Correction to a paper on modal logic

1955 ◽  
Vol 20 (2) ◽  
pp. 150-150 ◽  
Author(s):  
Alan Ross Anderson
Keyword(s):  
Modal Logic ◽  
Decision Procedure ◽  
The Way

In a recent paper (here referred to as “IDP”) the writer outlined a decision procedure for Lewis's system S4 of modal logic. One of the clauses in definition 3.1 of IDP requires correction. Clause II of 3.1 (2) should read as follows.II. Some constituent of the form ◊β, of degree n1 ≤ n, has the value T in Row (i), and some constituents of the forms ◊δ1, … ◊δh, and ◊η1, …, ◊ηm, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ1 ∨ … ∨ δh ∨ ◊η1 ∨ … ∨ ◊ηm) is an (n1 − 1)-tautology of S4.This change is required in order to carry out the proof of Metathcorcm 3.19. In particular, the change guarantees the following. If the expression η of the second paragraph of 3.20 is of degree n − 1, then the antecedent λ of formula ζ on page 210 of IDP is also of degree n − 1; and consequently the formula ζ of 3.19 is of degree n (since ◊λ is a constituent of ζ). (If we fail to make the correction, then it might be the case that both ζ and (3) are of degree n − 1, in which case Row (i) would not satisfy clause II as originally stated, contrary to the claim at the end of 3.20.) The proofs for the remaining cases of 3.20 can then be carried out, using the revised clause II, in the way originally indicated.The proof of Metatheorem 3.2 requires only trivial corrections for case 2.

Relational dual tableau decision procedure for modal logic K

2011 ◽  
Vol 20 (4) ◽  
pp. 747-756 ◽  
Author(s):  
J. Golinska-Pilarek ◽  
E. Munoz-Velasco ◽  
A. Mora-Bonilla
Keyword(s):  
Modal Logic ◽  
Decision Procedure

Heuristics all the way up?

2000 ◽  
Vol 23 (5) ◽  
pp. 758-759 ◽  
Author(s):  
Adam Morton
Keyword(s):  
Decision Procedure ◽  
Good Decision ◽  
Trade Off ◽  
Similar Kind ◽  
The Way

I investigate whether heuristics similar to those studied by Gigerenzer and his co-authors can apply to the problem of finding a suitable heuristic for a given problem. I argue that not only can heuristics of a very similar kind apply but they have the added advantage that they need not incorporate specific trade-off parameters for balancing the different desiderata of a good decision-procedure.

scholarly journals A Note on the Issue of Cohesiveness in Canonical Models

2019 ◽  
Vol 29 (3) ◽  
pp. 331-348
Author(s):  
Matteo Pascucci
Keyword(s):  
Modal Logic ◽  
Decision Procedure ◽  
Canonical Model ◽  
Canonical Models

Abstract In their presentation of canonical models for normal systems of modal logic, Hughes and Cresswell observe that some of these models are based on a frame which can be also thought of as a collection of two or more isolated frames; they call such frames ‘non-cohesive’. The problem of checking whether the canonical model of a given system is cohesive is still rather unexplored and no general decision procedure is available. The main contribution of this article consists in introducing a method which is sufficient to show that canonical models of some relevant classes of normal monomodal and bimodal systems are always non-cohesive.

Time and Modality

2011 ◽  
Author(s):  
Ulrich Meyer
Keyword(s):  
Modal Logic ◽  
Tense Logic ◽  
Linguistic Theory ◽  
Fourth Section ◽  
The Third ◽  
Metaphysical Theory ◽  
Verb Tense ◽  
The Way

This chapter discusses modal logic: the logic of possibility and necessity. After a brief review of modal logic in the second section, the third section presents basic results of propositional tense logic. The fourth section develops a system of quantified tense logic. With these technical preliminaries out of the way, the fifth section explains why tense logic ultimately fails as a linguistic theory of verb tense. The sixth section presents the main objection to tense primitivism: that tense logic has insufficient expressive resources to serve as a metaphysical theory of time. The seventh section argues that the tense primitivist can overcome these problems by treating times as maximally consistent sets of sentences. The eighth section discusses a key difference between time and modality: the lack of a temporal analogue of actualism.

A cut-free Gentzen-type system for the modal logic S5

1980 ◽  
Vol 45 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Masahiko Sato
Keyword(s):  
Modal Logic ◽  
Natural Number ◽  
Decision Procedure ◽  
Type System ◽  
Point Of View ◽  
Usual Sense ◽  
Cut Elimination ◽  
Strict Sense ◽  
Theoretic Point ◽  
Subformula Property

The modal logic S5 has been formulated in Gentzen-style by several authors such as Ohnishi and Matsumoto [4], Kanger [2], Mints [3] and Sato [5]. The system by Ohnishi and Matsumoto is natural, but the cut-elimination theorem in it fails to hold. Kanger's system enjoys cut-elimination theorem, but, strictly speaking, it is not a Gentzen-type system since each formula in a sequent is indexed by a natural number. The system S5+ of Mints is also cut-free, and its cut-elimination theorem is proved constructively via the cut-elimination theorem of Gentzen's LK. However, one of his rules does not have the so-called subformula property, which is desirable from the proof-theoretic point of view. Our system in [5] also enjoys the cut-elimination theorem. However, it is also not a Gentzen-type system in the strict sense, since each sequent in this system consists of a pair of sequents in the usual sense.In the present paper, we give a Gentzen-type system for S5 and prove the cut-elimination theorem in a constructive way. A decision procedure for S5 can be obtained as a by-product.The author wishes to thank the referee for pointing out some errors in the first version of the paper as well as for his suggestions which improved the readability of the paper.

Well-behaved modal logics

1984 ◽  
Vol 49 (4) ◽  
pp. 1393-1402
Author(s):  
Harold T. Hodes
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Classical Model ◽  
Classical Case ◽  
Recursion Theory ◽  
Modal Logics ◽  
Modal Model ◽  
The Right ◽  
Domain Property ◽  
The Way

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.

scholarly journals The interpretability logic of Peano arithmetic

1990 ◽  
Vol 55 (3) ◽  
pp. 1059-1089 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Modal Logic ◽  
Modal Analysis ◽  
Set Theory ◽  
Decision Procedure ◽  
Peano Arithmetic ◽  
Relevant Modal Logic ◽  
Binary Operator ◽  
Base Theory ◽  
Interpretability Logic ◽  
Work Done

AbstractPA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

A complete and consistent modal set theory

1967 ◽  
Vol 32 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Continuous Functions ◽  
Subsequent Paper ◽  
Object Language ◽  
Real Numbers ◽  
Consistent System ◽  
Detailed Treatment ◽  
The Way

1.1. The aim of this paper is the construction of a demonstrably consistent system of set theory that (1) contains roughly the same amount of mathematics as the writer's system K′ [3], including a theory of continuous functions of real numbers, and (2) provides a way for expressing in the object language various propositions which, in the case of K′, could be expressed only in the metalanguage, for example, general propositions about all real numbers. It was not originally intended that the desired system should be a modal logic, but the modal character of the system appears to be a natural outgrowth of the way it is constructed. A detailed treatment of the natural, rational, and real numbers is left for a subsequent paper.

Elements of a Decision Procedure for Christian Social Ethics

1972 ◽  
Vol 65 (4) ◽  
pp. 509-530 ◽  
Author(s):  
Charles H. Reynolds
Keyword(s):  
Ethical Theory ◽  
Decision Procedure ◽  
Social Ethics ◽  
Christian Ethics ◽  
Ideal Observer ◽  
Moral Judgments ◽  
Christian Social Ethics ◽  
The Way

Several philosophers have observed an affinity between a role that an understanding of God has in Christian ethics and a role of an ideal observer in their own ethical theory. R. M. Hare has even gone so far as to assert that, “Since for many Christians God occupies the role of ‘ideal observer,’ the moral judgments which they make may be expected to coincide with those arrived at by the method of reasoning which I am advocating.” Now, Hare is correct in observing that God and an ideal observer have certain characteristics in common. But God is not simply an ideal observer. And some of the differences between God and an ideal observer may be as important as the similarities for the way in which Christians make moral judgments. It is therefore somewhat hasty of Hare to assume that his method of reasoning is identical to the method of reasoning appropriate in Christian ethics.

Prior, Arthur Norman (1914–69)

2018 ◽  
Author(s):  
C.J.F. Williams
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Tense Logic ◽  
Central Importance ◽  
Change Events ◽  
Nature Of Time ◽  
Theory Of Descriptions ◽  
Time Existence ◽  
The Way

Prior is most often thought of as the creator of tense logic. (Tense logic examines operators such as ‘It will be the case that’ in the way that modal logic examines ‘It must be the case that’.) But his first book was on ethics, and his views on metaphysical topics such as determinism, thinking, intentionality, change, events, the nature of time, existence, identity and truth are of central importance to philosophy. Using methods akin to Russell’s in his Theory of Descriptions, he showed that times, events, facts, propositions and possible worlds were logical constructions. For example, we get rid of events by recognizing among other things that to say that the event of Caesar’s crossing the Rubicon took place later than the event of Caesar’s invading Britain is to say that it has been the case that both Caesar is crossing the Rubicon and it has been the case that Caesar is invading Britain. The title of the posthumous work, Worlds, Times and Selves (1977), indicates the breadth and depth of his thought. He is also fun to read. He died at the age of fifty-four, at the height of his powers.

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