Constructing formal semantics from an ontological perspective. The case of second-order logics

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view \(O\) and wants to interpret a formal language \(L\) , a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for \(L\) in such a way that its ontological commitment will be in perfect accordance with \(O\) . Basically, that is what I call constructing formal semantics from an ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. From those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, showing thus that we can perfectly quantify over properties and relations while being ontologically committed only to individuals. I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and thus why endorsing an ontological view should have an impact on the kind of logic one should use.     

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.     

On the notions of indiscernibility and indeterminacy in the light of the Galois–Grothendieck theory

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry.     

Divine thoughts and Fregean propositional realism

Anderson and Welty have recently advanced an argument for the claim that the laws of logic are ontologically dependent upon a necessarily existent mind, i.e. God. In this paper I argue that a key premise of Anderson and Welty’s argument—viz., a premise which asserts that \(x\) is intrinsically intentional only if \(x\) is mind-dependent—is false, for on a broadly Fregean account of propositions, propositions are intrinsically intentional but not mind-dependent.     

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.     

Teacher involvement as a protective factor from the association between race-based bullying and smoking initiation

Experiencing bullying as a victim is associated with negative health and health behavior outcomes, including substance use, among adolescents. However, understandings of protective factors—factors that enhance adolescents’ resilience to the negative consequences of bullying—remain limited. The current study investigates whether teacher involvement protects adolescent students from the association between being bullied due to race and smoking initiation. Students were recruited from 12 Kindergarten through 8th grade schools in an urban school district in the Northeast United States. The analytic sample included 769 students who responded to surveys in 5th or 6th grade (2009), and two years later in 7th or 8th grade (2011). Students primarily identified as Latino and/or Black, and 90 % were eligible for free or reduced lunch. Fifty-four (7 %) students initiated smoking between survey time points. Among students reporting lower teacher involvement, race-based bullying was associated with higher likelihood of smoking initiation (OR \(=\) 1.69, \(p =0.03\) ). In contrast, among students reporting higher teacher involvement, race-based bullying was not associated with higher likelihood of smoking initiation (OR \(=\) 0.95, \(p =0.81\) ). Results suggest that teacher involvement may protect students from the association between race-based bullying and smoking initiation. Enhancing teacher involvement among students experiencing race-based bullying in schools may limit smoking initiation.     

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 ${\mathcal {SI}}$ .     

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.     

Acquisition and decision in visual same-different search of letter displays

From 2 to 23 capital As, Bs, and Cs were positioned randomly over visual displays varying in sizefrom5 to 10deg square and in luminance from7 to 250cd/m², The task was to decide whether all letters were the same or one was different from the rest. Instructions stressed accuracy, and responses were 97% correct. Three experiments with 50 observers varied amount of practice, number of letters (N), and size and luminance of the display. All experiments produced a linear invariance between mean “same” \((\bar S)\) and mean “different” \((\bar D)\) response times in seconds with N as the parameter: \(\bar D\) ? \(\bar S\) /2+4. The data are consistent with Krueger’s same-different decision theory, and with the separation of acquisition from decision processes.     

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).     

Analytic Calculi for Circular Concepts by Finite Revision

The paper introduces Hilbert– and Gentzen-style calculi which correspond to systems ${\mathsf{C}_{n}}$ from Gupta and Belnap [3]. Systems ${\mathsf{C}_{n}}$ were shown to be sound and complete with respect to the semantics of finite revision. Here, it is shown that Gentzen-style systems ${\mathsf{GC}_{n}}$ admit a syntactic proof of cut elimination. As a consequence, it follows that they are consistent.     

Lost confidence and potential: a mixed methods study of underachieving college students’ sources of self-efficacy

This study investigated the sources of self-efficacy of college students enrolled in a learning frameworks course, whom we identified as underachievers. Through weekly journal entries, the group of underachievers ( \(n=13\) ) cited the factors that made them feel confident or not confident. Students’ responses to open-ended prompts were coded according to Bandura’s (Self-efficacy: the exercise of control, W.H. Freeman, New York, 1997) four hypothesized sources of self-efficacy and an additional emergent theme of self-regulated learning. After, through student responses on a self-report survey, we compared the group of underachievers to a group of achievers ( \(n=36\) ) on their self-efficacy and its sources. Both quantitative and qualitative results underscored the importance of mastery experiences for underachievers. The underachievers had significantly less mastery experiences and verbal persuasions despite having similar levels of self-efficacy. Implications for how to cultivate mastery and to provide efficacy-supportive feedback as well as psychological mechanisms regarding high potential underachievers are discussed.     

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.     

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 ${\preccurlyeq}$ .     

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.     

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.     

Margin for error semantics and signal perception

A joint modelling of objective worlds and subjective perceptions within two-dimensional semantics eliminates the margin for error principle and solves the epistemic sorites paradox. Two objective knowledge modalities can be defined in two-dimensional frames accounting for subjective perceptions: “necessary knowledge” (NK) and “possible knowledge” (PK), the latter being better suited to the interpretation of knowledge utterances. Two-dimensional semantics can in some cases be reduced to one-dimensional ones, by defining accessibility relations between objective worlds that reflect subjective perceptions: NK and PK are respectively equivalent to ${\square \square }$ and ${\lozenge \square }$ in some one-dimensional frame, and to ${\square }$ and another modality in some other.     

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.     

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.