The Category of Node-and-Choice Preforms for Extensive-Form Games

It would be useful to have a category of extensive-form games whose isomorphisms specify equivalences between games. Since working with entire games is too large a project for a single paper, I begin here with preforms, where a “preform” is a rooted tree together with choices and information sets. In particular, this paper first defines the category \(\mathbf {Tree}\), whose objects are “functioned trees”, which are specially designed to be incorporated into preforms. I show that \(\mathbf {Tree}\) is isomorphic to the full subcategory of \(\mathbf {Grph}\) whose objects are converging arborescences. Then the paper defines the category \(\mathbf {NCP}\), whose objects are “node-and-choice preforms”, each of which consists of a node set, a choice set, and an operator mapping node-choice pairs to nodes. I characterize the \(\mathbf {NCP}\) isomorphisms, define a forgetful functor from \(\mathbf {NCP}\) to \(\mathbf {Tree}\), and show that \(\mathbf {Tree}\) is equivalent to the full subcategory of \(\mathbf {NCP}\) whose objects are perfect-information preforms. The paper also shows that many game-theoretic entities can be derived from preforms, and that these entities are well-behaved with respect to \(\mathbf {NCP}\) morphisms and isomorphisms.     

Correia Semantics Revisited

Despite a renewed interest in Richard Angell’s logic of analytic containment (\({\mathsf{AC}}\)), the first semantics for \({\mathsf{AC}}\) introduced by Fabrice Correia has remained largely unexamined. This paper describes a reasonable approach to Correia semantics by means of a correspondence with a nine-valued semantics for \({\mathsf{AC}}\). The present inquiry employs this correspondence to provide characterizations of a number of propositional logics intermediate between \({\mathsf{AC}}\) and classical logic. In particular, we examine Correia’s purported characterization of classical logic with respect to his semantics, showing the condition Correia cites in fact characterizes the “logic of paradox” \({\mathsf{LP}}\) and provide a correct characterization. Finally, we consider some remarks on related matters, such as the applicability of the present correspondence to the analysis of the system \({\mathsf{AC}^{\ast}}\) and an intriguing relationship between Correia’s models and articular models for first degree entailment.     

Topological-Frame Products of Modal Logics

The simplest bimodal combination of unimodal logics \(\text {L} _1\) and \(\text {L} _2\) is their fusion, \(\text {L} _1 \otimes \text {L} _2\), axiomatized by the theorems of \(\text {L} _1\) for \(\square _1\) and of \(\text {L} _2\) for \(\square _2\), and the rules of modus ponens, necessitation for \(\square _1\) and for \(\square _2\), and substitution. Shehtman introduced the frame product \(\text {L} _1 \times \text {L} _2\), as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced the topological product \(\text {L} _1 \times _t \text {L} _2\), as the logic of the products of certain topological spaces. For almost all well-studies logics, we have \(\text {L} _1 \otimes \text {L} _2 \subsetneq \text {L} _1 \times \text {L} _2\), for example, \(\text {S4} \otimes \text {S4} \subsetneq \text {S4} \times \text {S4} \). Van Benthem et al. show, by contrast, that \(\text {S4} \times _t \text {S4} = \text {S4} \otimes \text {S4} \). It is straightforward to define the product of a topological space and a frame: the result is a topologized frame, i.e., a set together with a topology and a binary relation. In this paper, we introduce topological-frame products \(\text {L} _1 \times _ tf \text {L} _2\) of modal logics, providing a complete axiomatization of \(\text {S4} \times _ tf \text {L} \), whenever \(\text {L} \) is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include \(\text {T} , \text {S4} \) and \(\text {S5} \), but not \(\text {K} \) or \(\text {K4} \). We leave open the problem of axiomatizing \(\text {S4} \times _ tf \text {K} \), \(\text {S4} \times _ tf \text {K4} \), and other related logics. When \(\text {L} = \text {S4} \), our result confirms a conjecture of van Benthem et al. concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.     

On All Strong Kleene Generalizations of Classical Logic

By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the \({{\bf SK}^\mathcal{P}}\) and \({\bf SK}^{\mathcal{N}}\) calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, we also hint at its philosophical significance.     

Axiomatizing Jaśkowski’s Discussive Logic $$\mathbf {D_2}$$

We outline the rather complicated history of attempts at axiomatizing Ja?kowski’s discussive logic \(\mathbf {D_2}\) and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing \(\mathbf {D_2}\) in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of \(\mathbf {D_2}\) in the language Ja?kowski suggested in his second paper on discussive logic, by following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of \(\mathbf {D_2}\), introduced by Ciuciura, in which negation is also taken to be discussive.     

