The continuum and first-order intuitionistic logic

1992 ◽  
Vol 57 (4) ◽  
pp. 1417-1424 ◽  
Author(s):  
D. van Dalen
Keyword(s):  
Prima Facie ◽  
Order Logic ◽  
Point Of View ◽  
Ordered Sets ◽  
Practice Model ◽  
Original Proof ◽  
First Order ◽  
The One ◽  
Intuitionistic Mathematics ◽  
The Continuum

Ever since Cantor, we have known that the reals and the rationals are not isomorphic (as equality structures, i.e., sets). Logically speaking, however, they are not all that different; in first-order classical logic they are elementarily equivalent, since the theory of infinite sets is complete. The same holds for ℝ and ℚ as ordered sets; again the theory of dense linear order without end points is complete.From an intuitionistic point of view these matters are more complicated; e.g., the theory of equality of ℚ is decidable, whereas the one of ℝ patently is not. This, in a roundabout way, shows that ℚ and ℝ are not isomorphic; of course, there is no need for such a detour, as Cantor's original proof [2] is intuitionistically correct, and Brouwer's new proof [1] is another alternative intuitionistic argument.In view of the fact that ℚ and ℝ behave so strikingly differently with respect to first-order logic, one is easily tempted to look for elementary equivalences among the subsets of ℝ. Until quite recently most model theoretic investigations of intuitionistic theories made use of special (artificial) notions of “model”, e.g., Kripke models, sheaf models,…; but there is no prima facie reason why one should not practice model theory much the same way as traditional model theorists do. That is to say on the basis of a naive set theory, or, in our case, of naive intuitionistic mathematics.This paper uses the method of (k, p)-isomorphisms of Fraïssé, and it is briefly shown that one half of the Fraïssé theorem holds intuitionistically.

Gödel and Set Theory

2007 ◽  
Vol 13 (2) ◽  
pp. 153-188 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
Point Of View ◽  
First Order ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
Universe Of Sets ◽  
The Universe ◽  
The Continuum

Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

scholarly journals The Road to Modern Logic—An Interpretation

2001 ◽  
Vol 7 (4) ◽  
pp. 441-484 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Theoretical Point ◽  
Primary Role ◽  
Philosophical Discussion ◽  
Modern Logic ◽  
Core System ◽  
First Order ◽  
The Road

AbstractThis paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

Melissus And His Opponents: The Argument of DK 30 B 8

Phronesis ◽  
2005 ◽  
Vol 50 (4) ◽  
pp. 263-288 ◽  
Author(s):  
Stephen Makin
Keyword(s):  
Prima Facie ◽  
Point Of View ◽  
The Other ◽  
Good Argument ◽  
Orthodox Interpretation ◽  
Realist Position ◽  
The World ◽  
New Interpretation ◽  
The One

AbstractIn this paper I offer a new interpretation of Melissus' argument at DK 30 B8.In this passage Melissus uses an Eleatic argument against change to challenge an opponent who appeals to the authority of perception in order to support the view that there are a plurality of items in the world. I identify an orthodox type of approach to this passage, but argue that it cannot give a charitable interpretation of Melissus' strategy. In order to assess Melissus' overall argument we have to identify the opponent at whom it is aimed. The orthodox interpretation of the argument faces a dilemma: Melissus' argument is either a poor argument against a plausible opponent or a good argument against an implausible opponent.My interpretation turns on identifying a new target for Melissus' argument. I explain the position I call Bluff Realism (contrasting it with two other views: the Pig Headed and the Fully Engaged). These are positions concerning the dialectical relation between perception on the one hand, and arguments to counter-perceptual conclusions on the other. I argue that Bluff Realism represents a serious threat from an Eleatic point of view, and is prima facie an attractive position in its own right.I then give a charitable interpretation of Melissus' argument in DK 30 B8, showing how he produces a strong and incisive argument against the Bluff Realist position I have identified. Melissus emerges as an innovative and astute philosopher.

Logical Methods for Self-Configuration of Network Devices

2011 ◽  
pp. 189-216 ◽  
Author(s):  
Sylvain Hallé ◽  
Roger Villemaire ◽  
Omar Cherkaoui
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Declarative Languages ◽  
First Order ◽  
User Intervention

The goal of self-configuration consists of providing appropriate values for parameters that modulate the behaviour of a device. In this chapter, self-configuration is studied from a mathematical logic point of view. In contrast with imperative means of generating configurations, characterized by scripts and templates, the use of declarative languages such as propositional or first-order logic is argued. In that setting, device configurations become models of particular logical formulæ, which can be generated using constraint solvers without any rigid scripting or user intervention.

