Generalisation of First-Order Logic to Nonatomic

Generalisation of First-Order Logic to Nonatomic Domains Author(s): P. Roeper Source: The Journal of Symbolic Logic, Vol. 50, No. 3 (Sep., 1985), pp. 815-838 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2274334 . Accessed: 30/11/2014 23:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 50, Number 3, Sept. 1985

GENERALISATION OF FIRST-ORDER LOGIC TO NONATOMIC DOMAINS

P. ROEPER1

The quantifiers of standard predicate logic are interpreted as ranging over domains of individuals, and interpreted formulae beginning with a quantifier make claims to the effect that something is true of every individual, i.e. of the whole domain, or of some individuals, i.e. of part of the domain. To state that something is true of all or part of a totality seems to be the basic significance of universal and existential quantification, and this by itself does not involve a specification of the structure of the totality. This means that the notion of quantification by itself does not demand totalities of individuals, i.e. atomic totalities, as domains of quantification. Nonatomic domains, such as volumes of space, or surfaces, are equally in order. So one might say that a certain predicate applies "everywhere" or "somewhere"in such a domain. All that the concept of quantification requires is a totality which is structured in terms of a part-to-whole relation, and appropriate properties that apply to part or all of the totality. Quantification does not demand that the totality have smallest parts, or atoms. There is no conflict with the sense of universal or existential quantification if the domain is nonatomic, if every one of its parts has itself proper parts. The most general kind of quantification theory must then deal with totalities of any kind, atomic or not. The relationships among the parts of a domain are described by the theory of Boolean algebras, which we can regard as the most general characterization of a totality, of a domain of quantification. In this paper I shall be concerned with this generalised theory of quantification, which encompasses nonatomic domains as well as atomic and mixed domains, i.e. totalities consisting entirely or partly of individuals. ?1. Functions.One of the central concepts of the predicate calculus is the concept of a function. Most of the variables represent functions: truth functions, functions from the domain of individuals into the set of truth values and functions from the domain of individuals into that domain. The generalization of quantification theory to nonatomic domains has to begin with a reassessment of the role of functions. The notion of a function in the strict sense (i.e. as something that takes as arguments the Received July 23, 1981; revised August 16, 1984. lI am grateful to the referees for important corrections and suggestions. (C 1985, Association

for Symbolic Logic

0022-481 2/85/5003-0023/$03.40 815

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816

P. ROEPER

individual elements of a domain) may not be needed for quantification to make sense. What is sufficient for an analytic understanding of quantified statements is the concept of the image of a domain, and parts of a domain, under a mapping. For a function F from the domain of individuals D into the set {T, I } of truth values the associated set function F from YD into ?B'{T,I} has for any set E c D as argument the image of E under F as value: F(E) = {F(e): e E E}. So, interpreting a predicate P as a function from D into {T, 1}, to say that P is true of every member of D is to say that the image of D under P is the set {T}: ?(D) = {T}. The aim is then to generalise the functions F to go from any Boolean algebra into {T, I} (and similarly for other functions usually considered in logic), so that the universal truth of a predicate P in a domain 7t (which need not be atomic) amounts to ?(Zr)= {T }. q is now a function from W, the Boolean algebra associated with 7t, into Y1{T, I}, and may be called the image-function associated with P. The intended generalisation is achieved by specifying the conditions under which a mapping is the associated set function of some function F, and then generalising these conditions to all Boolean algebras. The following are the conditions that a mapping of ??D into P Y has to satisfy if it is to be the associated set function of a function from D into Y: A=0. (Al) F(A)=0 (A2) A c B => F(A) c F(B). (A3) F(A) = A' and 0 7&B' c A'=> there is a set B E? D, 0 =AB c A, with F(B) c B'. I be the Boolean algebras describing the mereological Now let Wi, and structures of it and it', so that it is the unit element of 1,,and it' that of W,. 0 is, in any Boolean algebra, the null element; p, T, p', T' are arbitrary elements. Then a mapping f of 9Z into W, is an i-functionif it satisfies the following conditions: (Bi) f(p) = 0r p = 0. (B2) T < p => f(T) < f(p). (B3) f (p) =p' and O = T' Ai(z) = {I1}. (B3*) ?(p) = {T, I} = there is an element z- E %,?0 : z1 < p, such that P(z1) = {1}. = {T} and there is an element ?2 E 91n 0 = T2 < p, such that 0(z2) (B2*) 0 :4z < p and ?(p) ={T}=

