The epistemology of absence-based inference

Our main aim in this paper is to contribute towards a better understanding of the epistemology of absence-based inferences. Many absence-based inferences are classified as fallacies. There are exceptions, however. We investigate what features make absence-based inferences epistemically good or reliable. In Section 2 we present Sanford Goldberg’s account of the reliability of absence-based inference, introducing the central notion of epistemic coverage. In Section 3 we approach the idea of epistemic coverage through a comparison of alethic and evidential principles. The Equivalence Schema–a well-known alethic principle–says that it is true that $p$ if and only if $p$ . We take epistemic coverage to underwrite a suitably qualified evidential analogue of the Equivalence Schema: for a high proportion of values of $p$ , subject $S$ has evidence that $p$ due to her reliance on source $S^{*}$ if and only if $p$ . We show how this evidential version of the Equivalence Schema suffices for the reliability of certain absence-based inferences. Section 4 is dedicated to exploring consequences of the Evidential Equivalence Schema. The slogan ‘absence of evidence is evidence of absence’ has received a lot of bad press. More elaborately, what has received a lot of bad press is something like the following idea: absence of evidence sufficiently good to justify belief in $p$ is evidence sufficiently good to justify belief in $\sim p$ . A striking consequence of the Evidential Equivalence Schema is that absence of evidence sufficiently good to justify belief in p is evidence sufficiently good to justify belief in $\sim p$ . We establish this claim in Section 4 and show how this supports the reliability of an additional type of absence-based inference. Section 4 immediately raises the following question: how can we make philosophically good sense of the idea that absence of evidence is evidence of absence? We address this question in Section 5. Section 6 contains some summary remarks.     

Actuality in Propositional Modal Logic

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5.     

Phenomenalist dogmatist experientialism and the distinctiveness problem

Phenomenalist dogmatist experientialism (PDE) holds the following thesis: if $S$ has a perceptual experience that $p$ , then $S$ has immediate prima facie evidential justification for the belief that $p$ in virtue of the experience’s phenomenology. The benefits of PDE are that it (a) provides an undemanding view of perceptual justification that allows most of our ordinary perceptual beliefs to be justified, and (b) accommodates two important internalist intuitions, viz. the New Evil Demon Intuition and the Blindsight Intuition. However, in the face of a specific version of the Sellarsian dilemma, PDE is ad hoc. PDE needs to explain what is so distinct about perceptual experience that enables it to fulfill its evidential role without being itself in need of justification. I argue that neither an experience’s presentational phenomenology, nor its phenomenal forcefulness can be used to answer this question, and that prospects look dim for any other phenomenalist account. The subjective distinctness of perceptual experience might instead just stem from a higher-order belief that the experience is a perceptual one, but this will only serve to strengthen the case for externalism: externalism is better suited to provide an account of how we attain justified higher-order beliefs and can use this account to accommodate the Blindsight Intuition.     

Inverse Images of Box Formulas in Modal Logic

We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which ${\square B}$ is provably equivalent to ${\square A}$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (‘cancellation’) rule taking us from ${\square C \leftrightarrow \square D}$ to ${C \leftrightarrow D}$ , there is only one formula B, to within equivalence, in this inverse image, as we shall call it, of ${\square A}$ (relative to the logic concerned); for logics for which the intended reading of “ ${\square}$ ” is epistemic or doxastic, failure to be closed under this rule indicates that from the proposition expressed by a knowledge- or belief-attribution, the propositional object of the attitude in question cannot be recovered: arguably, a somewhat disconcerting situation. More generally, the inverse image of ${\square A}$ may comprise a range of non-equivalent formulas, all those provably implied by one fixed formula and provably implying another—though we shall see that for several choices of logic and of the formula A, there is not even such an ‘interval characterization’ of the inverse image (of ${\square A}$ ) to be found.     

Perceptual experience and seeing that p

I open my eyes and see that the lemon before me is yellow. States like this—states of seeing that $p$ —appear to be visual perceptual states, in some sense. They also appear to be propositional attitudes (and so states with propositional representational contents). It might seem, then, like a view of perceptual experience on which experiences have propositional representational contents—a Propositional View—has to be the correct sort of view for states of seeing that $p$ . And thus we can’t sustain fully general non-Propositional but Representational, or Relational Views of experience. But despite what we might initially be inclined to think when reflecting upon the apparent features of states of seeing that $p$ , a non-propositional view of seeing that $p$ is, I argue, perfectly intelligible.     

Completeness in Hybrid Type Theory

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \(@_i\) in propositional and first-order hybrid logic. This means: interpret \(@_i\alpha _a\) , where \(\alpha _a\) is an expression of any type \(a\) , as an expression of type \(a\) that rigidly returns the value that \(\alpha_a\) receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual inhybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic.     

Dependence and Independence

We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using = ${(\vec{x}, \vec{y})}$ have.     

Dynamic Modalities

A new modal logic containing four dynamic modalities with the following informal reading is introduced: ${\square^\forall}$ – always necessary, ${\square^\exists}$ – sometimes necessary, and their duals – ${\diamondsuit^\forall}$ – always possibly, and ${\diamondsuit^\exists}$ – sometimes possibly. We present a complete axiomatization with respect to the intended formal semantics and prove decidability via fmp.     

Dynamics of lying

We propose a dynamic logic of lying, wherein a ‘lie that $\varphi $ ’ (where $\varphi $ is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula $\varphi $ . The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish two speaker perspectives: (Obs) an outside observer who is lying to an agent that is modelled in the system, and (Ag) an agent who is lying to another agent, and where both are modelled in the system. We distinguish three addressee perspectives: (Cred) the credulous agent who believes everything that it is told (even at the price of inconsistency), (Skep) the skeptical agent who only believes what it is told if that is consistent with its current beliefs, and (Rev) the belief revising agent who believes everything that it is told by consistently revising its current, possibly conflicting, beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or in the extension of that logic to belief revision.     

Semi-intuitionistic Logic

The purpose of this paper is to define a new logic ${\mathcal {SI}}$ called semi-intuitionistic logic such that the semi-Heyting algebras introduced in [4] by Sankappanavar are the semantics for ${\mathcal {SI}}$ . Besides, the intuitionistic logic will be an axiomatic extension of ${\mathcal {SI}}$ .     

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.     

The Church–Fitch knowability paradox in the light of structural proof theory

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$ , by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$ (KP). The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$ (OP). A Gentzen-style reconstruction of the Church–Fitch paradox is presented following a labelled approach to sequent calculi. First, a cut-free system for classical (resp. intuitionistic) bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism.     

Classical Modal De Morgan Algebras

In this note we introduce the variety ${{\mathcal C}{\mathcal D}{\mathcal M}_\square}$ of classical modal De Morgan algebras as a generalization of the variety ${{{\mathcal T}{\mathcal M}{\mathcal A}}}$ of Tetravalent Modal algebras studied in [11]. We show that the variety ${{\mathcal V}_0}$ defined by H. P. Sankappanavar in [13], and the variety S of Involutive Stone algebras introduced by R. Cignoli and M. S de Gallego in [5], are examples of classical modal De Morgan algebras. We give a representation theory, and we study the regular filters, i.e., lattice filters closed under an implication operation. Finally we prove that the variety ${{{\mathcal T}{\mathcal M}{\mathcal A}}}$ has the Amalgamation Property and the Superamalgamation Property.     

Reconstructing an Open Order from Its Closure, with Applications to Space-Time Physics and to Logic

In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation ${\preccurlyeq}$ can be uniquely reconstructed if we know the “interior” ${\prec}$ of the order relation. It is also known that in some cases, we can uniquely reconstruct ${\prec}$ (and hence, topology) from ${\preccurlyeq}$ . In this paper, we show that, in general, under reasonable conditions, the open order ${\prec}$ (and hence, the corresponding topology) can be uniquely determined from its closure ${\preccurlyeq}$ .     

Probability Propagation in Generalized Inference Forms

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach.     

Decidability of the Equational Theory of the Continuous Geometry CG(\Bbb {F})

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.     

The variance of d′ estimates obtained in yes—no and two-interval forced choice procedures

In studies of detection and discrimination, data are often obtained in the form of a 2 × 2 matrix and then converted to an estimate of d′, based on the assumptions that the underlying decision distributions are Gaussian and equal in variance. The statistical properties of the estimate of d′, $\hat d'$ , are well understood for data obtained using the yes—no procedure, but less effort has been devoted to the more commonly used two-interval forced choice (2IFC) procedure. The variance associated with $\hat d'$ is a function of trued′ in both procedures, but for small values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be less than the variance of $\hat d'$ obtained using yes—no; for large values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be greater than the variance of $\hat d'$ from yes—no. These results follow from standard assumptions about the relationship between the two procedures. The present paper reviews the statistical properties of $\hat d'$ obtained using the two standard procedures and compares estimates of the variance of $\hat d'$ as a function of trued′ with the variance observed in values of $\hat d'$ obtained with a 2IFC procedure.     

On the Algebraizability of the Implicational Fragment of Abelian Logic

In this paper we consider the implicational fragment of Abelian logic \({{{\sf A}_{\rightarrow}}}\) . We show that although the Abelian groups provide an semantics for the set of theorems of \({{{\sf A}_{\rightarrow}}}\) they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi (Mem Am Math Soc 77, 1989). In the second part of the paper, we investigate an extension of \({{{\sf A}_{\rightarrow}}}\) in the same language and having the same set of theorems and show that this new consequence relation is algebraizable with the Abelian groups as its equivalent algebraic semantics. Finally, we show that although \({{{\sf A}_{\rightarrow}}}\) is not algebraizable, it is order-algebraizable in the sense of Raftery (Ann Pure Appl Log 164:251–283, 2013).     

Indistinguishability, Choices, and Logics of Agency

This paper deals with structures ${\langle{\bf T}, I\rangle}$ in which T is a tree and I is a function assigning each moment a partition of the set of histories passing through it. The function I is called indistinguishability and generalizes the notion of undividedness. Belnap’s choices are particular indistinguishability functions. Structures ${\langle{\bf T}, I\rangle}$ provide a semantics for a language ${\mathcal{L}}$ with tense and modal operators. The first part of the paper investigates the set-theoretical properties of the set of indistinguishability classes, which has a tree structure. The significant relations between this tree and T are established within a general theory of trees. The aim of second part is testing the expressive power of the language ${\mathcal{L}}$ . The natural environment for this kind of investigations is Belnap’s seeing to it that (stit). It will be proved that the hybrid extension of ${\mathcal{L}}$ (with a simultaneity operator) is suitable for expressing stit concepts in a purely temporal language.