Approximations and Logic

Scientists work with approximations almost all the time and have to reason with these approximations. Therefore, it is very natural to wonder if approximations influence logical deduction and if so, how. Our goal in this paper is to present a class of lattices which captures some intuitions concerning propositions expressing numerical approximations and consider how they relate to classical logic. / Basic assumptions Our starting point is the relation between approximations and truth values. We assume that a proposition which expresses an approximation to a correct value receives a specific truth value reflecting the accuracy of this approximation. (We have proposed a formalization of this assumption in [8].) This is equivalent to saying that the notion of a partially true proposition makes sense. The next step in this framework consists in extending this attribution of partial truth-values to logically complex propositions. Formally, we want to construct a valuation function from propositions to a multivalent truth structure. In fact, we will do something more general. We will present a whole family of algebraic structures that we believe capture intuitively correct properties of partially true propositions the same way that the class of Boolean algebras captures essential properties of totally true (or false) propositions. Let us first present an informal motivation that will lead us naturally to a valuation system proposed by Bunge [2]. (A subvaluation system of this system was first presented independently and in a different context by Slupecki [15]. See also Weston [18,19] for some of the results with different proofs and a different point of view.) The originality and interest of Bunge's valuation system comes from his views about the (semantic) concept of negation (and implication, but the peculiarity of the latter is a consequence of the peculiarity of the former). There are different ways to extend the classical Boolean matrix for negation. After all, for a formal logical operation to qualify as a representation of the concept of negation, it has to satisfy two requirements: Received April 30, 1990; revised October 22, 1990 APPROXIMATIONS AND LOGIC 185 (i) it has to be a unary operator; (ii) it has to agree with classical negation in the "extreme" cases, i.e. when a proposition is totally true or totally false. Many of the extensions of the concept have taken the following algebraic form: if we denote the truth value of a proposition p by V(p), where the valuation function Fhas as target the real unit interval [0, 1], and the negation operator by *-»', then V(-.p) = 1 V(p), where T denotes the complete truth. The major advantage of this definition is algebraic. For the equation V( -;—/?) = V(p) is an immediate consequence of it. In other words, truth and falsity are symmetrical. This seems at first reasonable. But is it fair to put a proposition and its negation on a par? If the truth value of a proposition is 0.5, why would the negation also have to be 0.5? It is well known that a negated proposition most of the time "says" less than a nonnegated one and, hence, it is more secure. It is hard to find a nonalgebraic way to justify the above definition. It does not seem to reflect all of our uses of negation. Bunge himself, after having accepted it in [2], [3] and [4], rejected it in [5] and [6], He suggests that we consider negated propositions in a different way. Johnny turns 10 years old. His friend Peter believes Johnny to be 11, whereas his friend Jane suspects that he must be 9. Both are in error but not by much: their relative error is 1 in 10, so the truth value of each of their beliefs about Johnny's age can be taken to be 1.0-0.1 = 0.9. Charlie, a third friend of Johnny's, is uncertain about his age and, being a very cautious person, avoids any risky estimates and states "Johnny in not 9 years old". He is of course right, and would also be right if Johnny were 8 or 11, 7 or 12, and so on. So, it is a bad mistake to assign his statement the truth value 1.0 — 0.9 = 0.1, (Bunge [5], p. 88) In fact, the truth value of Charlie's statement should be 1, i.e. he is totally right. Clearly, most scientific propositions we are concerned with, namely, propositions expressing numerical approximations, behave the same way. Given the error involved in all verifiable propositions, we can safely claim that their negation is true. Thus, it is easier to hit on the truth with a negatively charged projectile than with a positive one. The only time a negated proposition is false is when the proposition is totally true. As soon as we move away from the truth, negation brings us right back to it. This is in a way the most relaxed reading of the negation operator. Notice immediately that even though the above situation seems to be natural in the case of factual propositions, a more restricted reading of negation seems more natural in some areas of mathematics. Indeed, when one claims that a recursive function is not computable, one has to give a proof in the same way as one would have to give a proof, which would undoubtedly look different than the previous one, that the given function is computable. Hence, to establish the truth value of this kind of negated proposition is as hard as establishing the truth value of a positive proposition. This seems to indicate that there might conceivably be many different kinds of negations, as Wittgenstein has already suggested. Moreover, the "strong" interpretation of negation we have just mentioned might be also justifiable in terms of approximations. A topos of sheaves can be thought 186 JEAN-PIERRE MARQUIS of as capturing an approximation of a "part-whole" type. We will explore this possibility elsewhere. We will now translate the above intuition concerning the behavior of the concept of negation in factual propositions in the following way. First, we will construct a valuation function from a set of propositions to the unit interval [0,1]. This valuation function will attribute to the atomic propositions an arbitrary value in the interval. This is the given (relative) truth value of the propositions. Then, a negated proposition is equal to 1, i.e. true if and only if the truth value of the proposition is strictly less than 1. When the proposition is fully true, then its negation is false. We will now formalize these facts. 2 First formalization: The unit interval Following Rasiowa and Sikorski [13] and Rasiowa [12], we define classical propositional logic in a purely syntactic way. Let L denote the structure L = <{/?,-:/ E N),-•, V,Λ,->>, where the last four symbols are the usual symbols for the propositional connectives and {/?, : / G N j is a denumerable set of propositional variables. The formulas of L are defined inductively as usual. Then L = with the usual axioms is the classical propositional logic, where *\-L' is the classical consequence or deducibility relation with its usual properties. Define TL = [A : h ^ } , the set of theorems of L and Ά 9 is a metavariable denoting well-formed formulas of L. This is all we need as far as the logic is concerned. Following McKinsey and Tar ski [10], we define a valuation system Mas an n + 2 tuple /«>> where Mis a nonempty set with at least two elements, D is a subset of M with at least one element (usually called the 'designated elements'), and them's are functions from M | ( / / ) to M, where \(ft) is the arity of/}. Let MB be defined by <[0, 1 ], {1 j,max,min,->,-»>, where (i) [0, 1] is the real unit interval; (ii) max and min are the usual binary functions defined on [0, 1 ] (iii) i : [0,1] {0,1} is defined by fo iffJC = 1 -.(*) = \ [l iff JC< 1, where x is an element of the real unit interval; (iv) -•: [0, I ] 2 -> [0, 1] is defined by