ABSTRACT. Reasoning rarely proceeds exclusively with categorical claims. Especially under uncertainty, the claims that enter into our reasoning are often qualified: they may carry, for example, a prevalence qualification such as "most", a likelihood qualification such as "likely", an evidential qualification such as "strongly supported", or a practical-normative qualification such as "provisionally recommended". Such qualifications matter inferentially: they can be strengthened or weakened, they may conflict, and conclusions carrying them can themselves become premises in further reasoning.

In this talk I focus on a basic but pervasive pattern of qualified reasoning: from the claim that an individual a is an F, together with a qualified generalization such as "most Fs are G", we infer a qualified singular conclusion such as "a is likely to be G".

Adaptive logics already provide a general framework for defeasible reasoning in which conclusions may be derived conditionally and later withdrawn when relevant abnormalities are established. They have been applied, among other things, to inductive generalization and abduction. The novelty explored here lies not primarily in this dynamic machinery, but in bringing the qualifications themselves into the object language and into the inferential relations represented by the logic. Rather than defeasibly deriving the unqualified conclusion that a is G, the aim is to derive a qualified conclusion such as "a is likely to be G" from a differently qualified premise such as "most Fs are G". This requires bridging principles between graded prevalence and graded likelihood, and allows the resulting qualified conclusion, including its degree of qualification, to affect subsequent reasoning.

I will explore how such reasoning can be represented within the framework of adaptive logics. This requires extending the underlying logics with generalized quantifiers and graded likelihood operators, and identifying appropriate abnormalities for the defeasible bridging inferences. Further information may rebut a qualified conclusion or undercut the inference supporting it, in which case the conclusion—and further conclusions depending on it—may have to be withdrawn or revised.

A simple running example will be used to discuss the required bridging principles, the logical form of abnormalities, and some complications raised by conflicts between differently qua

ABSTRACT. Scientific understanding (henceforth, understanding) is commonly characterized as an epistemic achievement involving the grasping of relations of dependence. Yet, comparatively little attention has been paid to its inferential character and to the possibility of subjecting it to logical analysis. Whatever the correct account of the relations between understanding, truth, justification, and belief may be, understanding seems intimately connected to our capacity to draw appropriate inferences: the better one understands a phenomenon, the better positioned one is to identify its consequences, assess the relevance of new evidence, and determine how changes in one’s epistemic commitments affect what follows from them. This talk argues that understanding should be treated as a saliently inferential epistemic phenomenon and, consequently, as a legitimate subject of logical investigation. The aim is not to propose a particular logic of understanding, but to establish why such a logic is needed and to articulate some of the desiderata that an adequate logical treatment should satisfy.

One central desideratum concerns the combination of stability and flexibility in the inferential dynamics of understanding. Understanding involves a form of inferential stability: the acquisition of new information need not invalidate previously supported inferences or undermine every established relation of dependence. At the same time, understanding must remain responsive to new evidence, including evidence that reveals exceptions, exposes previously unnoticed dependencies, or requires the revision of explanatory commitments. An adequate logical treatment must therefore account for both the preservation of inferential relations and their context-sensitive revision. This challenge cannot be addressed simply by characterizing understanding as the accumulation of propositions or as the possession of a set of valid inferences. It requires an account of how inferences are embedded in, and constrained by, a structured body of epistemic commitments.

The Network Model of Understanding (NMU) provides a framework for making this challenge explicit. It represents understanding through a network (N,E,L,W), whose nodes include beliefs, models, data, and theoretical principles, and whose edges represent relations such as inference, explanation, causation, and grounding. Understanding, on this account, depends not merely on possessing epistemic elements but on grasping how they are connected and how their relations support, constrain, and sometimes undermine one another. This framework highlights a further desideratum for a logic of understanding: it should distinguish the different relations that contribute to inferential competence without treating them as interchangeable. The challenge becomes particularly pressing in scientific understanding, where causal explanation is frequently regarded as central. Does understanding a causal explanation involve two distinct conditional relations: one capturing causal dependence and another capturing the explanatory relevance of that dependence to a particular phenomenon, given a model, background assumptions, and an evidential context? If so, a logic of understanding must account for how these relations interact without conflating causal dependence with explanatory adequacy.

The talk thus poses a challenge for the logical analysis of understanding. It seeks, first, to establish understanding as a distinctively inferential epistemic phenomenon deserving of logical treatment and, second, to identify desiderata for such a treatment, including the articulation of stability and flexibility, the sensitivity of inference to new evidence, and the differentiation of causal, inferential, and explanatory relations. The hypothesis of a double conditional logic serves as a focal question rather than a proposed solution: whether understanding requires one conditional logic, two interacting conditional relations, or a more complex logical architecture remains open. By shifting attention from the construction of a formal system to the epistemic features that any such system must capture, the talk invites logicians to reconsider what a logic of understanding should explain and what constraints the phenomenon itself places on its formalization.

ABSTRACT. Bertrand Russell is well known for his causal eliminativism in his celebrated On the Notion of Cause (1912), a view he famously expressed by his claim that causation is retained as an important scientific notion only because, like the monarchy, people assume it to do no harm.

On this view metaphysically fundamental scientific theories, such as mathematical physics, dispense with the notion of cause in their fundamental physical laws. However, years later in The Analysis of Matter (1927) and Human Knowledge (1948) Russell returned to the discussion of causation and argues for the following two views, which are prima facie incompatible with each other and incompatible with his earlier claim.

First, that there are separable causal lines which should be identified with physically distinguished geodesics in relativistic space-time (1927, 1948) and that, (2) causation is a fundamental postulate in non-demonstrative scientific inference (1948) in his epistemic structural realism. These two claims seem inconsistent with Russell’s main claim in On the Notion of Cause, but in this talk I will argue, after elaborating on these theses, that these are compatible with each other and with Russell’s earlier eliminativistic thesis. Furthermore, I explain how these are part of the logical atomist research program (Landini, 2014; Klement, 2017; Elkind, 2019).

ABSTRACT. This talk develops a semantic framework with two connected aims. First, it seeks a genuinely hyperintensional semantics for quantum informational systems, one capable of distinguishing not only which propositions are true but also the exact informational states by which they are made true. Second, it aims to isolate and systematically describe the most fundamental semantic and informational respects in which quantum structures differ from classical ones, independently of their microphysical realization and independently of probabilistic considerations. The aim is not to explain specifically physical phenomena, but to identify structural features of information itself that underlie quantum reasoning.

The framework builds on Fine’s exact truthmaker semantics [2, 3]. Exact truthmaker semantics already provides a hyperintensional account of content through the distinction between exact verification and exact falsification. However, it remains largely embedded within a classical conception of information. Informational states are assumed to combine coherently, and the resulting propositional structures are Boolean. Quantum information departs from these assumptions. At the same time, extensional semantics is too coarse to represent such departures because it identifies propositions with their extensions and thereby loses track of their exact informational grounds. Quantum Truthmaker Semantics (QTM) preserves the Finean distinction between exact verifiers and exact falsifiers while replacing the underlying classical informational geometry by a quantum one.

Three structural features are especially important. First, non-contrary incompatibility: there exist informational states that are not contrary yet cannot coherently obtain together. Classically, incompatibility and contradiction coincide; in quantum structures they come apart. Second, non-primeness: maximally informative states may verify a disjunction without verifying either disjunct. Third, non-specifiability: there exist informational questions that cannot be settled by any coherent refinement of a given state. These three features capture fundamental respects in which quantum informational organization differs from classical organization.

The primitive semantic structure is

S=⟨S,⊑,⊥g⟩,

where S is a collection of informational states, ⊑ is an informational parthood relation, and ⊥_g is a primitive relation of exact informational contrariety. States are pieces of information rather than possible worlds. Contrariety is not analysed in terms of negation, inconsistency, probability, or Hilbert-space orthogonality. Informational fusion is derived from the parthood structure.

For X ⊆S, let

X^⊥_g ={s∈S:∀t ∈X(s⊥_gt)}.

Fine-grained propositions are identified with the doubly closed sets P = P^⊥_g⊥_g. Consequently, propositions are not merely sets of states but informational contents determined by networks of contrariety.

The semantics employs Fine’s dual notions of exact verification and exact falsification. For every sentence A, let |A|⁺ and |A|⁻ denote its exact verifiers and exact falsifiers. The logical clauses for negation, conjunction, and disjunction are those of exact bivalent truthmaker semantics [2, 3]. Exact verifiers and exact falsifiers of the same sentence are connected through contrariety via principles of exclusion and exhaustivity. Negation, consequence, and orthocomplementation are thereby grounded in more primitive informational relations between states.

A central role is played by the incoherent states. Let s_Ø be the empty informational state. A state s is exactly incoherent iff s ⊥_g s_Ø. Contradiction is therefore treated as a structural feature of informational space rather than merely as a property of sentences. Exactly incoherent states include but do not coincide with those generated by combining exactly contrary informational components. States combining information that merely can never obtain together are also among the incoherent ones.

The resulting semantics induces a hyperintensional consequence relation that remains sensitive to distinctions between exact content and logical consequence. At the same time, passing from fine-grained propositions to suitable upward closures yields the familiar coarse-grained propositional structures of quantum logic [1]. The central representation theorem establishes that orthomodular lattices [4] arise as the lattices of coarse-grained propositions associated with appropriate QTM structures, while the fine-grained level retains hyperintensional distinctions that disappear after coarse-graining. Traditional quantum logic is thus recovered as a quotient of a richer truthmaker semantics.

Classical truthmaker semantics appears as a limiting case. When incoherence and contrariety collapse into the same relation, incompatibility reduces to contradiction, orthomodularity collapses into Booleanity, distributivity is restored, and the framework reduces to a classical exact truthmaker semantics.

QTM therefore provides a unified semantic framework for hyperintensional content, incompossibility, contradiction, contextuality, and orthomodular consequence. By locating the source of quantumlogical phenomena in the structure of informational states and their patterns of exclusion, it offers a truthmaker-theoretic foundation for the study of non-classical information independently of specifically physical or probabilistic considerations.

References

[1] Garrett Birkhoff & John von Neumann (1936): The Logic of Quantum Mechanics. Annals of Mathematics 37(4), pp. 823–843, doi:10.2307/1968621.

[2] Kit Fine (2014): A Theory of Truthmaker Content I: Conjunction, Disjunction and Negation. Journal of Philosophical Logic 43(4), pp. 625–674, doi:10.1007/s10992-014-9303-4.

[3] Kit Fine (2017): Truthmaker Semantics. In Bob Hale, Crispin Wright & Alexander Miller, editors: A Companion to the Philosophy of Language, 2 edition, Wiley–Blackwell, pp. 554–573.

[4] Gudrun Kalmbach (1983): Orthomodular Lattices. Academic Press, London.

ABSTRACT. It is commonly held that science provides a privileged—if not the best—path to understanding the world and its causal structure. Consequently, it is often—explicitly or implicitly—assumed that a scientific worldview is, or ought to be, the natural outcome of scientific knowledge. In my presentation, drawing on important distinction between logical, explicatory and scientific ontology, I seek to provide an analytic reconstruction of mechanistic theory of causation and to provide a formal analysis of this theory. In fact, one of the great merits of the mechanistic philosophy is that it has revived philosophical interest in the problem of causality, with close attention to explanatory issues in particular sciences. The main aim of my presentation is to provide an analytic reconstruction of Stuart Glennan’s theory of causation and to provide such a formal analysis of his theory in which are represented causal and constitutive relations, not excluding infinite chains of causal mechanisms. In fact, according to Stuart Glennan for the purpose of building the mechanistic theory of causation, it is not necessary to assume the existence of either fundamental laws (explicated by counterfactuals) or fundamental causal mechanisms. Finally, I will discuss further possible developments of my proposal so as to account for the connection between mechanisms and events, as well as their internal composition.

References:

AJDUKIEWICZ, KAZIMIERZ. 1978. The Scientific World-Perspective and Other Essays, 1931–1963. Edited and with an Introduction by Jerzy Giedymin. Dordrecht: D. Reidel Publishing Company.

CHAKRAVARTTY, ANJAN. 2017. Scientific ontology: Integrating naturalized metaphysics and voluntarist epistemology. Oxford University Press. https://doi.org/10.1093/oso/9780190651459.001.0001.

GLENNAN, STUART. 1996. Mechanisms and the Nature of causation, Erkenntnis 44: 49-71.

GLENNAN, STUART. 2017. The New Mechanical Philosophy. Oxford: Oxford University Press.

GLENNAN, STUART. 2021. Corporeal composition. Synthese 198, 11439–11462. https://doi.org/10.1007/s11229-020-02805-x.

GLENNAN, STUART. 2024. The Object-Activity Theory of Events. Erkenntnis 89: 503–519. https://doi.org/10.1007/s10670-022-00542-w.

OLEKSOWICZ, M. 2026. Causes, Explanations, and Mechanisms. A Philosophical Analysis of the Secret Connection. Synthese Library, Springer Cham.

ABSTRACT. Neil Tennant introduced two variants of the logic of classes with and without individuals. Both logics aim to reconstruct in the safe way the essential features of Fregean approach. In the talk cut-free sequent calculus for both variants will be presented.

ABSTRACT. We combine formal tools from free logic \cite{pavlovic2021more} and counterpart semantics for quantified modal logic \cite{hazen1979counterpart} to obtain a labelled sequent calculus in which definite descriptions are treated as binary quantifiers, as introduced by Kürbis in \cite{kurbis2022binary}, and terms that occur in them may denote an non-existent object. We prove that the calculus is cut-free, sound and complete.

The calculus was designed following the steps of Orlandelli's approach to quantified modal logic presented in \cite{orlandelli2024quantified}. However, the semantics on which our calculus is based differs from Orlandelli's in at least two respects. First, we take as our first order base for the calculus a positive free logic with outer quantifiers, as Priest does in \cite{priest2005towards}; although, unlike Priest, we couple each possible world with an inner and outer domain as done by Corsi in \cite{corsi2002unified}, which contain respectively the objects that exist and do not exist according to a specific world in the current model, and we consider definite descriptions as referential expressions, that can be paraphrased away through outer quantifiers mimicking Russell's formalization in \cite{russell1905denoting}.

Even though the paraphrase is inspired by Russell, it does not align with the usual truth conditions for definite descriptions, namely existence, uniqueness and maximality \cite{sep-descriptions}, since we just require uniqueness and a relaxed version of maximality: \begin{gather*} \dfrac{ \Gamma \seq \Delta,\, u \mathbin{:} F[t/x] \quad \Gamma \seq \Delta,\, u \mathbin{:} G[t/x] \quad u \mathbin{:} F[y/x],\, u \mathbin{:} G[y/x],\, \Gamma \seq \Delta,\, u \mathbin{:} y = t } { \Gamma \seq \Delta,\, u \mathbin{:} \I x[F,G] } \;(\I R) \\[1.6em] \dfrac{ u \mathbin{:} F[y/x],\, u \mathbin{:} G[y/x],\, \Gamma \seq \Delta } { u \mathbin{:} \I x[F,G],\, \Gamma \seq \Delta } \;(\I L) \\[1.6em] \dfrac{ \Gamma \seq \Delta,\, u \mathbin{:} F[t_1/x]\land G[t_1/x] \quad \Gamma \seq \Delta,\, u \mathbin{:} F[t_2/x]\land G[t_2/x] \quad \Gamma \seq \Delta,\, u \mathbin{:} C[t_2/x] } { u \mathbin{:} \I x[F,G],\, \Gamma \seq \Delta,\, u \mathbin{:} C[t_1/x] } \;(\I_{\mathrm{Repl}}) \end{gather*}

Moreover, both the inner and outer domain can grow, shrink or remain constant from world to world in the model. Secondly, we provide a new semantical clause for the modal box operator: a formula under the range of the box operator is true in a world if and only if it holds for all existing counterparts in all accessible worlds, where counterparts are related by what we call a transition relation, following Orlandelli's terminology. When the transition relation is universal, the left and right rules for the box operator look as follows: \noindent \[ \dfrac{(s,\vec{t})\mathrel{\mathcal{T}}(u,\vec{r}),\; s\mathbin{:}\modality{\vec{t}}{\vec{x}}{A},\; u\mathbin{:}E\,r_1,\,\ldots,\,u\mathbin{:}E\,r_n,\; u\mathbin{:}A[\vec{r}/\vec{x}],\;\Gamma\seq\Delta} {(s,\vec{t})\mathrel{\mathcal{T}}(u,\vec{r}),\; s\mathbin{:}\modality{\vec{t}}{\vec{x}}{A},\;\Gamma\seq\Delta} \;\mathsf{(L\Box)} \] \[ \dfrac{(s,\vec{t})\mathrel{\mathcal{T}}(u,\vec{y}),\; u\mathbin{:}E\,y_1,\,\ldots,\,u\mathbin{:}E\,y_n,\; \Gamma\seq\Delta,\;u\mathbin{:}A[\vec{y}/\vec{x}]} {\Gamma\seq\Delta,\;s\mathbin{:}\modality{\vec{t}}{\vec{x}}{A}} \;\mathsf{(R\Box)}\quad(u,\vec{y}\ \text{fresh}) \]

The guiding intuition behind this new semantical clause for the box operator is the following: since the transition relation tracks down if one or more counterparts of an object in a current world belong to the inner or outer domain of an alternative world, we can attribute a necessary property of something existing in a current world just in case it holds for all its counterparts that exist in other worlds. So, we slightly depart from Lewis' interpretation of the counterpart relation as a similarity relation between individuals, applied for example in \cite{lewis1971counterparts}.\\

To prove that the calculus is cut-free, sound and complete with respect to the semantics sketched above we provide a correspondence result between with free logic with outer quantifiers and a quantified version of $\mathrm{S5}$, which we call $\mathrm{GIM}.{\mathrm{S5}}$. Finally, we will explore how different systems are obtained when different restrictions are imposed on either the transition relation or the world-bounded domains.

ABSTRACT. In three brief passages Lambert and van Frassen describe a hierarchy, or rather a spectrum, of theories of definite descriptions in positive free logic: the minimal theory MFD is the weakest, the chosen object theory FD2 the strongest, and a number of theories are ordered along two dimensions between them. This was once a research project, but its results are somewhat obscure. The most detailed (but still extremely sketchy) account is in a two-page appendix to an article of Lambert’s on the foundations of the spectrum. I’ll present the axioms and rules found there and in a passage of van Fraassen’s, look at three of them more closely, and hope that listeners will be enticed to engage with this forgotten research project themselves.

ABSTRACT. In his book `Modal Logic for Philosophers', Garson proposed positive free modal logics with definite descriptions. His logics have a desirable property that two improper definite descriptions which fail to uniquely denote objects can denote different objects. However, some intuitively valid non-modal instances of the substitution axiom are not valid in the semantics characterizing his logics. In this talk, we propose a new S5-based positive free modal logic with definite descriptions. Our logic not only has the aforementioned property, but also validates all non-modal instances of the substitution axiom. We also present a sound and complete cut-free hypersequent calculus for our logic.

ABSTRACT. In our recent work [1], we proposed a tableau calculus for the description logic ALC augmented with definite description operators. The introduced tableau system addressed the problems of concept satisfiability and ontology consistency. While conceptually simple, with rules closely reflecting the semantics of the operators, the system exhibited suboptimal worst-case complexity: the concept satisfiability and ontology consistency problems in ALC with definite descriptions are known to be EXPTIME-complete, whereas the proposed system runs in 2-EXPTIME in the worst case.

In this talk, I propose a global caching mechanism that can be incorporated into the tableau calculus. The resulting system produces a so-called AND-OR graph in which information about the cardinalities of extensions of concepts occurring within the scope of definite description operators is globally cached. This avoids performing certain reasoning steps multiple times across different branches and thus eliminates the double-exponential blow-up. The resulting system runs in EXPTIME and thus matches the complexity bounds of the problems it solves.

References

[1] Sochański, M., Wałęga, P. A., & Zawidzki, M. Description Logics with Two Types of Definite Descriptions: Complexity, Expressiveness, and Automated Deduction. Proceedings of the AAAI Conference on Artificial Intelligence, 40(23), 2026, 19371–19379. https://doi.org/10.1609/aaai.v40i23.39014.

ABSTRACT. In this talk, I will present ongoing work, with several collaborators, on characterisation theorems for description logics extended with operators for definite descriptions. These theorems follow the tradition of van Benthem-like results, and aim to characterise the fragments of first-order logic that have the same expressive power as the aforementioned description logics.

ABSTRACT. In the talk I will present work-in-progress on the implementation of the tableau calculus for Quine’s logic of classes, also called virtual theory of classes (VTC). The latter aims to capture the most fundamental fraction of set theory, without commiting oneself to any of its specific axiomatically formulated versions. VTC allows, in particular, to reason about classes, which are represented by set abstracts. It is based on classical first-order logic, although variables are restricted for denoting sets only, which means that proper classes are not assumed to exist by VTC.