For the purpose of formulating semantic rules it proves advantageous to consider just the elements p of the Boolean algebra for which ?(p) is {T}. The set of these elements must conform to the following set of conditions. (DI) ?(p) ={T } p =O0. (D2) 0(p) = {T} and 0 =Az < p => 0(z) = {T} ?, with (D3) p =A0 and (for every z, 0 =Az < p, there exists an element po,0 = So< 0((p) = {T})-

=

(p) = {T}.

The value of P for those elements of 91 for which it is not {T} is determined in accordance with the clauses of (D4): {1} p :A0 and for every element z < p, 'P(z) = {T}. (D4) (i) P(p) = and for some z < p, (z)={T}. I} T, p=0,(p)={T} (ii) 0p(p) P = 0. (iii) P(p) = 0 For the proof that conditions (BI *)-(B3*) are equivalent to conditions (D1)-(D4) assume (i) that (B1*)-(B3*) are satisfied by '. Then (DI) and (D2) follow from (B1*) and (B2*). As to (D3), assume that p =A0 and for every z, 0 =Az < p, there exists a 9o, 0 = 9?, o < with l(9o) = {T}, but that 0(p) =A{T}. Then, by (B1*) and (B2*), 0(p) {I}.But then, according = {T, I}. By (B3*) there exists I, O :A / < p, withl(f) to (B2*), P(Z) = {I} for every X, 0 :A. < ?; and this contradicts the assumption. The first two parts of (D4) are easily derivable from (B2*) and (B3*). (ii) If (D 1)-(D4) are satisfied by ' then (B2*) must be true. For if 0 =Az < p, then if P(p) {T }, P(z) {I} and P(z) were differentfrom {I } then P(z) {T} or = {T} by (D2); if 'P(p) = there would be a 9o,0 =ASo< , with P(po)= {T}, by in case either I}; {T, O(T) (D4(ii)). But then there would be a 9o,0 0 So< p, with P(po)= {T} and so P(p) &{L}, in virtue of (D4(i)), contradicting the assumption. To prove (B3*) it has to be shown that if '(p) = {T, I} then there exist r1 andz2, ? T1 ? p. ? ? < p, with P(-r1) {T} and O((T2)= 1 }. The existence of Tl follows from (D4(ii)). Suppose I_ } for every T, 0 = T < p, which means, by (D4(i)), that there is no T2. Then P(z) ? , with P(po) = {T}. By (D3) it follows that P(p) = {T}, there is a So, ?0 o < contrary to the assumption. The reformulated conditions show that ( is completely determined by the class of arguments for which it takes the value {T}. The classes of elements satisfying (Dl)(D3) are the complete ideals (minus the null element) of the Boolean algebra ,. (An ideal A of a (not necessarily complete) Boolean algebra is complete if for any X c A, whenthe infinitejoin UX exists, UX e A.) (ii) Relations. If R is an n-place relation let the n-place i-function [7 - {T. I}] R be the associated image-function. The appropriate version of the conditions

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818

P. ROEPER

(C1)-(C3) can also be expressed as conditions under which a class of n-tuples of elements of 91 contains just the arguments for which the value of R is {T}: :A ? 0,a)fn (E 1) R(al akn) = {T}= al T, -, n) = T1 (E2) (R(al,..a, ain) = T} and T< am.)= R(al,.. (E3) (Pi =AO,.., =An 0 and for all a,, 0 U1 < Pl; U2' 0 = U2 < P2;... ;n, o =Aan < Pn, there exist z1, 0 =A-1 < ol; z2, 0 7 2 < O2; . ;zn, 0 : -Tn< an_ with Pn) = {TJ R(rl .... an = { TI) -> R(pl,., (E4) (i) R(c1,...,

= {} I I

an)

, , an

?

