Pairs, sets and sequences in first-order theories

Asuransi sebagai aktivitas bisnis diharuskan memenuhi prinsip-prinsip hukum asuransi. Salah satu prinsip yang harus dipegang teguh adalah principle of  utmost good faith, di samping prinsip yang lain. Prinsip ini berbunyi bahwa seorang tertanggung wajib memberi informasi secara jujur terhadap apa yang dipertanggungkan kepada penanggung. Dalam bisnis Islam, kejujuran merupakan prinsip yang harus dijunjung tinggi. Secara hukum, prinsip ini diatur dalam KUH Dagang. Persoalannya adalah apakah prinsip ini dianggap cukup dari sudut pandang hukum perjanjian syariah. Secara sekilas bahwa prinsip iktikad baik sempurna ini telah memenuhi asas perjanjian syariah, namun demikian tidak memiliki kriteria maksimal kejujuran. Ketiadaan kejujuran dalam bisnis asuransi akan berdampak pada batalnya perjanjian asuransi karena ada unsur cacat kehendak (‘uyub ar-ridla). Insurance as a business activity must fulfill principles of insurance law. One of the principles that must be hold on is the principle of  utmost good faith. The principle says that an endured person must honestly give information of  what should be given responsibility to the guarantor. In Islamic business, honesty is a principle that should be respected. From point of  view of  law, the principle is settled in commerce law. The problem is that whether the principle is represenative enough if it is viewed from law of  syariah agreement. At glance, the principle has fulfilled the basic of syariah agreement, however, it does not have maximum criteria of  honesty. Unavailability of honesty in insurance business will give effect of  invalidate of  insurance agreement, for there is a deformity of desire (‘uyub ar-ridla).</p

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.Comment: 23 pages, extension of conference paper published at FSCD 202

The Strength of Truth-Theories

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed in discussions of axiomatic theories of truth. These results are then used to address a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism. This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now

Mathematical Logic: Proof Theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of core mathematics and theoretical computer science as well as homotopy type theory and logical aspects of computational complexity

Some Notes on Truths and Comprehension

In this paper we study several translations that map models and formulae of the language of second-order arithmetic to models and formulae of the language of truth. These translations are useful because they allow us to exploit results from the extensive literature on arithmetic to study the notion of truth. Our purpose is to present these connections in a systematic way, generalize some well-known results in this area, and to provide a number of new results. Sections 3 and 4 contain some recursion- and proof-theoretic results about Kripke-style fixed-point theories of truth. Section 5 shows how to derive full second-order arithmetic from principles of truth. Section 6 investigates the proof-theoretic strength of disquotation without an arithmetical base theory