Paradoxes and many-valued set theory

ROBERT E. MAYDOLE

PARADOXES AND MANY-VALUED SET THEORY

I. INTRODUCTION

It is a common misconception that the paradoxes of Naive Set Theory (NST) can be dodged by simply giving up the Principle of Bivalence, and by shifting thereby from the classical two-valued logic to a many-valued logic. This dodge, if successful, would result in a many-valued set theory. The basis for this misconception, presumably, is the belief that a contradiction can be avoided by assigning the same intermediate truth-value to otherwise incompatible statements.

Consider, for example, the following proposal for dodging Russell’s Paradox. Let L be a logic with true, false, and middle as its truth-values. Suppose that the truth-rules of L are such that a statement is middle just in case its negation is middle, and that a biconditional is true just in case its immediate components have the same truth-value. Now Russell’s Paradox shows up in NST just at the point where it is derived that Russell’s class is a member of itself if and only if it is not a member of itself, since such a biconditional cannot be true in the classical two-valued logic. But the latter biconditional can be true in L. We simply force the statement that Russell’s class is a member of itself to be middle.

The dodging of paradoxes is not nearly as simple, however, as the above example might seem to suggest. It will be shown in this paper that the axioms of NST are inconsistent in sundry well known many-valued logics, infinite as well as finite-valued. A general method will also be developed for generating set-theoretic paradoxes in many-valued logics. In fact, only three of the well known many-valued logics described below even prima facie stand a chance of basing a set theory which has axioms at least as strong as those of NST, and all three of these logics are indenumerable- valued.

II. FIRST ORDER LOGICS

Set theory requires a first order logic for its formulation and development. A first order logic L is a quadruple @an(L), Tru(L), Des(L), Fun(L)), where

Journal of Philosophical Logic 4 (1975) 269-291. All Rights Reserved Copyright o 1975 by D. Retie1 Publishing Company, Dordrecht-Holland

270 ROBERT E. MAYDOLE

Lan(L) is a first order language, Tru(L) is a truth-space, Des(L) is a set of designated truth-values (a proper subset of Tru(L)), and Fun(L) is a set of semantic truth-functions.

First order languages are understood in the usual way: individual variables, predicate letters, sentential connectives, quantifiers, punctuation marks, atomic and molecular well formed formulas, etc. Unless stated otherwise, each fust order logic discussed in this paper has the same first order language. Each has a binary predicate letter ‘e’; a monadic sentential connective 1; the binary sentential connectives V, A, + and t+; and the quantifiers V and 3.

The semantic truth-functions of a logic L are functions that are assigned to the sentential connectives and quantifiers of Lan(L) in the following way: (i) if Q is an n-ary sentential connective, then the semantic truth-function fu(o, L) assigned to o is a member of the set of all functions from (Tru(L))” into Tru(L); (ii) if o is a quantifier, then fu(0, L) is a member of the set of all functions from the set of all non-empty subsets of Tru(L) into Tru(L).

The general semantics of a logic L are also understood in the usual way. Let d be a non-empty set - a so-called semantical domain. An assignment h on d is a function that assigns a function h(P) to every predicate letter P of Lan(L) as follows: if P is n-ary, then h(P) is a member of the set of all functions from d” into Tru(L). A model for L is then understood as an ordered pair (d, h).

A v-interpretation of L on a model M = (d, h) is a function that assigns a member of d to every individual variable of Lan(L).

(Note: Much of the subsequent discussion is conducted in a semi- formalized metalanguage. The reader is therefore urged to check the Appen- dix for a brief explanation of the symbols used.)

Given a model m = (d, h) for a logic L and a v-interpretation w of L on m, the truth-value of a wHffA of Lan(L) with respect to m and w is then defined inductively as follows:

(9

(3

tv(A, m, w, L) = h(P)(w(xl), . . . . w(x,)), if A is the atomic wffP(xr, . . ..X”).

tv(A, m, w, L) = fu(0, L)(tv(B,, m, w, L), . . . . tv(B,, m, w, L)), if o is an n-ary sentential connective and A is the wff o(Br, . . ., B,),

PARADOXES AND MANY-VALUED SET THEORY 271

(iii) tv(A, m, w, L) = fu(o, L)({g I (Ewr)(Eb)(b Ed & vin(wl, m, L) & wi(x) = b &g = tv(B, m, wl, L) & fre(B, x, w, wr)))), if 0 is a quantifier and A is the wff (sx)B.

A set of wffs A of Ian(L) is consistent in L if and only if there is a wff A of Ian(L), a model m for L, and a zrinterpretation w of L on m such that tv(A, m, w, L) $ Des(L), and for every wff B E A, tv(B, m, w, L) E Des(L). Derivatively, a single wff B is consistent if and only if the unit set of B is consistent. Remark: The rather unusual consistency condition that there be a wff A such that tv(A, m, w, L) 65 Des(L) with respect to some m and w such that tv(B, m, w, L) E Des(L), for B E A, is required for many-valued logics in order to prevent the necessity of having to say that a set of wffs A is consistent if its turns out that there is some ml and w1 such that tv(A1,m,,wl,L)EDes(L),foreverywffA,ofLan(L).

