arXiv subjects

Seppo Heikkilä

Publications and source records attributed to Seppo Heikkilä.

7 recordsLinked to original sources

arXiv2020

Theories of truth for countable languages which conform to classical logic

Every countable language which conforms to classical logic is shown to have an extension which conforms to classical logic, and has a definitional theory of truth. That extension has a semantical theory of truth, if every sentence of the object language is valuated by its meaning either as true or as false. These theories contain both a truth predicate and a non-truth predicate. Theories are equivalent when the sentences of the object language are valuated by their meanings.

math.LO↗

arXiv2017

A mathematically derived definitional/semantical theory of truth

Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation. This interpretation is equivalent to the interpretation by meanings of sentences if the object language is so interpreted. The added formula provides a truth predicate for the constructed language. The so obtained theory of truth satisfies the norms presented in Hannes Leitgeb's paper 'What Theories of Truth Should be Like (but Cannot be)'.

math.LO↗

arXiv2015

On the construction of fully interpreted formal languages which posses their truth predicates

We shall construct by ordinary recursion method subsets to the set $D$ of Gödel numbers of the sentences of a language $\mathcal L$. That language is formed by sentences of a fully interpreted formal language $L$, called an MA language, and sentences containing a monadic predicate letter $T$. From the class of the constructed subsets of $D$ we extract one set $U$ by transfinite recursion method. Interpret those sentences whose Gödel numbers are in $U$ as true, and their negations as false. These sentences together form an MA language. It is a sublanguage of $\mathcal L$ having $L$ as its sublanguage, and $T$ is its truth predicate.

math.LO↗

arXiv2015

On summability, multipliability, product integrability and parallel translation

In this paper we provide necessary and sufficient conditions for the existence of the Kurzweil, McShane and Riemann product integrals of step mappings with well-ordered steps, and for right regulated mappings with values in Banach algebras. Our basic tools are the concepts of summability and multipliability of families in normed algebras indexed by well-ordered subsets of the real line. These concepts also lead to the generalization of some results from the usual theory of infinite series and products. Finally, we present two applications of product integrals: First, we describe the relation between Stieltjes-type product integrals, Haahti products, and parallel translation operators. Second, we provide a link between the theory of strong Kurzweil product integrals and strong solutions of linear generalized differential equations.

math.FA↗

arXiv2015

A mathematical theory of truth and an application to the regress problem

In this paper a class of languages which are formal enough for mathematical reasoning is introduced. First-order formal languages containing natural numbers and numerals belong to that class. Its languages are called mathematically agreeable (shortly MA). Languages containing a given MA language L, and being sublanguages of L augmended by a monadic predicate are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of these languages. MTT makes them MA languages which posses their own truth predicates. MTT is shown to conform well with the eight norms presented for theories of truth in 'What Theories of Truth Should be Like (but Cannot be)', by Hannes Leitgeb. MTT is free from infinite regress, providing a proper framework to study the regress problem. Main tools used in proofs are Zermelo-Fraenkel (ZF) set theory and classical logic.

math.LO↗

arXiv2013

On summability, integrability and impulsive differential equations in Banach spaces

We shall first study summability of families in normed spaces indexed with well ordered sets of real numbers extended by infinity. Obtained results and a generalized iteration method are applied to derive necessary and sufficient conditions for HK, HL, Bochner and Riemann integrability of step mappings and for right regulated mappings from an interval of reals extended by infinity to a Banach space. Finally, solvability of impulsive differential equations are studied.

math.CA↗

arXiv2013

Distributions, their primitives and integrals with applications to differential equations

In this paper we will study integrability of distributions whose primitives are left regulated functions and locally or globally integrable in the Henstock--Kurzweil, Lebesgue or Riemann sense. Corresponding spaces of distributions and their primitives are defined and their properties are studied. Basic properties of primitive integrals are derived and applications to systems of first order nonlinear distributional differential equations and to an $m$th order distributional differential equation are presented. The domain of solutions can be unbounded, as shown by concrete examples.

math.CA↗