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**1 - 7**of**7**### Observational Truth as a Categorical Modality

"... Abstract We examine the notion of truth-up-to-observability in the setting of categorical logic and show that it can be captured by a modal operator. We view so-called `hidden algebras ' as set-valued functors on a category of observations. Our main results extend the Kripke-Beth-Joyal semantic ..."

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Abstract We examine the notion of truth-up-to-observability in the setting of categorical logic and show that it can be captured by a modal operator. We view so-called `hidden algebras ' as set-valued functors on a category of observations. Our main results extend the Kripke-Beth-Joyal semantics for the internal logics of functor toposes to the observational modality. We also give a categorical account of coinduction as a proof technique for establishing observational truth, and give a covariety theorem that characterises a class of algebras as equationally definable up to observability iff the class forms a slice topos over a particular form of subobject.

### Specifying, Programming and Verifying with Equational Logic

"... 1 Introduction Programming is difficult, as shown by the fact that debugging a program usually takes more time than creating it; moreover, the difficulty of debugging increases non-linearly with program size. One reason for such phenomena is the astonishing complexity and subtlety of the semantics o ..."

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1 Introduction Programming is difficult, as shown by the fact that debugging a program usually takes more time than creating it; moreover, the difficulty of debugging increases non-linearly with program size. One reason for such phenomena is the astonishing complexity and subtlety of the semantics of most widely used programming languages, due mainly to the desire for high efficiency on conventional processors. But rapid increases in the power and flexibility of hardware, and in the need for greater reliability and security in applications, suggest that it may be valuable to consider alternative approaches, based on higher level languages with much simpler semantics, despite the undoubted inertia of tradition, and the difficulty of learning new languages and new paradigms. This paper focuses on the OBJ family of languages, which have semantics based on various extensions of (first order) equational logic. The OBJ languages are logical programming languages, in which programs are theories, and computation is deduction, which makes it possible to do specification, programming and verification in a unified framework. This paper is mainly intended to introduce and motivate the material that it covers, rather than to provide a thorough mathematical exposition. Consequently, there are many references and several examples, but all proofs and many technical details are omitted.

### Observational Truth as a Categorical Modality

, 2002

"... This paper examines the notion of truth-up-to-observability in the setting of categorical logic and shows that it can be captured by a modal operator. Our main results extend the Kripke-Beth-Joyal semantics for the logics of presheaf toposes to the observational modality. We also give a categori ..."

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This paper examines the notion of truth-up-to-observability in the setting of categorical logic and shows that it can be captured by a modal operator. Our main results extend the Kripke-Beth-Joyal semantics for the logics of presheaf toposes to the observational modality. We also give a categorical account of coinduction (or bisimulation) as a proof-technique for establishing observational truth.

### Consciousness and the Decline of Cognitivism

"... This paper is not another absolutist attempt to #nd a #rm foundation or solid ground on which to base further speculations; it is also not an attempt to deny the possibility of making sense of social phenomena, or the existence of human values. Instead, it proposes a kind of open space of collective ..."

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This paper is not another absolutist attempt to #nd a #rm foundation or solid ground on which to base further speculations; it is also not an attempt to deny the possibility of making sense of social phenomena, or the existence of human values. Instead, it proposes a kind of open space of collective possibilities, within which many di#erent kinds of discourse can occur at many di#erent levels of granularity, including of course political discourse

### Reality and Human Values in Mathematics

"... Many mathematicians and philosophers say that mathematical objects have a real existence independent of any human activities or values. But do mathematicians behave as if this were true? This paper applies techniques from linguistics and sociology to show that mathematical discourse involves a highl ..."

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Many mathematicians and philosophers say that mathematical objects have a real existence independent of any human activities or values. But do mathematicians behave as if this were true? This paper applies techniques from linguistics and sociology to show that mathematical discourse involves a highly nuanced assignment of values to objects, which is then used in resolving references to objects; it also discusses the nature of abstraction, and shows how the appearance of reality for mathematical objects arises through the use of conventions from ordinary discourse, including narrative. Results in the paper have implications for the exposition and use of mathematics, for mathematics education, and for philosophy.

### Modeling and Verification of Distributed Real-Time Systems Based on CafeOBJ

"... CafeOBJ is a wide spectrum formal specification language based on multiple logical foundations: mainly initial and hidden algebra. A wide range of systems can be specified in CafeOBJ thanks to its multiple logical foundations. However, distributed real-time systems happen to be excluded from targets ..."

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CafeOBJ is a wide spectrum formal specification language based on multiple logical foundations: mainly initial and hidden algebra. A wide range of systems can be specified in CafeOBJ thanks to its multiple logical foundations. However, distributed real-time systems happen to be excluded from targets of CafeOBJ. In this paper, we propose a method of modeling and verifying such systems based on CafeOBJ, together with timed evolution of UNITY computational models.