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Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
Notre Dame Journal of Formal Logic A Simple Proof That SuperConsistency Implies Cut Elimination
"... Abstract We give a simple and direct proof that superconsistency implies the cut elimination property in deduction modulo. This proof can be seen as a simplification of the proof that superconsistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that ..."
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Abstract We give a simple and direct proof that superconsistency implies the cut elimination property in deduction modulo. This proof can be seen as a simplification of the proof that superconsistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cutfree calculus. As an application, we compare our work with the cut elimination theorems in higherorder logic that involve Vcomplexes. 1
Complete CutFree Tableaux for Equational Simple Type Theory
, 2009
"... We present a cutfree tableau system for a version of Church’s simple type normalization operator that completely hides the details of lambda conversion. We prove completeness of the system relative to Henkin models. The proof constructs Henkin models using the novel notion of a value system. 1 ..."
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We present a cutfree tableau system for a version of Church’s simple type normalization operator that completely hides the details of lambda conversion. We prove completeness of the system relative to Henkin models. The proof constructs Henkin models using the novel notion of a value system. 1
Transcendental syntax 2.0
, 2012
"... How come that finite language can produce certainty — at least a sufficiently certain certainty, sometimes apodictic — in the presence of infinity? The answer can by no means be found in external reality, quite the contrary: it seems that the very purpose of semantics is to make this question untrac ..."
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How come that finite language can produce certainty — at least a sufficiently certain certainty, sometimes apodictic — in the presence of infinity? The answer can by no means be found in external reality, quite the contrary: it seems that the very purpose of semantics is to make this question untractable with the help of an ad hoc analytical newspeak in which one cannot even formulate the above question. Transcendental syntax comes from the constatation that logic is better off when there is no�reality�at all and thus restores the priority of syntax over anything else. What follows is the present state of a new programme. 1 The conditions of possibility of language This first lecture is rather philosophical, too much indeed: I didn’t find a way to distillate the philosophical issues in the more technical chapters 2 — 4. For those allergic to philosphy, I swear that a change of philosophical background is absolutely necessary to achieve logical maturity, including — indeed, especially — at technical maturity. 1.1 Introduction: philosophy