End extensions of models of weak arithmetics

2016 ◽  
Author(s):  
Βασίλειος Πασχάλης
Keyword(s):  
Semantic Tableaux

Η διδακτορική διατριβή ασχολείται με τη μελέτη προβλημάτων που αφορούν τελικές επεκτάσεις μοντέλων υποσυστημάτων της πρωτοβάθμιας αριθμητικής Peano. Πιο συγκεκριμένα, το πρόβλημα του J. Paris: «Υπάρχει, για κάθε αριθμήσιμο μοντέλο της Σ_1 συλλογής γνήσια τελική επέκτασή του που ικανοποιεί την ∆_0 επαγωγή;» παραμένει ανοικτό.Το πρόβλημα μελέτησαν οι J. Paris και A. Wilkie (1989), οι οποίοι απέδειξαν ότι ικανή συνθήκη για θετική απάντηση είναι το μοντέλο να είναι I∆_0 -πλήρες (όπου με I∆_0 συμβολίζεται η θεωρία της ∆_0 -επαγωγής). Αποδεικνύουμε ότι η χρήση της έννοιας της I∆_0 -πληρότητας μπορεί να παρακαμφθεί και στη θέση της να χρησιμοποιηθεί η τυποποίηση του κλασικού επιχειρήματος του θεωρήματος πληρότητας (θεώρημα Hilbert-Bernays), με χρήση σημασιολογικών πινάκων (semantic tableaux).Επιπλέον, με την ίδια μεθοδολογία κατάλληλα τροποποιημένη αποδεικνύουμε τη γενίκευση του αποτελέσματος, δηλαδή ότι για κάθε αριθμήσιμο μοντέλο της Σ_n -συλλογής, n > 1, υπάρχει γνήσια Σ_n -στοιχειώδης τελική επέκτασή του που ικανοποιεί την ∆_0 -επαγωγή.

An algebraic study of Diodorean modal systems

1965 ◽  
Vol 30 (1) ◽  
pp. 58-64 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Discrete Time ◽  
Decision Procedure ◽  
John Locke ◽  
Original Model ◽  
Completeness Proof ◽  
Semantic Tableaux ◽  
Algebraic Treatment ◽  
Algebraic Result ◽  
Modal Systems

Attention was directed to modal systems in which ‘necessarily α’ is interpreted as ‘α. is and always will be the case’ by Prior in his John Locke Lectures of 1956. The present paper shows that S4.3, the extension of S4 withALCLpLqLCLqLp,is complete with respect to this interpretation when time is taken to be continuous, and that D, the extension of S4.3 withALNLpLCLCLCpLpLpLp,is complete with respect to this interpretation when time is taken to be discrete. The method employed depends upon the application of an algebraic result of Garrett Birkhoff's to the models for these systems, in the sense of Tarski.A considerable amount of work on S4.3 and D precedes this paper. The original model with discrete time is given in Prior's [7] (p. 23, but note the correction in [8]); that taking time to be continuous yields a weaker system is pointed out by him in [9]. S4.3 and D are studied in [3] of Dummett and Lemmon, where it is shown that D includes S4.3 andCLCLCpLpLpCMLpLp.While in Oxford in 1963, Kripke proved that these were in fact sufficient for D, using semantic tableaux. A decision procedure for S4.3, using Birkhoff's result, is given in my [2]. Dummett conjectured, in a conversation, that taking time to be continuous yielded S4.3. Thus the originality of this paper lies in giving a suitable completeness proof for S4.3, and in the unified algebraic treatment of the systems. It should be emphasised that the credit for first axiomatising D belongs to Kripke.

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

Sets of theorems with short proofs

1974 ◽  
Vol 39 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Daniel Richardson
Keyword(s):  
Proof Complexity ◽  
Predicate Calculus ◽  
Definable Set ◽  
Semantic Tableaux ◽  
Skolem Functions ◽  
Definition Of

R. Parikh has shown that in the predicate calculus without function symbols it is possible to decide whether or not a given formula A is provable in a Hilbert-style proof of k lines. He has also shown that for any formula A(x1, …, xn) of arithmetic in which addition and multiplication are represented by three-place predicates, {(a1, …, an): A(a1, …, an) is provable from the axioms of arithmetic in k lines} is definable in (N,′, +,0). See [2].In §1 of this paper it is shown that if S is a definable set of n-tuples in (N,′, +, 0), then there is a formula A(x1, …, xn) and a number k so that (a1 …, an) ∈ S if and only if A(a1 …, an) can be proved in a proof of complexity k from the axioms of arithmetic. The result does not depend on which formalization of arithmetic is used or which (reasonable) measure of proof complexity. In particular, this gives a converse to Parikh's result.In §II, a measure of proof complexity is given which is based on counting the quantifier steps in semantic tableaux. The idea behind this measure is that A is k provable if in the attempt to construct a counterexample one meets insurmountable difficulties in the definition of the appropriate Skolem functions over sets of cardinality ≤ k. A method is given for deciding whether or not a sentence A in the full predicate calculus is k provable. The question for the full predicate calculus and Hilbert-style systems of proof remains open.

scholarly journals INSTANTIAL NEIGHBOURHOOD LOGIC

2016 ◽  
Vol 10 (1) ◽  
pp. 116-144 ◽  
Author(s):  
JOHAN VAN BENTHEM ◽  
NICK BEZHANISHVILI ◽  
SEBASTIAN ENQVIST ◽  
JUNHUA YU
Keyword(s):  
Fine Structure ◽  
Normal Form ◽  
Axiom System ◽  
Topological Spaces ◽  
Relational Semantics ◽  
Basic Model ◽  
Semantic Tableaux ◽  
Dynamic Logics ◽  
A Current

AbstractThis paper explores a new language of neighbourhood structures where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. The resulting system of ‘instantial neighbourhood logic’ INL has a nontrivial mix of features from relational semantics and from neighbourhood semantics. We explore some basic model-theoretic behavior, including a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we relate INL to other modal logics by means of translations, and determine its precise SAT complexity. Finally, we discuss proof-theoretic fine-structure of INL in terms of semantic tableaux and some expressive fine-structure in terms of fragments, while discussing concrete illustrations of the instantial neighborhood language in topological spaces, in games with powers for players construed in a new way, as well as in dynamic logics of acquiring or deleting evidence. We conclude with some coalgebraic perspectives on what is achieved in this paper. Many of these final themes suggest follow-up work of independent interest.

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