An exact philosophy of inexactness

An Exact Philosophy of Inexactness Michael Katz

1. Frame

The title above reflects the fact that while this paper was presented at a meeting of the Society for Exact Philosophy, and indeed the philosophical device described in it is as

exact as can be, its subject matter, the theme to which this device is applied, is Inexactness, specifically the problem of

inexactness and approximation in science. It was in fact my work on inexactness, my proposal to handle it within the framework of a logical system developed in several recent papers of mine and called 'the logic of approximation' (the device mentioned above), that prompted the invitation extended to me to address the SEP 1983 Annual Meeting.* Accordingly, this paper is to be a general picture of the current state of my work on this topic.

This short section sets the frame of the picture, drawing a line around its content. The logic of approximation, to be considered in Section 2, is the ground of this picture. It lies behind those specific theories, the figure, studied in Section 3. Like almost any picture the one portrayed here has several interpretations (in sciences ranging from physics to psychology and economics), as we shall see in Section 4.

Generally speaking, our aim here is to bridge the gap, universally acknowledged by scientists, between the in. evitability of imprecision in research and the need for idealization, ignoring imprecision, in theory. To achieve this goal we use the logic of approximation, which is a multivalued logic with truth-values considered degrees of error (or inexactness) and with a special type of deduction based on the notion of being 'very nearly true' (or 'almost error-free'). A theory in the formal language of the logic is a list of deduction expressions, i.e., of idealized) assertions that certain formulae ('consequents') follow from other formulae ('antecedents'). Yet, in an actual structure for the language all formulae are subject to varying degrees of error ('truth-values'), and a deduction holds if and only if mini- mizing the error in the antecedents always leads to mini- mizing the error in the consequents (without necessarily eliminating it).

This idea seems to be intuitively plausible, and in the

next section we shall show that it can be made mathematical- ly and formally rigorous. In the remaining sections of this paper we shall try to demonstrate that it also works. For

this purpose we shall axiomatize in our logic theories of similarity and other predicates and look at their potential applications in psychological measurements, utility models and quantum physics.

2. Ground

In this paper we present the full version of the logic of approxima'tion, as in, e.g., [13] and [14], rather than the restricted versions of, e.g., [15] and [16].

We start with a first-order language L whose alphabet consists of a set V of variables, sets Pn of n-ary predicate symbols for various n's in w, the connectives 7 , ^, v and

and the quantifiers V and 3. Atomic formulae of L are of the form p~ where for some n E tg, p E Pn and ~ = (vl . . . . .

Vn)E V n. Compound formulae of L are of the forms 7 4 , ~ A 0, ~O V 0, ~O -+ 0, Vv~ and 3v~, where v E V and

and 0 are (atomic or compound) formulae of L. Commas and parentheses may be added to formulae of L for ease of reading, and the notion of a free variable in a formula of L is defined in the usual way.

A deduction in L is an expression of the form F F-A (read: "F entails A"), where F and A are finite sets of formulae of L. A (formal) theory of the language L is a list of deductions in L. In writing a specific deduction, we simply list the elements of P and/or A (on one or both sides

of the symbol t--, as the case requires ) omitting the set brackets. We write t ---A when r is empty, in agreement with common usage.

A (multi-valued) structure X for the language L consists of a non-empty set X (the 'domain' of X) and for each n E o~ and each p E P n a [0,1]-valued function p on X n

(the 'interpretation' of p in X). If ~ is a formula of L, U a finite subset of V containing all free variables of ~0 and ~ an element of X v, then ~k2 is the truth value of ~k a t e . These truth-values in X are defined inductively as follows:

Topoi3 (1984), 43-53. 0167-7411/84/0031-0043501.65. (~) 1984 by D. Reidel Publishing Company.

44 MICHAEL KATZ

(a) I fnEco , pEPn a n d ~ = ( v l . . . . . Vn)E V n then

p w = p (vl ), . . . , 2 ( v , ) ) .

(The term on the right side of this equation will sometimes

be written as p~.)

(b) If 4 and 0 are formulae of L then

(-q4)2 = 1 - 4 2 (4 ^ 0)2 = max(42, 02) (4 v 0)2 = min(42, 02)

(4 ~ 0)2 = max(0, 02 - 42) .

(e) I f 4 is a formula of L with free variables vl . . . . . Vn and 1 ~< i ~< n then

(Vvi4)2 = sup 42(i /x) x ~ X

(3vi4)2 = inf 42( i /x) x ~ X

where here, and in the sequel, i f 2 E X ~r then also 2(i/x) E X U and for each v E V and each x E X

2(i /x)v= { 2(v) if v:/~-vi x i f v = v i

(If the variable v of L is not free in 4 we set ( g v 4 ) ~ =

(3 v4)2 = 42). So far there is nothing very special about the logic of

approximation. The semantic rules above are simply reversals of the well-known rules of the ~Lukasiewicz Logic (see, e.g., pp. 38 -59 in Tarski's [25]), extended to a first-order

[0,1]-valued language. The reason for the reversal is that, as

we explained in Section 1 above, we think of truth-values

as representing degrees of error. Thus 0 (no error) stands

for absolute truth, and 1 (maximal error) for absolute false-

hood. The feeling behind rules (b) and (c) above is that a

conjunction maximizes the error while a disjunction mini-

mizes it (and similarly for the quantifiers), that the error

in 4 -+ 0 at 2 is the degree to which the error in 0 at exceeds that in 4 at 2 (while there is no error in 4 -> 0

wherever 0 is 'truer than' 4), and that the error in -14 at 2 is just the error in 4 ~ "absolute falsehood" at 2. Similar ideas can be found in Scott's [22] and [23], as well as in several papers of Giles (e.g., [6], [7] and his Comment on [23]).

