Menschen, maschinen und g?dels theorem

Mechanism is the thesis that men can be considered as machines, that there is no essential difference between minds and machines.John Lucas has argued that it is a consequence of Gödel's theorem that mechanism is false. Men cannot be considered as machines, because the intellectual capacities of men are superior to that of any machine. Lucas claims that we can do something that no machine can do-namely to produce as true the Gödel-formula of any given machine. But no machine can prove its own Gödel-formula.In order to discuss and evaluate this argument, the author makes a distinction between formal and informal proofs, and between proofs given by men and proofs given by machines. It is argued that the informal proof capacities of machines are possibly greater and the formal proof capacities of men are possibly smaller than the anti-mechanist claims. So the argument from Gödel's theorem against mechanism fails.Though Gödel's theorem does not prove that minds are different from machines, it is not irrelevant to the analysis of thought and to the mind/machine controversy. It points to the importance of informal methods even within formal sciences and to the need for an analysis of the notion of informal thinking in cognitive science.     

Mathematical Method and Proof

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.     

Connection-Driven Inductive Theorem Proving

We present a method for integrating rippling-based rewriting into matrix-based theorem proving as a means for automating inductive specification proofs. The selection of connections in an inductive matrix proof is guided by symmetries between induction hypothesis and induction conclusion. Unification is extended by decision procedures and a rippling/reverse-rippling heuristic. Conditional substitutions are generated whenever a uniform substitution is impossible. We illustrate the integrated method by discussing several inductive proofs for the integer square root problem as well as the algorithms extracted from these proofs.     

Efficiently checking propositional refutations in HOL theorem provers

This paper describes the integration of zChaff and MiniSat, currently two leading SAT solvers, with Higher Order Logic (HOL) theorem provers. Both SAT solvers generate resolution-style proofs for (instances of) propositional tautologies. These proofs are verified by the theorem provers. The presented approach significantly improves the provers' performance on propositional problems, and exhibits counterexamples for unprovable conjectures. It is also shown that LCF-style theorem provers can serve as viable proof checkers even for large SAT problems. An efficient representation of the propositional problem in the theorem prover turns out to be crucial; several possible solutions are discussed.     

Proof Theory for Reasoning with Euler Diagrams: A Logic Translation and Normalization

Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs.     

Continuum,name and paradox

The article deals with Cantor’s argument for the non-denumerability of reals somewhat in the spirit of Lakatos’ logic of mathematical discovery. At the outset Cantor’s proof is compared with some other famous proofs such as Dedekind’s recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are “ontologically” safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on diagonal construction are caused by superficial understanding of what a name is.     

On the Proof-Theory of two Formalisations of Modal First-Order Logic

We introduce a Gentzen-style modal predicate logic and prove the cut-elimination theorem for it. This sequent calculus of cut-free proofs is chosen as a proxy to develop the proof-theory of the logics introduced in [14, 15, 4]. We present syntactic proofs for all the metatheoretical results that were proved model-theoretically in loc. cit. and moreover prove that the form of weak reflection proved in these papers is as strong as possible.     

Proving quadratic reciprocity: explanation,disagreement, transparency and depth

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late eighteenth century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

     

Syntax-directed discovery in mathematics

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures.     

Integrating external deduction tools with ACL2

We present an interface connecting the ACL2 theorem prover with external deduction tools. The ACL2 logic contains several mechanisms for proof structuring, which are important to the construction of industrial-scale proofs. The complexity induced by these mechanisms makes the design of the interface challenging. We discuss some of the challenges, and develop a precise specification of the requirements on the external tools for a sound connection with ACL2. We also develop constructs within ACL2 to enable the developers of external tools to satisfy our specifications. The interface is available with the ACL2 theorem prover starting from Version 3.2, and we describe several applications of the interface.     

On the Proof Theory of the Modal mu-Calculus

We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary. Presented by Heinrich Wansing     

Constructions of classical models by means of Kripke models (survey)

It is demonstrated how Kripke models for intuitionistic predicate logic can be applied in order to prove classical theorems. As examples proofs of the independence of the axiom of constructibility, of the omitting types theorem and of Shelah's ultrapower theorem are sketched.     

Normal Natural Deduction Proofs (in classical logic)

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? This informal question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local "detours", but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation.

     

A Generalized Syllogistic Inference System based on Inclusion and Exclusion Relations

We introduce a simple inference system based on two primitive relations between terms, namely, inclusion and exclusion relations. We present a normalization theorem, and then provide a characterization of the structure of normal proofs. Based on this, inferences in a syllogistic fragment of natural language are reconstructed within our system. We also show that our system can be embedded into a fragment of propositional minimal logic.     

Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic

We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.     

TPS: A hybrid automatic-interactive system for developing proofs

The theorem proving system Tps provides support for constructing proofs using a mix of automation and user interaction, and for manipulating and inspecting proofs. Its library facilities allow the user to store and organize work. Mathematical theorems can be expressed very naturally in Tps using higher-order logic. A number of proof representations are available in Tps, so proofs can be inspected from various perspectives.     

Lightweight relevance filtering for machine-generated resolution problems

Irrelevant clauses in resolution problems increase the search space, making proofs hard to find in a reasonable amount of processor time. Simple relevance filtering methods, based on counting symbols in clauses, improve the success rate for a variety of automatic theorem provers and with various initial settings. We have designed these techniques as part of a project to link automatic theorem provers to the interactive theorem prover Isabelle. We have tested them for problems involving thousands of clauses, which yield poor results without filtering. Our methods should be applicable to other tasks where the resolution problems are produced mechanically and where completeness is less important than achieving a high success rate with limited processor time.     

Reasoning Continuously: A Formal Construction of Continuous Proofs

We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism.

     

Free-Variable Tableaux for Propositional Modal Logics

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.