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)

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)

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)

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)

A formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. The absolute consistency of the system is proved. A completeness proof for the system is also constructed. This proof is relative to a concept of logical validity provided by a semantics, which assumes as its philosophical background an approach to sortals from a modern form of conceptualism.

System image is an extension of multiplicative linear logic with the rules mix, nullary mix, and a self-dual, noncommutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of image. Due to the operator seq, system image extends the applications of image to those where the sequential composition is crucial, e.g., concurrency theory. System image is an extension of image with the rules mix and (...) nullary mix. In this paper, by relying on the fact that system image is a conservative extension of system image, I show that system image is NP-complete by encoding the 3-Partition problem in image. I provide a simple completeness proof of this encoding by resorting to a novel proof theoretical method for reducing nondeterminism in proof search, which is also of independent interest. (shrink)

For each truth-functionally complete set of connectives, we construct a sound and complete natural deduction system containing no axioms and the smallest possible number of inference rules, namely one.

In this paper, we extend the system AF2 in order to have the subject reduction for the $betaeta$-reduction. We prove that the types with positive quantifiers are complete for models that are stable by weak-head expansion.

It is argued that behavior analysis is an actual or potential axiomatic system based upon the schedules of reinforcement which are behavioral, causative laws. Gödel proved that all axiomatic systems are complete or consistent, but not both at the same time. The point is made that behavior analysis is an incomplete, consistent system. The system's incompleteness is compensated for by the concept of the behavioral repertoire which, although in part lying outside of the axiomatic core of behavior (...) analysis, both extends and strengthens it. (shrink)

In this paper we consider a type system with a universal type $omega$ where any term (whether open or closed, $beta$-normalising or not) has type $omega$. We provide this type system with a realisability semantics where an atomic type is interpreted as the set of $lambda$-terms saturated by a certain relation. The variation of the saturation relation gives a number of interpretations to each type. We show the soundness and completeness of our semantics and that for different (...) notions of saturation (based on weak head reduction and normal $beta$-reduction) we obtain the same interpretation for types. Since the presence of $omega$ prevents typability and realisability from coinciding and creates extra difficulties in characterizing the interpretation of a type, we define a class ${mathbb U}^+$ of the so-called positive types (where $omega$ can only occur at specific positions). We show that if a term inhabits a positive type, then this term is $beta$-normalisable and reduces to a closed term. In other words, positive types can be used to represent abstract data types. The completeness theorem for ${mathbb U}^+$ becomes interesting indeed since it establishes a perfect equivalence between typable terms and terms that inhabit a type. In other words, typability and realisability coincide on ${mathbb U}^+$. We give a number of examples to explain the intuition behind the definition of ${mathbb U}^+$ and to show that this class cannot be extended while keeping its desired properties. (shrink)

Dmitrii Mendeleev’s periodic system is known for its predictive accuracy, but talk of its completeness is rarer. This is surprising because completeness was a quality that Mendeleev saw as important for a systematization of the chemical elements. Here, I explain how Mendeleev’s valuing of completeness influenced the development of his periodic system. After introducing five indicators of its completeness, I zoom into one in particular: Mendeleev’s inclusion of a schematic row of oxides. I then (...) show how it guided Mendeleev’s predictions of indium and ekaboron, which suggests that the valuing of completeness was instrumental for making predictions. (shrink)

Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined by \ is derivable in \. It is known (...) that any logic can be defined by a structurally complete deductive system—its structural completion. The main goal of the paper is to study the following problem: given a superintuitionistic logic L, is the structural completion of L hereditarily structurally complete? It is shown that, on the one hand, there is continuum many of such logics, including \, and many of its standard extensions. On the other hand, there is continuum many superintutitionistic logics structural completion of which is not hereditarily structurally complete. It is observed that the class of hereditarily structurally complete superintuitionistic consequence relations does not have the smallest element and it contains continuum many members lacking the finite model property. The following statement is instrumental in obtaining negative results: if a Lindenbaum algebra of formulas on one variable is finite and has more than 15 elements, then a structural completion of such a logic is not hereditarily structurally complete. (shrink)

The article concerns two axiom systems of Słupecki for the functionally complete three-valued propositional logic: W1–W6 and A1–A9. The article proves that both of them are inadequate—W1–W6 is semantically incomplete, on the other hand, A1–A9 governs a functionally incomplete calculus, and thus, it cannot be a semantically complete axiom system for the functionally complete three-valued logic.

This book provides the first complete, literal English translation of Alexis de Tocqueville’s and Gustave de Beaumont’s first edition of On the Penitentiary System in the United States and Its Application to France. The work contains a critical comparison of two competing American penitentiary disciplines known as the Auburn and Philadelphia systems, an evaluation of whether American penitentiaries can successfully work in France, a detailed description of Houses of Refuge as the first juvenile detention centers, and an argument against (...) penal colonization. The work provides valuable insights into understanding Tocqueville as a statesman, as well as a comparative look at civic engagement in early American and French penal reform movements. The Translator’s Introduction provides historical context for understanding Tocqueville’s work in French penal reform and the major themes of the report. The book thus fills a void in Tocquevillian studies and extrapolates the roots of American and French criminal justice systems in the nineteenth century. (shrink)