The usual completeness theorem for first-order logic is extended in order to allow for a natural incorporation of real analysis. Essentially, this is achieved by building in the set of real numbers into the structures for the language, and by adjusting other semantical notions accordingly. We use many-sorted languages so that the resulting formal systems are general enough for axiomatic treatments of empirical theories without recourse to elements of set theory which are difficult to interprete empirically. Thus we provide (...) a way of applying model theory to empirical theories without tricky detours. Our frame is applied to axiomatizations of three empirical theories: classical mechanics, phenomenological thermodynamics, and exchange economics. (shrink)

Ken Pledger devised a one-sorted approach to the incidence relation of plane geometries, using structures that also support models of propositional modal logic. He introduced a modal system 12g that is valid in one-sorted projective planes, proved that it has finitely many non-equivalent modalities, and identified all possible modality patterns of its extensions. One of these extensions 8f is valid in elliptic planes. These results were presented in his 1980 doctoral dissertation, which is reprinted in this issue (...) of the Australasian Journal of Logic. Here we show that 12g and 8f are strongly complete for validity in their intended one-sorted geometrical interpretations, and have the finite model property. The proofs apply standard technology of modal logic together with a step-by-step procedure introduced by Yde Venema for constructing two-sorted projective planes. (shrink)

We study the embedding property in the category of sorted profinite groups. We introduce a notion of the sorted embedding property (SEP), analogous to the embedding property for profinite groups. We show that any sorted profinite group has a universal SEP-cover. Our proof gives an alternative proof for the existence of a universal embedding cover of a profinite group. Also our proof works for any full subcategory of the sorted profinite groups, which is closed under taking (...) finite quotients, fibre products, and inverse limits. We also show that any sorted profinite group having SEP has a sorted complete system whose theory is $\omega $ -categorical and $\omega $ -stable under the assumption that the set of sorts is countable. (shrink)

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. Hybrid logics are here divided into two groups: standard hybrid logics involving nominals as expressions of a separate sort, and non-standard hybrid logics, which do not involve nominals but whose expressive power matches the expressive power of binder-free standard hybrid logics.The original results of this book are split into two parts. This division reflects the division of (...) the book itself. The first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, the efforts focused on hybrid logics referred to as non-standard in this book. Non-standard hybrid logics are understood as modal logics with global counting operators ) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise: 1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators ), which can be easily extended onto other frame classes, 2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames ), MT), MD), MB), MK4) with numerical subscripts coded in binary. Establishing the exponential-size model property for this logic defined over the classes of Euclidean and equivalential frames ), MS5).Results of the second type consist of designing concrete deductive systems for standard and non-standard hybrid logics. More precisely, they include: 1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics. An interesting feature of indicated calculi is the nonbranching character of the rule, 2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics. The internalization technique applied to a tableau calculus for the modal logic with global counting operators is novel in the literature, 3. Devising the first hybrid algorithm involving an inequality solver for modal logics with global counting operators. Transferring the arithmetical part of reasoning to an inequality solver turned out to be sufficient in ensuring termination.The book is directed to philosophers and logicians working with modal and hybrid logics, as well as to computer scientists interested in deductive systems and decision procedures for logics. Extensive fragments of the first part of the book can also serve as an introduction to hybrid logics for wider audience interested in logic.The content of the book is situated in the areas of formal logic and theoretical computer science with some elements of the theory of computational complexity. (shrink)

As noted in 1962 by Timothy Smiley, if Aristotle’s logic is faithfully translated into modern symbolic logic, the fit is exact. If categorical sentences are translated into many-sorted logic MSL according to Smiley’s method or the two other methods presented here, an argument with arbitrarily many premises is valid according to Aristotle’s system if and only if its translation is valid according to modern standard many-sorted logic. As William Parry observed in 1973, this result can be proved (...) using my 1972 proof of the completeness of Aristotle’s syllogistic. (shrink)

This paper is about how to give human-like powers to complete agents. For this the most important design choice concerns the overall architecture. Questions regarding detailed mechanisms, forms of representations, inference capabilities, knowledge etc. are best addressed in the context of a global architecture in which different design decisions need to be linked. Such a design would assemble various kinds of functionality into a complete coherent working system, in which there are many concurrent, partly independent, partly mutually (...) supportive, partly potentially incompatible processes, addressing a multitude of issues on different time scales, including asynchronous, concurrent, motive generators. Designing human like agents is part of the more general problem of understanding design space, niche space and their interrelations, for, in the abstract, there is no one optimal design, as biological diversity on earth shows. (shrink)

This paper is about how to give human-like powers to complete agents. For this the most important design choice concerns the overall architecture. Questions regarding detailed mechanisms, forms of representations, inference capabilities, knowledge etc. are best addressed in the context of a global architecture in which different design decisions need to be linked. Such a design would assemble various kinds of functionality into a complete coherent working system, in which there are many concurrent, partly independent, partly mutually (...) supportive, partly potentially incompatible processes, addressing a multitude of issues on different time scales, including asynchronous, concurrent, motive generators. Designing human like agents is part of the more general problem of understanding design space, niche space and their interrelations, for, in the abstract, there is no one optimal design, as biological diversity on earth shows. (shrink)

The United States has long been a model for accessible, affordable education, as exemplified by the country's public universities. And yet less than 60 percent of the students entering American universities today are graduating. Why is this happening, and what can be done? Crossing the Finish Line provides the most detailed exploration ever of college completion at America's public universities. This groundbreaking book sheds light on such serious issues as dropout rates linked to race, gender, and socioeconomic status. Probing graduation (...) rates at twenty-one flagship public universities and four statewide systems of public higher education, the authors focus on the progress of students in the entering class of 1999--from entry to graduation, transfer, or withdrawal. They examine the effects of parental education, family income, race and gender, high school grades, test scores, financial aid, and characteristics of universities attended. The conclusions are compelling: minority students and students from poor families have markedly lower graduation rates--and take longer to earn degrees--even when other variables are taken into account. Noting the strong performance of transfer students and the effects of financial constraints on student retention, the authors call for improved transfer and financial aid policies, and suggest ways of improving the sorting processes that match students to institutions. An outstanding combination of evidence and analysis, Crossing the Finish Line should be read by everyone who cares about the nation's higher education system. (shrink)

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.

In this paper we integrate a sorted unification calculus into free variable tableau methods for logics with term declarations. The calculus we define is used to close a tableau at once, unifying a set of equations derived from pairs of potentially complementary literals occurring in its branches. Apart from making the deduction system sound and complete, the calculus is terminating and so, it can be used as a decision procedure. In this sense we have separated the complexity (...) of sorts from the undecidability of first order logic. (shrink)

A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive (...) coherence criterion for such systems, and show that a quasi-canonical system of the type we investigate is coherent iff it is strongly paraconsistent or strongly paracomplete (in a sense defined in the paper), iff it has a trivalent, non-deterministic semantics of a special type (also defined in the paper) for which it is sound and complete. Finally, we determine when a system of this sort admits cut-elimination, and provide a simple procedure for transforming one which does not into one which does. (shrink)

A grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed. A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame (...) is not in the form of $\langle W, R, R'\rangle$ , or more generally, in the form of $\langle W, R, R', R'',...\rangle$ , which consists of a non-empty set with several relations defined on it. Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger (see [4] and [3]). On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames. However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment V M and some relevance truth-value assignments V R . V M (V R ) defined in this paper is not the same as the assignment contained in a modal (relevance) model. Second, in a fibred model each relevance world is associated (or fibred) with a modal model and each modal world with a relevance model. To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality □, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic. (shrink)

A grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed. A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame (...) is not in the form of $\langle W, R, R'\rangle$, or more generally, in the form of $\langle W, R, R', R'',...\rangle$, which consists of a non-empty set with several relations defined on it. Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger. On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames. However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment V$_M$ and some relevance truth-value assignments V$_R$. V$_M$ defined in this paper is not the same as the assignment contained in a modal model. Second, in a fibred model each relevance world is associated with a modal model and each modal world with a relevance model. To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality $\square$, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic. (shrink)

The purpose of this paper is threefold. First, I visit the Fogelin–Geach-dispute, criticizeMiller''s interpretation of the Geachian notationN(x:N(fx)) and conclude that Fogelin''s argumentagainst the expressive completeness of the Tractariansystem of logic is unacceptable and that the adoptionof the Geachian notation N(x:fx) would not violate TLP5.32. Second, I prove that a system of quantificationtheory with finite domains and with N as the solefundamental operation is expressively complete. Lastly, I argue that the Tractarian system is apredicate-eliminated many-sorted theory (...) (withoutidentity) with finite domains and with N as the solefundamental operation, and thus is expressivelycomplete. (shrink)

Medical classification systems aim to provide a manageable taxonomy for sorting diagnoses into their proper classes. The question, this paper wants to critically examine, is how to correctly systematise diseases within classification systems that are applied in a variety of different settings. ICD and DSM , the two major classification systems in medicine and psychiatry, will be the main subjects of this paper; however, the arguments are not restricted to these classification systems but point out general methodological and epistemological challenges (...) of classifying diseases for differing purposes. Deciding what qualifies as a disease to be included into a classification system as well as choosing a specific validator for correctly systematising diseases is complicated because the broad applicability of medical classification systems simultaneously appears as aim and challenge. Drawing upon the case study of classifying Alzheimer’s disease, this paper will address three dilemmas in designing ‘good’ medical classification systems. They are due to general epistemological problems of medicine, such as the relationship between individual manifestations of diseases and the necessity of building groups in order to scientifically elucidate causes of diseases. Moreover, they involve pragmatic issues of designing usable classifications that allow for easily discriminating between classes of diseases, restricting, however, the completeness of disease representations. This paper wants to trace how the choice of certain validators is unavoidably value-laden and deeply intertwined with epistemological assumptions of how different uses relate to each other, resulting either in a prioritisation of (constrained) coherence or of (vague) pluralistic connectibility. (shrink)

We investigate the formal theory of binary quantifiers, that is, quantifiers that take seriously the surface structure of natural language quantifier phrases. We show how to develop a natural deduction system for logics of this sort and demonstrate soundness and completeness results.

Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A (...) duality result is established between the category of sober algebras and the category of general Kripke structures. A simple enrichment of the proposed sequent calculi is proved to be complete over standard Kripke structures. The calculi are shown to be analytic in a useful sense. (shrink)

Fusion is a well-known form of combining normal modal logics endowed with a Hilbert calculi and a Kripke semantics. Herein, fusion is studied over logic systems using sequent calculi labelled with truth values and with a semantics based on a two-sorted algebra allowing, in particular, the representation of general Kripke structures. A wide variety of logics, including non-classical logics like, for instance, modal logics and intuitionistic logic can be presented by logic systems of this kind. A categorical approach of (...) fusion is defined in the context of these logic systems. Preservation of soundness and completeness by fusion is studied. Soundness is preserved without further requirements, completeness is preserved under mild assumptions. (shrink)

In this paper we first show that Robin Smith’s ecthetic system SE for Aristotle’s assertoric syllogistic is not complete, despite what is claimed by Smith. SE is then not adequate to establish that ecthesis allows one to dispense with indirect or per impossibile deductions in Aristotle’s assertoric logic. As an alternative to SE, we then present a stronger system EC which is adequate for this purpose. EC is a nonexplosive ecthetic system which is shown to be (...) sound and complete with respect to all valid syllogistic arguments with a consistent set of premises. (shrink)

We develop a set-theoretic semantics for Cocchiarella's second-order logical system . Such a semantics is a modification of the nonstandard sort of second-order semantics described, firstly, by Simms and later extended by Cocchiarella. We formulate a new second order logical system and prove its relative consistency. We call such a system and construct its set-theoretic semantics. Finally, we prove completeness theorems for proper normal extensions of the two systems with respect to certain notions of validity provided by (...) the semantics. (shrink)

This essay continues my investigation of `syntactic semantics': the theory that, pace Searle's Chinese-Room Argument, syntax does suffice for semantics (in particular, for the semantics needed for a computational cognitive theory of natural-language understanding). Here, I argue that syntactic semantics (which is internal and first-person) is what has been called a conceptual-role semantics: The meaning of any expression is the role that it plays in the complete system of expressions. Such a `narrow', conceptual-role semantics is the appropriate sort (...) of semantics to account (from an `internal', or first-person perspective) for how a cognitive agent understands language. Some have argued for the primacy of external, or `wide', semantics, while others have argued for a two-factor analysis. But, although two factors can be specified–-one internal and first-person, the other only specifiable in an external, third-person way–-only the internal, first-person one is needed for understanding how someone understands. A truth-conditional semantics can still be provided, but only from a third-person perspective. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 42.1 (2004) 110-112 [Access article in PDF] Terry Pinkard. German Philosophy, 1670-1860: The Legacy of Idealism. New York: Cambridge University Press, 2002. Pp. x + 382. Cloth, $65.00. Paper, $23.00. In one respect, the story related in Terry Pinkard's new book on German idealism is a very old-fashioned one of the "from Kant to Hegel" sort, inasmuch as Hegel's system is here (...) presented as the logical culmination of a single movement of thought, initiated by Kant and carried forward, first by Fichte and then by Schelling; and post-Hegelian philosophy is interpreted as a desperate reaction to the apparent failure of the grand idealist project. In other respects, however, Pinkard offers a welcome variation on this familiar tale by including chapter on less familiar figures and by interpreting the idealist project in the context of some current philosophical concerns.A distinctive feature of this new history is its powerful narrative thrust. Pinkard organizes his tale around a single, central issue, which he calls the "Kantian paradox." The paradox in question is that of self-legislation or self-authorization, a paradox implicit in Kant's claim that one can be bound by laws of which one is also the free author. Though this paradox obviously pertains to Kant's moral philosophy, Pinkard shows that it also extends to the application of any rational norms whatsoever. Pinkard's object is to show how each of the thinkers he discusses responded to this "Kantian paradox," culminating in Hegel's alleged recognition that this paradox is "the basic problem that all post-Kantian philosophies had to solve" (226) and in his effort to provide an essentially social solution to the problem of reason's self-authorization.The book commences with a superb essay on "'Germany' and German Philosophy," examining the relation of "German philosophy" to "Germany" during the period in question, an essay that would be an ideal first assignment for any course on nineteenth-century philosophy. Each of the four major parts of the book begins with a similarly useful, short historical/contextual introduction.Part One is a brilliant summary of the Kantian revolution, not just in epistemology and metaphysics, but also in moral philosophy, aesthetics, and teleology. Without bogging down in unnecessary details or arcane scholarly disputes, Pinkard manages to provide a nuanced account of Kant's arguments and conclusions, emphasizing those tensions and problems in Kant's writings that occupied the attention of the philosophers who followed.Part Two is devoted to the pre-Hegelian "post-Kantians," not just Fichte and the early Schelling, but also Jacobi, Reinhold, Hölderlin, Novalis, Schleiermacher, Frederich Schlegel, and Fries. Pinkard's discussions of these lesser-known figures is welcome and illuminating. The same cannot, unfortunately, be said of his discussion of Fichte. Though the paradox of self-authorization is obviously a central Fichtean concern, Pinkard's account of the Wissenschaftslehre seriously distorts Fichte's project and accomplishments—beginning with Pinkard's failure to distinguish between the Wissenschaftslehre itself, as the name for a complete system of philosophy, and the 1794/95Foundations of the Entire Wissenschaftslehre, which is concerned only with the rudimentary foundations of the same. (This leads him to the mistaken view that Fichte's treatises on right and ethics are "revisions" of the Wissenschaftslehre, [End Page 110] rather than systematic branches of the same.) The account of the Foundations is itself skewed, inasmuch as it fails to take account of how Part III of this text provides a "practical" solution to the "theoretical" impasse of Parts I and II. Instead, Pinkard offers an interpretation of Fichte's early system that simply ignores the centrality thereto of willing and striving, makes no mention of the key role of productive imagination, and characterizes Fichtean intellectual intuition as "an insight into the necessary structure of things themselves" (110). Though Pinkard is aware that the overwhelming majority of Fichte scholars reject such an interpretation, he nevertheless advances it, claiming the right to "argue the opposite view." Unfortunately, he keeps his "argument" to himself... (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations (...) are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$ -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Combining modalities has proven to have interesting applications and many approaches that combine time with other types of modalities have been developed. One of these approaches uses accessibility functions between flows of time to study the basic properties of the functions, such as being total or partial, injective, surjective, etc. The completeness of certain systems expressing many of these properties, with the exception of surjectivity, has been proven. In this paper we propose a language with nominals to denote the initial (...) and final sets of each function in a model in order to obtain a complete system for reasoning about surjective partial functions. (shrink)

