Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.     

Fibred Security Language

We study access control policies based on the says operator by introducing a logical framework called Fibred Security Language (FSL) which is able to deal with features like joint responsibility between sets of principals and to identify them by means of first-order formulas. FSL is based on a multimodal logic methodology. We first discuss the main contributions from the expressiveness point of view, we give semantics for the language both for classical and intuitionistic fragment), we then prove that in order to express well-known properties like ‘speaks-for’ or ‘hand-off’, defined in terms of says, we do not need second-order logic (unlike previous approaches) but a decidable fragment of first-order logic suffices. We propose a model-driven study of the says axiomatization by constraining the Kripke models in order to respect desirable security properties, we study how existing access control logics can be translated into FSL and we give completeness for the logic.     

Modal Logic for Other-World Agnostics: Neutrality and Halldén Incompleteness

The logic of ‘elsewhere,’ i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions (explaining the use of the word ‘elsewhere’), as well as in the case in which they represent moments of time. This logic is applied here to the alethic modal case, in which the points are thought of as possible worlds, with the suggestion that its deployment clarifies aspects of a position explored by John Divers un-der the name ‘modal agnosticism.’ In particular, it makes available a logic whose Halldén incompleteness explicitly registers the agnostic element of the position – its neutrality as between modal realism and modal anti-realism.     

Conditionals in reasoning

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent semantics and epistemic states are represented via preferential models.     

Common knowledge of payoff uncertainty in games

Using epistemic logic, we provide a non-probabilistic way to formalise payoff uncertainty, that is, statements such as ‘player i has approximate knowledge about the utility functions of player j.’ We show that on the basis of this formalisation common knowledge of payoff uncertainty and rationality (in the sense of excluding weakly dominated strategies, due to Dekel and Fudenberg (1990)) characterises a new solution concept we have called ‘mixed iterated strict weak dominance.’     

Modelling simultaneous games in dynamic logic

We make a proposal for formalizing simultaneous games at the abstraction level of player’s powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of ‘concurrent game logic’ CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.     

SUPERVALUATION FIXED-POINT LOGICS OF TRUTH

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived.     

Formalizing common sense: an operator-based approach to the Tibbles–Tib problem

The paper argues, that a direct formalization of the way common sense thinks about the numerical identity of enduring entities, requires that traditional predicate logic is developed. If everyday language mirrors the world, then persons, organisms, organs, cells, and ordinary material things can lose some parts but nonetheless remain numerically exactly the same entity. In order to formalize this view, two new logical operators are introduced; and they bring with them some non-standard syntax. One of the operators is called ‘the instantiation operator’; it is needed because the existential quantifier and its traditional relatives cannot do the job required. The other operator is called ‘the form-on-matter operator’, and it allows an individual (an instance of a form) to stay the same even though some of its parts (its constituting matter) is taken away from it. Also, a certain kind of predicates, called ‘nature terms’, is needed in order to represent what gives a particular its kind of identity. Both the operators and the nature terms introduced can be used in constructions of formal languages and formal systems, but no such constructions are made in the paper. The paper is structured as a comment on the philosophical problem called ‘the problem of the cats Tibbles and Tib’.     

The Rationale Behind Revision-Rule Semantics

According to Gupta and Belnap, the “extensional behavior” of ‘true’ matches that of a circularly defined predicate. Besides promising to explain semantic paradoxicality, their general theory of circular predicates significantly liberalizes the framework of truth-conditional semantics. The authors’ discussions of the rationale behind that liberalization invoke two distinct senses in which a circular predicate’s semantic behavior is explained by a “revision rule” carrying hypothetical information about its extension. Neither attempted explanation succeeds. Their theory may however be modified to employ a relativized notion of extension. The resulting contextualist semantics for ‘true’ construes circularity as a pragmatic phenomenon.     

