false" means "does not correspond to reality." This statement says that it is false. If that doesn't correspond to reality, it must mean that the statement is true, right? On the other hand, if the statement is true it means that what it says corresponds to reality. But it says that it is false. Therefore the sta
ON INTERPRETING CHAITIN’S INCOMPLETENESS THEOREM
, 1998
"... The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number ..."
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The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.
Proving FirstOrder Equality Theorems with HyperLinking
, 1995
"... Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving p ..."
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Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving problems. However, Lee's roundbyround implementation of hyperlinking is not particularly well suited for the addition of special methods in support of equality. In this dissertation, we describe, as alternative to the roundbyround hyperlinking implementation of Lee, a smallest instance first implementation of hyperlinking which addresses many of the inefficiencies of roundbyround hyperlinking encountered when adding special methods in support of equality. Smallest instance first hyperlinking is based on the formalization of generating smallest clauses first, a heuristic widely used in other automated theorem provers. We prove both the soundness and logical completeness of smallest instance first hyperlinking and show that it always generates smallest clauses first under
Programming Data Structures In Logic
, 1992
"... : Current programming languages that are grounded in a formal logic  such as pure Lisp (based on the lambda calculus) and Prolog (based on Horn clause logic)  do not support the use of complex, pointerbased data structures. The lack of this important feature in logically grounded languages co ..."
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: Current programming languages that are grounded in a formal logic  such as pure Lisp (based on the lambda calculus) and Prolog (based on Horn clause logic)  do not support the use of complex, pointerbased data structures. The lack of this important feature in logically grounded languages contrasts sharply with its strong support in the imperative programming languages that have enjoyed wide application, of which C is a prime example. Unfortunately, the formal methods for reasoning about imperative languages have not proved broadly useful for reasoning about programs that manipulate complex, pointerbased data structures. Between these two camps resides an open question: How can we verify programs involving complex, pointerbased data structures? This work gives an answer to this question. It describes a programming language in which a programmer can define logical predicates on data structures and pointers, and use these predicates to specify programs that manipulate complex, ...
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
Models, Rules, Deductive Reasoning
, 1999
"... We formulate a simple theory of deductive reasoning based on mental models. One prediction of the theory is experimentally tested and found to be incorrect. The bearing of our results on contemporary theories of mental models is discussed. We then consider a potential objection to current ruletheor ..."
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We formulate a simple theory of deductive reasoning based on mental models. One prediction of the theory is experimentally tested and found to be incorrect. The bearing of our results on contemporary theories of mental models is discussed. We then consider a potential objection to current ruletheories of deduction. Such theories picture deductive reasoning as the successive application of inferenceschemata from #rstorder logic. Relying on a theorem due to George Boolos, we show that under weak hypotheses #rstorder schemata cannot account for many people's abilitytoverify the validity of #rstorder arguments. The hypothesis that deductive reasoning is mediated by the construction of mental models has enjoyed predictive success across several studies. It has also proven to be a fertile source of ideas about other kinds of judgment, for example, temporal, spatial, and probabilistic. At the same time, the theory has su#ered from persistent criticism for ambiguity about the details of ...
THE BERRY PARADOX
, 1994
"... videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa ..."
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videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa
A model theory of induction
, 2010
"... Publication details, including instructions for authors and subscription information: ..."
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Publication details, including instructions for authors and subscription information: