Reflection in conditional rewriting logic

AbstractWe recall general metalogical axioms for a reflective logic based on the notion of a universal theory, that is, a theory that can simulate the deductions of all other theories in a class of theories of interest, including itself. We then show that conditional rewriting logic is reflective, generalizing in two stages: first to the unsorted conditional case, and then to the many-sorted conditional case, the already known result for unconditional and unsorted rewriting logic (Reflection in Rewriting Logic: Metalogical Foundations and Metaprogramming Applications. CSLI Publications, 2000). This work should be seen as providing foundations for many useful applications of rewriting logic reflection. The results presented here have greatly influenced the design of the Maude language, which implements rewriting logic and supports its reflective capabilities, and have been used as a theoretical foundation for applications such as internal rewrite strategies, reflective design of theorem proving tools, module algebra and metaprogramming, and metareasoning in metalogical frameworks

A Computational Approach to Reflective Meta-Reasoning about Languages with Bindings

We present a foundation for a computational meta-theory of languages with bindings implemented in a computer-aided formal reasoning environment. Our theory provides the ability to reason abstractly about operators, languages, open-ended languages, classes of languages, etc. The theory is based on the ideas of higher-order abstract syntax, with an appropriate induction principle parameterized over the language (i.e. a set of operators) being used. In our approach, both the bound and free variables are treated uniformly and this uniform treatment extends naturally to variable-length bindings. The implementation is reflective, namely there is a natural mapping between the meta-language of the theorem-prover and the object language of our theory. The object language substitution operation is mapped to the meta-language substitution and does not need to be defined recursively. Our approach does not require designing a custom type theory; in this paper we describe the implementation of this foundational theory within a general-purpose type theory. This work is fully implemented in the MetaPRL theorem prover, using the pre-existing NuPRL-like Martin-Lof-style computational type theory. Based on this implementation, we lay out an outline for a framework for programming language experimentation and exploration as well as a general reflective reasoning framework. This paper also includes a short survey of the existing approaches to syntactic reflection

The point of view: towards a social psychology of relativity

The explanation of social behaviour requires an understanding of individual orientations to social issues as these exist relative to others. This thesis argues that whilst the attitude concept and social representations have illuminated certain aspects of social behaviour, both are handicapped by a restricted focus. The former’s focus on the evaluation of attitude objects excludes a reference to wider societal processes. The latter provides an account of societal contingencies, but excludes an explanation of individual orientations towards objects and issues in the social environment. This thesis postulates the point of view concept to bridge this gap, that provides an explanation of social behaviour at the situational level. This complements attitude and social representations in a nested, multilevel explanation of social behaviour. The point of view is defined as an outlook towards a social event, expressed as a claim, which can be supported by an argument of opinion based on a system of knowledge from which it derives its logic. It reflects an individual’s orientation towards a social object, relative to others. This thesis has demonstrated, in a series of empirical studies, that the point of view can be typified in three categories. A monological point of view is closed to another’s perspective. A dialogical point of view acknowledges another’s perspective but dismisses it as wrong. A metalogical point of view acknowledges the relativity of its’ perspective, and concedes to an alternative the possibility of being right. These different types were demonstrated to be characterised by differences in positioning and in individuals’ capacity to fit a given social reality. Such relational outcomes accrue as a function of the socio-cognitive structure of points of view in relation with another perspective. This thesis demonstrates that points of view, alongside attitudes and social representations, provides a multilevel explanation of social behaviou

Twenty years of rewriting logic

AbstractRewriting logic is a simple computational logic that can naturally express both concurrent computation and logical deduction with great generality. This paper provides a gentle, intuitive introduction to its main ideas, as well as a survey of the work that many researchers have carried out over the last twenty years in advancing: (i) its foundations; (ii) its semantic framework and logical framework uses; (iii) its language implementations and its formal tools; and (iv) its many applications to automated deduction, software and hardware specification and verification, security, real-time and cyber-physical systems, probabilistic systems, bioinformatics and chemical systems