The implementation is done in Python, and currently supports only checking validity of formulas with selected set theoretic constructs. Its preparation is joint work with Andrzej Indrzejczak, who is the author of the tableau calculus for VTC as well as a corresponding sequent calculus.

ABSTRACT. We study finite local-unit-aligned totally ordered monoids: finite totally ordered monoids in which, for each element, its greatest right local unit and greatest left local unit coincide. We prove that every such monoid has a canonical rigid representation as a finite chain-indexed direct system of component monoids, and that every rigid direct system of the corresponding kind reconstructs such a monoid. The representation is induced intrinsically by the local-unit map $\tau$, through the finest $\tau$-multiplication-coherent partition of the positive idempotent skeleton. The components, transition maps, directed lexicographic order, and product formula recover the order and multiplication.

ABSTRACT. The paper [4] proposes the logic DR. This logic is an attempt to improve Standard Deontic logic, which is associated with various puzzles and paradoxes, widely discussed in the literature (for example in [1], [2]). The logical system of DR was defined by applying possible world seman- tics in combination with relating semantics. This type of semantics allows for modelling various relations between sentences in the object language (such as: relatedness relation, analytical relationships, causalities, temporal orderings, preference orderings, logical consequences of some logic, etc. [3], [7], [6]). Combining relating semantics with possible worlds is the realization of a postulate posted during the 1st Workshop of Relating Logic: Problem η: combining relating semantics with other kinds of formal semantics [5]. The problems that arise in logic D stem from the semantic definition of deontic functors. Since possible, accessible worlds are interpreted as deontically ideal worlds, so obligation O means the realization of the object of the command in every accessible world, while permission P means at least in one. In the semantics of the logic DR, possible worlds are endowed with relating relations that determine whether a given sentence is related to or is not related to the normative system on the basis of which the command or permission is stated. The definition of obligation now requires that the object of the command be realized not only in every possible world, but also related to the normative system under consideration. For permissibility, however, it suffices only that in some ideal world its object remains unrelated to the normative system un- der consideration. In the logic DR, permissiveness is therefore weaker than D, whereas obligation is stronger. The logic DR is significantly weaker than D, removing its various paradoxes and puzzles while simultaneously preserving 1 uncontroversial forms of inference. It is worth mentioning that by imposing con- ditions on the relating relation in models, one can strengthen DR, obtaining systems of deontic logic between DR and D. In the presentation, we will present three aspects related to the logic DR: philosophical motivations, examples of applications to legal reasoning, and moti- vations for its extensions based on imposing additional conditions on the relating relation in possible worlds. We also discuss the outline of proof theory for deontic relating logic and the prospects for combining relating semantics with possible worlds for other modal logics.

References

[1] J. Carmo and A. Jones. Deontic logic and contrary-to-duties. In M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 8, page 147–264. Springer Science+Business Media.

[2] Risto Hilpinen and Paul McNamara. Deontic logic. A historical survey and introduction. In D. M. Gabbay and et al., editors, Handbook of deontic logic and normative systems, pages 195–240. College Publications, 2013.

[3] Tomasz Jarmużek. Relating semantics as fine-grained semantics for inten- sional logics. In A. Giordani and J. Malinowski, editors, Logic in High Definition, volume 56 of Trends in Logic, pages 13–30. Springer, 2021. doi:10.1007/978-3-030-53487-5_2.

[4] Tomasz Jarmużek and Mateusz Klonowski. On logics of strictly-deontic modalities. A semantic and tableau approach. Logic and Logical Philosophy, 29(3):335–380, 2020. doi:10.12775/LLP.2020.010.

[5] Tomasz Jarmużek and Francesco Paoli. Relating logic and relating seman- tics. History, philosophical applications and some of technical problems. Logic and Logical Philosophy, 30(4):563–577, 2021.

[6] Mateusz Klonowski. Axiomatization of some basic and modal Boolean connexive logics. Logica Universalis, 15:517–536, 2021. doi:10.1007/ s11787-021-00291-4.

[7] Mateusz Klonowski. History of relating logic. The origin and research direc- tions. Logic and Logical Philosophy, 30(4):579–629, 2021. doi:10.12775/ LLP.2021.021.