Robert A. Herrmann
Mixed logic-systems contain both standard and nonstandard coordinate values. In this paper, it is shown that each of the specifically defined mixed logic-systems generates an internal nonstandard consequence operator.
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Halo Semantics for Modal Logic
利用非标准分析中的晕概念定义模态算子族，发现四个规范实例，其中第四个实例对应ω-聚点算子，在无分离公理下将任意集映射为闭集，并证明K4和GL的完全性。
In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out -mathematically speaking- for its challenge of Hilbert's formalist philosophy of mathematics and rejection of the law of excluded middle from the 'classical' logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer-Heyting-Kolmogorov interpretation by which 'there exists x' intuitively means 'an algorithm to compute x is given'. A number of schools of constructive mathe…
Weak, strong and mixed extensions of relations to spaces of ultrafilters
本文用非标准方法研究了关系扩展到超滤子空间的特性，提出了一种新的分析框架。
In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.
A mixed logic with binary operators
本文提出了一种包含可能性和充分性二元运算的逻辑系统，并证明了与弱混合代数相关的嵌入定理，解决了之前研究中的开放问题。
Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the Löwenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the situations in the models obtained by mixed support iteration of a supercompact length and then collapsing another supercompact cardinal to make it $(2^{\aleph_0})^+$. We show, among other things, that the reflection down to $< 2^{\aleph_0}$ of the non-metrizability of topological spaces with small character is independent from the reflection properties studied in [Fu…
In 1990 J-L. Krivine introduced the notion of storage operators. They are $λ$-terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF2 type system for storage operators using Gödel translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF2 to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage op…
In this paper, we present a propositional sequent calculus containing disjoint copies of classical and intuitionistic logics. We prove a cut-elimination theorem and we establish a relation between this system and linear logic.
Nonstandard polynomials: algebraic properties and elementary equivalence
本文解决了无限域或整数环上多项式环与洛朗多项式环的一阶分类问题，通过正则双向解释方法证明其一阶逻辑等价于弱二阶逻辑，从而描述了所有初等等价环的代数结构及其非标准模型性质。
The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical …
Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (includin…
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is "certainty" and "architecture" yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lession, namely that t…
Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Susz…
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