Public announcement logic with distributed knowledge: expressivity,completeness and complexity

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with distributed knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (\(\mathcal{PAL }\)) with distributed knowledge, in particular their expressivity, axiomatisations and complexity. \(\mathcal{PAL }\) extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on \(\mathcal{PACD }\), the result of adding both common and distributed knowledge to \(\mathcal{PAL }\), which is more expressive than each of its component logics. We introduce an axiomatisation of \(\mathcal{PACD }\), which is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with \(S5\) knowledge, distributed knowledge, common knowledge and public announcements at the same time. We furthermore show that \(\mathcal{PACD }\) is decidable, more precisely that it is \(\textsc {exptime}\)-complete. This result also carries over to \(\mathcal{S 5\mathcal CD }\) with common and distributed knowledge operators for all coalitions (and not only the grand coalition). Finally, we propose a notion of a trans-bisimulation to generalise certain results and give deeper insight into the proofs.     

Arithmetical Completeness Theorem for Modal Logic $$\mathsf{}$$

We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \(\Sigma _2\) provability predicate of T whose provability logic is precisely the modal logic \(\mathsf{K}\). For this purpose, we introduce a new bimodal logic \(\mathsf{GLK}\), and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \(\mathsf{GLK}\).     

Praise,blame, and demandingness

Consequentialism has been challenged on the grounds that it is too demanding. I will respond to the problem of demandingness differently from previous accounts. In the first part of the paper, I argue that consequentialism requires us to distinguish the justification of an act \(\varphi\) from the justification of an act \(\psi\), where \(\psi\) is an act of praise or blame. In the second part of the paper, I confront the problem of demandingness. I do not attempt to rule out the objection; instead, I argue that if certain plausible empirical claims about moral motivation are true, we morally ought not to blame people for failing to meet certain very demanding obligations. With this theory, we create a space in consequentialism for intuitions questioning the plausibility of demanding obligations. I conclude the paper by showing that separate justifications for \(\varphi\) and \(\psi\) may also give us a theoretical niche for intuitions about supererogation.     

Sequent Calculi for Semi-De Morgan and De Morgan Algebras

A contraction-free and cut-free sequent calculus \(\mathsf {G3SDM}\) for semi-De Morgan algebras, and a structural-rule-free and single-succedent sequent calculus \(\mathsf {G3DM}\) for De Morgan algebras are developed. The cut rule is admissible in both sequent calculi. Both calculi enjoy the decidability and Craig interpolation. The sequent calculi are applied to prove some embedding theorems: \(\mathsf {G3DM}\) is embedded into \(\mathsf {G3SDM}\) via Gödel–Gentzen translation. \(\mathsf {G3DM}\) is embedded into a sequent calculus for classical propositional logic. \(\mathsf {G3SDM}\) is embedded into the sequent calculus \(\mathsf {G3ip}\) for intuitionistic propositional logic.     

A Theory of Infinitary Relations Extending Zermelo’s Theory of Infinitary Propositions

An idea attributable to Russell serves to extend Zermelo’s theory of systems of infinitely long propositions to infinitary relations. Specifically, relations over a given domain \({\mathfrak{D}}\) of individuals will now be identified with propositions over an auxiliary domain \({\mathfrak{D}^{\mathord{\ast}}}\) subsuming \({\mathfrak{D}}\). Three applications of the resulting theory of infinitary relations are presented. First, it is used to reconstruct Zermelo’s original theory of urelements and sets in a manner that achieves most, if not all, of his early aims. Second, the new account of infinitary relations makes possible a concise characterization of parametric definability with respect to a purely relational structure. Finally, based on his foundational philosophy of the primacy of the infinite, Zermelo rejected Gödel’s First Incompleteness Theorem; it is shown that the new theory of infinitary relations can be brought to bear, positively, in that connection as well.     

Turing–Taylor Expansions for Arithmetic Theories

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \({\Pi_{n+1}}\) proof-theoretic ordinal \({|U|_{\Pi^0_{n+1}}}\) also denoted \({|U|_n}\). As such, to each theory U we can assign the sequence of corresponding \({\Pi_{n+1}}\) ordinals \({\langle |U|_n\rangle_{n > 0}}\). We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev’s universal model for the closed fragment of the polymodal provability logic \({\mathsf{GLP}_\omega}\). In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev’s model.     