=A0

and

for

every

-c1 < ,

= 7&{TI) = {T. cn)

tn< an) R(Tl ....n)

{ v ?,un and for 0; R(l,. + 7&fn) =A Tj (ul < = *,) {T}). some-Tn on_ R(c1,.. (iii) R(ul, din) = 0 . = 0 for some m, I < m < n. If there are denumerably let a, I, p be infinite sequences of many arguments arguments; let a be a positive sequence if u.. =A0 for every m; let -c < a mean that -T. < u_ for every m. Then the image function R associated with a relation R with denumerably many arguments conforms to the following conditions which extend (E1)-(E4) to infinitely many arguments. T} =:{ oais positive. (F 1) R(o) (ii) R(ul) some z1 < os,...,

(F2) (R(o) = {T}, - is positive and - < C) =- R(-) = {T}. (F3) (p is positive and for every positive a < p there exists a positive - < a with { T} R(-) = {T}) z, R(p) is positive and for every positive - < C, R(-) : {T}. (F4) (i) R(o) = {} I (o is positive; R(o) = {T} and for some positive - < C, R(-) (ii) R() = {T, I} (iii) R(o) = 0 o is not positive. This completes the characterization of the images of properties and relations. It remains to mention the images of truth functions. Both the domain and the counterdomain are YP{T, I}; so they are i-functions [{T, I} -+ {T, I}], and as associated set functions there is no doubt that they satisfy (B1)-(B3), or (C1)-(C3). The matrices for negation and conjunction are:

{T,} {T} {1} 0

{T, I} {I}{T}I{TI {T} 0

&

{T, lI}

{T}

{l}

0

{T, I} {1}

{T, I} } {1}

{T } {T} {1}

{} {I} {I}

0 0 0

0

0

0

0

0

?2. Operations on functions. The next question that has to be considered is what effect operations on functions have on the corresponding images. The operations that are relevant for quantification theory are functional composition, identification of argument places and quantifier application. The functional product is unproblematic. For ordinary functions F(x) and G(x) the image under FG is the same as the image under F of the image under G; therefore there corresponds to the functional product of F and G the functional product of F and G. It is easy to verify that if fand g are i-functions and so satisfy (B 1)-(B3) then fg does the same. When f or g or both are many-place functions we obtain an analogous result as long as

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GENERALISATION OF FIRST-ORDER LOGIC

819

identification of argument places in avoided. If both f(1,. and an) ram, g~1, .. ., -,.. .) satisfy conditions (C1)-(C3) and the variables . .1.... are different from the variables a,. . .,Sm... (except possibly then an) m ... Iam,I...g(T I l, ...T . ) also satisfies (C1)-(C3). (If the prohibition on f(, variable identification is not observed, (C3) cannot be guaranteed.) If G(x) is a function that has been obtained from the two-place function F(x, y) by identification of the two argument places, G(x) = F(x, x), then H(A) = F(A, A) is, in general, not the image of A under G, i.e. H(A) = G(A). Still, in the particular case where the values of F(x, y) are T and I, G(A) is definable in terms of F(A, B) in a way that admits of generalisation to nonatomic domains: G(A) = {T} (A = 0 and for every set B, 0 = B ' A, there exists a set C, 0 : C ' B, with F(C, C) = {T }). G(A) {I} { (A = 0 and for every set B, 0 = B ' A, F(B, B) : {T}). A = 0. G(A)= 02 G(A) = { T, I } otherwise. This suggests that one can define generally for two-place i-functions whose values are subsets of {T, I} an operation which corresponds to the identification of variables at the level of individuals when the domain is atomic: From R(p, X)obtain ?P(z)as follows: ?P(z)= {T } (z =A0 and for all elements So, o = So