In this paper I investigate whether certain substructural theories are able to dodge paradox while at the same time containing what might be viewed as a naive validity predicate. To this end I introduce the requirement of internalization, roughly, that an adequate theory of validity should prove that its own metarules are validity-preserving. The main point of the paper is that substructural theories fail this requirement in various ways.

The need for expressing temporal constraints in conceptual models is well-known, but it is unclear which representation is preferred and what would be easier to understand by modellers. We assessed five different modes of representing temporal constraints, being the formal semantics, Description logics notation, a coding-style notation, temporal EER diagrams, and (pseudo-)natural language sentences. The same information was presented to 15 participants in an experimental evaluation. Principally, it showed that 1) there was a clear preference for diagrams and natural language versus a dislike for other representations; 2) diagrams were preferred for simple constraints, but the natural language rendering was preferred for more complex temporal constraints; and 3) a multi-modal modelling tool will be needed for the data analysis stage to be effective.

I discuss a game-theoretic model in which scientists compete to finish the intermediate stages of some research project. Banerjee et al. (2014) have previously shown that if the credit awarded for intermediate results is proportional to their difficulty, then the strategy profile in which scientists share each intermediate stage as soon as they complete it is a Nash equilibrium. I show that the equilibrium is both unique and strict. Thus rational credit-maximizing scientists have an incentive to share their intermediate results, as long as this is sufficiently rewarded.

In models for paraconsistent logics, the semantic values of sentences and their negations are less tightly connected than in classical logic. In “American Plan” logics for negation, truth and falsity are, to some degree, independent. The truth of ∼p is given by the falsity of p, and the falsity of ∼p is given by the truth of p. Since truth and falsity are only loosely connected, p and ∼p can both hold, or both fail to hold. In “Australian Plan” logics for negation, negation is treated rather like a modal operator, where the truth of ∼p in a situation amounts to p failing in certain other situations. Since those situations can be different from this one, p and ∼p might both hold here, or might both fail here.

In the context of superintelligent AI systems, the term “oracle” has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety.

We give a precise semantics for a proposed revised version of the Knowledge Interchange Format. We show that quantification over relations is possible in a first-order logic, but sequence variables take the language beyond first-order.

We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition.

The standard propositional account of necessary and sufficient conditions in many introductory logic textbooks is based on the material conditional. Some examples include (Barker-Plummer, Barwise, and Etchemendy 2011: 181-182), (Churchill 1986: 391-392), (Forbes 1994: 20-25), (Gabbay 2002: 68), (Haight 1999: 187-189), (Halverson 1984: 285- 286), (Hardegree 2011: 129), (Layman 2002: 250-251), (Leblanc and Wisdom 1976: 16-18), (Salmon 1984: 47-48), (P. Smith 2003: 132), (Suppes 1957: 8-10) and (Watson and Arp 2015: 149). In the appendix, pertinent excerpts from some of these resources are provided. In general, the typical exposition goes along the following lines (again, cf. the appendix): • “A is sufficient for B” is best rendered as “if A, then B”, or symbolically, (A ⊃ B). • “A is necessary for B” is best rendered as ”if not A, then not B”, or symbolically, (¬A ⊃ ¬B). This is equivalent to (B ⊃ A).

Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin’s [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane’s indicative conditional via a full and faithful computable translation from their logic to the modal logic K45.

In 1986 David Gauthier proposed an arbitration scheme for two player cardinal bargaining games based on interpersonal comparisons of players’ relative concessions. In Gauthier’s original arbitration scheme, players’ relative concessions are defined in terms of Raiffa-normalized cardinal utility gains, and so it cannot be directly applied to ordinal bargaining problems. In this paper I propose a relative benefit equilibrating bargaining solution (RBEBS ) for two and n-player ordinal and quasiconvex ordinal bargaining problems with finite sets of feasible basic agreements based on the measure of players’ ordinal relative individual advantage gains. I provide an axiomatic characterization of this bargaining solution and discuss the conceptual relationship between RBEBS and ordinal egalitarian bargaining solution (OEBS ) proposed by Conley and Wilkie (2012). I show the relationship between the measurement procedure for ordinal relative individual advantage gains and the measurement procedure for players’ ordinal relative concessions, and argue that the proposed arbitration scheme for ordinal games can be interpreted as an ordinal version of Gauthier’s arbitration scheme.

Algebra is a branch of mathematics sibling to geometry, analysis (calculus), number theory, combinatorics, etc. Although algebra has its roots in numerical domains such as the reals and the complex numbers, in its full generality it differs from its siblings in serving no specific mathematical domain. Whereas geometry treats spatial entities, analysis continuous variation, number theory integer arithmetic, and combinatorics discrete structures, algebra is equally applicable to all these and other mathematical domains. Elementary algebra, in use for centuries and taught in secondary school, is the arithmetic of indefinite quantities or variables \(x, y,\ldots\).

Suszko’s problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski’s structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko’s problem and on mixed consequence.

Rational choice theorists and deontic logicians both study actions, yet using very different approaches and tools. This paper introduces some choice-theoretic concepts – feasible options, choice contexts, choice functions, rankings of options, and reasons structures – into deontic logic. These concepts are used to define a simple ‘choice-theoretic’ language for deontic logic, and four ‘choice-theoretic’ semantics for that language, called basic, behavioural, ranking-based and reason-based semantics, respectively. We compare these semantics in terms of the strength of their entailment relations, and characterize precisely the ‘gaps’ in strength between weaker and stronger ones of these semantics.

