Toward a Restriction-Centered Theory of Truth and Meaning (RCT)

Abstract

What is truth? The question does not admit a simple, precise answer. A dictionary-style definition is: The truth value of a proposition, p, is the degree to which the meaning of p is in agreement with factual information, F. A precise definition of truth will be formulated at a later point in this paper. The theory outlined in the following, call it RCT for short, is a departure from traditional theories of truth and meaning. In RCT, truth values are allowed to be described in natural language. Examples. Quite true, more or less true, almost true, largely true, possibly true, probably true, usually true, etc. Such truth values are referred to as linguistic truth values. Linguistic truth values are not allowed in traditional logical systems, but are routinely used by humans in everyday reasoning and everyday discourse. The centerpiece of RCT is a deceptively simple concept—the concept of a restriction. Informally, a restriction, R(X), on a variable, X, is an answer to a question of the form: What is the value of X? Possible answers:  $\mathrm{{X}}=10$, X is between 3 and 20, X is much larger than 2, X is large, probably X is large, usually X is large, etc. In RCT, restrictions are preponderantly described in natural language. An example of a fairly complex description is: It is very unlikely that there will be a significant increase in the price of oil in the near future. The canonical form of a restriction, R(X), is X isr R, where X is the restricted variable, R is the restricting relation, and r is an indexical variable which defines the way in which R restricts X. X may be an n-ary variable and R may be an n-ary relation. The canonical form may be interpreted as a generalized assignment statement in which what is assigned to X is not a value of X, but a restriction on the values which X can take. A restriction, R(X), is a carrier of information about X. A restriction is precisiated if X, R and r are mathematically well defined. A key idea which underlies RCT is referred to as the meaning postulate, MP. MP postulates that the meaning of a proposition drawn from a natural language, p—or simply p—may be represented as a restriction, $\mathrm{{p}}\rightarrow \mathrm{{X}}$ isr R. This expression is referred to as the canonical form of p, CF(p). Generally, the variables X, R and r are implicit in p. Simply stated, MP postulates that a proposition drawn from a natural language may be interpreted as an implicit assignment statement. MP plays an essential role in defining the meaning of, and computing with, propositions drawn from natural language. What should be underscored is that in RCT understanding of meaning is taken for granted. What really matters is not understanding of meaning but precisiation of meaning. Precisiation of meaning is a prerequisite to reasoning and computation with information described in natural language. Precisiation of meaning is a desideratum in robotics, mechanization of decision-making, legal reasoning, precisiated linguistic summarization with application to data mining, and other fields. It should be noted that most—but not all—propositions drawn from natural language are precisiable. In RCT, truth values form a hierarchy. First order (ground level) truth values are numerical, lying in the unit interval. Linguistic truth values are second order truth values and are restrictions on first order truth values $\mathrm{{n}}\mathrm{th}$ order truth values are restrictions on (n-1) order truth values, etc. Another key idea is embodied in what is referred to as the truth postulate, TP. The truth postulate, TP, equates the truth value of p to the degree to which X satisfies R. This definition of truth value plays an essential role in RCT. A distinguishing feature of RCT is that in RCT a proposition, p, is associated with two distinct truth values—internal truth value and external truth value. The internal truth value relates to the meaning of p. The external truth value relates to the degree of agreement of p with factual information. To compute the degree to which X satisfies R, it is necessary to precisiate X, R and r. In RCT, what is used for this purpose is the concept of an explanatory database, ED. Informally, ED is a collection of relations which represent the information which is needed to precisiate X and R or, equivalently, to compute the truth value of p. Precisiated X, R and p are denoted as $\mathrm{{X}}^{*}$, $\mathrm{{R}}^{*}$ and $\mathrm{{p}}^{*}$, respectively. X and R are precisiated by expressing them as functions of ED. The precisiated canonical form, $\mathrm{{CF}}^{*}$(p), is expressed as $\mathrm{{X}}^{*}$ $\mathrm{{isr}}^{*}$ $\mathrm{{R}}^{*}$. At this point, the numerical truth value of p, $\mathrm{{nt}}_\mathrm{p}$, may be computed as the degree to which $\mathrm{{X}}^{*}$ satisfies $\mathrm{{R}}^{*}$. In RCT, the factual information, F, is assumed to be represented as a restriction on ED. The restriction on ED induces a restriction, t, on $\mathrm{{nt}}_\mathrm{{p}}$ which can be computed through the use of the extension principle. The computed restriction on $\mathrm{{nt}}_\mathrm{{p}}$ is approximated to by a linguistic truth value, $\mathrm{{lt}}_\mathrm{{p}}$. Precisiation of propositions drawn from natural language opens the door to construction of mathematical solutions of computational problems which are stated in natural language.

This article was published in “Information Sciences, Lotfi A. Zadeh, Toward a restriction-centered theory of truth and meaning (RCT), 1-14, Copyright Elsevier (2014)”.

Research supported in part by ONR N00014-02-1-0294, Omron Corporation Grant, Tekes Grant, Azerbaijan Ministry of Communications and Information Technology Grant, Azerbaijan University of Azerbaijan Republic Grant and the BISC Program of UC Berkeley.