Arguments by Parallels in the Epistemological Works of Phya pa Chos kyi seng ge

The works of the Tibetan logician Phya pa Chos kyi seng ge (1109–1169) make abundant use of a particular type of argument that I term ‘argument by parallels’. Their main characteristic is that the instigator of the argument, addressing a thesis in a domain A, introduces a parallel thesis in an unrelated domain B. And in the ensuing dialogue, each of the instigator’s statements consists in replicating his interlocutor’s previous assertion, mutatis mutandis, in the other domain (A or B). I show that such a dialogue involves two parallel arguments that develop in an intersecting zigzag pattern, and discuss the principles involved in the establishment of the conclusion from the perspective of parity of reasoning and analogical argument. I examine the overall rhetorical strategy directing the use of arguments by parallels and the pedagogical and explanatory functions they can serve. I also evaluate the plausibility of their use in Phya pa Chos kyi seng ge’s works mirroring a contemporary practice of oral debate, and reflect on the status of such arguments in the framework of Indo-Tibetan logic.

儒家智者的逻辑——析荀子正名论

荀子适应当时社会需要 ,从儒家智者的视角 ,创发孔子正名的逻辑内涵 ,建构以概念论为中心的逻辑体系 ,继墨辩之后把中国古代逻辑推向又一高峰。荀子对概念论和语言论、本体论、认识论、判断论、推理论、诡辩论、语言规范化、华夏大一统的相关论述 ,是当前仍有积极价值的学术精华。以现代方法揭示荀子正名论的逻辑意义 ,是中国逻辑元研究的课题。     

Logic programs, iterated function systems, and recurrent radial basis function networks

Graphs of the single-step operator for first-order logic programs—displayed in the real plane—exhibit self-similar structures known from topological dynamics, i.e., they appear to be fractals, or more precisely, attractors of iterated function systems. We show that this observation can be made mathematically precise. In particular, we give conditions which ensure that those graphs coincide with attractors of suitably chosen iterated function systems, and conditions which allow the approximation of such graphs by iterated function systems or by fractal interpolation. Since iterated function systems can easily be encoded using recurrent radial basis function networks, we eventually obtain connectionist systems which approximate logic programs in the presence of function symbols.     

Conflicting imperatives and dyadic deontic logic

Often a set of imperatives or norms seems satisfiable from the outset, but conflicts arise when ways to fulfill all are ruled out by unfortunate circumstances. Semantic methods to handle normative conflicts were devised by B. van Fraassen and J.F. Horty, but these are not sensitive to circumstances. The present paper extends these resolution mechanisms to circumstantial inputs, defines dyadic deontic operators accordingly, and provides a sound and (weakly) complete axiomatic system for such deontic semantics.     

Logic,Rules and Intention: The Principal Aim Argument

Stephen Mumford develops his view of sport spectatorship partly through a rejection of an argument he attributes to Best, which distinguishes between two categories of sports, the ‘purposive’ and the ‘aesthetic’, on the basis of the claim that they have different principal aims. This paper considers the principal aim argument and one feature of Mumford’s rejection of that argument, namely, Best’s observation that the distinctions to which he draws attention are based on logical differences. The paper argues that Mumford misconstrues Best’s argument by taking it to be about the intentions of players and athletes, while it is actually about a specific feature of the rules of each sport.     

Reduction and emergence in the physical sciences: Reply to Rueger

I analyse Rueger’s application of Kim’s model of functional reduction to the relation between the thermal conductivities of metal bars at macroscopic and atomic scales. 1) I show that it is a misunderstanding to accuse the functional reduction model of not accounting for the fact that there are causal powers at the micro-level which have no equivalent at the macro-level. The model not only allows but requires that the causal powers by virtue of which a functional predicate is defined, are only a subset of the causal powers of the properties filling the functional specification. 2) The fact that the micro-equation does not converge to the macro-equation in general but only under the constraint of a “solvability condition” does not show that reduction is impossible, as Rueger claims, but only that reduction requires inter-level constraints. 3) Rueger tries to analyse inter-level reduction with the conceptual means of intra-level reduction. This threatens the coherence of his analysis, given that it makes no sense to ascribe macroproperties such as thermal conductivity to entities at the atomic level. Ignoring the distinction between theses two senses of “reduction” is especially confusing because they have opposite directions: in intra-level reduction, the more detailed account reduces to the less detailed one, whereas in inter-level reduction, the less detailed theory is reduced to the more detailed one. 4) Finally I criticize Rueger’s way of using Wimsatt’s criteria for emergence in terms of non-aggregativity, to construct a concept of synchronic emergence. It is wrong to require, over and above non-aggregativity, irreducibility as a criterion for emergence.     

Innovations in computational type theory using Nuprl

For twenty years the Nuprl (“new pearl”) system has been used to develop software systems and formal theories of computational mathematics. It has also been used to explore and implement computational type theory (CTT)—a formal theory of computation closely related to Martin-Löf's intuitionistic type theory (ITT) and to the calculus of inductive constructions (CIC) implemented in the Coq prover.

This article focuses on the theory and practice underpinning our use of Nuprl for much of the last decade. We discuss innovative elements of type theory, including new type constructors such as unions and dependent intersections, our theory of classes, and our theory of event structures.

We also discuss the innovative architecture of Nuprl as a distributed system and as a transactional database of formal mathematics using the notion of abstract object identifiers. The database has led to an independent project called the Formal Digital Library, FDL, now used as a repository for Nuprl results as well as selected results from HOL, MetaPRL, and PVS. We discuss Howe's set theoretic semantics that is used to relate such disparate theories and systems as those represented by these provers.     

Model checking propositional dynamic logic with all extras

This paper presents a model checking algorithm for Propositional Dynamic Logic (PDL) with looping, repeat, test, intersection, converse, program complementation as well as context-free programs. The algorithm shows that the model checking problem for PDL remains PTIME-complete in the presence of all these operators, in contrast to the high increase in complexity that they cause for the satisfiability problem.     

Hybrid Probabilistic Logic Programs as Residuated Logic Programs

In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kowalski. The importance of this result is that for the first time a framework encompassing several quite distinct logic programming semantics is described, namely Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic Logic Programming. Moreover, the embedding provides a more general semantical structure paving the way for defining paraconsistent probabilistic reasoning with a logic programming semantics.     

Old Foundations for the Logic of Agency and Action

The paper presents an infinite hierarchy of sound and complete axiomatic systems for Two-Dimensional Modal Tense Logic with Historical Necessity, Agents and Acts. A main novelty of these logics is their capacity to represent formally (i) basic action-sentences asserting that such and such an act is performed/omitted by an agent, as well as (ii) causative action-sentences asserting that by performing/omitting a certain act, an agent causes that such and such a state-of-affairs is realized (e.g. comes about/ceases/remains/remains absent). We illustrate how the formal machinery of our systems can be used to reconstruct a number of interesting ideas in the Logic of Agency and Action that have been proposed by authors like von Wright, von Kutschera, Belnap and Segerberg.