Despite the undeniable epistemic progress of developmental biology from the second half of the twentieth century to the present day, there still is widespread disagreement on defining the biological term of ‘development’. This scientific field epistemologically is neither unsuccessful nor immature, thus the persistent lack of agreement on its most central concept raises some important questions: is there any need for an explicit definition of biological development, and if so, what content should the definition have? My central thesis (...) is twofold. First, that current definitions of biological development are conceptually or empirically inadequate. Second, that an explicit definition of biological development is very much needed (a) for the practical purposes of science textbooks, but more importantly is needed (b) epistemically for exposing or overcoming problematic assumptions and for partially guiding scientific research by coding the appropriate assumptions. To support this thesis, initially I will show the deficiencies of the dominant definitions of biological development; and subsequently I will provide two arguments: an Argument from Practical Purposes and an Argument from Epistemic Purposes. Finally, for accommodating practical and epistemic purposes and as a response to the inadequacy of the available definitions of development, I will propose and defend an operational definition of biological development which is aimed to be broader than the received ones, while being more precise and fruitful to conduct empirical research. (shrink)

Despite numerous and increasing attempts to define what life is, there is no consensus on necessary and sufficient conditions for life. Accordingly, some scholars have questioned the value of definitions of life and encouraged scientists and philosophers alike to discard the project. As an alternative to this pessimistic conclusion, we argue that critically rethinking the nature and uses of definitions can provide new insights into the epistemic roles of definitions of life for different research practices. This paper examines the possible (...) contributions of definitions of life in scientific domains where such definitions are used most (e.g., Synthetic Biology, Origins of Life, Alife, and Astrobiology). Rather than as classificatory tools for demarcation of natural kinds, we highlight the pragmatic utility of what we call operational definitions that serve as theoretical and epistemic tools in scientific practice. In particular, we examine contexts where definitions integrate criteria for life into theoretical models that involve or enable observable operations. We show how these definitions of life play important roles in influencing research agendas and evaluating results, and we argue that to discard the project of defining life is neither sufficiently motivated, nor possible without dismissing important theoretical and practical research. (shrink)

Despite numerous and increasing attempts to define what life is, there is no consensus on necessary and sufficient conditions for life. Accordingly, some scholars have questioned the value of definitions of life and encouraged scientists and philosophers alike to discard the project. As an alternative to this pessimistic conclusion, we argue that critically rethinking the nature and uses of definitions can provide new insights into the epistemic roles of definitions of life for different research practices. This paper examines the possible (...) contributions of definitions of life in scientific domains where such definitions are used most. Rather than as classificatory tools for demarcation of natural kinds, we highlight the pragmatic utility of what we call operational definitions that serve as theoretical and epistemic tools in scientific practice. In particular, we examine contexts where definitions integrate criteria for life into theoretical models that involve or enable observable operations. We show how these definitions of life play important roles in influencing research agendas and evaluating results, and we argue that to discard the project of defining life is neither sufficiently motivated, nor possible without dismissing important theoretical and practical research. (shrink)

There is currently no consensus from the relevant stakeholders regarding the operational and construct definitions of child assent for research. As such, the requirements for assent are often construed in different ways, institutionally disparate, and often conflated with those of parental consent. Development of a standardized operational definition of assent would thus be important to ensure that investigators, institutional review boards, and policy makers consider the assent process in the same way. To this end, we describe a Delphi study (...) that provided consensus from a panel of expert stakeholders regarding the definitions of child assent for research. Based on current guidelines, a preliminary definition of assent was generated and sent out for review to a Delphi panel including pediatric bioethicists and researchers, Institutional Review Board members, parents, and individuals with regulatory/legal expertise. For each subsequent review, the process of summarizing and revising responses was repeated until consensus was achieved. Panelists were also required to rank order elements of assent that they believed were most important in defining the underlying constructs of the assent process. In providing these rankings, panelists were asked to frame their responses in the contexts of younger and adolescents/older children in non-therapeutic and therapeutic trials. Summary rankings of the most important identified elements were then used to generate written construct definitions which were sent out for iterative reviews by the expert panel. Consensus regarding the operational definition was reached by 14/18 of the panel members. Seventeen panelists agreed with the definitions of capacity for assent, elements of disclosure for younger children, and the requirements for meaningful assent, respectively. Fifteen members agreed with the elements of disclosure for adolescents/older children. It is hoped that this study will positively inform and effect change in the way investigators, regulators, and IRBs operationalize the assent process, respect children’s developing autonomy, and in concert with parental permission, ensure the protection of children who participate in research. (shrink)

The philosophy of Confucius has often been accused of lacking classical definitions of its core concepts. However, as I shall argue, Confucius systematically used nonclassical definitions—to be precise, operational ones. The notion of operational definition comes from Percy Bridgman’s The Logic of Modern Physics and means that the definiendum is defined by a set of operations that results in determining the meaning of the term in question. In the case of Confucian argumentation, operational definitions are mostly nominal and, in (...) contrast to unambiguous methods of measurement, also context-dependent. This results in there being various yet not mutually inconsistent definitions of one term, and in “paradigmatic examples” playing a crucial role. As I show, this mode of defining things had major implications for the content of Confucius’ thought. In fact, many of its forms could be traced back to other Chinese philosophers, including those of non-Confucian provenance. (shrink)

The distinction between clinical practice and surgical research may seem trivial, but this distinction can become a complex issue when innovative surgeries are substituted for standard care without patient knowledge. Neither the novelty nor the risk of a new surgical procedure adequately defines surgical research. Some institutions tacitly allow the use of new surgical procedures in series of patients without informing individuals that they are participating in a scientific study, as long as no written protocol or hypothesis exists. Institutions can (...) justify this practice by viewing human research in narrow terms as an activity outlined in a formal protocol. Application of limited definitions, however, erodes patients' rights and risks losing public confidence in how biomedical research is conducted. I propose an operational definition of human research also be recognised. Enforcing more rigid and less ambiguous guidelines of human research may curtail enrolment into some studies, but it will also protect patients from being used as subjects without their knowledge. (shrink)

The increasing focus to go beyond assessing cognitive constructs around the globe and in Singapore such as cultural awareness, grit and moral reasoning and the call for schools to prepare students for citizenship the development of corresponding instruments applicable in a larger scale. Given the significance of moral reasoning and its assessment are key elements within the current Singapore Character and Citizenship Education curriculum -syllabus.pdf, 2012), there is room to explore how moral reasoning can be assessed practically in a large-scale (...) setting. Undergirding this exploration requires a review of the epistemology of moral reasoning, corresponding theoretical frameworks, and how they might apply to moral education in Singapore schools. By reviewing perspectives of moral reasoning from relevant literature beyond educational research to business, medical and philosophical domains, this paper aims to propose an operational definition of moral reasoning suitable for use in Singapore schools, and to mitigate possible issues brought about by educators’ disparate or assumed understanding of moral reasoning. This can consequently support the enactment of the CCE curriculum and be applied to the development of assessment instruments suitable for use within Singapore schools and similar contexts. (shrink)

Tarski 1968 makes a move in the course of providing an account of ?definitionally equivalent? classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader. In ?1 we present this move as a response to a certain difficulty in the received account of what it is to define a function symbol (or ?operation symbol?). This difficulty, which presents itself as a minor technicality needing to be got around (...) especially for the case of symbols for zero-place functions (for ?distinguished elements?), has repercussions?not widely recognised?for the account of functional completeness in sentential logic. A similarly stark comment in Church 1956 reveals an appreciation of this difficulty, though not every subsequent author on the topic has taken the point. We fill out this side of the picture in ?2. The discussion of functional completeness in ?2 is supplemented by some remarks on what is involved in defining a connective, which have been included in an Appendix. The emphasis throughout is on conceptual clarification rather than on proving theorems, and the main body of the paper may be regarded as an elaboration on the remarks just mentioned by Tarski and Church. The Appendix (?3) is intended to be similarly clarificatory, though this time with some corrective intent, of remarks made in and about Makinson 1973. (shrink)

In this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.

Several important terms in biology have recently been criticized for not being "operational." In this paper the course of operationism in physics, psychology and genetics is sketched to show what effect this particular view on the meaning of scientific terms had on these disciplines. Then the biological species concept and the concept of homology are examined to see in what respects they are or are not "operational." One of the primary conclusions of this investigation is that few terms in science (...) are completely operational or completely nonoperational. Some scientific terms, especially theoretical terms, are a good deal less operational than others; but, far from being regrettable, this situation is essential if theoretical terms are to fulfill their systematizing function and if scientific theories are to be capable of growth. (shrink)

This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means ‘The F is G’. The system is shown to have desirable proof-theoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions recommended (...) here and the more usual approach that uses a term-forming operator ι, where ιxF means ‘the F’. (shrink)

It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the pre-assumption that the equations of electrodynamics are covariant against these---unknown---transformation rules. There are several problems to be raised concerning these derivations. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following fundamental question: Are the so-obtained transformation rules indeed identical with the true transformation (...) rules of the fundamental electrodynamical quantities? In other words, is it indeed the case that the values calculated from the quantities in one inertial frame by means of the transformation rules we derived are equal to the values of the same quantities obtained by the same operations with the same measuring equipments when they are co-moving with the other inertial frame? This is of course an empirical question. In this paper, we will investigate the problem in a purely theoretical framework by applying what J. S. Bell calls “Lorentzian pedagogy”---according to which the laws of physics in any one reference frame account for all physical phenomena. We will show that the transformation rules of the electrodynamical quantities are indeed identical with the ones obtained by presuming the covariance of the equations of electrodynamics, and that the covariance is indeed satisfied. Beforehand, however, we need to clarify the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. (shrink)

This paper distinguishes between definiteness and determinacy. Definiteness is seen as a morphological category which, in English, marks a uniqueness presupposition, while determinacy consists in denoting an individual. Definite descriptions are argued to be fundamentally predicative, presupposing uniqueness but not existence, and to acquire existential import through general type-shifting operations that apply not only to definites, but also indefinites and possessives. Through these shifts, argumental definite descriptions may become either determinate or indeterminate. The latter option is observed in examples like (...) ‘Anna didn’t give the only invited talk at the conference’, which, on its indeterminate reading, implies that there is nothing in the extension of ‘only invited talk at the conference’. The paper also offers a resolution of the issue of whether possessives are inherently indefinite or definite, suggesting that, like indefinites, they do not mark definiteness lexically, but like definites, they typically yield determinate readings due to a general preference for the shifting operation that produces them. (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the (...) validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices. (shrink)

The properties of the ${\forall^{1}}$ quantifier defined by Kontinen and Väänänen in [13] are studied, and its definition is generalized to that of a family of quantifiers ${\forall^{n}}$ . Furthermore, some epistemic operators δ n for Dependence Logic are also introduced, and the relationship between these ${\forall^{n}}$ quantifiers and the δ n operators are investigated.The Game Theoretic Semantics for Dependence Logic and the corresponding Ehrenfeucht- Fraissé game are then adapted to these new connectives.Finally, it is proved that the ${\forall^{1}}$ (...) quantifier is not uniformly definable in Dependence Logic, thus answering a question posed by Kontinen and Väänänen in the above mentioned paper. (shrink)

Epistemicism associates vagueness with ignorance produced by semantic plasticity: the shiftiness of intensions in our language resulting from small changes in usage. The recent literature (Caie 2012; Magidor 2018; Yli-Vakkuri 2016) points to a missing piece in the epistemicist theory of vagueness, namely a clear account of the semantics of the definiteness operator Δ. The fundamentals of the epistemicist theory are well understood. However, the technical work of defining the definiteness operator has proven difficult. There are several desiderata that we (...) would like Δ to satisfy. For instance, we would like the epistemicist notion of ‘definiteness’ to interact well with modal operators and validate intuitive principles like ‘necessarily, if φ is definitely true, then φ is true’. Providing an account that would meet all such desiderata has eluded the epistemicists so far. In this paper, I present a novel version of a multidimensional model inspired by the work of Robert Stalnaker and David Kaplan. Using this model, I provide an account of epistemicist definiteness that meets our desiderata. (shrink)

The official definition of ‘pain’ by the International Association for the Study of Pain (IASP) hasn’t seen much revision since its publication in 1979. There have been various criticisms of the definition in the literature from different quarters: that the definition implies a dubious metaphysical dualism, that it requires a strong form of consciousness as well as linguistic abilities, that it excludes many vulnerable groups that are otherwise perfectly capable of experiencing pain, that it has therefore unacceptable (...) practical as well as ethical consequences, that it is unhelpful to the health-care professional as an operational definition, that it is too narrow, and many others. In my view, most of these criticisms depend on misunderstandings or on uncharitable interpretations. My aim is to go over the definition, clarify some potential ambiguities, and argue that these criticisms don’t cut much ice. At the end, I will present a few slightly reworded versions of the definition that I claim are more felicitous and capture the original intended meaning by the members of the taxonomic committee in a much better and transparent way that avoids all the major extant criticisms in the literature. (Written for interdisciplinary audiences.). (shrink)

definitions and explanations frequently come together and permeate almost all fields of knowledge. This does not exclude mathematics, even when these definitions hold clear links and close connections with our physical world. Here we propose a rather different perspective. Making operational physical assumptions, we show how it is possible to rigorously reconstruct some features of both geometry and topology. Broadly speaking, assuming this operational and more concrete philosophy we not only are capable of defining primitive concepts like points, straight lines, (...) planes, angles and spaces, but we also go all the way up to more complex constructions. We show that within our method it is also possible to come up with what we call a compass-based topology as well as a normed and a metric space. We hope this operationalist point-of-view can assist other researchers and students to have a clearer understanding of these quite abstract concepts. (shrink)

Storage operators have been introduced by J. L. Krivine in [5] they are closed λ-terms which, for a data type, allow one to simulate a "call by value" while using the "call by name" strategy. In this paper, we introduce the directed λ-calculus and show that it has the usual properties of the ordinary λ-calculus. With this calculus we get an equivalent--and simple--definition of the storage operators that allows to show some of their properties: $\bullet$ the stability of the (...) set of storage operators under the β-equivalence (Theorem 5.1.1); $\bullet$ the undecidability (and semidecidability) of the problem "is a closed λ-term t a storage operator for a finite set of closed normal λ-terms?" (Theorems 5.2.2 and 5.2.3); $\bullet$ the existence of storage operators for every finite set of closed normal λ-terms (Theorem 5.4.3); $\bullet$ the computation time of the "storage operation" (Theorem 5.5.2). (shrink)

. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a principle for reasoning (...) on the basis of rule-based atomic definitions, proposed by Hallnäs and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω- principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself. (shrink)

Using covariant derivatives and the operator definitions of quantum mechanics, gauge invariant Proca and Lehnert equations are derived and the Lorenz condition is eliminated in U(1) invariant electrodynamics. It is shown that the structure of the gauge invariant Lehnert equation is the same in an O(3) invariant theory of electrodynamics.

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency and by the logics of formal undeterminedness. LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of (...) the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices. (shrink)

Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented in (...) (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (shrink)

In the most fundamental scientific sense, to define is to delimit. Thus definitions serve to fix boundaries of phenomena or the range of applicability of terms or concepts. That whose range is to be delimited is called the definiendum, and that which delimits the definiens. In practice, the hard sciences tend to be more concerned with delimiting phenomena, and definitions are frequently informal, given on the fly, as in “Therefore, a layer of high rock strength, called the lithosphere, exists near (...) the surface of planets.” Social science practice tends to focus on specifying application of concepts through formal operational definitions. Philosophical discussions have concentrated almost exclusively on articulating definitional forms for terms. (shrink)

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use (...) the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS. (shrink)

In this paper we reconsider the issue of free choice and the role of the wh-morphology employed in it. We show that the property of being an interrogative wh-word alone is not sufficient for free choice, and that semantic and sometimes even morphological definiteness is a pre-requisite for some free choice items (FCIs) in certain languages, e.g. in Greek and Mandarin Chinese. We propose a theory that explains the polarity behaviour of FCIs cross-linguistically, and allows indefinite (Giannakidou 2001) as well (...) as definite-like FCIs. The difference is manifested as a lexical distinction in English between any (indefinite) and wh-ever (definite); in Greek it appears as a choice between a FCI nominal modifier (taking an NP argument), which illustrates the indefinite option, and a FC free relative illustrating the definite one. We provide a compositional analysis of Greek FCIs in both incarnations, and derive in a parallel manner the Chinese FCIs. Here the definite versus indefinite alternation is manifested in the presence or absence of dōu, which we take to express the maximality operator. It is thus shown that what we see in the morphology of FCIs in Greek is reflected in syntax in Chinese. Our analysis has important consequences for the class of so-called wh-indeterminates. In the context of current proposals, free choiceness is taken to come routinely from interrogative semantics, and wh-indeterminates are treated as question words which can freely become FCIs (Kratzer and Shimoyama 2002). Our results from Mandarin and Greek emphasize that wh-indeterminates do not form a uniform class in this respect, and that interrogative semantics alone cannot predict either sensitivity of free choice to definiteness, or the polarity behaviour of FCIs. (shrink)

Previous theorists have claimed that Russell’s theory of definite descriptions gives the wrong truth conditions to sentences in which definite descriptions are embedded under certain other operators; but the other operators used, such as conditionals and propositional attitude verbs, have introduced intensional and hyperintensional complications that might be thought to obscure the point against Russell. This paper shows that the same kind of problem arises when the operator in question allows the context to be extensional. It is further argued that (...) presuppositional theories of definite descriptions give intuitively satisfying analyses of the novel data. (shrink)

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of "and", "or", (...) "not" and "conditioning" are then defined on these conditional events with variable conditions. The probabilities of the disjunction and of the conjunction of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed. (shrink)