Many Concepts and Two Logics of Algorithmic Reduction

Within the program of finding axiomatizations for various parts of computability logic, it was proven earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting’s intuitionistic calculus. That sort of reduction permits unlimited reusage of the computational resource represented by the antecedent. An at least equally basic and natural sort of algorithmic reduction, however, is the one that does not allow such reusage. The present article shows that turning the logic of the first sort of reduction into the logic of the second sort of reduction takes nothing more than just deleting the contraction rule from its Gentzen-style axiomatization. The first (Turing) sort of interactive reduction is also shown to come in three natural versions. While those three versions are very different from each other, their logical behaviors (in isolation) turn out to be indistinguishable, with that common behavior being precisely captured by implicative intuitionistic logic. Among the other contributions of the present article is an informal introduction of a series of new — finite and bounded — versions of recurrence operations and the associated reduction operations. Presented by Robert Goldblatt     

Mismatches between ‘Scientific’ and ‘Non-Scientific’ Ways of Knowing and Their Contributions to Public Understanding of Science

As differentiation within scientific disciplines increases, so does differentiation between the sciences and other ways of knowing. This distancing between ‘scientific’ and ‘non-scientific’ cultures reflects differences in what are considered valid and reliable approaches to acquiring knowledge and has played a major role in recent science-oriented controversies. Scientists’ reluctance to actively engage in science communication, coupled with journalists’ reliance on the norms of balance, conflict, and human interest in covering scientific issues, have combined to exacerbate public mistrust of science on issues like the measles-mumps-rubella (MMR) vaccine. The failure of effective communications between scientists and non-scientists has hindered the progress of both effective science and effective policy. In order to better bridge the gap between the ‘scientific’ and ‘non-scientific’ cultures, renewed efforts must be made to encourage substantive public engagement, with the ultimate goal of facilitating an open, democratic policy-making process.     

Conquering the quarters: Religion and politics in Hinduism

Conclusion Our understanding of South Asian society and history is sometimes muddled by the rigid distinctions we make between ‘religion’ and ‘politics.’ The resurgent appeal of Hindu nationalism, the involvement of Hindu renouncers in contemporary Indian politics, and the continuing relevance of religious issues to political discourse throughout South Asia, show that such a distinction is of limited utility. In this essay, I have examined the notion of digvijaya in some detail, in an attempt to show that this ‘most important Indian concept with regard to sovereignty’ was always both a ‘religious’ and a ‘political’ phenomenon. When it was performed by Hindu kings in the classical period, the ‘political’ dimension of digvijaya was foregrounded, while in the medieval and modern periods, when it was associated primarily with Hindu renouncers, its ‘religious’ aspects were paramount. But neither ‘political’ nor ‘religious’ aspects were ever absent from any of the digvijayas discussed here because religion and politics were mutually entailed in the digvijaya at all times, just as kings and renouncers were—and still are—alter-egos of each other. I am tempted to conclude that the digvijaya melded religious and political domains. Yet perhaps even to speak of ‘melding’ religion and politics is a peculiarly modern kind of discourse. Perhaps we need to rethink our categories and recognize that politics always has a religious element, while religion is always a political force.     

The ‘Gray’s Elegy’ Argument, and The Prospects for the Theory of Denoting Concepts

Russell’s new theory of denoting phrases introduced in “On Denoting” in Mind 1905 is now a paradigm of analytic philosophy. The main argument for Russell’s new theory is the so-called ‘Gray’s Elegy’ argument, which purports to show that the theory of denoting concepts (analogous to Frege’s theory of senses) promoted by Russell in the 1903 Principles of Mathematics is incoherent. The ‘Gray’s Elegy’ argument rests on the premise that if a denoting concept occurs in a proposition, then the proposition is not about the concept. I argue that the premise is false. The ‘Gray’s Elegy’ argument does not exhaust Russell’s ammunition against the theory of denoting concepts. Another reason Russell rejects the theory is, as he says, that it cannot provide an adequate account of non-uniquely denoting concepts. In the last section of the paper, I argue that even though Russell was right in thinking that the theory of denoting concepts cannot provide an adequate account of non-uniquely denoting concepts, Russell’s new theory does not succeed in eliminating the occurrence of all denoting concepts, as it requires a commitment to the existence of variables that indirectly denote their values. However, the view that variables are denoting concepts is unproblematic once the ‘Gray’s Elegy’ argument is blocked.     

Trustworthiness in explanation: The obligation to explain well

‘Scientific integrity’ certainly requires that data and references be beyond reproach. However, issues within the theory of scientific explanation suggest that there may be more to it than just this. While it is true that some contemporary, pragmatic analyses of explanation suffer from the ‘problem of relevance’ (an inability to ensure that explanations which are paradigmatic technically are relevant to the question being posed), it does not seem to be true that the addition of formal, metaphysical constraints is necessary to solve this problem. I argue that, when viewed as requests for help with an epistemic problem, explanation-seeking questions reveal the existence of a set of moral criteria centered in trust which, when satisfied, prevent trivial or irrelevant explanations from being offered, thereby broadening the concept of ‘scientific integrity’.     

