Albert Visser

A propositional logic with explicit fixed points

This paper studies a propositional logic which is obtained by interpreting implication as formal provability. It is also the logic of finite irreflexive Kripke Models.A Kripke Model completeness theorem is given and several completeness theorems for interpretations into Provability Logic and Peano Arithmetic.

Semantics and the Liar Paradox

The semantical paradoxes are not a scientific subject like Inductive Definitions, Algebraic Geometry or Plasma Physics. At least not yet. On the other hand the paradoxes exert a strong fascination and many a philosopher or logician has spent some thought on them, mostly in relative isolation. The literature on the paradoxes is vast but scattered, repetitive and disconnected. This made it impossible to give a presentation in which all ideas in the literature receive their due.

The provability logics of recursively enumerable theories extending peano arithmetic at arbitrary theories extending peano arithmetic

Solovay’s proof of the arithmetical completeness theorem for provability logic shows more than is stated in the theorem. The idea is roughly this. Let cp be a modal propositional formula. Suppose cp is not derivable in L, i.e., Lab’s Logic (also known as G, for ‘Giidel’). There is a finite, transitive, irreflexive Kripke model K such that K F p. Solovay provides a specific arithmetical interpretation connected with K of the atoms of cp to show that not all arithmetical interpretations of p are derivable in PA, Peano Arithmetic. It turns out that Solovay’s interpretation does more: it connects the interpretation of cp with sentences of the form otAl, reflecting the way cp is connected with sentences of the form q k 1 in the specific model K. We employ this fact to state a lemma that captures the content of the proof better. Consequences:

Peano's smart children: a provability logical study of systems with built-in consistency.

A pnovubW y 2.acgtica,2 ztudy o6 zysteni with built-in cones tevccy Alb ent V,iss etc 1 Introduction Consistency can be built into a system in various ways. The two best known constructions are Rossers and Fefermans. Both constructions take a given formal system in the usual sense as initial data. Consider for example Peano Arithmetic (PA). A proof in the Peano System will count as a proof in the Rosser System based on PA, if there is no shorter Peano proof of the negation of its conclusion. The Feferman System can be described in various interesting ways-modulo provable equivalence in PA of the formulas defining the set of theorems. One such way is: a proof in the Peano System will count as a proof in the Feferman System based on PA, if the finite set of arithmetical Peano axioms smaller than or equal to the largest arithmetical Peano axiom used in the proof is consistent. The reasons such constructions occur in the literature are various: i) They serve as counterexamples in the study of the relation between Godels first and second Incompleteness Theorem (see Feferman [19601). ii) They serve as didactical examples in philosophical discussions, like the debate on intensionality in Mathematics (see e.g. Auer-bach[ 1985]) and the discussion on the possible bearing of the In-completeness Theorems on the Minds & Machines problem (see e.g.

On the completenes principle: A study of provability in heyting's arithmetic and extensions

In this paper extensions of HA are studied that prove their own completeness, i.e. they prove A → □ A, where □ is interpreted as provability in the theory itself. Motivation is three-fold: (1) these theories are thought to have some intrinsic interest, (2) they are a tool for producing and studying provability principles, (3) they can be used to proved independence results. Work done in the paper connected with these motivations is respectively: 1. (i) A characterization is given of theories proving their own completeness, including an appropriate conservation result. 2. (ii) Some new provability principles are produced. The provability logic of HA is not a sublogic of the of PA. A provability logic plus completeness theorem is given for a certain intuitionistic extension of HA. De Jonghs theorem for propositional logic is a corollary. 3. (iii) FP-realizability in Beesons proof that ∦HA KLS is replaced by theories proving their own completeness. New consequences are ∦HA+−MPR KLS, ∦HA+DNS KLS.

Can We Make the Second Incompleteness Theorem Coordinate Free

Is it possible to give a coordinate free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EA-provable equivalence, (ii) consistency for finitely axiomatized sequential theories can be uniquely characterized modulo EA-provable equivalence. The case of infinitely axiomatized ce theories is more delicate. We carefully discuss this in the paper.

Pairs, sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.

Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic

Abstract This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ10-sentences over Heyting arithmetic ( HA ). On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ10-substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic ( IPC ). Here NNIL is the class of propositional formulas with no nestings of implications to the left. The identical embedding of IPC -derivability (considered as a preorder and, thus, as a category) into a consequence relation (considered as a preorder) has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA .

Growing Commas. A Study of Sequentiality and Concatenation

In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding.

Problems in the Logic of Provability

In the first part of the paper we discuss some conceptual problems related to the notion of proof. In the second part we survey five major open problems in Provability Logic as well as possible directions for future research in this area.