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.

Modal operators and functional completeness, II

1977 ◽  
Vol 42 (3) ◽  
pp. 391-399 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Kripke Semantics ◽  
Functional Completeness ◽  
Possible World ◽  
States Of Affairs ◽  
Propositional Modal Logic ◽  
Modal Operators ◽  
Fixed Frame ◽  
State Of Affairs

In the Kripke semantics for propositional modal logic, a frame W = (W, ≺) represents a set of “possible worlds” and a relation of “accessibility” between possible worlds. With respect to a fixed frame W, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a way of forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fw: (P(W))n → P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term “states of affairs” for our “possible worlds”.}In a broader sense, an n-ary connective is represented by an n-ary operatorf = {fw∣ W ∈ Fr}, where Fr is the class of all frames and each fw: (P(W))n → P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(P1,…, Pn) is true in a possible world depends only upon which modal combinations of P1,…,Pn are true in that world. (A modal combination of P1,…,Pn is the result of applying a modal connective to P1,…, Pn.) A connective C is strongly coherent if whether C(P1, …, Pn) obtains in a state of affairs depends only upon which modal combinations of P1,…, Pn obtain in that state of affairs.

On Necessary Gratuitous Evils

2020 ◽  
Vol 12 (3) ◽  
pp. 117
Author(s):  
Michael James Almeida
Keyword(s):  
Necessary Condition ◽  
Possible World ◽  
States Of Affairs ◽  
Gratuitous Evil ◽  
Moral Perfection ◽  
Standard Position ◽  
State Of Affairs

The standard position on moral perfection and gratuitous evil makes the prevention of gratuitous evil a necessary condition on moral perfection. I argue that, on any analysis of gratuitous evil we choose, the standard position on moral perfection and gratuitous evil is false. It is metaphysically impossible to prevent every gratuitously evil state of affairs in every possible world. No matter what God does—no matter how many gratuitously evil states of affairs God prevents—it is necessarily true that God coexists with gratuitous evil in some world or other. Since gratuitous evil cannot be eliminated from metaphysical space, the existence of gratuitous evil presents no objection to essentially omnipotent, essentially omniscient, essentially morally perfect, and necessarily existing beings.

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.

Probabilities over rich languages, testing and randomness

1982 ◽  
Vol 47 (3) ◽  
pp. 495-548 ◽  
Author(s):  
Haim Gaifman ◽  
Marc Snir
Keyword(s):  
Formal Language ◽  
Possible Worlds ◽  
Random Sequence ◽  
A Priori ◽  
Mathematical Structure ◽  
Mathematical Framework ◽  
The Family ◽  
Logical Connectives ◽  
State Of Affairs ◽  
General Abstract

The basic concept underlying probability theory and statistics is a function assigning numerical values (probabilities) to events. An “event” in this context is any conceivable state of affairs including the so-called “empty event”—an a priori impossible state. Informally, events are described in everyday language (e.g. “by playing this strategy I shall win $1000 before going broke”). But in the current mathematical framework (first proposed by Kolmogoroff [Ko 1]) they are identified with subsets of some all-inclusive set Q. The family of all events constitutes a field, or σ-field, and the logical connectives ‘and’, ‘or’ and ‘not’ are translated into the set-theoretical operations of intersection, union and complementation. The points of Q can be regarded as possible worlds and an event as the set of all worlds in which it takes place. The concept of a field of sets is wide enough to accommodate all cases and to allow for a general abstract foundation of the theory. On the other hand it does not reflect distinctions that arise out of the linguistic structure which goes into the description of our events. Since events are always described in some language they can be indentified with the sentences that describe them and the probability function can be regarded as an assignment of values to sentences. The extensive accumulated knowledge concerning formal languages makes such a project feasible. The study of probability functions defined over the sentences of a rich enough formal language yields interesting insights in more than one direction.Our present approach is not an alternative to the accepted Kolmogoroff axiomatics. In fact, given some formal language L, we can consider a rich enough set, say Q, of models for L (called also in this work “worlds”) and we can associate with every sentence the set of all worlds in Q in which the sentence is true. Thus our probabilities can be considered also as measures over some field of sets. But the introduction of the language adds mathematical structure and makes for distinctions expressing basic intuitions that cannot be otherwise expressed. As an example we mention here the concept of a random sequence or, more generally, a random world, or a world which is typical to a certain probability distribution.