Gödel, Kurt (1906–78)

2018 ◽  
Author(s):  
John W. Dawson
Keyword(s):  
Twentieth Century ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Mathematical Platonism ◽  
The Axiom Of Choice ◽  
Fundamental Theorems ◽  
The Continuum

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.

scholarly journals Conservative Intensional Extension of Tarski's Semantics

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Zoran Majkić
Keyword(s):  
Free Algebra ◽  
Possible Worlds ◽  
Order Logic ◽  
Point Of View ◽  
Relational Algebra ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
Possible World ◽  
Logic Formula ◽  
First Order

We considered an extension of the first-order logic (FOL) by Bealer's intensional abstraction operator. Contemporary use of the term “intension” derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the “extension” of an idea consists of the subjects to which the idea applies, and the “intension” consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of standard FOL, we obtain a commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of the FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from Cylindric algebras). Then we show that this composition corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

Cognitive Modelling Applied to Aspects of Schizophrenia and Autonomic Computing

2011 ◽  
pp. 220-234
Author(s):  
Lee Flax
Keyword(s):  
Autonomic Computing ◽  
Signs And Symptoms ◽  
Order Logic ◽  
First Order Logic ◽  
Object Language ◽  
Cognitive Modelling ◽  
Belief Contraction ◽  
Algebraic Version ◽  
First Order ◽  
The One

We give an approach to cognitive modelling, which allows for richer expression than the one based simply on the firing of sets of neurons. The object language of the approach is first-order logic augmented by operations of an algebra, PSEN. Some operations useful for this kind of modelling are postulated: combination, comparison, and inhibition of sets of sentences. Inhibition is realised using an algebraic version of AGM belief contraction (Gärdenfors, 1988). It is shown how these operations can be realised using PSEN. Algebraic modelling using PSEN is used to give an account of an explanation of some signs and symptoms of schizophrenia due to Frith (1992) as well as a proposal for the cognitive basis of autonomic computing. A brief discussion of the computability of the operations of PSEN is also given.

Formal topologies on the set of first-order formulae

2000 ◽  
Vol 65 (3) ◽  
pp. 1183-1192 ◽  
Author(s):  
Thierry Coquand ◽  
Sara Sadocco ◽  
Giovanni Sambin ◽  
Jan M. Smith
Keyword(s):  
Order Logic ◽  
Macneille Completion ◽  
Real Numbers ◽  
Algebraic Form ◽  
Completeness Proof ◽  
First Order ◽  
Existence Property ◽  
The One ◽  
Dedekind Cuts ◽  
Formal Topology

The completeness proof for first-order logic by Rasiowa and Sikorski [13] is a simplification of Henkin's proof [7] in that it avoids the addition of infinitely many new individual constants. Instead they show that each consistent set of formulae can be extended to a maximally consistent set, satisfying the following existence property: if it contains (∃x)ϕ it also contains some substitution ϕ(y/x) of a variable y for x. In Feferman's review [5] of [13], an improvement, due to Tarski, is given by which the proof gets a simple algebraic form.Sambin [16] used the same method in the setting of formal topology [15], thereby obtaining a constructive completeness proof. This proof is elementary and can be seen as a constructive and predicative version of the one in Feferman's review. It is a typical, and simple, example where the use of formal topology gives constructive sense to the existence of a generic object, satisfying some forcing conditions; in this case an ultrafilter satisfying the existence property.In order to get a formal topology on the set of first-order formulae, Sambin used the Dedekind-MacNeille completion to define a covering relation ⊲DM. This method, by which an arbitrary poset can be extended to a complete poset, was introduced by MacNeille [9] and is a generalization of the construction of real numbers from rationals by Dedekind cuts. It is also possible to define an inductive cover, ⊲I, on the set of formulae, which can also be used to give canonical models, see Coquand and Smith [3].

scholarly journals Sharpened lower bounds for cut elimination

2012 ◽  
Vol 77 (2) ◽  
pp. 656-668
Author(s):  
Samuel R. Buss
Keyword(s):  
Lower Bounds ◽  
Order Logic ◽  
Maximum Depth ◽  
First Order Logic ◽  
Cut Elimination ◽  
Original Proof ◽  
First Order

AbstractWe present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depthdof the formulas in the original proof. Our results remove the constant of proportionality, giving an exponential stack of height equal tod−O(1). The proof method is based on more efficiently expressing the Gentzen–Solovay cut formulas as low depth formulas.

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