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions have been considered and axiomatic systems which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we (...) use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are (...) closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation φ/Tφ and conecessitation T φ/φ and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is w-inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and (...) give an analysis of its proof theory. (shrink)

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation and conecessitation T and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is -inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give an (...) analysis of its proof theory. (shrink)

In the domain of information systems ontologies, the notion of completeness refers to ontological contents by demanding that they be exhaustive with respect to the domain that the ontology aims to represent. The purpose of this paper is to analyze such a notion, by distinguishing different varieties of completeness and by questioning its consistency with the open-world assumption, which formally assumes the incompleteness of conceptualizations on information systems ontologies.

This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language ℒ with a distinguished unary predicate c(x), function-symbols +, · and - and constants 0 and 1 is defined. An interpretation ℜ for ℒ is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as 'region x is connected' and the function-symbols and (...) constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation ������ base on the real closed plane which turns out to be isomorphic to ℜ. A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation. (shrink)

“The great field for new discoveries,” said a scientific friend to me the other day, “is always the Unclassified Residuum.” Round about the accredited and orderly facts of every science there ever floats a sort of dust-cloud of exceptional observations, of occurrences minute and irregular, and seldom met with, which it always proves less easy to attend to than to ignore. The ideal of every science is that of a closed and completed system of truth. The charm of most (...) sciences to their more passive disciples consists in their appearing, in fact, to wear just this ideal form. Each one of our various ologies seems to offer a definite head of classification for every possible phenomenon of the sort which it professes to cover; and, so far from free is most men’s fancy, that when a consistent and organized scheme of this sort has once been comprehended and assimilated, a different scheme is unimaginable. No alternative, whether to whole or parts, can any longer be conceived as possible. Phenomena unclassifiable within the system are therefore paradoxical absurdities, and must be held untrue. When, moreover, as so often happens, the reports of them are vague and indirect, when they come as mere marvels and oddities rather than as things of serious moment, one neglects or denies them with the best of scientific consciences. Only the born geniuses let themselves be worried and fascinated by these outstanding exceptions, and get no peace till they are brought within the fold. Your Galileos, Galvanis, Fresnels, Purkinjes, and Darwins are always getting confounded and troubled by insignificant things. Anyone will renovate his science who will steadily look after the irregular phenomena. And when the science is renewed, its new formulas often have more of the voice of the exceptions in them than of what were supposed to be the rules. (shrink)