### General revealed Preference Theory

, 2010

"... We provide general conditions under which an economic theory has a universal axiomatization: one that leads to testable implications. Roughly speaking, if we obtain a universal axiomatization when we assume that unobservable parameters (such as preferences) are observable, then we can obtain a unive ..."

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We provide general conditions under which an economic theory has a universal axiomatization: one that leads to testable implications. Roughly speaking, if we obtain a universal axiomatization when we assume that unobservable parameters (such as preferences) are observable, then we can obtain a universal axiomatization purely on observables. The result "explains" classical revealed preference theory, as applied to individual rational choice. We obtain new applications to Nash equilibrium theory and Pareto optimal choice.

### The usual model construction for NFU preserves information

, 2009

"... The “usual ” model construction for NFU (Quine’s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). “Most ” elements of the resulting model of NFU are urelements ..."

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The “usual ” model construction for NFU (Quine’s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). “Most ” elements of the resulting model of NFU are urelements (it appears that information about their extensions is discarded). The surprising result of this paper is that this information is not discarded at all: the membership relation of the original model (restricted to the domain of the model of NFU) is definable in the language of NFU. A corollary of this is that the urelements of a model of NFU obtained by the “usual ” construction are inhomogeneous: this was the question the author was investigating initially. Other aspects of the mutual interpretability of NFU and a fragment of ZFC are discussed in sufficient detail to place

### IFSA-EUSFLAT 2009 Non-commutative EQ-logics and their extensions

"... Abstract — We discuss a formal many-valued logic called EQlogic which is based on a recently introduced special class of algebras called EQ-algebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated ..."

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Abstract — We discuss a formal many-valued logic called EQlogic which is based on a recently introduced special class of algebras called EQ-algebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated lattices. The goal of EQ-logics is to present a possible direction in the development of mathematical logics in which axioms are formed as identities. In this paper we propose a basic EQ-logic and three extensions which end up with a logic equivalent to the MTL-logic. Keywords — EQ-algebra, fuzzy equality, residuated lattice, MTLlogic, fuzzy logic. 1

### GÖDELIAN PLATONISM

, 2009

"... The members of the Comittee appointed to examine the thesis of TEPPEI HAYASHI ..."

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The members of the Comittee appointed to examine the thesis of TEPPEI HAYASHI

### Journal of the IGPL

"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositions-as-types and proofs-as-constructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."

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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositions-as-types and proofs-as-constructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logical-operations-as-adjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9g-fragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and single-valued relations. However, when enriched with proofs-as-arrows, this familiar concept must be supplied with an additional conversion rule, conne...

### Minimalism and Paradoxes ∗

"... Abstract. This paper argues against minimalism about truth. It does so by way of a comparison of the theory of truth with the theory of sets, and consideration of where paradoxes may arise in each. The paper proceeds by asking two seemingly unrelated questions. First, what is the theory of truth abo ..."

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Abstract. This paper argues against minimalism about truth. It does so by way of a comparison of the theory of truth with the theory of sets, and consideration of where paradoxes may arise in each. The paper proceeds by asking two seemingly unrelated questions. First, what is the theory of truth about? Answering this question shows that minimalism bears important similarities to naive set theory. Second, why is there no strengthened version of Russell’s paradox, as there is a strengthened Liar paradox? Answering this question shows that like naive set theory, minimalism is unable to make adequate progress in resolving the paradoxes, and must be replaced by a drastically different sort of theory. Such a theory, it is shown, must be fundamentally non-minimalist. Why is there no strengthened version of Russell’s paradox, as there is a Strengthened Liar paradox? This question is rarely asked. It does have a fairly standard answer, which I shall not challenge. But I shall argue that asking the question helps to point out something important about the theory of truth. In particular, it raises a serious challenge to

### EQ-algebra-based Fuzzy Type Theory and Its Extensions

"... In this paper, we introduce a new algebra called ‘EQ-algebra’, which is an alternative algebra of truth values for formal fuzzy logics. It is specified by replacing implication as the main operation with a fuzzy equality. Namely, EQ-algebra is a semilattice endowed with a binary operation of fuzzy e ..."

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In this paper, we introduce a new algebra called ‘EQ-algebra’, which is an alternative algebra of truth values for formal fuzzy logics. It is specified by replacing implication as the main operation with a fuzzy equality. Namely, EQ-algebra is a semilattice endowed with a binary operation of fuzzy equality and a binary operation of multiplication. Implication is derived from the fuzzy equality and it is not a residuation with respect to multiplication. Consequently, EQ-algebras overlap with residuated lattices but are not identical with them. We choose one class of suitable EQ-algebras (good EQ-algebras) and develop a formal theory of higher-order fuzzy logic called ‘basic fuzzy type theory ’ (FTT). We develop in detail its syntax and semantics, and we prove some basic properties, including the completeness theorem with respect to generalized models. The paper also provides an overview of the present state of the art of FTT.

### EQ-logics: non-commutative fuzzy logics based on fuzzy equalityI

"... In this paper, we develop a specific formal logic in which the basic connective is fuzzy equality and the implication is derived from the latter. Moreover, the fusion connective (strong conjunction) is non-commutative. We call this logic EQ-logic. First, we formulate the basic EQ-logic which is rich ..."

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In this paper, we develop a specific formal logic in which the basic connective is fuzzy equality and the implication is derived from the latter. Moreover, the fusion connective (strong conjunction) is non-commutative. We call this logic EQ-logic. First, we formulate the basic EQ-logic which is rich enough to enjoy the completeness property. Furthermore, we introduce two extensions which seem to us interesting. The first one is IEQ-logic which is EQ-logic with double negation. The second one adds prelinearity that enables us to prove a stronger variant of the completeness property. Finally, we extend the latter logic by three more axioms including the residuation one (importation–exportation law) and prove that the resulting logic is equivalent with MTL-logic. Formal proofs in this paper proceed mostly in an equational style.