Law, Politics and Paradox : Orientations in Legal Formalism

The aim of this dissertation is to analyze the significance of the logical phenomenon of paradox for law and its relation to politics. I examine a selection of formal legal and political theories that in different ways understand law as a totality of norms, communications or behaviors, how paradox emerges in these theories, and what implications their understanding of paradox has for the relationship between law and politics. I argue that these legal and political theories can be meaningfully and in a novel way grouped according to their orientation to legal totality and paradox. To my knowledge, there is no research systematically mapping orientations to paradox in legal theory. It is the objective of this dissertation to fill this lack. Paradox presents challenges for formal thought, i.e. thought that analyzes the logic of totalities. Law, considered as a totality or form, gathers a plurality of entities under a common denominator and into a legal order. It is in reflecting on such formalization that we encounter paradoxes. This work aims to contribute to a growing literature on the implications of formalism for contemporary social and political thought by providing a legal theoretical perspective hitherto missing in these discussions. I use as a heuristic device a grouping of formal thought presented by the philosopher Paul M. Livingston. According to this grouping, there are three main orientations in contemporary formal thought to totality: the constructivist-criteriological, the paradoxico-critical and the generic orientation. These orientations arise on grounds of the “metalogical choice”: they prefer to view totality (such as law as a system or order) either as complete but inconsistent (the paradoxico-criticism), or as consistent but incomplete (the constructivist-criteriological and the generic orientation). I will apply, and modify when necessary, this categorization in order to analyze the theories of Hans Kelsen, Niklas Luhmann, Giorgio Agamben, Alain Badiou and Hans Lindahl, and to provide a systematic mapping of how the nature of law as a totality is understood in contemporary formal legal-political thought. Accounts of modern law encounter a paradox, I argue, if they observe law as an autonomous, self-referential totality that claims for itself the right to draw a distinction between itself and non-law. The paradox of autonomous law is that it cannot consistently show that it is itself legal as a totality. The basic problem that this implies is that the legal system or collective is unable to legitimate its existence and identity in response to challenges in any other way than by drawing on its own resources – which precisely is what the challenge targets in the first place. If we think of law as offering a framework within which questions of justice and injustice can be answered, the paradox emerges when we question the justice of this framework itself. The dissertation defends the paradoxico-critical orientation. It argues that the legal system is a paradoxical totality, which implies that there is no neutral metalanguage, such as natural law, that could solve the problem of law’s self-reference for good. This challenges legal theory to show how the problem of nihilistic relativism, the mere perpetuation of the self-referential legal system, can be mitigated and law’s normative authority in society rethought. In Chapter 1, I define the notion of paradox, explicate its meaning and role in formal thought and motivate its application to legal theory. In Chapter 2, I show that in his theory of the basic norm, Kelsen can be understood as oscillating between the constructivist-criteriological position and the paradoxico-criticism, between an attempt at guaranteeing legal order’s consistency in a metalanguage, i.e. legal science, and an acknowledgement of law as an inconsistent totality. In Chapter 3, I interpret Luhmann as a paradoxico-evolutionary thinker: he observes the legal system as constitutively inconsistent but emphasizes the ways in which the system seeks to make this inconsistency unproblematic for functional reasons. In Chapter 4, I show that in systems theory, just like in Kelsen’s pure theory, the politics of the paradox remains unarticulated. I also show that, for Agamben, a paradoxico-critical thinker, the paradoxical articulation of law and politics is exposed in the state of exception, which, in his analysis, has become the new normal, requiring “messianic” politics to deactivate the whole nihilistic sovereign-legal apparatus. For Badiou, the representative of the generic orientation, which I discuss in Chapter 5, what can be said within a language, and by implication a legal system, is pre-determined by that language. Politics, the desire to say the unsayable, is thrown fully outside the language and the legal system to a position from which law’s incompleteness, its incapacity to offer space for justice and politics, can only be disclosed. Both Agamben and Badiou, thus, think about politics as “post-juridical.” In Chapter 6, I show that the very inconsistency and paradox at the heart of the legal order is, for Lindahl’s paradoxico-criticism, the site of the politics of its limits. This dissertation, then, concludes that the paradoxical limits of the legal totality can be understood as the site of politics in law. Taking law’s paradox into account allows for a non-nihilistic conception of politically contestable law and legal authority.Väitöskirja selvittää paradoksin käsitteen merkitystä oikeudelle ja oikeuden ja politiikan väliselle suhteelle. Analysoin oikeusfilosofian alaan kuuluvassa tutkimuksessani, miten valikoimani oikeus- ja politiikan teoreetikot ymmärtävät oikeuden normeista, kommunikaatioista tai toiminnasta koostuvana kokonaisuutena, miten paradoksi ilmenee heidän teorioissaan ja mitä seurauksia sillä on heidän käsitykselleen oikeuden ja politiikan suhteesta. Väitän, että oikeus- ja politiikan teorian kenttää voi uudella tavalla hahmottaa selvittämällä suhtautumistapoja oikeuden paradoksiin. Aiemmin oikeusteoriassa ei ole systemaattisesti selvitetty käsityksiä oikeuden paradoksista, ja väitöskirjan tavoitteena on täyttää tämä aukko. Se osallistuu kasvavaan filosofiseen keskusteluun formaalin ajattelun merkityksestä yhteiskunta- ja poliittiselle teorialle ja tarjoaa oikeusteoreettisen näkökulman, joka keskustelusta vielä puuttuu. Paradoksi hahmottuu oikeusteoreettisena ongelmana, kun oikeutta teoretisoidaan kokonaisuutena eli oikeusjärjestyksenä. Käytän tutkimuksessani heuristisena apuna filosofi ja loogikko Paul M. Livingstonin kehittämää formaalin ajattelun jaottelua kolmeen, konstruktivistis-kriteriologiseen, paradoksis-kriittiseen ja geneeriseen suuntaukseen. Nämä suuntaukset käsittävät kokonaisuuksien luonteen eri tavoin ja siten tekevät erilaisen “metaloogisen valinnan”: ne käsittävät kokonaisuudet, kuten oikeuden systeeminä tai normijärjestyksenä, joko täydellisinä mutta paradoksaalisina tai konsistentteinä mutta epätäydellisinä. Sovellan tutkimuksessani tätä jaottelua ja analysoin sen avulla Hans Kelsenin, Niklas Luhmannin, Giorgio Agambenin, Alain Badioun ja Hans Lindahlin oikeus-poliittista ajattelua. Tavoitteena on systemaattisesti selvittää, miten nykyaikaisessa formaalissa oikeus-poliittisessa ajattelussa ymmärretään oikeuden luonne kokonaisuutena. Väitöskirja puolustaa paradoksis-kriittistä suuntausta. Väitän, että moderni oikeus voidaan ymmärtää paradoksaalisena, jos se käsitetään autonomisena, itseensä viittaavana kokonaisuutena, joka pidättää itselleen oikeuden vetää raja oikeuden ja ei-oikeuden välille. Autonomisen oikeuden paradoksi on se, ettei oikeusjärjestys pysty itse ristiriidattomasti oikeuttamaan itseään. Oikeusjärjestys mahdollistaa riidanratkaisun sekä oikean ja väärän, laillisen ja laittoman erottamisen toisistaan, mutta oikeuden yritykset ratkaista tarjoamansa riidanratkaisun oma oikeutus ja laillisuus johtavat paradoksiin. Seurauksena on, että oikeusjärjestelmä ja -yhteisö kykenee vastamaan kohtaamaansa kritiikkiin vain omasta näkökulmastaan, mikä juuri on kritiikin kohteena. Väitöskirjassa esitetään, että oikeusjärjestelmän ymmärtäminen paradoksaalisena kokonaisuutena merkitsee sekä ”metakielen”, kuten itsenäisen luonnonoikeuden, hylkäämistä ratkaisuna oikeuden itseensä viittaavuuden ongelmaan että luopumista täydellisen ja konsistentin oikeusjärjestyksen ideasta. Tästä seuraa, että oikeusteoria joutuu kohtaamaan oikeuden poliittisuuden, nihilistisen relativismin ongelman sekä etsimään uusia tapoja käsittää oikeuden normatiivisuus ja auktoriteetti yhteiskunnassa

