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By
Minari, Pierluigi
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4 Citations
We give completeness results — with respect to Kripke's semantic — for the negationfree intermediate predicate calculi:
(1)
$$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$
(2)
$$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$
and the superintuitionistic predicate calculus:
(3)
$$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$
The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3].
For a general account on negationfree intermediate predicate calculi — see CasariMinari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].
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By
Cooke, Roger M.; Lambalgen, Michiel
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2 Citations
Gaisi Takeuti has recently proposed a new operation on orthomodular latticesL,
$$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$
:P(L)»L. The properties of
$$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$
suggest that the value of
$$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$
(A) (A)
$$ \subseteq $$
L) corresponds to the degree in which the elements ofA behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular latticesL and the existence of twovalued homomorphisms onL.
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By
Borga, Marco
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5 Citations
This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cutelimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.
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By
Zawadowski, Marek
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1 Citations
The topos theory gives tools for unified proofs of theorems for model theory for various semantics and logics. We introduce the notion of power and the notion of generalized quantifier in topos and we formulate sufficient condition for such quantifiers in order that they fulfil downward SkolemLöwenheim theorem when added to the language. In the next paper, in print, we will show that this sufficient condition is fulfilled in a vast class of Grothendieck toposes for the general and the existential quantifiers.
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By
Bellissima, Fabio; Mirolli, Massimo
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1 Citations
We find a short way to construct a formula which axiomatizes a given finite frame of the modal logicK, in the sense that for each finite frameA, we construct a formula ωA which holds in those and only those frames in which every formula true inA holds.
To obtain this result we find, for each finite model
$$\mathfrak{A}$$
and each natural numbern, a formula ω
$$\mathfrak{A}$$
which holds in those and only those models in which every formula true in
$$\mathfrak{A}$$
, and involving the firstn propositional letters, holds.
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By
Czelakowski, Janusz
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1 Citations
The classesMatr(
$$ \subseteq $$
) of all matrices (models) for structural finitistic entailments
$$ \subseteq $$
are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for
$$ \subseteq $$
, thenMatr(
$$ \subseteq $$
) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms and strict homomorphic preimages. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments.
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By
Tembrowski, Bronisław
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This paper deals with Boolean algebras supplied with an additional binary operation, calledBalgebras for short.
The aim of the paper is to generalize some theorems concerning topological Boolean algebras to more comprehensive classes ofBalgebras, to formulate fundamental properties ofBalgebras, and to find more important relationships of these algebras to other known algebras.
The paper consists of two parts. At the beginning of the first one, several subclasses ofBalgebras are distinguished, and then, their basic properties, connections between them as well as certain relationships with other algebras, are investigated. In particular, it is shown that the class of Boolean algebras together with an arbitrary unary operation is polynomially equivalent to the class ofB_{1}algebras.
The second part of the paper is concerned with the theory of filters and congruences inBalgebras.
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By
Kalman, J. A.
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3 Citations
Condensed detachment is usually regarded as a notation, and “defined” by example. In this paper it is regarded as a rule of inference, and rigorously defined with the help of the Unification Theorem of J. A. Robinson. Historically, however, the invention of condensed detachment by C. A. Meredith preceded Robinson's studies of unification. It is argued that Meredith's ideas deserve recognition in the history of unification, and the possibility that Meredith was influenced, through Łukasiewicz, by ideas of Tarski going back at least to 1939, and possibly to 1930 or earlier, is discussed. It is proved that a term is derivable by substitution and ordinary detachment from given axioms if and only if it is a substitution instance of a term which is derivable from these axioms by condensed detachment, and it is shown how this theorem enables the ideas of Łukasiewicz and Tarski mentioned above to be formalized and extended. Finally, it is shown how condensed detachment may be subsumed within the resolution principle of J. A. Robinson, and several computer studies of particular Hilberttype propositional calculi using programs based on condensed detachment or on resolution are briefly discussed.
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By
Guenoche, A.; Hesnard, A.