The distinctive feature of the logic of approximation is its treatment of deductions in structures for L. To clarify this let X be a structure for L with domain X, let F and A be finite sets of formulae of L and let U be the set of all variables of L appearing free in members o f F u A. We shall denote by ^F the conjunction in L of all members of F,

by vA the disjunction in L of all members of A and by =}

implication outside L. The structure X is said to be a model of the deduction

P }-- A (and P I-- A is said to hold in X) if for every positive

real number e there is a positive real number 8 such that for every 2 E X tr

( ^ r ) 2 < 8 (va)2< e.

Thus P [ -A holds in X if we can make vA as 'nearly

true' as we wish in X by making Ar 'true enough' in X.

This resembles natural deductions in classical logic, where

r 1- A holds in X if at least one member of A is true in X

whenever all members of r are true in X. Resemblance with

classical logic is maintained also in the case of deductions where r is empty (i.e., deductions of the form t -A) by

adopting the convention that in this case (Ar )2 = 0 for every 2 E X U (and hence (vA)2 = 0 for every 2 EXtr) .

It is interesting at this point to contrast the logic of approximation with two other, more conventional, multi-

valued logics (based on the same language, L, and the same structures and semantic rules as above). The first

of these logics will be called the Lukasiewicz Logic, as it is,

according, e.g., to Smiley's Comment on [23], most faith-

ful to the original intention of Lukasiewicz. The second

was called 'the logic of inexactness' in my [10] and [13],

and is an extension, and a reformulation, of Scott's system

in [221. With notations as before, X is said to be a model of

F ~- A in the Lukasiewicz Logic if for every 2 E X U

( ^ r ) 2 = 0 ~ (va)2 = O.

And X is said to be a model of P t - A in the logic of inexactness if for every 2 e X U

(At )2 1> (VA)s

Thus the Lukasiewicz Logic is very much like classical (two-valued) logic when it comes to deductions: at least one element of A must be fully true in X whenever all elements of F are fully true in X if F [- A is to hold in X. In the logic of inexactness, on the other hand, for P t-- A to hold in X the maximal error in F must, throughout X, exceed the minimal error in A. I t seems that as far as deductions are concerned, the Lukasiewicz Logic ignores errors altogether, while the logic of inexactness, in contrast,

ignores the possibility of controlling errors (without com- pletely eliminating them). Our logic of approximation takes

the middle road of admitting both the existence of errors and the ability to control them. That the logic of approximation is indeed 'in the middle' can also be seen from the

AN EXACT PHILOSOPHY OF INEXACTNESS 45

following theorem (proved in [15], and in a slightly different form in [14]).

THEOREM 1. The structure X for L with domain X is

a model of the deduction F F- A (in the sense of the logic of approximation) if and only ff for every sequence {Xrn : 1 ~<m < ~ } of elements o f X tr (where U is the set of free variables in F u A)

lim = 0 lira (vA)x m = 0. m - - + o o m ----~ o o

We close this section with another theorem which will be needed later on. The proof of this theorem is easy, and in fact some parts of it are immediate corollaries of other parts. We write here F ~AX ~ to say that X is a model of p [---A in the sense of the logic of approximation, and similarly we write I" B-LX A for the Lukasiewicz Logic and 1-' ~/X A for the logic of inexactness.

THEOREM 2. For every pair of finite sets, P and A, of formulae of L and every structure X for L:

(i) r (ii) A

(tU) PAX^r-'VA = r

(iv) r�89 (v) r �89 vzx,*

(vi) xZX,*

Let us call ~ ^ I ~ -+ VA the one-sided transform o f f 1- A. Then, loosely speaking, the theorem says that any deduction is equivalent to its one-sided transform in the logic of inexactness (i) and weaker than its one-sided transform in the Lukasiewicz Logic (ii) and in the logic of approximation (iii); that the set of deductions holding in the logic of inexactness is contained in the set of deductions holding in the logic of approximation, which itself is contained in the set of deductions holding in the Lukasiewicz Logic (iv); that every deduction in the logic of inexactness is equivalent to its one-sided transform in both the logic of approximation and the Lukasiewicz Logic (v); that all three logics agree on deductions from the empty set (vi).

For our purposes in this paper the important clauses of this theorem are (iv) and (v). They tell us in.particular that every deduction holding in the logic of inexactness holds also in the logic of approximation, and moreover, every deduction in the logic of inexactness can be replaced by an equivalent deduction in the logic of approximation. Thus our earlier works (e.g., [10], [11], [12])on applications to

science of the logic of inexactness can also be carried out within the logic of approximation. For instance, the symmetry and transitivity deductions in the theory of similarity in the following section, where we return to deal solely with the logic of approximation, are the one-sided transforms of the corresponding deductions in the theory of metric equality, formulated in the logic of inexactness in [11] and [121.

3. Figure

Let us now pull into the foreground of our picture some binary predicate which we want to consider as representing similarity. We treat similarity as an inexact analogue of identity, and accordingly we shall axiomatize it as an approximation to identity. In Section 4 we shall see that the role played by similarity in many scientific theories is as important as the role played by identity (or equality) in mathematical theories.

