WWW.ABSTRACT.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Abstract, dissertation, book
 
<< HOME
CONTACTS



Pages:   || 2 |

«Mariusz Urbański SOME REMARKS CONCERNING MODAL PROPOSITIONAL LOGIC OF QUESTIONS Abstract. Recently, it has become a custom to treat questions (or, ...»

-- [ Page 1 ] --

Logic and Logical Philosophy

Volume 6 (1998), 187–196

Mariusz Urbański

SOME REMARKS CONCERNING

MODAL PROPOSITIONAL

LOGIC OF QUESTIONS

Abstract. Recently, it has become a custom to treat questions (or, better,

questioning) as a game between two subjects. Unfortunately, one rarely goes

beyond the scheme of Questioner-Scientist and Answerer-Nature, although the Interlocutor so conceived displays some undesirable features. This paper argues for the idea that logic of questions can be build as a logic of the game between “knowledge resources” persons or theories, rather than errant Scientist and omniscient Nature. To this end the concept of epistemically-possible worlds is discussed, which is conceived as analogous to that of possible worlds in modal logic. And, furthermore, the concepts of relation of epistemic alternativeness and of epistemically-alternative worlds are introduced. On this basis a version of semantics for propositional, three-valued logic of questions is offered and semantic proofs of some theses are given.

Received June 20, 1998; Revised October 15, 1998 The aim of this paper is to construct a certain logic of yes–no questions.

This logic will be based on three-valued propositional logic. We will also make use of some concepts which are similar to those which are often used in modal logic.

In the first section of this paper, we shall introduce some auxiliary concepts, at the first place the concept of epistemically-possible worlds. To this end, let us consider three kinds of utterance expression: propositions, assertions, and questions. Unlike propositions, assertions and questions involve a subject. Any assertion presupposes a person who asserts (an “asserter”), any question — a questioner. Yet, there is no such an animal as a “propositioner”. Assertions differ from questions in that they involve only one subject, whereas questions have both questioners and addressees. So in the case of questions we can talk about “bi-subjectivity”.

In short, a question is always asked by someone to someone, by someone, who wants to get an answer to someone, who is supposed to know the answer.

So questioning (or, better, questioning-answering) is a game between two subjects, and there is nothing new in that. Unfortunately, one rarely goes beyond the scheme of a Questioner-Scientist (explorer, discoverer) and an Answerer-Nature (or whatever name we want to call the Second Side). The Interlocutor so conceived displays some undesirable features: it is secretive and taciturn, and not always answers our questions, although it is supposed to be omniscient and frank.

In fact, we hardly deal with such interlocutors. Usually our questioninganswering games are played between “knowledge resources”: between persons (e.g. when John asks Peter whether it is raining), or persons and theories (e.g. when John asks about the cause of snowfall on the basis of a certain meteorological theory); perhaps even between theories themselves.1 We can express this intuitions by introducing a semantic construction which employs an epistemological counterpart of the ontological idea of possible worlds.

Possible worlds so conceived will be called epistemically-possible worlds.

As a matter of fact, in the case of modal logic, the notion of possible worlds (“ontically-possible” from now on) is used to introduce the alWe mean “knowledge” in the objective sense, e.g. as a set of propositions. Therefore, in the case of “knowledge resources” of persons one is in need of some logic of assertions or acknowledgements, which avoids the omniscience paradox. Theories are unprotected against that (we use the term “theory” in the wide sense).

... Logic of Questions 189 Some Remarks concerning ternativeness relation and, furthermore, the notion of alternative worlds (“ontically-alternative”, resp.). Similarly, we are mainly interested here in epistemically- alternative worlds. And we shall try to characterize this notion on the analogy of certain modal logic constructions.

By “ontically-possible worlds” one usually means certain (actual or possible, concrete or abstract) temporal and spatial systems — arrangements of objects, courses of events (local or global) — related in some way to the structure which is usually called “the real world”. Similarly, epistemicallypossible worlds will also be related to some kind of reality. Yet, in this particular context by “reality” we mean the true-and-complete knowledge about states of affairs rather than the states of affairs themselves. One can say that an epistemically-possible world is a world of knowledge about what obtains and what does not. In this sense, epistemic constructions are founded on ontical ones: the former are the look down on the latter. Or perhaps a multitude of looks, for the true-and-complete one — God’s one — is beyond our reach.

Of course, we will not try to define the notion of “epistemically-possible worlds” in detail. Such a definition will obviously depend on the concept of knowledge. For example, “epistemologically-possible worlds” can be conceived as sets of propositions accepted by some subject(s) — provided that the concept of knowledge is understood in terms of a mere acceptance.





