Results 1  10
of
20
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 ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
: 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, ...
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 ..."
Abstract
 Add to MetaCart
videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa
PROOFS AND ARGUMENTS THE SPECIAL CASE OF MATHEMATICS
"... Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are lost out of view entirely. As I will defend, it is precisely in those aspects that similarities can be found between pract ..."
Abstract
 Add to MetaCart
Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are lost out of view entirely. As I will defend, it is precisely in those aspects that similarities can be found between practices in the exact sciences and in mathematics. Hence, if we are looking for a (more) unified treatment of science and mathematics, it is necessary to incorporate these elements into our view of what mathematics is about. As a helpful tool I will introduce the notion of a mathematical argument as a more liberalized version of the notion of mathematical proof.
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 ..."
Abstract
 Add to MetaCart
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 ...