124 research outputs found
Grilliot's trick in Nonstandard Analysis
The technique known as Grilliot's trick constitutes a template for explicitly
defining the Turing jump functional in terms of a given
effectively discontinuous type two functional. In this paper, we discuss the
standard extensionality trick: a technique similar to Grilliot's trick in
Nonstandard Analysis. This nonstandard trick proceeds by deriving from the
existence of certain nonstandard discontinuous functionals, the Transfer
principle from Nonstandard analysis limited to -formulas; from this
(generally ineffective) implication, we obtain an effective implication
expressing the Turing jump functional in terms of a discontinuous functional
(and no longer involving Nonstandard Analysis). The advantage of our
nonstandard approach is that one obtains effective content without paying
attention to effective content. We also discuss a new class of functionals
which all seem to fall outside the established categories. These functionals
directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page
Constructive set theory and Brouwerian principles
The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF
A deductive-reductive form of logic: Intuitionistic S4 modalities
The paper is a continuation of A deductive-reductive form of logic: general theory and intuitionistic case ([1]) and considers the problem of definability of modal operators on the intuitionistic base. Contrary to the classical case, it seems that the fact whether the connective is Heyting’s or Brouwerian is essential for the intuitionistic logic. The connective of possibility has the classical interpretation, if it is defined on the base of the logic with Brouwerian connective of coimplication
The computational content of Nonstandard Analysis
Kohlenbach's proof mining program deals with the extraction of effective
information from typically ineffective proofs. Proof mining has its roots in
Kreisel's pioneering work on the so-called unwinding of proofs. The proof
mining of classical mathematics is rather restricted in scope due to the
existence of sentences without computational content which are provable from
the law of excluded middle and which involve only two quantifier alternations.
By contrast, we show that the proof mining of classical Nonstandard Analysis
has a very large scope. In particular, we will observe that this scope includes
any theorem of pure Nonstandard Analysis, where `pure' means that only
nonstandard definitions (and not the epsilon-delta kind) are used. In this
note, we survey results in analysis, computability theory, and Reverse
Mathematics.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Singly generated quasivarieties and residuated structures
A quasivariety K of algebras has the joint embedding property (JEP) iff it is
generated by a single algebra A. It is structurally complete iff the free
countably generated algebra in K can serve as A. A consequence of this demand,
called "passive structural completeness" (PSC), is that the nontrivial members
of K all satisfy the same existential positive sentences. We prove that if K is
PSC then it still has the JEP, and if it has the JEP and its nontrivial members
lack trivial subalgebras, then its relatively simple members all belong to the
universal class generated by one of them. Under these conditions, if K is
relatively semisimple then it is generated by one K-simple algebra. It is a
minimal quasivariety if, moreover, it is PSC but fails to unify some finite set
of equations. We also prove that a quasivariety of finite type, with a finite
nontrivial member, is PSC iff its nontrivial members have a common retract. The
theory is then applied to the variety of De Morgan monoids, where we isolate
the sub(quasi)varieties that are PSC and those that have the JEP, while
throwing fresh light on those that are structurally complete. The results
illuminate the extension lattices of intuitionistic and relevance logics
Reverse Mathematics and parameter-free Transfer
Recently, conservative extensions of Peano and Heyting arithmetic in the
spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been
proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis
restricted to formulas without parameters. Based on this axiom, we formulate a
base theory for the Reverse Mathematics of Nonstandard Analysis and prove some
natural reversals, and show that most of these equivalences do not hold in the
absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi
Equivalence relations and operators on ordered algebraic structures with difference.
This work concerns algebraic models of fuzzy and many-valued propositional logics, in particular Boolean Algebras, Heyting algebras, GBL-algebras and their dual structures, and partial algebras.
The central idea is the representation of complex structures through simpler structures and equivalence relations on them: in order to achieve this, a structure is often considered under two points of view, as total algebra and partial algebra. The equivalence relations which allow the representation are congruences of partial algebras.
The first chapter introduces D-posets, the partial algebraic structures used for this representation, which generalize Boolean algebras and MV-algebras.
The second chapter is a study of congruences on D-posets and the structure of the quotients, in particular for congruences induced by some kinds of idempotent operators, here called S-operators. The case of Boolean algebras and MV-algebras is studied more in detail.
The third chapter introduces GBL-algebras and their dual, and shows how the interplay of an S-operator with a closure operator gives rise to a dual GBL-algebra. Other results about the representation of finite GBL-algebras and GBL*algebras (GBL-algebras with monoidal sum), part of two papers previously published, are summarized and put in relation with the other results of this work
Admissibility of Π<sub>2</sub>-inference rules: Interpolation, model completion, and contact algebras
We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus S2IC, where we also supply a finite axiomatization of the model completion of its algebraic counterpart, via the equivalent theory of contact algebras. Using this result we obtain a finite basis for admissible Π2-rules
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