So let L be a language as above, with variables u, v, w (with or without subscripts), assume that the set/'2 of binary predicate symbols of L is non-empty and pick an element, s say, of this set. We shall now write a theory in L involving s, so that in models of this theory the formula s(u, v), in accordance with our interpretation of similarity as approxi- mating identity, can be read as asserting that the elements to which u and v refer are similar.

In the sequel if ~ is a formula of L with free variables Vl . . . . . v n, for some n E co, and 1 ~< i ~< n, we shall denote by ~(i/v) the formula of L obtained from ff by substituting the variable v of L for each free occurrence of vi in if, and similarly for ~ (i/u). Our theory of similarity is the following list of three deductions and one deduction scheme.

(re) [- s(u, u) (sy) I-s(u, v ) ~ s ( v , u) (tr) [- s(u, v) ~ (s(v, w) ~ s(u, w)) (su) s(u, v) ~- p~(i /u) -+ p~(i/v)

Here (re) stands for reflexivity, (sy) for symmetry, (tr) for transitivity and (su) for substitutability. The scheme (su) applies to every n E co such that Pn is non-empty, every p EPn, every ~ = (vl . . . . . Vn) E V n and every i such that l<~i<~n.

According to our semantic rules and the interpretation of deductions in the logic of approximation, a structure X for L is a model of the above theory of similarity if and only if

(A) for every x, y, z E X

46 MICHAEL KATZ

(re) s(x, x) = 0 (sy) s(x, y) = s(y, x) (tr) s(x, y) + s(y, z) t> s(x, z)

(B) for every p, ~ and i as above and for every positive real number e there is a positive real number 6 such that for allx, y E X a n d all2 E X v

(su) s(x, y) < 6 ~ Ip~(i/x) - p~(i/y)l < e

Thus in the (idealized) formal language L, where imprecision is ignored, the predicate s has all the usual properties of identity. Yet, in an actual model X of the theory inexactness turns the interpretation s of s into a (pseudo-)- metric on the domain X of X so that the further apart two objects the greater the error (truth-value) in asserting that they are similar. In addition, with respect to the metric s all interpretations of predicates of L in X are uniformly continuous in every argument (in fact, each of them is equi-continuous over all its arguments). So a statement about x and the same statement about y (x, y E X) are about equally true if x and y are very similar. This is true not only for atomic statements but also for compound ones as the following theorem shows:

THEOREM 3: Let ff be a formula of L with free variables vl . . . . . Vn, let u and v be variables of L and let the structure X for L with domain X be a model of the theory of similarity. Then for every i such that 1 ~< i ~< n X is a model of the deduction

s(u, v) p ~ ( i / u ) ~ t~(i/v).

In other words, for every i as above and every positive real number e there is a positive real number 8 such that for all

- i

E X v (~ = (vl, ..., vn)) and for all x, y E X

s(x, y) < 8 =~ [ ~2( i /x ) - t~Y(i/y) l < e.

The proof of this theorem, by induction on the structure of 4, can be found in [14]. It is also shown there that the theorem can be reformulated so as to assert uniform continuity of ~b over the product space X n with the obvious extension of the metric s. To clarify this let us write CF for

and ~ (where ~ = (ul . . . . , Un) E V n) for the formula of L obtained from ~ by substituting, for each i such that 1 <~ i <~ n, ui for every occurrence of vi in ~b. Then the structure X of the above theorem is a model of the deduction

S(Ul, ~)1), ..., S(Un, Vn) ~ ~ K + ~)V

In other words, for every e > 0 we can find a 6 > 0 such

that for every ~ E X ~ and y ~ X ~

m a x ( s ( x l , y l ) . . . . . S(Xn,Yn))<8 ~ I ~ - r < e ,

where for 1 ~ i~ n

xi =~(ut) Yi = Y(vi).

Having axiomatized an inexact analogue of equality we turn now to the second most frequently encountered predicate in mathematical theories, namely, ordering. Our analogue of ordering will be an inexact binary predicate which we shall call 'dominance'. We shall also axiomatize an inexact ternary predicate of non-directed dominance, a 'betweenness' predicate. (An object, y, is between two other objects, x and z, if either it dominates x and is dominated by z, or it dominates z and is dominated by x).

Let the set P2 of binary predicates of L contain a predicate d in addition to the predicate s (of similarity). The theory of dominance in L is the following list of deductions, where (co) stands for connectedness, (tit) for dominance transitivity and (as) for asymmetry.

(co) b- d(u, v), d(v, u) (tr) d(u, v) [-- d(v, w) ~ d(u, w) (as) d(u, v), d(v, u) t-- s(u, v).

The reader can easily write down the interpretation of these deductions in a structure X for L and see that they seem to be quite plausible. For instance, (as) says that for any two objects of the domain of X if each of them 'practically' dominates the other then they are 'practically' similar. It is also easy to see that any model of the theories of similarity and dominance is also a model of the following deductions of weak reflexitivity and strong reflexitivity.

(wr) b- d(u, u) (sr) s(u, v) ~-d(u, v).

(Clearly, (wr) follows from (sr) + (re) in any structure X; but to show that these deductions hold in a model of our theories, we first have to obtain (wr), using (co), and then we can use (wr), together with (su), to obtain (sO).

Assume now that the set P3 of ternary predicate symbols of L is non-empty and let b EP3. The theory of betweenness in L is the following list of deduction expressions.

(bl) b(u, v, w) I--- b(w, v, u) (b2) I-- b(u, v, w), b(v, w, u), b(w, u, v) (b3) b(u, ul , v) ,b(u, u2, v) l---b(u, ul , u2) ,b(u , u2, ua) (b4) b(u, v, u) t-s(u, v)

AN E X A C T PHILOSOPHY OF I N E X A C T N E S S 47

(b5) (s(v, w)-~s(w, u))-,s(u, v) ~-b(u, r, w) (b6) b(ul, u2, v),b(u2, u3, V)~-b(ul , u2, ua),S(U2, V) A

s(u~, v).

Once again it is easy to see, with or without translation

to actual structures for L, that most of these deductions are meaningful and plausible. For instance, (b~) says that betweenness is 'symmetric', while (b~) says that of each three objects one is between the two others, so that be-

tweenness is 'connected'. In the remaining parts of this section we want to consider

one more type of predicates which are also essential to any mathematical or scientific theory. We refer to predicates representing functions. To say that the predicate p, belong- ing to Pn for some n ~ r represents in the structure X (with domain X) a function / : X n - ~ ~ X is to say that if ~ = (v~ . . . . . vn)~ V n, then for some i (1 ~<i~<n) p~

asserts that for every ~ E X ~

~(v~) = f ( ~ ( v ~ ) , . . . , ~(v~ _ 1), ~(v~ + 1) . . . . , ~ ( v ~ ) ) .

For this to be the case a value of f has to exist in X for every list of arguments of f in X, and it has to be unique (up to similarity). In the logic of approximation we want these conditions to be satisfied in an approximate manner. So we shall say that p represents a function in X for some i as above if with this i X is a model of the following deductions, where (ex) stands for existence and (un) for unique-

hess.

We shall say that a model X (with domain X) of our theory of similarity is a complete structure if for every n E co, every ~ = (vl, ..., Vn) E V n, every p E P n representing a function in X for some i (1 ~< i ~< n), and every ~ E X v

there is an x o ~ X such that p~(i/xo)=O. In [14] we explained how this notion of completeness is related to other notions of completeness in the literature on various non-classical logics, and we proved (equivalents of) the following two theorems which relate it to the analytic notion of completeness of metric spaces (namely, that every Cauchy sequence attains a limit). In these theorems we assume, for simplicity, that the interpretation s of s in a model X of our theory of similarity for s is a metric

(rather than a pseudo-metric) on the domain of X.

THEOREM 4. Let the structure X for L with domain X be a model of the theory of similarity. Obtain a language L* by adding to L, for every Cauchy sequence (xm: 1 < m <oo) of elements of X, a unary predicate symbol

p whose interpretation p in X is given by

p(x) = lira s(x, Xm) / T / - -4 oo

for every x EX. Then X is a complete structure of L* if and only if (X, s) is a complete metric space. (In particular, X is a complete structure of L if(X, s) is a complete metric

space).

(ex) t-- 3 vip~ (un) p~(i/u), p~(i/v) P- s(u, v).

These deductions capture the intentions of the require- ments above, and their translations in X nicely fit our intuition about approximation. First, the meaning of the assertion that for every e > 0 there is a 8 > 0 such that for every x , y E X and every ~ E X ~

(un) max(p~(i/x), p~(i/y)) < 8 =~ s(x, y) < e

can be that any element of X which is 'practically' the value, at a given ~ ' E X n -1, of the function described b y p inX, is unique up to 'near' similarity. Second, the assertion that for every ~ E X ~

(~x) inf p~(i/x) = 0 x ~ X

can be interpreted as implying that for every ~ ' E X n -1 there are elements of X which get closer and closer to being the value at ~ ' of the function described by p in X. This does not guarantee that there is always a definite value in X for this function, i.e., that for every ~ E X ~ there is an x0 E X such that p~(i/xo) = 0, which will make p~ fully true at~(i/xo).

THEOREM 5. For every structure X for L which is a

model of the theory of similarity there is a complete structure X for L such that X is elementarily imbeddable inX-

To build the structure X, which we call the completion of X, we first take the metric completion of the domain X of X to be the domain ~" of X. Then, identifying each element x of X with the equivalence class of (x, x, ...) modulo ~ = 0 where ~ is the extension of s to ~', thus considering X a subset of ~', we let for each n E r and each n-ary predicate symbol p of L, the interpretation

o fp in X be defined for every ~ E .~n by

~ = lira P ~ m

where p is the interpretation ofp inX and (Xrn : 1 ~< m < ~) is a sequence of elements o f X n converging to ~. It can now be proved, by induction on the structure of formulae of L, that for every formula tp with a list ~ of free variables, and

for every ~ ~ X ~

m --~ oo

48 MICHAEL KATZ

where ~2 is the truth-value X attaches to ~ a t2 (under the semantic rules of Section 2 above) and {Xm : 1 < m ~< oo) is a sequence of elements of X e converging to 2. In particular, if ~ E X e then ~2 = ~ , which is the claim of Theo-

rem 5.

4. Interpretations

'Meanings' of predicates and their properties may well vary from one scientific discipline to another. We illustrate this here by looking at three such disciplines, of which the last two, utility theory and measurement theory, are closely related to each other, while the first, quantum theory, stands apart. In each discipline we provide specific interpretations to some or all of the predicates studied above, some or all of their properties, and some or all of the deductions governing them in our formal theories. We also add some predicates and theorems of particular relevance to these three disciplines. Mostly, however, this section consists not of additional theorems or other mathematical 'technicalities', but of informal discussions intended to show that the logic of approximation is readily applicable to various areas of scientific inquiry.

4.1 Quantum theory

In this subsection we shall be concerned with the so-called modal interpretations of quantum logic. Our similarity predicate will refer here to the basic notion of these interpretations, namely, the notion of similarity, or accessibility, between states of a mechanical system.

In most works on modal quantum logic (see, e.g., Gold- blatt [8], or Dalla Chiara [4]), as well as in works on modal logic in general (see, e.g., Hughes & Cresswell [9]), accessibility between states (or 'worlds') is a yes-or-no relation. For instance, in the famous $5 system of modal logic it is an equivalence relation.

Yet, it has been argued, e.g., by Bigelow in [ 1], that, at least in the context of quantum logic, accessibility should be a multi-valued function, related to a metric (on the set of states). Bigelow used this idea to construct probability measures which give rise, under certain restrictions on states and obserables, to the usual structure of quantum statistics.

In [15] we suggested our axiomatization of similarity in the logic of approximation as a compromise between the two views of accessibility. It allows us to say that while in

principle accessibility is a relation and any state is either

accessible or non-accessible from any other state, in practice there may well be degrees of (error concerning) accessibility,

linked to distances between states. Indeed in our formal language L, under the deductions (re), (sy) and (tr), the predicate s, which will be called the accessibility predicate here, is an equivalence relation (as in $5). Yet in an actual structure X for L, under the conditions (re), (sy) and (tr), the interpretation s of s is a metric on the domain X of X

(the set of states). In the context of quantum logic and quantum theory a

one-place atomic formula of L may be interpreted as asserting that a certain event occurs (specifically, that some observable takes a value in a Borel set of real numbers) when a given mechanical system is in a given state. Such a formula will be called a quantum proposition (or a q-proposition), and according to van-Fraassen [28] propositions of this type are necessary, in the sense that if they are true in one state, they are true in every state accessible from it. In other words, whenever the state to which the variable v of

L refers is accessible from the state to which the variable u of L refers, i.e., wherever s(u, v) is true, the truth of a q- proposition p(u) implies the truth of the q-proposition p(v). In the language L this can be expressed by the deduction

s(u, v) I--- p(u) -+ p(v)

which is just our substitutability axiom (su) for the unary predicate p.

So in this context substitutability in the formal language is interpreted as necessity. And in a model X of our theory of similarity this condition generalizes from atomic formulae to compound ones as we saw in Theorem 3 above. More- over, for any two elements x and y in the domain of such a model necessity says not only that if y is accessible from x then the truth of p at x implies its truth in y , but also that if y is 'nearly' accessible from x then the truth-value of p at x is 'about the same' as its truth-value at y. For as we already know the deduction above translates in X to uniform continuity over X of the interpretation p of p w.r.t, to the (pseudo-)metric interpretation s of s. That is to say, X is a model of this deduction if and only if for every e > 0 we can find a 6 > 0 s.t. for all x, y E X

s(x, y) < 6 =~ Ip(x) - P(Y)I < e.

Following our interpretation, for q-propositions, of the substitutability deduction (su) as expressing necessity, we shall now interpret, for such propositions, the existence deduction (ex) as expressing possibility. This deduction


tells us that there is, or there might be, a state in which the proposition is (fully) true. The uniqueness deduction (un) adds to this that any state in which the proposition under

consideration is 'possible' is unique up to (near) accessibility. If for some one-place atomic formula of L all three deductions (su), (ex) and (un), hold in a certain structure X for L, we shall refer to this formula as a possible quantum proposition, or a pq-proposition, in X. It should be clear that we can apply this definition, and the following results, to general formulae of L, and the restriction to one-place atomic formulae is a matter of convenience.

In the following theorem, part (i) which says in fact that for certain q-propositions the logic of approximation reduces to the Lukasiewicz Logic (see Section 2 above), can be proved using Theorem 1. Part (ii) can either be proved directly or shown to be a special case of part (i).

THEOREM 6. Let s be the accessibility predicate in the language L and let the structure X for L with domain X and interpretation s of s be a model of the theory of similarity for s. In addition let p(v) and q(v) be q-propositions of L and let p and q be the interpretations o fp and q in X. Then

(i) If every sequence (Xm : 1 ~< m < oo} of elements of X

for which

l imp (Xm) = 0 m --+ oo

attains a limit in (X, s) then we have (with V denoting here universal quantification in the meta-language)

[*] V x ~ X . p ( x ) = O ~ q ( x ) = O i f fp(v) F-q(v)holds in X

(ii) In particular we have [*] in X if X is a complete structure and p (v) is a pq-proposition.

The importance of this theorem is two-fold. First, consider the following assumption, where Px(P) and Px(q) denote the probabilities attached to p(v) and q(v) in the state x (and V belongs again to the meta-language):

Vx EX 'Px(p) = 1 ~ex(q) = 1 iff Vx E X .ex(p) <<, Px(q).

This assumption is forced in quantum logic (and is true in the standard quantum theory) and yet it may be doubted, as noted, e.g., by van Fraassen in [28]. The analogue of this assumption in the logic of approximation, with truth-values (i.e., degrees of error) replacing probabilities, is exactly [*] above, which intuitively appears to be much less doubted.

Second, recall that in the usual Hilbert space presenta-

tion of a quantum mechanical system, rays, i.e., one-dimensional subspaces, represent (pure) states of the system, and a metric is defined on the set of these rays with angles serving as distances. This metric, when made to range in [0, 1], is a very natural interpretation of the accessibility predicate, since if the thus defined distance between two states is maximal, so that the corresponding accessibility statement is (fully) false, then the two states (or rays) are orthogonal; and indeed in many works on modal quantum logic accessibility and orthogonality are considered com- plementary. In addition the set of rays is complete in this metric, so that we have here (in view of Theorem 4) a complete structure and thus [*] holds at least for our pq-propositions as part (ii) of the theorem above asserts. Finally, to this metric on rays quantum probabilities are closely related as shown in, e.g., Giles [5]. Tying these probabilities to those obtained by Bigelow [1] from a general metric on states may lead to a quantum probability theory based on the logic of approximation.

4.2 Utility theory

The three predicates (similarity, dominance, betweenness) axiomatized in Section 3 above will be studied now in connection to certain unary predicates representing 'gain' or 'utility'. In this context we interpret similarity and dominance as 'indifference' and 'preference', respectively, between actions, options, games or objects. A look at any work relating to utility theory (e.g., Luce & Raiffa's classical [19]) will immediately reveal the centrality of indifference and preference in axiomatic treatments of utility. It will also expose three of the basic, not unrelated, difficulties in such treatments.

(a) Axiomatic theories of utility and decision tend to ignore errors and inconsistencies which are inherent in human behavior. Hence such theories should be considered theories of 'normative' or 'idealized' behavior which is at best an approximation to real-life behavior ([19], p. 25).

(b) In axiomatic theories indifference and preference are usually treated as yes-or-no relations, but in practice intermediate degrees are often attached to the two predi-

cates. On many occasions these degrees arise from the inconsistencies mentioned above. For instance, a subject may sometimes be indifferent and sometimes not indifferent between the same two objects, x and y say, or he may sometimes prefer x to y and sometimes y to x. To over- come this difficulty it:is customary to say that the subject is indifferent between x and y if he finds them indifferent

50 MICHAEL KATZ

in more than half of the cases, and similarly, that he (definitely) prefers x to y if he does so in at least half of the cases (see, e.g., Scott & Suppes [24], and we shall return to this in the next subsection). But it is also possible, and perhaps more natural, to admit that there are degrees of indifference and preference, which may be proportional to the number of times x andy are perceived as indifferent, or to the number of times one of them is preferred to the other. So, interestingly, there is an interplay of relation and function concerning indifference (and preference) in utility theory, much like the interplay concerning accessibility in quantum modal logic.

(c) As yes-or-no relations indifference and preference are sometimes (e.g., in [19], pp. 28, 332) considered transitive and sometimes (e.g., in Roberts' [20] and in Tversky's [27]) intransitive. In practice they are more often intransitive than transitive. While this is surely accept- able in the case of indifference, it may be considered an 'error' in the case of preference (although it is not always SO).

Our logic of approximation, with the theories of similarity and dominance viewed as theories of indifference and preference, offers a way of handling the three problems above together. It allows us to treat the two predicates as error free, dichotomous and transitive in the formal language L, but as error-bound, multi-valued and only approximately transitive (e.g., up to a triangle inequality) in an actual structure for L.

Accordingly, in [16] we axiomatized elementary utility theory in the logic of approximation, taking as our axiom system the theories reassembled in Section 3 of this paper. But instead of similarity and dominance we spoke of indifference and preference, and we treated betweenness as undirected preference. We called certain unary predicates of L 'gain predicates', interpreting the formula g(v), where g is such a predicate, as asserting that the object to which

v refers is gainful. These predicates play in structures for L a role similar to that usually played by utility functions in models of choice behavior. (The truth-values, or degrees of error, attached to the assertion g(v) in elements of a given structure are inversely related to perceived gains, or utilities, in these elements). Our system, however, is unique in bringing utility into the formal language itself, and axio- matizing it as an integral part of the formal theory.

Like other predicates of L, a gain predicate g is expected to satisfy the substitutability deduction (su) with respect to indifference, s, in any model X of our axiomatic utility theory. That is to say, the interpretation g ofg in X is to be uniformly continuous over the domain X of X with respect

to the interpretation s of s in X. Any unary predicate satisfying this condition will be considered a gain predicate in X. Thus, for a gain predicate g, as for a quantum proposition in the preceding subsection, X is a model of the deduction

s(u, v) t- g(u) ~ g(v)

so that for every e > 0 there is a 8 > 0 such that for all

x, y E X

s(x, y) < 6 =~ Ig(x) - g(Y)l < e.

And this fits well the feeling that we cannot be practically indifferent between two objects which are not close enough to each other in terms of expected gain.

A specific meaning is given in our utility theory to the uniqueness property (un) when it relates to gain predicates. It is interpreted here as expressing unidimensionality. We say that the gain predicate g of L is unidimensional in a model X of our formal theory i fX is a model of the deduction

g(u), g(v) ~- s(u, v)

so that for every e > 0 there is a ~ > 0 such that for all x, y E X

max(g(x), g(y)) < 6 ~ s(x, y) < e.

The idea is that if gain has several dimensions then it may be very nearly true that two objects are gainful and yet they are far from indifferent (presumably they are gainful along different dimensions; e.g., one is f'mancially beneficial, the other spiritually rewarding). Hence if it is the case that every two objects which are very close to being gainful must also be quite indifferent it makes sense to say that gain is unidimensional.