What is Nominalistic Mereology?

Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language $\mathcal {H}_{\textsf {m}}$ is maximally acceptable for nominalistic mereology. In an extension $\mathcal {H}_{\textsf {gem}}$ of $\mathcal {H}_{\textsf {m}}$ , a modal analog for the classical systems of Leonard and Goodman (J Symb Log 5:45–55, 1940) and Le?niewski (1916) is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for $\mathcal {H}_{\textsf {gem}}$ -bisimulations.     

Arithmetical Soundness and Completeness for $$\varvec{\Sigma }_{\varvec{2}}$$ Numerations

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \(n \ge 2\), there exists a \(\Sigma _2\) numeration \(\tau (u)\) of T such that the provability logic of the provability predicate \(\mathsf{Pr}_\tau (x)\) naturally constructed from \(\tau (u)\) is exactly \(\mathsf{K}+ \Box (\Box ^n p \rightarrow p) \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.     

Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures

Every transformation monoid comes equipped with a canonical topology, the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever \({\mathbf {A}}\) is a countable \(\aleph _0\)-categorical G-finite structure whose automorphism group has a trivial center and if \({\mathbf {B}}\) is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism.     

Foundationalism with infinite regresses of probabilistic support

There is a long-standing debate in epistemology on the structure of justification. Some recent work in formal epistemology promises to shed some new light on that debate. I have in mind here some recent work by David Atkinson and Jeanne Peijnenburg, hereafter “A&P”, on infinite regresses of probabilistic support. A&P show that there are probability distributions defined over an infinite set of propositions {\(p_{1}, p_{2}, p_{3}, {\ldots }, p_{n}, {\ldots }\}\) such that (i) \(p_{i}\) is probabilistically supported by \(p_{i+1}\) for all i and (ii) \(p_{1}\) has a high probability. Let this result be “APR” (short for “A&P’s Result”). A&P oftentimes write as though they believe that APR runs counter to foundationalism. This makes sense, since there is some prima facie plausibility in the idea that APR runs counter to foundationalism, and since some prominent foundationalists argue for theses inconsistent with APR. I argue, though, that in fact APR does not run counter to foundationalism. I further argue that there is a place in foundationalism for infinite regresses of probabilistic support.     

Factor analysis models via I-divergence optimization

Given a positive definite covariance matrix \(\widehat{\Sigma }\) of dimension n, we approximate it with a covariance of the form \(HH^\top +D\), where H has a prescribed number \(k<n\) of columns and \(D>0\) is diagonal. The quality of the approximation is gauged by the I-divergence between the zero mean normal laws with covariances \(\widehat{\Sigma }\) and \(HH^\top +D\), respectively. To determine a pair (H, D) that minimizes the I-divergence we construct, by lifting the minimization into a larger space, an iterative alternating minimization algorithm (AML) à la Csiszár–Tusnády. As it turns out, the proper choice of the enlarged space is crucial for optimization. The convergence of the algorithm is studied, with special attention given to the case where D is singular. The theoretical properties of the AML are compared to those of the popular EM algorithm for exploratory factor analysis. Inspired by the ECME (a Newton–Raphson variation on EM), we develop a similar variant of AML, called ACML, and in a few numerical experiments, we compare the performances of the four algorithms.     

Resampling-Based Inference Methods for Comparing Two Coefficients Alpha

The two-sample problem for Cronbach’s coefficient \(\alpha _C\), as an estimate of test or composite score reliability, has attracted little attention compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistical procedures of comparing two coefficients \(\alpha _{C,1}\) and \(\alpha _{C,2}\). The null hypothesis of interest is \(H_0 : \alpha _{C,1} = \alpha _{C,2}\), which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed for two-group multivariate non-normal models under the general asymptotically distribution-free (ADF) setting. These statistical tests ensure a better control of the type-I error, in finite or very small sample sizes, when the state-of-affairs ADF large-sample test may fail to properly attain the nominal significance level. By proper choice of a studentized test statistic, the resampling tests are modified in order to be valid asymptotically even in non-exchangeable data frameworks. Moreover, extensions of this approach to other designs and reliability measures are discussed as well. Finally, the usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.     

Infinite Populations,Choice and Determinacy

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy (\(\mathsf {AD}\)), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \(\mathsf {AD}\) from the point of view of the research area in question is its game-theoretic character.