Why There Can\u27t be a Logic of Induction

Carap\u27s attempt to develop an inductive logic has been criticized on a variety of grounds, and while there may be some philosophers who believe that difficulties with Carnap\u27s approach can be overcome by further elaborations and modifications of his system, I think it is fair to say that the consensus is that the approach as a whole cannot succeed. In writing a paper on problems with inductive logic (and with Carnap\u27s approach in particular), I might therefore be accused of beating a dead horse. However, there are still some (e.g., Spirtes, Glymour and Scheines 1993) who seem to believe that purely formal methods for scientific inference can be developed. It may still then be useful to perform an autopsy on a dead horse when establishing the cause of death can shed light on issues of current concern. My intention in this paper is to point out a problem in Carnap\u27s inductive logic which has not been clearly articulated, and which applies generally to any inductive logic. My conclusion will be that scientific inference is inevitably and ineliminably guided by background beliefs and that different background beliefs lead to the application of different inductive rules and different standards of evidentiary relevance. At the end of this paper I will discuss the relationship between this conclusion and the problem of justifying induction

Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

A System of Interaction and Structure

This paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae submitted to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allow the design of BV, yield a modular proof of cut elimination.Comment: This is the authoritative version of the article, with readable pictures, in colour, also available at . (The published version contains errors introduced by the editorial processing.) Web site for Deep Inference and the Calculus of Structures at <http://alessio.guglielmi.name/res/cos

Distilling Abstract Machines (Long Version)

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between big-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.Comment: 63 page

There are no universal rules for induction

In a material theory of induction, inductive inferences are warranted by facts that prevail locally. This approach, it is urged, is preferable to formal theories of induction in which the good inductive inferences are delineated as those conforming to universal schemas. An inductive inference problem concerning indeterministic, nonprobabilistic systems in physics is posed, and it is argued that Bayesians cannot responsibly analyze it, thereby demonstrating that the probability calculus is not the universal logic of induction. Copyright 2010 by the Philosophy of Science Association.All right reserved