Logic and Reasoning: do the facts matter?

Modern logic is undergoing a cognitive turn, side-stepping Frege’s ‘antipsychologism’. Collaborations between logicians and colleagues in more empirical fields are growing, especially in research on reasoning and information update by intelligent agents. We place this border-crossing research in the context of long-standing contacts between logic and empirical facts, since pure normativity has never been a plausible stance. We also discuss what the fall of Frege’s Wall means for a new agenda of logic as a theory of rational agency, and what might then be a viable understanding of ‘psychologism’ as a friend rather than an enemy of logical theory. Edited by Hannes Leitgeb     

What is <Emphasis Type="Italic">the</Emphasis> Logic of Inference?

The topic of this paper is the question whether there is a logic which could be justly called the logic of inference. It may seem that at least since Prawitz, Dummett and others demonstrated the proof-theoretical prominency of intuitionistic logic, the forthcoming answer is that it is this logic that is the obvious choice for the accolade. Though there is little doubt that this choice is correct (provided that inference is construed as inherently single-conclusion and complying with the Gentzenian structural rules), I do not think that the usual justification of it is satisfactory. Therefore, I will first try to clarify what exactly is meant by the question, and then sketch a conceptual framework in which it can be reasonably handled. I will introduce the concept of ‘inferentially native’ logical operators (those which explicate inferential properties) and I will show that the axiomatization of these operators leads to the axiomatic system of intuitionistic logic. Finally, I will discuss what modifications of this answer enter the picture when more general notions of inference are considered. Presented by Hannes Leitgeb     

Statues and Lumps: A Strange Coincidence?

Puzzles about persistence and change through time, i.e., about identity across time, have foundered on confusion about what it is for ‘two things’ to be have ‘the same thing’ at a time. This is most directly seen in the dispute over whether material objects can occupy exactly the same place at the same time. This paper defends the possibility of such coincidence against several arguments to the contrary. Distinguishing a temporally relative from an absolute sense of ‘the same’, we see that the intuition, ‘this is only one thing’, and the dictum, ‘two things cannot occupy the same place at the same time’, are individuating things at a time rather than absolutely and are therefore compatible with coincidence. Several other objections philosophers have raised ride on this same ambiguity. Burke, originating what has become the most popular objection to coincidence, argues that if coincidence is possible there would be no explanation of how objects that are qualitatively the same at a time could belong to different sorts. But we can explain an object’s sort by appealing to its properties at other times. Burke’s argument to the contrary equivocates on different notions of ‘cross-time identity’ and ‘the statue’. From a largely negative series of arguments emerges a positive picture of what it means to say multiple things coincide and of why an object’s historical properties explain its sort rather than vice versa – in short, of how coincidence is possible.     

Local contexts and local meanings

Stalnaker (1978) made two seminal claims about presuppositions. The most influential one was that presupposition projection is computed by a pragmatic mechanism based on a notion of ‘local context’. Due to conceptual and technical difficulties, however, the latter notion was reinterpreted in purely semantic terms within ‘dynamic semantics’ (Heim 1983). The second claim was that some instances of presupposition generation should also be explained in pragmatic terms. But despite various attempts, the definition of a precise ‘triggering algorithm’ has remained somewhat elusive. We discuss possible extensions of both claims. First, we offer a reconstruction of ‘local contexts’ which circumvents some of the difficulties faced by Stalnaker’s original analysis. We preserve the idea that local contexts are computed by a pragmatic mechanism that aggregates the information that follows from an incomplete sentence given the global context; but we crucially rely on a modified notion of entailment (‘R-entailment’), whose plausibility should be assessed on independent grounds. Second, we speculate that local contexts might prove necessary (though by no means sufficient) to understand how some presuppositions are triggered. In a nutshell, we suggest that a presupposition is triggered when the semantic contribution of an expression to its local context is in some sense ‘heterogeneous’. Without giving an analysis of the latter notion, we note that this architecture implies that presuppositions should be triggered on the basis of the meaning that an expression has relative to its local context (what we call its ‘local meaning’); we sketch some possible consequences of this analysis.