Constructive Metatheory Research Articles

Formalizing Constructive Quantifier Elimination in Agda

In this paper a constructive formalization of quantifier elimination is presented, based on a classical formalization by Tobias Nipkow. The formalization is implemented and verified in the programming language/proof assistant Agda. It is shown that, as in the classical case, the ability to eliminate a single existential quantifier may be generalized to full quantifier elimination and consequently a decision procedure. The latter is shown to have strong properties under a constructive metatheory, such as the generation of witnesses and counterexamples. Finally, this is demonstrated on a minimal theory on the natural numbers.

The Univalence Axiom in Cubical Sets

In this note we show that Voevodsky’s univalence axiom holds in the model of type theory based on cubical sets as described in Bezem et al. (in: Matthes and Schubert (eds.) 19th international conference on types for proofs and programs (TYPES 2013), Leibniz international proceedings in informatics (LIPIcs), Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, vol 26, pp 107–128, 2014. https://doi.org/10.4230/LIPIcs.TYPES.2013.107. http://drops.dagstuhl.de/opus/volltexte/2014/4628) and Huber (A model of type theory in cubical sets. Licentiate thesis, University of Gothenburg, 2015). We will also discuss Swan’s construction of the identity type in this variation of cubical sets. This proves that we have a model of type theory supporting dependent products, dependent sums, univalent universes, and identity types with the usual judgmental equality, and this model is formulated in a constructive metatheory.

A Sheaf Model of the Algebraic Closure

In constructive algebra one cannot in general decide the irreducibility of a polynomial over a field K. This poses some problems to showing the existence of the algebraic closure of K. We give a possible constructive interpretation of the existence of the algebraic closure of a field in characteristic 0 by building, in a constructive metatheory, a suitable site model where there is such an algebraic closure. One can then extract computational content from this model. We give examples of computation based on this model.

Automating the synthesis of decision procedures in a constructive metatheory

We present an approach to the automatic construction of decision procedures, via a detailed example in propositional logic. The approach adapts the methods of proofdplanning and the heuristics for induction to a new domain, that of metatheoretic procedures. This approach starts by providing an alternative characterisation of validitys the proofs of the correctness and completeness of this characterisation, and the existence of a decision procedure, are then amenable to automation in the way we describe. In this paper we identify a set of principled extensions to the heuristics for induction needed to tackle the proof obligations arising in the new problem domain and discuss their integration within the clamdOyster system.

A model for intuitionistic non-standard arithmetic

This paper provides an explicit description of a model for intuitionistic non-standard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice.

Origins of george kelly's constructivism in the work of korzybski and moreno

Abstract Although Mahoney (1988) has traced the heritage of general constructive metatheory and 7 Lelhart and Jackson (1983) have examined the influences of Kelly's Kansas environment on his developing theory, there has been relatively little investigation of the origins of Kelly's constructivism. Although Kelly (1955) was undoubtedly influenced by many philosophers and psychologists as he developed the psychology of personal constructs, the roles of these people have not been extensively investigated. However, Kelly (1955, 1969) cited, in a general way, the works of Korzybski and Moreno several times in describing the origins of his theory. Lecture notes taken by one of Kelly's students (Barry, 1948) reveal more specifically the sources (Korzybski, 1933, 1943; Moreno, 1937) that seemed influential as Kelly developed his theories. Kelly borrowed ideas of Korzybski and Moreno, among many others, in creating parts of his role therapy and personality theory. In adapting Korzybski's notion that semantic and l...

Constructive Metatheory: II. Implications for Psychotherapy

Abstract The implications of constructive metatheory for the conceptualization and practice of psychotherapy are briefly outlined. It is argued that the constructivist therapist construes the client as an active and developing process, and that psychological problems are approached not as piecemeal flaws or deficiencies, but as expressions of current discrepancies between an individual's adaptive capacities and the challenges she or he faces. Likewise, the process of psychotherapy is portrayed as one of trial-and-error experimentation with different (novel) ways of “being in the world.” Ideally, the client and therapist create an intimate and emotionally charged alliance in and from which the client can explore and experiment with self and world relationships. These practical features dovetail with many assertions of the major me-tatheories of psychotherapy, and it is argued that constructive metatheory may be uniauely suited to facilitating attempts at conceptual integration

Constructive Metatheory: 1. Basic Features and Historical Foundations

Abstract Constructive metatheory is playing an increasingly apparent role in the evolution of theories of human personality and psychotherapy. Three basic features of constructivism are outlined: (1) proactive cognition, (2) morphogenic nuclear structure, and (3) self-organizing development. Parallels to and relationships with evolutionary epistemology and autopoiesis are briefly noted. The emergence of constructive metatheory is traced from Vico, Kant, and Vaihinger to its diverse contemporary expressions.

Epistemic set theory is a conservative extension of intuitionistic set theory

In [6] Gödel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Ščedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.There is one question quite central to the above program that has remained open. Namely:Does Ščedrov's extension of the Gödel translation to set theory provide a faithful interpretation of intuitionistic set theory into epistemic set theory?In the present paper we give an affirmative answer to this question.The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Gödel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.