Whether the world of the true-and-complete knowledge exists at all, and what the relation of epistemic alternativeness is, are metaphysical problems.

The relation under consideration holds between those epistemically-possible worlds which can enter the game of questioning-answering. One problem is that saying that a world α is epistemically alternative to a world β if α is cognitively-possible with respect to β (or perhaps commutable with it in some sense) is as instructive as saying that “ontically-possible” means “that one, which can happen instead”. But it is not our ambition to solve this problem at this point. And following the tradition here, we shall confine ourselves to what is without doubt namely, that every epistemically-possible world is alternative to itself, and this is justified by the intuition that we can ask questions to ourselves.

The next question is who and what can ask about. Two problems are worthy of attention here. First, the problem of reason: in what epistemic situation it is reasonable (or who is epistemically entitled) to ask a question;

second, the problem of goal: what is the epistemical aim of posing questions.

Let us start from the latter.

The goal of asking questions is to get answers — true or false, but always direct (both possible and just-sufficient). We can distinguish two basic kinds 190 Mariusz Urbański of questions: correct questions and incorrect questions. At a first approximation we may say that a correct question is a question which has at least one direct answer with a determinate truth value (i.e. true or false). An incorrect question, in turn, is simply a question which is not correct. (Note that a question can be incorrect for two reasons: it has no direct, i.e. possible and just-sufficient, answer at all, or it has direct answer(s), but there is no direct answer to the question which has a determinate truth value.) Yet, the above definitions are not complete: the phenomenon of “bi-subjectivity” of questions must be taken into account. The goal of asking questions is to get answers, and it is impossible to achieve this goal if in the epistemic world of an addressee no direct answer has a determinate truth value. So it is the epistemic world of the addressee which counts here. Moreover, the epistemic world of the addressee must be compatible with that of the questioner. And finally, there must be at least one epistemic world which fulfils the above conditions. So the correctness of a question Q posed in an epistemic world α (of the questioner) is dependent upon the existence of an epistemically alternative world β (of a possible addressee) in which at least one direct answer to Q is true or false.

In turn, there are two satisfactory solutions of the problem of reason. One can argue that it is pointless to ask about what is known to the questioner.2 It is reasonable to ask somebody if an addressee can fill a truth-value gap in the questioner‘s knowledge: the informativeness of an answer with respect to the epistemically-possible world of the questioner is what is important here. Of course this approach would of impose some additional conditions on our (informal, as yet) definition of the correctness of a question.

The second possible solution of the problem of reason is to admit rhetorical and “examination” questions as a solid kind of questions. In this case it does not matter whether the questioner himself knows the answer in advance or not. In what follows we will adopt this latter solution.

Let us now analyze the expression “It is a question whether”. It is easily seen that this expression belongs to the same syntactic category as “It is possible that”, “It is necessary that”, “It is not the case that”, etc. On the other hand, as far as (simple) yes-no questions are concerned, the expression “It is a question whether A” seems to be true just in case the question “Is it the case that A?” is correct. Taking this intuition for granted, let us now construct a certain logic of the operator “It is a question whether”.

Aristotle, for example, in Analytica Posteriora, 90a, says that we do not ask about what is self-evident.

... Logic of Questions 191 Some Remarks concerning Let L0 be the language of Classical Propositional Calculus (CPC). Let us now supplement the vocabulary of L0 with the operator ‘?’, which can be interpreted as an erotetic one-place operator “It is a question whether”.

The vocabulary of L0 enriched with the operator ‘?’ is the vocabulary of

a new language L. There are two categories of well-formed formulas of L:

declarative well-formed formulas (d-wffs) and erotetic well-formed formulas (e-wffs). The d-wffs of L are simply the well-formed formulas of L0. The set

ΦE of e-wffs is the smallest set which fulfils the following conditions:

(i) if A is a d-wff, then ? A is in ΦE, (ii) if A, B are in ΦE, then ¬A, ? A, (A ∧ B), (A ∨ B), (A → B), (A ↔ B) are in ΦE.

Note that among the meaningful expressions of L there are no “mixed” formulas, i.e. there are no expressions of the form ? A ∗ B and A ∗ ? B, where ∗ stands for a propositional connective and A, B are d-wffs. Yet, we allow iterations of the operator ‘?’. And among e-wffs we can distinguish (i) simple e-wffs (that is e-wffs of the form ? A, where A is a d-wff) and (ii) compound e-wffs, which are the remaining e-wffs.

Let us now supplement the language L with some semantics.

Definition 1. An interpretation of the language L is an ordered quadruple I = U, A, η, η∗, where U is a non-empty set, A is a dyadic, reflexive relation defined over the members of U (A ⊆ U × U), and η, η∗ are mappings of the set Var of propositional variables into the set of all subsets of U which fulfil the following conditions: for any propositional variable pi, η(pi ) ∩ η∗ (pi ) = ∅ and η(pi ) ∪ η∗ (pi ) is a proper subset of U.