Calculus of Qualia 7: Equations vs. Qualations, Assertions with non-referential terms, Proofs, Logic

General Introduction: In [1] a Calculus of Qualia (CQ) was proposed. The key idea is that, for example, blackness is radically different than █. The former term, “blackness” refers to or is about a quale, whereas the latter term, “█” instantiates a quale in the reader’s mind and is non-referential; it does not even refer to itself. The meaning and behavior of these terms is radically different. All of philosophy, from Plato through Descartes through Chalmers, including hieroglyphics and emojis, used referential terms up until CQ. This paper in this series of papers address equations vs. qualations (which contain non-referential terms), proofs, and logic. An example of an equation is x+2=4, which uses exclusively referential terms, even if some of them are numbers. A Qualation is like an equation except it uses actual non-referential terms (qualia), like █ ≠ ▲, and can be clearly non-trivial, like ¬((█ ≠ ▲) ∧ (▲ ≠ ■) → (█ ≠ ■)). (see below). This has implications for assertions, proofs, truth, etc. Incompleteness and truth are addressed in a different paper in this series

Robot Consciousness: Physics and Metaphysics Here and Abroad

Interest has been renewed in the study of consciousness, both theoretical and applied, following developments in 20th and early 21st-century logic, metamathematics, computer science, and the brain sciences. In this evolving narrative, I explore several theoretical questions about the types of artificial intelligence and offer several conjectures about how they affect possible future developments in this exceptionally transformative field of research. I also address the practical significance of the advances in artificial intelligence in view of the cautions issued by prominent scientists, politicians, and ethicists about the possible dangers of such sufficiently advanced general intelligence, including by implication the search for extraterrestrial intelligence

Modelling Value-Oriented Legal Reasoning in LogiKEy

The logico-pluralist LogiKEy knowledge engineering methodology and framework is applied to the modelling of a theory of legal balancing, in which legal knowledge (cases and laws) is encoded by utilising context-dependent value preferences. The theory obtained is then used to formalise, automatically evaluate, and reconstruct illustrative property law cases (involving the appropriation of wild animals) within the Isabelle/HOL proof assistant system, illustrating how LogiKEy can harness interactive and automated theorem-proving technology to provide a testbed for the development and formal verification of legal domain-specific languages and theories. Modelling value-oriented legal reasoning in that framework, we establish novel bridges between the latest research in knowledge representation and reasoning in non-classical logics, automated theorem proving, and applications in legal reasoning

Grounding Reichenbach’s Pragmatic Vindication of Induction

This paper has three interdependent aims. The first is to make Reichenbach’s views on induction and probabilities clearer, especially as they pertain to his pragmatic justification of induction. The second aim is to show how his view of pragmatic justification arises out of his commitment to extensional empiricism and moots the possibility of a non-pragmatic justification of induction. Finally, and most importantly, a formal decision-theoretic account of Reichenbach’s pragmatic justification is offered in terms both of the minimax principle and the dominance principle