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2 Citations
For classifying wine amphoras used at the end of the Roman Republic and the beginning of the Empire (the socalled Dressel 2–4), we present a typological approach which combines a classification algorithm with the archeological reasoning. At the first step, clusters contain only nuclei based on the different production areas. To assign a corpus of artifacts to them, it is divided for each cluster into a context (artifacts which certainly do not belong to the cluster) and a residue. For each cluster, we built characteristic definitions with logical discriminant function of morphological attributes. Each definition cuts the residue in two classes: one containing the artifacts assigned to the cluster by the definition and the complementary one in the residue. Assignment and choices of cluster definitions and context remain with the archeological expert, who submits those typological constructions to a validation process founded on archeological knowledge. Such an approach focuses on a very common situation in human sciences: the construction of a cognitive typology beginning with a partially clustered set. Clustering must be done with descriptive attributes, without knowing if they can be connected with the wanted cluster.
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By
Bocharov, V. A.
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Two subjectpredicate calculi with equality,SP_{=} and its extensionUSP′_{=}, are presented as systems of natural deduction. Both the calculi are systems of free logic. Their presentation is preceded by an intuitive motivation.
It is shown that Aristotle's syllogistics without the laws of identitySaP andSiP is definable withinSP_{=}, and that the firstorder predicate logic is definable withinUSP′_{=}.
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By
Hautamäki, Antti
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4 Citations
In this paper a propositional logic of viewpoints is presented. The language of this logic consists of the usual modal operatorsL (of necessity) andM (of possibility) as well as of two new operatorsA andR. The intuitive interpretations ofA andR are “from all viewpoints” and “from some viewpoint”, respectively. Semantically the language is interpreted by using Kripke models augmented with sets of “viewpoints” and with a new alternativeness relation for the operatorA. Truth values of formulas are evaluated with respect to a world and a viewpoint. Various axiomatizations of the logic of viewpoints are presented and proved complete. Finally, some applications are given.
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By
Pörn, Ingmar
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In the paper I investigate aspects of adverbial modification as an operation applying an adverb or adverbial phrase to a predicate and thereby creating a new predicate. The logic of adverbial modification, on this view, belongs to the logic of predicate modifiers. The theory I present is intended to cover not only adverbial modification but also attributive modification, but problems concerning the latter will not be given any special attention.
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By
Gumański, Leon
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The chief aim of the paper is to extend the calculusDSC_{1} (see [4]) in such a way as to satisfy all the requirements listed in [4] as well as a further stipulation — called ‘the principle of uninvolvement’ — to the effect that neither deontic compatibility nor deontic incompatibility of codes (see [2]) should be presupposed in deontic logic.
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By
Oikkonen, Juha
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We discuss an abstract notion of a logical operation and corresponding logics. It is shown that if all the logical operations considered are implicitely definable in a logic ℒ*, then the same holds also for the logic obtained from these operations. As an application we show that certain iterated forms of infinitely deep languages are implicitely definable in game quantifier languages. We consider also relations between structures and show that Karttunen's characterization of elementary equivalence for the ordinary infinitely deep languages can be generalized to hold for the iterated infinitely deep languages. An early version of this work was presented in the Abstracts Section of ICM '78.
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By
Sidorenko, E. A.
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The aim of this paper is to present a modified version of the notion of strong proof from hypotheses (definition D2), and to give three deduction theorems for the relevant logicsR (theoremsT1, andT2) andE (theoremT3).
By
Talja, Jari
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This paper discusses refinements of the natural ordering of themdegrees (1degrees) of strong recursive reducibility classes. Such refinements are obtained by posing complexity conditions on the reduction function. The discussion uses the axiomatic complexity theory and is hence very general. As the main result it is proved that if the complexity measure is required to be linearly bounded (and spacelike), then a natural class of refinements forms a lattice with respect to a natural ordering upon them.
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By
Smirnov, V. A.
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1 Citations
LetEO be the elementary ontology of Leśniewski formalized as in Iwanuś [1], and letLS be the monadic secondorder calculus of predicates. In this paper we give an example of a recursive function ϕ, defined on the formulas of the language ofEO with values in the set of formulas of the language of LS, such that ⊢_{EO}A iff ⊢_{LS}ϕ(A) for each formulaA.
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By
Karpenko, A. S.
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In this note we prove that some familiar systems of finitely manyvalued logics havefactor semantics, and establish necessary conditions for a system of manyvalued logic having semantics of this kind.
By
PrzeŁęcki, Marian
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The approach adopted in the paper is based on the theory known as Montague grammar. Accepting, in general, that theory — especially in its modified version, which is due to Thomason and Kaplan — the author points out certain inadequacy in its treatment of the meaning of some indexical expressions and suggests some modification of its theoretical framework in order to avoid that shortcoming. It is claimed that to do justice to the meaning of socalled indefinite indexicals (such as “we”, “you”, “now”) two kinds of dependence of their semantic values upon the context of use must be taken into account — a semantic (or lexical) and a pragmatic (or extralexical) one.
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By
Wright, G. H.
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2 Citations
Summary
The paper is based upon a conception of norms as prescriptions which are neither true nor false. Two norms may be said to contradict one another when the conjunction of (the descriptions of) their contents is a logical contradiction. A norm is said to entail another norm when the first norm and the negationnorm of the second contradict one another. By the negationnorm of an obligation is understood a permission “to the contrary”, and by the negationnorm of a permission an obligation “to the contrary”. On the basis of these definitions it can be shown that the axioms and theorems of first order standard deontic logic are “normative tautologies”. Norms of higher order may be viewed as prescriptions “transmitting the will” of a higher or sovereign authority through a lower or subordinate authority to the normsubjects. Under this conception of the orders, it is provable that a “complete” system of deontic logic isS4like.
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By
Karttunen, Maaret
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1 Citations
We define a subhierarchy of the infinitely deep languagesN_{ℵλ} described by Jaakko Hintikka and Veikko Rantala. We shall show that some model theoretic results wellknown in the model theory of the ordinary infinitary languages ℒ_{ℵλ} can be generalized for these new languages. Among these are the downward LöwenheimSkolem and Łoś's theorems as well as some compactness properties.
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By
Benthem, Johan
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1 Citations
Exact philosophy consists of various disciplines scattered and separated. Formal semantics and philosophy of science are good examples of two such disciplines. The aim of this paper is to show that there is possible to find some integrating bridge topics between the two fields, and to show how insights from the one are illuminating and suggestive in the other.
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By
Vojshvillo, E. K.
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A system of natural deduction is presented whose set of theses is identical with that of the systemE of entailment. For that system a decision procedure is described proving the decidability ofE.
By
Furmanowski, Tomasz
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In this paper we start an investigation of a logic called the logic of algebraic rules. The relation of derivability of this logic is defined on universal closures of special disjunctions of equations extending the relation of derivability of the usual equational logic. The paper contains some simple theorems and examples given in justification for the introduction of our logic. A number of open questions is posed.
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By
Dziobiak, Wiesław
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We prove that each proper ideal in the lattice of axiomatic, resp. standard strengthenings of the intuitionistic propositional logic is of cardinality 2^{ℵ0}. But, each proper ideal in the lattice of structural strengthenings of the intuitionistic propositional logic is of cardinality 2^{2ℵ0}. As a corollary we have that each of these three lattices has no atoms.
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By
Niiniluoto, Ilkka
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2 Citations
The recent theories of truthlikeness have not paid attention to the distinction between lawlike and accidental generalizations. L.J. Cohen has expressed this by saying that science aims at “legisimilitude” rather than “verisimilitude”. G. Oddie has given a reply to Cohen by defining the notion of legisimilitude in terms of higherorder logics. This paper gives a different reply to Cohen by treating laws as physically necessary generalizations and by defining the notion of legisimilitude as closeness to a suitably chosen lawlike sentence.
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By
Pearce, David; Rantala, Veikko
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4 Citations
This essay is an attempt to consider dynamic aspects of scientific theorising from a formal perspective. Our emphasis will be on the aims and methods for constructing formal models of theory dynamics which will be conceived from a general or ‘theoretical’ rather than ‘applied’ standpoint.
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By
Tuomela, Raimo
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The paper discusses eliminative explanation in which a (social) successor theory correctively explains and, as a consequence, eliminates its predecessor theory. Technical concepts and results from general logic are applied to the explication of corrective explanation, especially to the notion of framework translation that it involves.
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By
Pearce, David; Rantala, Veikko
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3 Citations
In this paper we give the gist of our reconstructed notion of (limiting case) correspondence. Our notion is very general, so that it should be applicable to all the cases in which a correspondence has been said to exist in actual science.
By
Wolniewicz, Bogusław
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The paper applies the theory presented in “A Formal Ontology of Situations” (this journal, vol. 41 (1982), no. 4) to obtain a typology of metaphysical systems by interpreting them as different ontologies of situations. Four are treated in some detail: Hume's diachronic atomism, Laplacean determinism, Hume's synchronic atomism, and Wittgenstein's logical atomism. Moreover, the relation of that theory to the “situation semantics” of Perry and Barwise is discussed.
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By
Czelakowski, Janusz
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9 Citations
We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's wellknown result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based.
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By
Piochi, Brunetto
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2 Citations
In the present paper, we study some properties of matrices for nonstructural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcicki's ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one.
Such matrices are used to characterize lattices of nonstructural consequence operators, by constructing lattices, antiisomorphic to them (see section 5.). In the last section, a sufficient condition is given for a nonstructural operator to be finite.
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By
Georgescu, George
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Chang algebras as algebraic models for Chang's modal logics [1] are defined. The main result of the paper is a representation theorem for these algebras.
By
Päppinghaus, Peter; Wirsing, Martin
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1 Citations
Nondeterministic programs occurring in recently developed programming languages define nondeterminate partial functions. Formulas (Boolean expressions) of such nondeterministic languages are interpreted by a nonempty subset of {T (“true”), F (“false”), U (“undefined)}. As a semantic basis for the propositional part of a corresponding nondeterministic threevalued logic we study the notion of a truthfunction over {T, F, U} which is computable by a nondeterministic evaluation procedure. The main result is that these truthfunctions are precisely the functions satisfying four basic properties, called
$$ \subseteq $$
isotonic,
$$ \subseteq $$
^{−}isotonic, hereditarily guarded, and hereditarily guardusing, and that a function satisfies these properties iff it is explicitly definable (in a certain normal form) from “if..then..else..fi”, binary choice, and constants.
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By
Bugajski, Sławomir
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2 Citations
A new approach to semantics, based on ordered Banach spaces, is proposed. The Banach spaces semantics arises as a generalization of the four particular cases: the Giles' approach to belief structures, its generalization to the nonBoolean case, and “fuzzy extensions” of Boolean as well as of nonBoolean semantics.
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By
Goldberg, Moshe S.
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6 Citations
In this note, we give a representation of distributive Ockham algebras via natural homfunctors. In order to do this, we describe two different structures (one algebraic, and the other ordertopological) on the set of subsets of the natural numbers. The topological duality previously obtained by A. Urquhart is used throughout.
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By
Shehtman, V. B.
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14 Citations
This paper concerns modal logics appearing from the temporal ordering of domains in twodimensional Minkowski spacetime. As R. Goldblatt has proved recently, the logic of the whole plane isS4.2. We consider closed or open convex polygons and closed or open domains bounded by simple differentiable curves; this leads to the logics:S4,S4.1,S4.2 orS4.1.2.
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By
Hajičová, Eva
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The notion of presupposition still belongs to one of the widely discussed notions in the presentday linguistic writings. We tried to show elsewhere (Hajičová, 1971; 1974) that the difficulties concerning the test of negation that is used as an operational criterion for the determination of presuppositions are closely connected with the position of the relevant element of the sentence either in the topic, or in the focus of the sentence. (1) is a presupposition of (2), because it is entailed both by (2) and by (3); (3) can be followed by (4) rather than by (5). The bearer of the intonation centre in all our examples is denoted by capitals.
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By
Benthem, J. F. A. K.
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1 Citations
In this chapter the main ingredients are surveyed that go into our temporal structures: individuals, relations and operations. Some of the more interesting ones will be selected for further logical investigation.
By
Benthem, J. F. A. K.
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Now that the notions of’point structure’ and ‘period structure’ have been developed to some extent, it becomes of interest to relate the two in a systematic fashion. For this purpose, once more, here are the relevant notions as they evolved in the previous discussions.
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By
Benthem, J. F. A. K.
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The Priorean language of the preceding chapter is not inextricably tied up with the usual point ontology: it may also be interpreted in period structures. Indeed, the same truth definition II.2.1.2. works as well in the latter context.