We now want to consider the case where a binary dominance (i.e., preference) predicate d of L is reducible in a model X of (certain fragments of) our utility theory to a unary predicate g of L. We shall see that g is then a unidimensional gain predicate in X. The reduction of d to g is such that it is approximately true that d(u, v), i.e., that the object to which u refers is preferred to the one to which v refers, if and only if it is approximately true that g(v)-+g(u), i.e., that the object to which v refers being gainful implies the object to which u refers being gainful. In a similar manner, and with similar intentions and results, we reduce the ternary betweenness predicate b to a binary predicate d (and then to a unary predicate g).

The reductions mentioned above, and their outcomes, are summarized in the following theorem, which is an


easy to prove reformulation of several theorems of [16]. In this theorem we refer to formal theories of Section 3 using the terminology (names for predicates) of the present subsection.

Theorem 7. (i) Let the language L contain binary predicate symbols s and d, and for these predicates let the structure X for L be a model of the theories of indifference and preference. If there is a unary predicate symbol g of L such that X is a model of the deductions

(dgl) d(u, v) [-g(v)~g(u) (dg:) g(v) +g(u) }--d(u, v)

then g is a unidimensional gain predicate in X, i.e., X is a model of (su) and (un) for g and s.

(ii) Let the languageL contain a binary predicate symbol s and a ternary predicate symbol b, and for these predicates let the structure X for L be a model of the theories of indifference and betweenness. If there is a binary predicate symbol d of L such that X is a model of the deductions

(bdl) b(u, v, w) t-d(u, v)Ad(v, w),d(w, v)^d(v, u) (bd2) d(u, v), d(v, w) }--b(u, v, w)

and such that d is transitive in X, i.e., X is a model of (dt) for d, then d is a preference predicate in X, i.e., X is a model of (co), (as) and (su) for d and s.

(iii) Let L and X be as in (ii). If there are a binary predicate symbol d and a unary predicate symbol g in L such that X is model of (d 0, (dgl), (dg2), (bdl) and (bd2) then d is a preference predicate in X, g is a unidimensional gain predicate in X, and X is a model of the deductions

(bgl) b(u, v, w)[--(g(u)-+g(v))A(g(v)~g(w)), (g(w)-~ g(v)) ^ (g(v) + g(u))

(bg2) g(u)~g(v),g(v)~g(w) I--b(u, v, w).

4.3. Behavioral measurement

Utility theory to us, as presented in the preceding subsection, is a formal axiomatic theory of utility predicates ('gain predicates') in relation to other choice behavior predicates such as indifference and preference. To other authors (e.g., Krantz et al. in Chapter 8 of [17]), utility theory is a theory of measurement. In such a theory there is usually no reference to utilities in the axiom system, but the existence of a utility function, or in general of a measurement function, is derived from the axioms (formal or informal) governing, e.g., preference or indifference.

By a measurement function we mean, following Scott & Suppes [24], a function from 'objects' to numbers pre- serving certain relations between objects (i.e., representing them by appropriate relations between numbers). Intuitive- ly, the numbers can sometimes be thought of as describing magnitudes of the objects to which they are assigned by the measurement function, and then, for example, an ordering of objects by magnitudes must be reflected by the usual ordering of numbers.

Orderings of objects by real or perceived magnitudes (or Scalues', or 'marks') are examples of what Coombs in his influential theory of data [2] calls dominance relations. Other examples are various types of preference relations, like those discussed in connection to utility theory in the preceding subsection, which are ubiquitous in sciences such as psychology, sociology, economics, ecology and operations research. It is not surprising that Coombs refers to dominance as one of the two basic predicates of behavioral inquiry.

The other basic predicate is that of proximity (or consonance, or similarity). Our claim in Section 3 above that similarity, which we Shall here call proximity (as Coombs does), is as central in science as equality is in mathematics, is nowhere truer than in the domain of the social and behavioral sciences. Examples of proximity are (i) indifference between options in utility theory as studied above, (ii) indifference between stimuli in theories of perception as defined by the gap between the stimuli being below the differential threshold, (iii) closeness of nouns in psycholinguistic research which is inversely related to the difference between evaluations of the nouns on one or more dimensions (or attributes), (iv) congruence of attitudes in social psychology as expressed by some correlation coefficient, (v) perceived similarity of concepts as assessed either directly through paired comparisons or indirectly through a variety of judgment processes.

Getting back to measurement theories (in the behavioral sciences), it should be clear now that the relations (between objects) to which these theories refer ought to be, and are indeed, associated with proximity and dominance. To this we add that the relations between numbers which, through a measurement function, represent proximity or dominance, are usually linked to distances. For instance, in Chapter 4 of Krantz et al. [17], although it is not explicitly said so there, orderings of absolute distances represent orderings of proximity while orderings of algebraic or positive distances represent orderings of dominance. In works involving multidimensional scaling techniques (see, e.g., Kruskal & Wish, [18]) this idea is extended to representations by

52 MICHAEL KATZ

distances between points in multidimensional spaces. So we have here, just as we had in the preceding sub-

sections, an interplay of relations and distances. Moreover, as the examples above show, the relations themselves are sometimes two-valued, sometimes multi-valued, and sometimes two-valued in principle but possibly multi-valued in practice (due to errors and inconsistencies). This enables us to apply again our logic of approximation as a bridge between relations and functions. And we shall do it in a

way which will facilitate the construction of measurement functions.

Before doing so we note that the presence of errors and inconsistencies led Scott & Suppes in [24] and Roberts in [20] to suggest that proximity and dominance be represented by up-to-e relations between numbers. In this context the positive real number e is the 'size' of the

differential threshold, or the just noticeable gap, or the maximal admissible error. We say that two objects are proximate (or similar, or indifferent) if the gap between (the numbers assigned to) them is below e (e.g., if they are indifferent in more than half the times they are presented together). Similarly, we say that one object dominates (or is definitely preferred to) the other if the gap in its favor exceeds e (e.g., if it is preferred in at least half the cases). This is extended in a natural way by Roberts in [21] to the case of betweenness (which we consider an undirected dominance). So if for the elementsx, y, z of some set Xwe write S(x, y) for "x and y are proximate (or similar)", D(x, y) for "x dominates y " and B(x, y, z) for "y is between x and z", the papers cited in this paragraph provide conditions guaranteeing the existence, for every e > 0, of a real-valued measurement function f on X s.t. for all x,y, z E X

S(x,y) iff I f (x ) - f (y ) l<e D(x,y) iff f(x)-f(y)>.--e B(x, y, z) iff If(x) - fCv) l + If(y) - f(z)l -

If(x) - / ( z ) l < 2e

Our suggestion is to replace the fixed error, or threshold,

by a varying one, which seems to be more appropriate to experimental and real-life situations. Then the two-valued, up-to-e, relations, determined by, e.g., the half-cases cut- point, are replaced by multi-valued relations. And for these new relations, we seek measurement functions yielding representations analogous to those the f above yields for S, D and B. Conditions for the existence of such functions are provided in our [10], [11] and [12]. They are based on theories like those in Section 3 of this paper, but formulated in the logic of inexactness. However, in view of Theorem 2

above, and the remarks following it, deductions in the logic of inexactness can easily be converted to ones in the logic of approximation. Thus we have the following theorem, where we return to our logic of approximation with its language and structures as throughout this paper.

THEOREM 8. Let the language L contain binary predicate symbols s and d and a ternary predicate symbol b. Let X be a structure for L with domain X and interpretations s, d, b of s, d, b (respectively). Then X is a model of the

one-sided transforms of the deductions in the theories (in Section 3 above) of similarity, dominance and betweenness (for s, d, and b, respectively) and of the following deduction for d

I-- d(u, v) ^ a(u, w) ^ d(w, v), ((d(v, w) ~ d(u, w)) ~ d(u, v)) ^ d(w, v), ((d(w, u) ~ a(w, v)) ~ d(u, v)) ^ d(u, w), ((d(w, v) --, d(u, v))--, d(u, w)) ^ d(w, u) ^ d(v, w)

if and only if there is a real-valued measurement funct ionf on X such that for all x, y, z E X

s(x, y ) = I f ( x ) - f ( y ) l d(x, y) = max(0, f(x) -fO')) b(x, y, z) = Y~(If(x) - / ( Y ) I + If(y) - f ( z ) l -

If(x) - / ( z ) l).

The intuitive meaning of the above deduction for d, when interpreted is a structure X for L, is that every three elements in the domain of X are 'ordered' from the most to the least dominant (with errors allowed). Deductions for

s with similar meanings, yielding some or all of the representations in the theorem above, can be found in the three papers cited before the theorem. In addition, the between-

ness theory of this theorem is interpreted in [10] as an axiom system for the geometry of visual perception. It is a multi-valued adaptation of Roberts' 'tolerance' geometry [21] (tolerating 'small' errors), which is, in turn, an up-to-e adaptation of (the restriction to one dimension of) Tarski's classical elementary geometry [26].

We note that in the special case of our utility theory, when the function / of Theorem 8 ranges in [0, 1] and happe/ls to be the interpretation in X of some unary predicate g of L (so that we can write g for f ) , it is easy to see that g is a unidimensional gain predicate in X and all the deductions of Theorem 7 (in fact their one-sided transforms) hold in X. These (one-sided) deductions can be considered representations of dominance and betweenness by the gain predicate g in the formal language L. Proximity can also be represented by g in L, using one-sided transforms of


the substi tutabil i ty deduction and its converse, both of

which hold in X in the special case discussed here.

Finally we mention Coombs' observation in [3] that in

psychology measurement theories are behavior theories.

They describe behavioral regularities concerning proximity

and dominance and, in view of the centrali ty of these two

predicates, can be applied, as theories of substance (not

only of tools), to many areas of behavioral inquiry. The

handling of such theories in the logic of approximation,

where inexactness is taken into account, adds intuitive

appeal to their potent ial applications.

Note

* For this invitation I am very grateful to the meeting's organizers, and in particular to Ray Jennings and Peter Schotch.

References

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of Michigan Press, Ann Arbor. [4] Dalla Chiara, M.L.: 1977, 'Quantum logic and physical

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[17] Krantz, D. H., Lute, R. D., Suppes, P. & Tversky, A.: 1974, Foundations of Measurement, Academic Press, New York.

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[23] Scott, D.S.: 1976, 'Does many valued logic have any use?', in S. K6rner (ed.), Philosophy of Logic, Basil Blackwell, Oxford, pp. 64-95 (including Comments by T.J. Smiley, J. P. Cleave and R. Giles).

[24] Scott, D. S. & Suppes, P.: 1958, 'Foundational aspects of theories of measurement', Journal of Symbolic Logic 23, 113-128.

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[28] van-Fraassen, B.C.: 1973, 'Semantic analysis of quantum logic', in C. A. Hooker (ed.), Contemporary Research in the Foundations and Philosophy of Quantum Theory, Reidel, Dordrecht, pp. 80-113.

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