Intuitively, U is the set of epistemically-possible worlds (and thus can be infinite) and A is the relation of epistemic alternativeness.

We can now define a valuation function V for the declarative part of the language L (i.e. for the d-wffs of L). The initial conditions are as follows (we

assume that pi ∈ Var and α ∈ U):

Definition 2. V(pi, α) = 1 iff α ∈ η(pi );

V(pi, α) = 0 iff α ∈ η∗ (pi );

V(pi, α) = n iff α ∈ η(pi ) and α ∈ η∗ (pi ).

The clauses for formulas with propositional connectives (i.e. negation, conjunction, disjunction, implication and equivalence) can be built as semantic counterparts of the following truth tables of the Kleene‘s “strong”

three valued logic:

192 Mariusz Urbański

–  –  –

But this was a preliminary step. The next one is as follows. Using the above definition we will define a second valuation function V ′ for simple e-wffs. This function will only be a two valued one, according to the intuitions that any question is correct or incorrect. Moreover, its definition will involve two elements of the set U, due to the phenomenon of the “bi-subjectivity” of questions mentioned above (an expression ‘? A, α/β’ can be read as: “it is a question whether A in a world α with respect to a world β”).

Definition 3. If pi ∈ Var, α, β ∈ U, α A β and A, B are d-wffs of L, then:

1) V ′ (? pi, α/β) = 1 iff V(pi, β) = 1 or V(pi, β) = 0 (i.e. iff V(pi, β) is determinate), V ′ (? pi, α/β) = 0 iff V(pi, β) = n (i.e. iff V(pi, β) is undeterminate);

2) V ′ (? A, α/β) = 1 iff V(A, β) = 1 or V(A, β) = 0, V ′ (? A, α/β) = 0 iff V(A, β) = n;

3) V ′ (? ¬A, α/β) = 1 iff V(¬A, β) = 1 or V(¬A, β) = 0 V ′ (? ¬A, α/β) = 0 iff V(¬A, β) = n;

4) V ′ (?(A ∧ B), α/β) = 1 iff V((A ∧ B), β) = 1 or V((A ∧ B), β) = 0 V ′ (?(A ∧ B), α/β) = 0 iff V((A ∧ B), β) = n;

and so on for other connectives.

Let us now introduce the definition of the concept of truth of a formula

of L in a world α of an interpretation I:

–  –  –

3. if A is a compound e-wff of the form ? B, then A ∈ VerL (I, α) iff B ∈ VerL (I, α) or B ∈ VerL (I, α);

4. if A is a compound e-wff of the form ¬B, then A ∈ VerL (I, α) iff B ∈ VerL (I, α);

5. if A is a compound e-wff of the form (B ∧ C), then A ∈ VerL (I, α) iff B ∈ VerL (I, α) and C ∈ VerL (I, α);

6. if A is a compound e-wff of the form (B ∨ C), then A ∈ VerL (I, α) iff B ∈ VerL (I, α) or C ∈ VerL (I, α);

7. if A is a compound e-wff of the form (B → C), then A ∈ VerL (I, α) iff B ∈ VerL (I, α) or C ∈ VerL (I, α);

8. if A is a compound e-wff of the form (B ↔ C), then A ∈ VerL (I, α) iff B ∈ VerL (I, α) just in case that C ∈ VerL (I, α).



Pages:   || 2 |


Similar works:

«The Discursive Construction of the Ivorian Nation in the Period of Ivoirité by Cary Keith Campbell BA French, Linguistics, Brigham Young University, 1999 MA French Linguistics and Literature, University of Pittsburgh, 2002 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh UNIVERSITY OF PITTSBURGH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Cary Keith Campbell It...»

«DISSERTATION Titel der Dissertation „Brand Positioning Through the Consumers’ Lens“ Verfasserin / Verfasser Mag. rer. soc. oec. Christoph Fuchs Angestrebter akademischer Grad Ph. D. (Doctor of Philosophy) Wien, im April 2008 Studienkennzahl lt. Studienblatt: A 094 146 Dissertationsgebiet lt. Studienblatt: Management Betreuer / Betreuerin: Univ.-Prof. Dr. Adamantios Diamantopoulos ii ACKNOWLEDGMENTS I would like to thank Univ. Prof. Adamantios Diamantopoulos, Ph. D., for the supervision of...»

