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1.
In [8] Jakowski defined by means of an appropriate interpretation a paraconsistent calculus D
2
. In [9] J. Kotas showed that D
2
is equivalent to the calculus M( S5) whose theses are exactly all formulas a such that Ma is a thesis of S5. The papers [11], [7], [3], and [4] showed that interesting paraconsistent calculi could be obtained using modal systems other than S5 and modalities other than M. This paper generalises the above work. Let A be an arbitrary modality (i.e. string of M's, L's and negation signs). Then the A-extension of a set of formulas X is {¦A X}}. Various properties of A-extensions of normal modal systems are examined, including a problem of their axiomatizability 相似文献
2.
In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a
purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language.
It is the logic K
t
4 plus a new axiom ( P): ‘ p Λ G
p ⊃ P G
p’. This formula was used by Prior in his original analysis of Master Argument. ( P) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is
a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear
time. We show what properties of the set of moments are exactly forced by ( P) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction.
Presented by Jacek Malinowski 相似文献
3.
A first order uncountably valued logic L
Q(0,1) for management of uncertainty is considered. It is obtained from approximation logics L
T
of any poset type ( T, ) (see Rasiowa [17], [18], [19]) by assuming ( T, )=( Q(0, 1), ) — where Q(0, 1) is the set of all rational numbers q such that 0< q<1 and is the arithmetic ordering — by eliminating modal connectives and adopting a semantics based on LT-fuzzy sets (see Rasiowa and Cat Ho [20], [21]). Logic L
Q(0,1) can be treated as an important case of LT-fuzzy logics (introduced in Rasiowa and Cat Ho [21]) for ( T, )=( Q(0, 1), ), i.e. as LQ(0, 1)-fuzzy logic announced in [21] but first examined in this paper. L
Q(0,1) deals with vague concepts represented by predicate formulas and applies approximate truth-values being certain subsets of Q(0, 1). The set of all approximate truth-values consists of the empty set ø and all non-empty subsets s of Q(0, 1) such that if qs and qq, then qs. The set LQ(0, 1) of all approximate truth-values is uncountable and covers up to monomorphism the closed interval [0, 1] of the real line. LQ(0, 1) is a complete set lattice and therefore a pseudo-Boolean (Heyting) algebra. Equipped with some additional operations it is a basic plain semi-Post algebra of type Q(0, 1) (see Rasiowa and Cat Ho [20]) and is taken as a truth-table for L
Q(0,1) logic. L
Q(0,1) can be considered as a modification of Zadeh's fuzzy logic (see Bellman and Zadeh [2] and Zadeh and Kacprzyk, eds. [29]). The aim of this paper is an axiomatization of logic L
Q(0,1) and proofs of the completeness theorem and of the theorem on the existence of LQ(0, 1)-models (i.e. models under the semantics introduced) for consistent theories based on any denumerable set of specific axioms. Proofs apply the theory of plain semi-Post algebras investigated in Cat Ho and Rasiowa [4].Presented by Cecylia Rauszer 相似文献
4.
Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions', ``substances' or ``substantial forms'. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances' in a boolean universe V( C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae
N
of formulae of a normal modal theory T m based on T, such that the counterpart ` i' of a the modal variable `x i' of L(T m) in this translation-scheme ranges over elements of V( C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T).The article's basic completeness-result (2.13) then establishes that varphi; is a theorem of T m iff [[
N
() is a consequenceof
N
(T m) for each extension N of T which is a subtheory of the canonical generic theory (ultrafilter) u]] = 1 – or equivalently, that T m is consistent iff[[there is anextension N of T such that N is a subtheory of the canonical generic theory u, and N() for all in T m]] > 0.The proof of thiscompleteness-result also shows that an N which provides a countermodel for a modally unprovable – or equivalently, a closed set in the Stone space St(T) in the sense of V( C) – is intertranslatable with an ` accessibility'- relation of a closely related Kripke-semantics whose `worlds' are generic extensions of an initial universe V via C.This interrelation providesa fairly precise rationale for the semantics' recourse to C-valued structures, and exhibits a sense in which the boolean-valued context is sharp. 相似文献
5.
Lindemann's melting criterion remains useful. However, one prediction it makes for liquid metals (our focus here) is that the long-wavelength limit of the structure factor S( q) at freezing, S T m (0), where T m is the melting temperature, is a universal constant. For 34 metals we have calculated S T m (0) from input data, which is essentially the measured T m and the surface thickness L, defined near freezing as the product of isothermal compressibility and surface tension. To complete the characterization of S T m (0) we fit to one metal, chosen as Rb, for which S T m (0) is well established experimentally. For a wide variety of metals considered, S T m (0) is then found to vary by a factor of 10. 相似文献
6.
We describe a class of MV-algebras which is a natural generalization of the class of “algebras of continuous functions”. More
specifically, we're interested in the algebra of frame maps Hom
(Ω( A), K) in the category T of frames, where A is a topological MV-algebra, Ω( A) the lattice of open sets of A, and K an arbitrary frame.
Given a topological space X and a topological MV-algebra A, we have the algebra C ( X, A) of continuous functions from X to A. We can look at this from a frame point of view. Among others we have the result: if K is spatial, then C( pt( K), A), pt( K) the points of K, embeds into Hom
(Ω( A), K) analogous to the case of C ( X, A) embedding into Hom
(Ω( A), Ω ( X)).
1991 Mathematics Subject Classification: 06F20, 06F25, 06D30
Presented by Ewa Orlowska 相似文献
7.
Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L
*
1( H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L
*
1( H) with respect to Boolean operations, and obtain the language L
1( H). At the next level, we consider an extension L
*
2( H) of L
1( H) in which every sentence is an L
1( H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996). 相似文献
8.
We present a class of normal modal calculi P FD, whose syntax is endowed with operators M
r (and their dual ones, L
r), one for each r [0,1]: if a is sentence, M
r is to he read the probability that a is true is strictly greater than r and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model. Every such a model is a kripkean model, enriched by a family of regular (see below) probability evaluations with range in a fixed finite subset F of [0,1]: there is one such a function for every world w, P
F(w,-), and this allows to evaluate M
ra as true in the world w iff p
F(w, ) r.For every fixed F as before, suitable axioms and rules are displayed, so that the resulting system P
FD is complete and compact with respect to the class of all the F-restricted probabilistic kripkean models. 相似文献
9.
Using a recently introduced entropy-like measure of uncertainty of quantum mechanical states, the problem of hidden variables is redefined in operator algebraic framework of quantum mechanics in the following way: if A, , E( A), E() are von Neumann algebras and their state spaces respectively, (, E()) is said to be an entropic hidden theory of ( A, E( A)) via a positive map L from onto A if for all states E( A) the composite state ° L E() can be obtained as an average over states in E() that have smaller entropic uncertainty than the entropic uncertainty of . It is shown that if L is a Jordan homomorphism then (, E()) is not an entropic hidden theory of ( A, E( A)) via L. 相似文献
10.
Vanadium dioxide (VO 2) thin films deposited on (1010 ) sapphire are composed of two mixed monoclinic phases, namely M1 and M2. The M1 phase is unstable because of the existence of a larger misfit strain in the (102 ) VO 2 film. The reduction of misfit strain in the film favours the formation of the M2 phase. The X-ray diffraction and pole figure results show that both M1 and M2 phases are well aligned with the substrate and both contain twinned structures. Therefore, the microstructure of the film can be regarded as being a transversely modulated heterophase polydomain. A higher electrical resistivity ratio of the semiconductor phase to the metallic phase (rho s/rho m) can be achieved only in single-phase VO 2 thin films, either the M2 or M1 phase. Phase mixing degrades the ratio of rho s/rho m. The film with a single M2 phase exhibits a lower transition temperature of 58 C without any degradation of the rho s/rho m ratio. 相似文献
11.
The main result of this paper is the following theorem: each modal logic extending K4 having the branching property below m and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property below m and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logics K4, K4.1, K4.2, K4.3, S4.1, S4.2, GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logics D
n; to all intermediate Gabbay-De-Jong logics D
n; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well.The results of this paper were obtained by the author during a stay at the Free University of Berlin with support of the Alexander von Humboldt Foundation in 1992 – 1993.Presented by Wolfgang Rauntenberg 相似文献
12.
We show that every proper normal extension of the bi-modal system S5
2 has the poly-size model property. In fact, to every proper normal extension L of S5
2 corresponds a natural number b( L) - the bound of L. For every L, there exists a polynomial P(·) of degree b( L) + 1 such that every L-consistent formula is satisfiable on an L-frame whose universe is bounded by P(||), where || denotes the number of subformulas of . It is shown that this bound is optimal. 相似文献
13.
Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establish the minimum system
C2L
m of primary-conditionals and secondary-conditionals, and then prove some of the formal theorems of the system which have important
intuitive meanings. Secondly, the authors constructed the neighborhood semantics, prove the soundness of C2L
m, introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of C2L
m by the canonical model. Finally, according to the technical results of the minimum system C2L
m, the authors discuss some of the important problems concerning primary-conditionals and secondary-onditionals.
__________
Translated from Luoji Yu Renzhi 逻辑与认知 (Logic and Cognition) (online journal), 2004 (3) 相似文献
14.
The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. With each (consistent) derivation there is associated a Buridan-Volpin (or BV) structure [ M, { r( x)}] which is simply a Tarski structure M for the language and a map giving the range r( x) of each variable x in the derivation. If LK* is (approximately) the classical sequent calculus LK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas has a Tarski model M, then from any normal derivation D in LK* of can be constructed a BV model M
D=[ M, { r( x)}] of where each range r( x) is finite.Presented by Melvin Fitting; 相似文献
15.
The Logic R4 is obtained by adding the axiom (A v B( Av B) to the modal relevant logic NR. We produce a model theory for this logic and show completeness. We also show that there is a natural embedding of a Kripke model for S4 in each R4 model structure.We are indebted to several people for discussions relating to the topic of this paper, in particular, Kit Fine, John Slaney, J. M. Dunn, Jacques Riche, M. A. McRobbie, and Jill LeBlanc. We would also like to thank the Automated Reasoning Project for material assistance. Mares would like to thank The Social Sciences and Humanities Research Council of Canada for fellowships 456-89-0128 and 457-90-0081, which supported him while writing this paper. 相似文献
16.
For a Euclidean space
, let L
n denote the modal logic of chequered subsets of
. For every n 1, we characterize L
n using the more familiar Kripke semantics, thus implying that each L
n is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics L
n form a decreasing chain converging to the logic L
of chequered subsets of
. As a result, we obtain that L
is also a logic over Grz, and that L
has the finite model property. We conclude the paper by extending our results to the modal language enriched with the universal modality. 相似文献
17.
Structurally complicated ξ′- and ξ-phases have been found, for the first time, in as-cast Al 73Ni 5Rh 22 and Al 75Ni 15Rh 10 alloys. The lattice parameters of these two phases were determined by means of electron diffraction and high-resolution transmission electron microscopy (HREM). These two phases have similar orthorhombic structures but with different lattice parameters of a?=?23.2?Å, b?=?16.4?Å, c?=?12.0?Å for the ξ′-phase and a?=?20.3?Å, b?=?16.4?Å, c?=?14.8?Å for the ξ-phase. A new two-dimensional domain-boundary network has also been observed in these two phases. Domain boundaries with normals closely parallel to the [001] direction are actually phason planes represented by a translation vector of r?=(1/2) a?+(1/2τ) c in the ξ′-phase and r=±(1/2τ 2) a?+(τ/2) c in the ξ-phase, whereas the newly-found wide and zigzag boundaries perpendicular to the above set were attributed from the step-like boundary structures of domains related by a translation vector of r?=(1/τ)((1/2) a?+(1/2τ) c). The structural difference between the two types of planar faults is discussed. 相似文献
18.
This paper introduces the logic QLETF, a quantified extension of the logic of evidence and truth LETF, together with a corresponding sound and complete first-order non-deterministic valuation semantics. LETF is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (FDE) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘A entails that A behaves classically, ∙A follows from A’s violating some classically valid inferences. The semantics of QLETF combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of FDE, K3, and LP, we show how these tools, which we call here the method of anti-extensions + valuations, can be naturally applied to a number of non-classical logics. 相似文献
19.
Summary The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined
by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular lattices P
L or partial-Boolean algebras P
A; according to whether the semantic mappings are required to be truth-functional
or truth-functional
; and according to whether two-or-higher dimensional Hilbert space P structures or three-or-higher dimensional Hilbert space P structures are being considered.
If the quantum P structures are taken to be orthomodular lattices P
L, then bivalent mappings which preserve the operations and relations of a P
L must be truth-functional
. Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements
in a P
L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for any two-or-higher dimensional Hilbert space P
L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and
the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and
propositions.
However, the weaker truth-functionality
condition can instead be imposed upon the semantic or hidden-variable mappings on the P
L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean
algebra structural features of the P
L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebras P
A, where bivalent mappings which preserve the operations and relations of a P
A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden
variable proposal which imposes this truth-functionality (c) condition is impossible for any three-or-higher dimensional Hilbert space P
A or P
L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert space P
A or P
L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty
Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean
in the Piron sense.
The present paper is a sequel of the proceedings of the Society of Exact Philosophy annual meeting, published in our issue 9:2, pp. 187–278. — Ed.
I am indebted to my supervisor, Dr. Edwin Levy, for many hours of helpful discussion on the drafts leading to this paper. 相似文献
20.
In this paper we consider the structure of the class FGMod S of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACS s A of all algebraic closed-set systems C on A such that ( A, C) ε FGMod S. We relate some properties of these structures with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGMod S under substructures and under reduced products, and the property that for any A the lattice FACS s A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence
of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems
in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples
and counterexamples.
Dedicated to the memory of Willem Johannes Blok 相似文献
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