Lewis' Indexical Argument for World-Relative Actuality

Dialogue ◽  
1989 ◽  
Vol 28 (2) ◽  
pp. 289-304 ◽  
Author(s):  
Richard M. Gale
Keyword(s):  
David Lewis ◽  
Cost Benefit ◽  
The Other ◽  
Abstract Entity ◽  
Modal Realism ◽  
Possible World ◽  
States Of Affairs ◽  
Systematic Analysis ◽  
Reductio Ad Absurdum ◽  
Critical Scrutiny

David Lewis has shocked the philosophical community with his original version of extreme modal realism according to which “every way that a world could possibly be is a way that some world is”. Logical Space is a plenitude of isolated physical worlds, each being the actualization of some way in which a world could be, that bear neither spatiotemporal nor causal relations to each other. Lewis has given independent, converging arguments for this. One is the argument from the indexicality of actuality, the other an elaborate cost-benefit argument of the inference-to-the-best explanation sort to the effect that a systematic analysis of a number of concepts, including modality, causality, propositions and properties, fares better under his theory than under any rival one that takes a possible world to be either a linguistic entity or an ersatz abstract entity such as a maximal compossible set of properties, propositions or states of affairs. Lewis' legion of critics have confined themselves mostly to attempts at a reductio ad absurdum of his theory or to objections to his various analyses. The indexical argument, on the other hand, has not been subject to careful critical scrutiny. It is the purpose of this paper to show that this argument cannot withstand such scrutiny. Its demise, however, leaves untouched his argument from the explanatory superiority for his extreme modal realism.

scholarly journals Mínimo e máximo em estruturas nominais – uma análise semântica

2019 ◽  
pp. 250-264
Author(s):  
Rui Marques
Keyword(s):  
Possible Worlds ◽  
Modal Operator ◽  
Possible World

This paper is concerned with the semantics of the portuguese phrases with the form o mínimo/máximo N (‘the minimum N’) and o mínimo/máximo de N (‘the minimum/maximum of N’). Some nouns may occur in both of these constructions, while others might occur in only one of them, and still other nouns might occur only if accompanied by a modal operator. The proposal is made that these facts can be straightforwardly explained by the hypothesis that the first and the second of these syntactic constructions have, respectively, an extensional and an intensional meaning, together with the fact that some nouns have the same denotation in any possible world, while others denote different sets of entities in different possible worlds.

scholarly journals Nonexistence, Vague Existence, Merely Possible Existence

Disputatio ◽  
2012 ◽  
Vol 4 (33) ◽  
pp. 427-443
Author(s):  
Iris Einheuser
Keyword(s):  
Possible Worlds ◽  
Target Object ◽  
Possible World ◽  
Plausible Assumption ◽  
De Re ◽  
Modal Semantics ◽  
Vague Existence

Abstract This paper explores a new non-deflationary approach to the puzzle of nonexistence and its cousins. On this approach, we can, under a plausible assumption, express true de re propositions about certain objects that don’t exist, exist indeterminately or exist merely possibly. The defense involves two steps: First, to argue that if we can actually designate what individuates a nonexistent target object with respect to possible worlds in which that object does exist, then we can express a de re proposition about “it”. Second, to adapt the concept of outer truth with respect to a possible world – a concept familiar from actualist modal semantics – for use in representing the actual world.

Metaphysics

2017 ◽  
Author(s):  
Colin McGinn
Keyword(s):  
Conceptual Analysis ◽  
Possible Worlds ◽  
A Priori ◽  
A Posteriori ◽  
De Re ◽  
Necessary Truth ◽  
Possible Worlds Theory ◽  
De Dicto

This chapter explores philosophical issues in metaphysics. It begins by distinguishing between de re and de dicto necessity. All necessity is uniformly de re; there is simply no such thing as de dicto necessity. Indeed, in the glory days of positivism, all necessity was understood as uniformly the same: a necessary truth was always an a priori truth, while contingent truths were always a posteriori. The chapter then assesses the concept of antirealism. Antirealism is always an error theory: there is some sort of mistake or distortion or sloppiness embedded in the usual discourse. The chapter also considers paradoxes, causation, conceptual analysis, scientific mysteries, the possible worlds theory of modality, the concept of a person, the nature of existence, and logic and propositions.

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