Related papers: No speedup for geometric theories
A theorem, usually attributed to Barr, yields that (A) geometric implications deduced in classical L_{\infty\omega} logic from geometric theories also have intuitionistic proofs. Barr's theorem is of a topos-theoretic nature and its proof…
We describe a method for inverting Gentzen's cut-elimination in classical first-order logic. Our algorithm is based on first computign a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize…
In this paper we show that an intuitionistic theory for fixed points is conservative over the Heyting arithmetic with respect to a certain class of formulas. This extends partly the result of mine. The proof is inspired by the quick…
This paper explores the connection between two central results in the proof theory of classical logic: Gentzen's cut-elimination for the sequent calculus and Herbrands "fundamental theorem". Starting from Miller's expansion-tree-proofs, a…
Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from…
This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting…
General relativity required the abandonment of Euclidean geometry. Here we show that quantum theory requires the abandonment of classical logic. We show that the Hilbert space representation of quantum theory is logically inevitable. There…
Plural (or multiple-conclusion) cuts are inferences made by applying a structural rule introduced by Gentzen for his sequent formulation of classical logic. As singular (single-conclusion) cuts yield trees, which underlie ordinary natural…
Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are (1) to consider extensions of…
The Curry-Howard correspondence is often described as relating proofs (in intutionistic natural deduction) to programs (terms in simply-typed lambda calculus). However this narrative is hardly a perfect fit, due to the computational content…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
This paper represents classical propositional proofs as *combinatorial proofs*, which are more abstract than proof nets: superposition (contraction/weakening) is modelled mathematically, as a lax form of fibration, rather than syntactically…
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…
A Henkin-style proof of completeness of first-order classical logic is given with respect to a very small set (notably missing cut rule) of Genzten deduction rules for intuitionistic sequents. Insisting on sparing on derivation rules,…
"Physical theories of fundamental significance tend to be gauge theories. These are theories in which the physical system being dealt with is described by more variables than there are physically independent degree of freedom. The…
Elimination of quantifiers is shown to fail dramatically for a group of well-known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by…
The coupling between the electromagnetic and gravitational fields results in "faster than light" photons and invalids the Lorentz invariance and some laws of physics. A typical example is that the first and third laws of geometric optics…
Almost from the inception of Hilbert's program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various…
Prawitz suggested expanding a natural deduction system for intuitionistic logic to include rules for classical logic constructors, allowing both intuitionistic and classical elements to coexist without losing their inherent characteristics.…
Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely…