Systems of modal logic which are not unreasonable in the sense of Halldén

1953 ◽  
Vol 18 (2) ◽  
pp. 109-113 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Modal Logic ◽  
Present Note ◽  
The Other ◽  
Normal Extension ◽  
Sentential Calculus ◽  
Other Hand ◽  
Negation Sign

Halldén, in [1], has recently pointed out that it is highly undesirable, in a system of sentential calculus, for there to exist two formulas α and β such that: (i) α and β contain no variable in common; (ii) neither α nor β is provable; (iii) α ∨ β is provable. We shall call a system unreasonable (in the sense of Halldén) if there exists a pair of formulas α and β having properties (i), (ii), and (iii). Halldén shows (in [1]) that the Lewis systems S1 and S3 are unreasonable in this sense; and that the same is true of any system which is between S1 and S3, as well as of every system which is stronger than S3 but weaker than both S4 and S7. In the present note we shall show that this defect does not occur in S4, nor in S5, nor in any “quasi-normal” extension of S5; we give an example, on the other hand, of an unreasonable system which lies between S4 and S5.When we speak, in what follows, of a system of modal logic, we shall mean a system having the same class of well-formed formulas as have the various Lewis calculi. Thus the well-formed formulas of a system of modal logic, when written in unabbreviated form, are just those formulas which can be built up from sentential variables by use of the binary connective ‘·’ (conjunction sign), and the two unary connectives ‘˜’ (negation sign) and ‘◇’ (possibility sign). We shall, however, also make use of some of the defined signs of Lewis.

scholarly journals Well-definedness and observational equivalence for inductive–coinductive programs

2019 ◽  
Vol 29 (4) ◽  
pp. 419-468
Author(s):  
Henning Basold ◽  
Helle Hvid Hansen
Keyword(s):  
Fixed Point ◽  
Modal Logic ◽  
The Other ◽  
Observational Equivalence ◽  
Other Hand ◽  
Inductive Types ◽  
Point Types ◽  
Usual Topology

Abstract We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

Amalgamation in relation algebras

1998 ◽  
Vol 63 (2) ◽  
pp. 479-484 ◽  
Author(s):  
Maarten Marx
Keyword(s):  
Modal Logic ◽  
Equational Theory ◽  
Weak Form ◽  
Boolean Algebras ◽  
The Other ◽  
Relation Algebras ◽  
Boolean Algebras With Operators ◽  
Other Hand ◽  
Arrow Logic ◽  
Representable Relation

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].Let K be a class of relational type algebras such that(i) composition is associative,(ii) K is a class of boolean algebras with operators, and(iii) K contains the representable relation algebras RRA.Then the equational theory of K is undecidable.On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.

Note on a theorem of R. Baer

1949 ◽  
Vol 45 (3) ◽  
pp. 321-327 ◽  
Author(s):  
G. Higman
Keyword(s):  
Finite Number ◽  
Present Note ◽  
The Other ◽  
Fixed Integer ◽  
Other Hand

It is trivial that a group all of whose elements except the identity have order two is Abelian; and F. Levi and B. L. van der Waerden(1) have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R. Baer(3) has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.

The problem of interpreting modal logic

1947 ◽  
Vol 12 (2) ◽  
pp. 43-48 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Material Objects ◽  
Quantification Theory ◽  
Other Hand

There are logicians, myself among them, to whom the ideas of modal logic (e. g. Lewis's) are not intuitively clear until explained in non-modal terms. But so long as modal logic stops short of quantification theory, it is possible (as I shall indicate in §2) to provide somewhat the type of explanation desired. When modal logic is extended (as by Miss Barcan1) to include quantification theory, on the other hand, serious obstarles to interpretation are encountered—particularly if one cares to avoid a curiously idealistic ontology which repudiates material objects. Such are the matters which it is the purpose of the present paper to set forth.

On axiomatising products of Kripke frames

2000 ◽  
Vol 65 (2) ◽  
pp. 923-945 ◽  
Author(s):  
Ágnes Kurucz
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Modal Language ◽  
First Order ◽  
Kripke Frames ◽  
Other Hand ◽  
The Many ◽  
By Products

AbstractIt is shown that the many-dimensional modal logic Kn, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any n > 2. On the other hand, Kn is determined by a class of frames satisfying a single first-order sentence.

Infinitary combinatorics and modal logic

1990 ◽  
Vol 55 (2) ◽  
pp. 761-778 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Modal Logic ◽  
Propositional Logic ◽  
The Other ◽  
Ordinal Numbers ◽  
Mahlo Cardinal ◽  
Other Hand ◽  
Modal Propositional Logic

AbstractWe show that the modal propositional logic G, originally introduced to describe the modality “it is provable that”, is also sound for various interpretations using filters on ordinal numbers, for example the end-segment filters, the club filters, or the ineffable filters. We also prove that G is complete for the interpretation using end-segment filters. In the case of club filters, we show that G is complete if Jensen's principle □κ holds for all κ < ℵω; on the other hand, it is consistent relative to a Mahlo cardinal that G be incomplete for the club filter interpretation.

scholarly journals On Some Divisibility Properties of

1964 ◽  
Vol 7 (4) ◽  
pp. 513-518 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Simple Proof ◽  
Present Note ◽  
The Other ◽  
Other Hand ◽  
Image Position ◽  
Divisibility Properties

L. Moser [3] recently gave a very simple proof that1.has no solutions. In the present note we shall first of all prove that for , which by the fact that there is a prime p satisfying n < p ≤ 2n immediately implies that2.has no solutions. It is easy to see on the other hand that3.has infinitely many non-trivial solutions.

scholarly journals Founded (Auto)Epistemic Equilibrium Logic Satisfies Epistemic Splitting

2019 ◽  
Vol 19 (5-6) ◽  
pp. 671-687 ◽  
Author(s):  
JORGE FANDINNO
Keyword(s):  
Modal Logic ◽  
Logic Programming ◽  
Epistemic Logic ◽  
The Other ◽  
Stable Model ◽  
Logic Programs ◽  
Splitting Property ◽  
Other Hand ◽  
The Many ◽  
Programming Semantics

AbstractIn a recent line of research, two familiar concepts from logic programming semantics (unfounded sets and splitting) were extrapolated to the case of epistemic logic programs. The property of epistemic splitting provides a natural and modular way to understand programs without epistemic cycles but, surprisingly, was only fulfilled by Gelfond’s original semantics (G91), among the many proposals in the literature. On the other hand, G91 may suffer from a kind of self-supported, unfounded derivations when epistemic cycles come into play. Recently, the absence of these derivations was also formalised as a property of epistemic semantics called foundedness. Moreover, a first semantics proved to satisfy foundedness was also proposed, the so-called Founded Autoepistemic Equilibrium Logic (FAEEL). In this paper, we prove that FAEEL also satisfies the epistemic splitting property something that, together with foundedness, was not fulfilled by any other approach up to date. To prove this result, we provide an alternative characterisation of FAEEL as a combination of G91 with a simpler logic we called Founded Epistemic Equilibrium Logic (FEEL), which is somehow an extrapolation of the stable model semantics to the modal logic S5.

scholarly journals Note on the Law of Transformation of Energy and its Applications

1893 ◽  
Vol 19 ◽  
pp. 253-259 ◽  
Author(s):  
W. Peddie
Keyword(s):  
Present Note ◽  
The Other ◽  
Mathematical Methods ◽  
Other Hand ◽  
Elementary Methods ◽  
The Law ◽  
Physical Quantities

The investigation of the relations which subsist amongst various physical quantities, and which are exemplified in the different transformations of energy, is one of the most interesting investigations in the whole range of physics. On the other hand, the mathematical methods which are usually employed for this purpose are of such a nature as to be unavailable to any who are unacquainted with higher mathematical methods. It is one of the objects of the present note to point out how these relations may be worked out by means of the most elementary methods. For the notation of the calculus, though used below as a matter of convenience, is quite unnecessary, since we might use, e.g., small and capital type of the same letter to denote quantities of the same kind of different magnitudes.

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