Whither relevant arithmetic?

1992 ◽  
Vol 57 (3) ◽  
pp. 824-831 ◽  
Author(s):  
Harvey Friedman ◽  
Robert K. Meyer
Keyword(s):  
Relevant Logic ◽  
Peano Arithmetic ◽  
Central Question ◽  
Proof Methods ◽  
Classical Proof ◽  
Natural Way

AbstractBased on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R#, which does formalize arithmetic on relevant principles, but also admits γ in a natural way?

scholarly journals Arithmetic Formulated Relevantly

2021 ◽  
Vol 18 (5) ◽  
pp. 154-288
Author(s):  
Robert Meyer
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Critical Problem ◽  
First Order ◽  
Order Form ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Natural Way

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that (1) it is trivial that relevant arithmetic is absolutely consistent, but (2) classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under (1), I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under (2), I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment.

scholarly journals Inconsistent models for relevant arithmetics

1984 ◽  
Vol 49 (3) ◽  
pp. 917-929 ◽  
Author(s):  
Robert K. Meyer ◽  
Chris Mortensen
Keyword(s):  
Classical Logic ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
The Absolute

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano's axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R#, was absolutely consistent. It was pointed out that such a result escapes incautious formulations of Gödel's second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not ordinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle.The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein constitutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R#. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P#. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here).

scholarly journals Arithmetic Formulated in a Logic of Meaning Containment

2021 ◽  
Vol 18 (5) ◽  
pp. 447-472
Author(s):  
Ross Brady
Keyword(s):  
Relevant Logic ◽  
Peano Arithmetic ◽  
Recursive Functions ◽  
Primitive Recursive

We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.

scholarly journals Inconsistent Models for Relevant Arithmetics

2021 ◽  
Vol 18 (5) ◽  
pp. 380-400
Author(s):  
Robert Meyer ◽  
Chris Mortensen
Keyword(s):  
Classical Logic ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
Intrinsic Interest ◽  
Recursively Enumerable ◽  
Classical Quantum ◽  
The Absolute ◽  
Quantum Arithmetic ◽  
Inconsistent Theories

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest.

Ruler Visibility and Popular Belonging in the Ottoman Empire, 1808-1908

2018 ◽  
Author(s):  
Darin Stephanov
Keyword(s):  
Public Space ◽  
Ottoman Empire ◽  
The Other ◽  
Group Consciousness ◽  
Central Question ◽  
Point Model ◽  
The Everyday ◽  
National Movements ◽  
Tanzimat Reforms ◽  
The One

‘What do we really speak of when we speak of the modern ethno-national mindset and where shall we search for its roots?’ This is the central question of a book arguing that the periodic ceremonial intrusion into the everyday lives of people across the Ottoman Empire, which the annual royal birthday and accession-day celebrations constituted, had multiple, far-reaching, and largely unexplored consequences. On the one hand, it brought ordinary subjects into symbolic contact with the monarch and forged lasting vertical ties of loyalty to him, irrespective of language, location, creed or class. On the other hand, the rounds of royal celebration played a key role in the creation of new types of horizontal ties and ethnic group consciousness that crystallized into national movements, and, after the empire’s demise, national monarchies. The book discusses the themes of public space/sphere, the Tanzimat reforms, millet, modernity, nationalism, governmentality, and the modern state, among others. It offers a new, thirteen-point model of modern belonging based on the concept of ruler visibility.

The Tribal Impact on Political Stability in Sudan

2018 ◽  
Vol 11 (1-2) ◽  
pp. 167-188
Author(s):  
Abdu Mukhtar Musa
Keyword(s):  
Urban Areas ◽  
Third World ◽  
Political Process ◽  
Political Stability ◽  
The Political ◽  
Central Question ◽  
Conflict Escalation ◽  
Islamic Movement ◽  
Patterns Of Behavior ◽  
Cultural Structure

As in most Arab and Third World countries, the tribal structure is an anthropological reality and a sociological particularity in Sudan. Despite development and modernity aspects in many major cities and urban areas in Sudan, the tribe and the tribal structure still maintain their status as a psychological and cultural structure that frames patterns of behavior, including the political behavior, and influence the political process. This situation has largely increased in the last three decades under the rule of the Islamic Movement in Sudan, because of the tribe politicization and the ethnicization of politics, as this research reveals. This research is based on an essential hypothesis that the politicization of tribalism is one of the main reasons for the tribal conflict escalation in Sudan. It discusses a central question: Who is responsible for the tribal conflicts in Sudan?

INEVITABILITY OF REVIVAL: PRO AND COUNTER ANTINATALISM

2019 ◽  
Vol 19 (1) ◽  
pp. 80-88
Author(s):  
Nikolay S. Savkin
Keyword(s):  
Social Relations ◽  
Systematic Approach ◽  
Human Life ◽  
Laws Of Nature ◽  
Arthur Schopenhauer ◽  
Expected Effect ◽  
Philosophy Of Life ◽  
Active Subject ◽  
Over Time ◽  
Natural Way

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.

Who’s holding up the sky? On Rymba, a novel by A. Bushkovsky

2020 ◽  
Vol 1 (1) ◽  
pp. 150-158
Author(s):  
A. V. Zhuchkova
Keyword(s):  
Gender Politics ◽  
The State ◽  
Christian Faith ◽  
The Novel ◽  
Central Question ◽  
Men And Women ◽  
Flood Zone ◽  
The Social ◽  
Russian North ◽  
And Gender

The article deals with A. Bushkovsky’s novel Rymba that goes beyond the topics typical of Russian North prose. Rather than limiting himself to admiring nature and Russian character, the author portrays the northern Russian village of Rymba in the larger context of the country’s mentality, history, mythology, and gender politics. In the novel, myth clashes with reality, history with the present day, and an individual with the state. The critic draws a comparison between the novel and the traditions of village prose and Russian North prose. In particular, Bushkovsky’s Rymba is discussed alongside V. Rasputin’s Farewell to Matyora [ Proshchanie s Matyoroy ] and R. Senchin’s The Flood Zone [ Zona zatopleniya ]. The novel’s central question is: what keeps the Russian world afloat? Depicting the Christian faith as such a bulwark, Bushkovsky links atheism with the social and spiritual roles played by contemporary men and women. The critic argues, however, that the reliance on Christianity in the novel verges on an affectation. The book’s main symbol is a drowning hawk: it perishes despite people’s efforts to save it.

The Fight to Save Welfare for Low-Income Older Asian Immigrants: The Role of National Asian American Organizations

2003 ◽  
Vol 1 (1) ◽  
pp. 85-101 ◽  
Author(s):  
Grace Yoo
Keyword(s):  
Welfare Reform ◽  
Asian American ◽  
Low Income ◽  
Bill Clinton ◽  
Asian Immigrants ◽  
Central Question ◽  
Political Debate ◽  
Welfare Reform Law ◽  
Selection Of

The welfare reform law of August 1996 signed by President Bill Clinton put an end to immigrants’ eligibility of federal means tested entitlements. The rollbacks on welfare are the most drastic for older, low-income Asian immigrants who are on Supplemental Security Income. The article’s focus is in on national Asian American organizations who are involved in this political debate. The central question discuss is how did national Asian American organizations characterize and affect the 1996 federal welfare reform and immigrant debate. The selection of organizations that was studied and the findings of that investigation, along with the assessment of its effectiveness and the resources barriers they face are discussed.

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