«Einleitender Überblick und Zusammenfassung 12. Es anders tun! Entwicklung einer anderen kulturellen Lebenspraxis Von grundlegender Bedeutung auf dem Weg in eine Kultur der Zukunftsfähigkeit ist auch das Möglichwerden und Sichtbarmachen von Alternativen, um dem derzeitigen Paradigma von der gestaltungsfeindlichen Alternativlosigkeit entgegenzuwirken und den Menschen ihre Gestaltungsmacht (wieder) bewusst zu machen. Zentral ist die Einsicht, dass es immer auch anders sein könnte und dass zu...»

«! ! ! ! ! ! !Second Nature and Ethical Life: Habit, Culture, and Critique in Hegel’s Science of Right Andreja Novakovic Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY © 2012 Andreja Novakovic All Rights Reserved ABSTRACT Second Nature and Ethical Life: Habit, Culture, and Critique in Hegel’s Science of Right Andreja Novakovic My dissertation investigates the status of reflection in...»

«Todd M. Frye, PhD, LCPC, LCMFT, NCC, SATP-S Home Address: Work Address: 267 N. Sumac St. MidAmerica Nazarene Olathe, KS 66061 University Phone: 913/626-1387 2030 E. College Way Cell: 913/626-1387 Olathe, KS 66062 Email: tmfrye@mnu.edu 913/971-3731 EDUCATIONAL BACKGROUND Idaho State University, Pocatello, Idaho • Doctorate of Philosophy: Counselor Education and Counseling. Accredited by the Council for the Accreditation of Counseling and Related Programs (CACREP) • CACREP specialty focus:...»

«Trans-Scripts 3 (2013) Animals Are Not Things: A View on Animal Welfare Based on Neurological Complexity* Temple Grandin* Since I am autistic I do not understand purely Abstract concepts that are based only in language. To understand a word I have to make a picture in my imagination and define words with concrete examples. When I think of the phrase “buy a car” I immediately get images of past experiences of buying cars. Some purely philosophical arguments I do not understand because I...»

«Naturalismus bei Foot und Hursthouse ANTON LEIST 1. Wie passt die Moral in die übrige Welt? Die gegenwärtige Ethik ist unter anderem mit zwei Problemen konfrontiert, deren Zusammenhang auf den ersten Blick undurchsichtig ist. Das erste Problem ergibt sich aus dem schwierigen Verhältnis von Ethik und Wissenschaft. Die Wissenschaften besitzen die letzte öffentliche Autorität bei fast allen Themen, die in der Gesellschaft von Bedeutung sind, und das gilt zweifellos auch für menschliche...»

«Essays in Heterogeneous Agent Macroeconomics A Dissertation Submitted to the Faculty of the Graduate School of the University of Minnesota by Andrew S. Glover In Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy Jos´-V´ e ıctor R´ ıos-Rull, Adviser August 2011 c Andrew S. Glover, 2011 Acknowledgements I would like to thank my adviser, Jos´-V´ e ıctor R´ ıos-Rull, for his guidance and encouragement. This dissertation would not exist without his assistance....»

«Qui A Vol A Les Tartes You may make requesting these japanese soul pdf to people through when to have met hotel objectives bosses. Philosophy someone invited thin, financial, and here limited. A that starts Qui A Volé Les Tartes? released in you to recognize is to rise out this way mornings having everywhere in you. Term, special people and income numbers back not learn our state. The higher interview the store is, the more unfortunately some CAGR shall jeopardize. A success is increased...»

«BEHIND AND BEYOND THE IVY: HOW SCHOOLS PRODUCE ELITES THROUGH THE BODIES OF RACIAL OTHERS by Leila Angod A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Social Justice Education Ontario Institute for Studies in Education University of Toronto © Copyright by Leila Angod 2015 BEHIND AND BEYOND THE IVY: HOW SCHOOLS PRODUCE ELITES THROUGH THE BODIES OF RACIAL OTHERS Doctor of Philosophy 2015 Leila Angod Graduate Department of...»

«Optimizing Information Mediators By Selectively Materializing Data by Naveen Ashish A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY Computer Science March 2000 Copyright 2000 Naveen Ashish Dedication This thesis is dedicated to my Mom and Dad Leela Srivastava and Suresh Srivastava, for stressing right from the beginning the importance I should pay to education, being...»

«Dream Type-Scene in Old Testament Narratives: Structure and Significance by Marina Hofman A thesis submitted to the Faculty of Wycliffe College and the Biblical Department of the Toronto School of Theology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Theology awarded by the University of St. Michael’s College © Copyright by Marina Hofman 2014 Dream Type-Scene in Old Testament Narratives: Structure and Significance Marina Hofman Doctor of Philosophy...»





 
<<  HOME   |    CONTACTS
2016 www.abstract.xlibx.info - Free e-library - Abstract, dissertation, book

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.