Modal semantics without possible worlds

1981 ◽  
Vol 46 (1) ◽  
pp. 77-86 ◽  
Author(s):  
John T. Kearns
Keyword(s):  
Possible Worlds ◽  
The Other ◽  
Modal Operator ◽  
Logical Operators ◽  
Semantic Account ◽  
Necessary Truth ◽  
Modal Semantics ◽  
Propositional Language ◽  
Image Position ◽  
Plain Truth

In this paper I will develop a semantic account for modal logic by considering only the values of sentences (and formulas). This account makes no use of possible worlds. To develop such an account, we must recognize four values. These are obtained by subdividing (plain) truth into necessary truth (T) and contingent truth (t); and by subdividing falsity into contingent falsity (f) and necessary falsity (impossibility: F). The semantic account results from reflecting on these concepts and on the meanings of the logical operators.To begin with, we shall consider the propositional language L0. The language L0 has (1) infinitely many atomic sentences, (2) the two truth-functional connectives ∼, ∨, and the modal operator □. (Square brackets are used for punctuation.) The other logical expressions are defined as follows:D1 [A & B] = (def)∼[∼A ∨ ∼ B],D2 [A ⊃ B] = (def)[∼A ∨ B],D3 ◊ A =(def)∼□∼A.I shall use matrices to give partial characterizations of the significance of logical expressions in L0. For negation, this matrix is wholly adequate:Upon reflection, it should be clear that this matrix is the obviously correct matrix for negation.

Computational Principles

1990 ◽  
Author(s):  
John L. Pollock
Keyword(s):  
Mathematical Theory ◽  
Possible Worlds ◽  
Mathematical Structure ◽  
Possible World ◽  
States Of Affairs ◽  
Classical Probability ◽  
Probability Calculus ◽  
Logical Operators ◽  
Necessary Truth ◽  
State Of Affairs

Much of the usefulness of probability derives from its rich logical and mathematical structure. That structure comprises the probability calculus. The classical probability calculus is familiar and well understood, but it will turn out that the calculus of nomic probabilities differs from the classical probability calculus in some interesting and important respects. The purpose of this chapter is to develop the calculus of nomic probabilities, and at the same time to investigate the logical and mathematical structure of nomic generalizations. The mathematical theory of nomic probability is formulated in terms of possible worlds. Possible worlds can be regarded as maximally specific possible ways things could have been. This notion can be filled out in various ways, but the details are not important for present purposes. I assume that a proposition is necessarily true iff it is true at all possible worlds, and I assume that the modal logic of necessary truth and necessary exemplification is a quantified version of S5. States of affairs are things like Mary’s baking pies, 2 being the square root of 4, Martha’s being smarter than John, and the like. For present purposes, a state of affairs can be identified with the set of all possible worlds at which it obtains. Thus if P is a state of affairs and w is a possible world, P obtains at w iff w∊P. Similarly, we can regard monadic properties as sets of ordered pairs ⧼w,x⧽ of possible worlds and possible objects. For example, the property of being red is the set of all pairs ⧼w,x⧽ such that w is a possible world and x is red at w. More generally, an n-place property will be taken to be a set of (n+l)-tuples ⧼w,x1...,xn⧽. Given any n-place concept α, the corresponding property of exemplifying a is the set of (n + l)-tuples ⧼w,x1,...,xn⧽ such that x1,...,xn exemplify α at the possible world w. States of affairs and properties can be constructed out of one another using logical operators like conjunction, negation, quantification, and so on.

Completeness of global intuitionistic set theory

1997 ◽  
Vol 62 (2) ◽  
pp. 506-528 ◽  
Author(s):  
Satoko Titani
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Mathematical Object ◽  
The Other ◽  
Logical System ◽  
Logical Operators ◽  
Logical Symbol ◽  
Image Position ◽  
Intuitionistic Set Theory

Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol → and the other is the metalogical symbol ⇒ in sequentsConsidering the logical system LJ as a mathematical object, we understand that the logical symbols ∧, ∨, →, ¬, ∀, ∃ are operators on formulas, and ⇒ is a relation. That is, φ ⇒ Ψ is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence.The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.Let Ω be a cHa. Ω can be a truth value set of a model of LJ. Then the logical symbols ∧, ∨, →, ¬, ∀x, ∃x are interpreted as operators on Ω, and the sentence φ ⇒ Ψ is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol ⇒ also can be expressed as a logical operators such that φ ⇒ Ψ is interpreted as 1 or 0.

scholarly journals The Dissociation of the Compound of Iodine and Thio-urea

1904 ◽  
Vol 24 ◽  
pp. 233-239 ◽  
Author(s):  
Hugh Marshall
Keyword(s):  
Nitric Acid ◽  
General Formula ◽  
Free Acid ◽  
The Other ◽  
Image Position

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

Extremely undecidable sentences

1982 ◽  
Vol 47 (1) ◽  
pp. 191-196 ◽  
Author(s):  
George Boolos
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Propositional Modal Logic ◽  
Image Position

Let ‘ϕ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P0, P1, … Pn, … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define Aϕ by[and similarly for other nonmodal propositional connectives]; andwhere Bew(x) is the standard provability predicate for PA and ⌈F⌉ is the PA numeral for the Gödel number of the formula F of PA. Then for any ϕ, (−□⊥)ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ(p0) = S. If ψ(p0) is the undecidable sentence constructed by Gödel, then ⊬PA (−□⊥→ −□p0 & − □ − p0)ψ and ⊢PA(P0 ↔ −□⊥)ψ. However, if ψ(p0) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬PA(P0 ↔ −□⊥)ψ and ⊢PA (−□⊥→ −□−p0 & −□−p0)ψ. We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p0, if for some ψ, ⊬PAAψ, then ⊬PAAϕ, where ϕ(p0) = S. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

3. Laboratory Notes by Professor Tait (a.) Measurement of the Potential, required to produce Sparks of various lengths, in Air at different pressures, by a Holtz Machine

1878 ◽  
Vol 9 ◽  
pp. 332-333
Author(s):  
Messrs Macfarlane ◽  
Paton
Keyword(s):  
General Result ◽  
The Other ◽  
Considerable Distance ◽  
Image Position

The general result of these strictly preliminary experiments appears to show that for sparks not exceeding a decimetre in length (L), taken in air at different pressures (P), between two metal balls of 7mm·5 radius, the requisite potential (V), is expressed by the formulaThe Holtz machine employed is a double one, made by Ruhmkorff, and it was used with its small Leyden jars attached. The measurements had to be made with a divided-ring electrometer, so that two insulated balls, at a considerable distance from one another, were connected, one with the machine, the other with the electrometer.

scholarly journals Certain Mathematical Instruments for Graphically Indicating the Direction of Refracted and Reflected Light Rays

1906 ◽  
Vol 25 (2) ◽  
pp. 806-812
Author(s):  
J.R. Milne
Keyword(s):  
Optical System ◽  
The Other ◽  
Reflected Light ◽  
Light Rays ◽  
Image Position

The refraction equation sin i == μ sin r, though simple in itself, is apt to give rise, in problems connected with refraction, to formulæ too involved for arithmetical computation. In such cases it may be necessary to trace the course through the optical system in question of a certain number of arbitrarily chosen rays, and thence to find the course of the other rays by interpolation. Thelinkage about to be described affords a rapid and accurate means of determining the paths of the rays through any optical system.

Note on a Theorem regarding a Series of Convergents to the Roots of a Number

1893 ◽  
Vol 19 ◽  
pp. 15-19
Author(s):  
Thomas Muir
Keyword(s):  
The Other ◽  
Image Position

If the positive integral powers of be taken, and the expansion of each be separated into two parts, rational and irrational, thus—then the ratio of the rational portion to the coefficient of in the other portion is approximately equal to , the convergence being perfect when the power of the binomial is infinite. This is the simplest case of a theorem discovered by the late Dr Sang, and enunciated by him as the result of a process of induction in his paper “On the Extension of Brouncker's Method to the Comparison of several Magnitudes” (Proc. Roy. Soc. Edin., vol. xviii. p. 341, 1890–91).

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