In his seminal address delivered in 1945 to the Royal Society Gilbert Ryle considers a special case of knowing-how, viz., knowing how to reason according to logical rules. He argues that knowing how to use logical rules cannot be reduced to a propositional knowledge. We evaluate this argument in the context of two different types of formal systems capable to represent knowledge and support logical reasoning: Hilbert-style systems, which mainly rely on axioms, and Gentzen-style systems, which mainly rely on rules. We build a canonical syntactic translation between appropriate classes of such systems and demonstrate the crucial role of Deduction Theorem in this construction. This analysis suggests that one’s knowledge of axioms and one’s knowledge of rules under appropriate conditions are also mutually translatable. However our further analysis shows that the epistemic status of logical knowing-how ultimately depends on one’s conception of logical consequence: if one construes the logical consequence after Tarski in model-theoretic terms then the reduction of knowing-how to knowing-that is in a certain sense possible but if one thinks about the logical consequence after Prawitz in proof-theoretic terms then the logical knowledge-how gets an independent status. Finally we extend our analysis to the case of extra-logical knowledge-how representable with Gentzen-style formal systems, which admit constructive meaning explanations. For this end we build a typed sequential calculus and prove for it a “constructive” Deduction Theorem interpretable in extra-logical terms. We conclude with a number of open questions, which concern translations between knowledge-how and knowledge-that in this more general semantic setting.

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation.

The periodic table of elements represents and organizes all known chemical elements on the basis of their properties. While the importance of this table in chemistry is uncontroversial the role that it plays in scientific reasoning remains heavily disputed. Many philosophers deny the explanatory role of the periodic table, while insisting that it is “merely” classificatory (Shapere 1977, 534-5) (Scerri 1997a, 239). In particular, it has been claimed that the table doesn’t figure in causal explanation because it “does not reveal causal structure” (Woody 2014, 143). This paper argues that the modern periodic table does reveal causal structure in the sense of containing causal information that figures in explanations in chemistry. However, this analysis suggests that the earliest versions of the table did serve more of a classificatory role, as they lack the causal structure present in modern versions.

There is an argument based on sentences that describe pictures in favor of a viewpoint-centered possible worlds semantics for pictures, over a propositional semantics (J. Ross 1997). The argument involves perspectival lexical items such as “front”. We show that when a projective possible worlds semantics for pictures is employed, there is a problem with the argument coming from propositional contents being strong. The argument is reconstructed in a model modal space involving linear worlds, and it is shown that it works there, by computing the possible worlds semantics. The construction involves propositions and centered propositions that are regular sets of strings. Finally, by manipulating the marking parameter in a projective semantics for pictures, the argument is reconstructed also for 3D models.

What we call the Hilbert-Bernays (HB) Theorem establishes that for any satisfiable first-order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine’s claim that the HB theorem licenses us to define the first-order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine’s substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians.

The aim of this paper is to argue that models in cognitive science based on probabilistic computation should not be restricted to those procedures that almost surely (with probability 1) terminate. There are several reasons to consider non-terminating procedures as candidate components of cognitive models. One theoretical reason is that there is a perfect correspondence between the enumerable semi-measures and all probabilistic programs, as we demonstrate here (generalizing a better-known fact about computable measures and almost-surely halting programs). One practical reason is that the line between almost sure termination and non-termination is elusive, as well as arbitrary. We argue that this matters not only for theorists, but also potentially for a learner faced with the task of inducing programs from experience.

Lawrence S. Moss Indiana University Bloomington, IN, USA lsm@cs.indiana.edu tures of natural language is the prevalence of “upward” and “downward” inferences involving determiners and other functional expressions. These inferences are associated with negative and positive polarity positions in syntax, and they also feature in computer implementations of textual entailment. Formal treatments of these phenomena began in the 1980’s and have been refined and expanded in the last 10 years. This paper takes a large step in the area by extending typed lambda calculus to the ordered setting. Not only does this provide a formal tool for reasoning about upward and downward inferences in natural language, it also applies to the analysis of monotonicity arguments in mathematics more generally.

In the literature over the Ramsey-sentence approach to structural realism, there is often debate over whether structural realists can legitimately restrict the range of the second-order quantifiers, in order to avoid the Newman problem. In this paper, I argue that even if they are allowed to, it won’t help: even if the Ramsey sentence is interpreted using such restricted quantifiers, it is still an implausible candidate to capture a theory’s structural content. To do so, I use the following observation: if a Ramsey sentence did encode a theory’s structural content, then two theories would be structurally equivalent just in case they have logically equivalent Ramsey sentences. I then argue that this criterion for structural equivalence is implausible, even where frame or Henkin semantics are used.

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the “quantum logic” of subspaces of a general vector space–which is then specialized to the closed subspaces of a Hilbert space. But there is a “dual” progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space–which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.

This essay is about how the notion of “structure” in ontic structuralism might be made precise. More specifically, my aim is to make precise the idea that the structure of the world is (somehow) given by the relations inhering in the world, in such a way that the relations are ontologically prior to their relata. The central claim is the following: one can do so by giving due attention to the relationships that hold between those relations, by making use of certain notions from algebraic logic. In the remainder of this introduction, I sketch two motivations for structuralism, and make some preliminary remarks about the relationship between structuralism and ontological dependence; in the next section, I outline my preferred way of unpacking the notion of “structure”; in §3, I compare this view to Dasgupta’s algebraic generalism; and finally, I evaluate the view, by considering how well it can be defended against objections, and how well it discharges the motivations with which we began.