III. A METHOD FOR PROVING THE INCONSISTENCY OF COMPREHENSION

NST has one proper axiom, the Principle of Extensionality, and one proper axiom schema, the Principle of Comprehension. Since it is the Principle of Comprehension that is most directly tied to the generation of set-theoretic paradoxes, I will not discuss the Principle of Extensionality further in this paper. The Principle of Comprehension, which I hereafter refer to by ‘COM’, is expressed in first order languages as follows:

(Vz) . . . (Vz,)(Vy)(Vx)(x E y * F(x, Zl, . . . zn));

where F(x,zi, . . . . z,) is any wff with at most x, z i, . . ., z, as free variables. Note that COM is really a set of wffs.

In order to be able to show how one might go about proving that COM is inconsistent in a logic L, we first need a few definitions.

DEFINITION: (A A B) for (,4 + B).

DEFINITION: (A 1: B) for (A 5 B)).

DEFINITION: COM(A, n) for @y)(Vx)(x EJJ * (x E x $4)).

272 ROBERTE.MAYDOLE

DEFINITION: A” for (Wc,) . . . (tlxr)A, (where x1, . . ..x. are the free variables of A in order of their first left most occurrence in A).

A” is the so-called universal closure of A. Suppose now that either CURY (1, n, L) or CURY(2, n, L) below are

true. CURY(l,n, L): (A)(m)(w)(tv(COM(A*, n), m, w, L) E Des(L) 3

tv(A, m, w, L) E Des(L)). CURY(2, n, L): (Es)(EB)(s C Tru(L) & (m)(w)(tv(B, m, w, L) f$ s)

& (A)(m)(w)(tv(COM(A*, n), m, w, L) E Des(L) 3 tv(A, m, w, L) E s)). Then COM will be inconsistent in L; for COM is consistent in L only if

(1) below is true.

(1) (EB)(Em)(Ew)(A)(tv(B, m, w, L) B Des(L) & tv(COM(A*, n), m, w, L) E Des(L)).

But (1) contradicts both CURY (1, n, L) and CURY (2, n, L). Thus, if either CURY (1, n, L) or CURY (2, n, L) is true, then (1) is false and COM is inconsistent in L.

How then might we go about proving that there is a n such that either CURY (1, n, L) or CURY (2, n, L) is true? Again it will be helpful to have some definitions.

DEFINITION: Exis(s, t, L) for (A)(B)(m)(w)(tv((3y)(Vx) (A cf B), m, w, L) E s 3 (Ew)(tv((tlx)(A ++B), m, w, L) E f).

DEFINITION: Nof(s, L) for (A)(x)((- fr(x, A) & (Ew)(tv(A, m, w, L) Es)) 3 (y)(Ew)(w(x> = w(y) & tv(A, m, w, L) Es)).

DEFINITION: Univ(s, L) for (A)(m)(w)(tv((vx)A, m, w, L) Es 3 tv(A, m, w, L) Es).

DEFINITION: Var(s, L) for (A)(E)(m)(w)(x)(y)((w(x) = w(y) & lk(A, B,x,y)) 3 (tv(A, m, w, L) Es = tv(B, m, w, L) E s)).

DEFINITION: Equi(s, L) for (A)(B)(m)(w)(tv((A *B), 112, w, L) Es 3 (tv((A + B), 112, w, L) Es & tv((B +A), m, w, L) E s)).


DEFINITION: Ab(n, s, L) for (A)(B)(m)(w)(tv((A %I), m,w,L)Es>tv((&B),m,w,L)Es).

DEFINITION: MP(s, L) for (A)(B)(w)((tv(A + B), m, w, L) E s & tv(A, m, w, L) Es) > tv(B, 112, w, L) Es).

DEFINITION: Clo(s, L) for (A)(m)(cl(A) > ((Ew)(tv(A, m, w, L) E s) 3 (w)(tv(A, m, w, L) Es))).

Now consider THESIS(l, n, L) and THESIS(2, n, L).

THESIS(l, n, L): Exis(Des(L), Des(L), L) & Nof(Des(L), L) & Univ(Des(L), L) & Var(Des(L), L) & Equi(Des(L), L) & Ab(n, Des(L), L) & MP(Des(L), L) & Clo(Des(L), L).

THESIS(2, n, L): (EB)(Es)(s C Tru(L) & (m)(w)(tv(B, m, w, L) $ s) & Exis(Des(L), s, L) & Nof(s, L) & Var(s, L) & Equi(s, L) & Ab(n, s, L) & MP(s, L) & Clo(s, L)).

Clearly, THESIS(1, n, L) entails CURY(l, n, L), and THESIS(2, n, L) entails CURY (2, n, L). Since the proofs are about identical, I shall only prove the latter.

PROOF that THESIS(2, n, L) entails CURY(2, n, L). Assume (1): (EB)(Es)(s C Tru(L) & (m)(w)(tv(B, m, w, L) es) & Exis(Des(L), s, L) & Nof(s, L) & Univ(s, L) & Var(s, L) 8c Equi(s, L) & Ab(n, s, L) & MP(s, L) & Clo(s, L)). From (1) we get (2): s C Tru(L) & (m)(w)(tv(B, m, w, L) $ s); and (3): Exis(Des(L), s, L); and (4): Nof(s, L); and (5): Univ(s, L); and (6): Var(s, L); and (7): Equi(s, L); and (8): Ab(n, s, L); and (9): MP(s, L); and (10): Clo(s, L). Assume (11): tv(COM(A*, n), m, w, L) E Des(L). From (11) and (3) we get (12): (Ew)(tv((tlx)(x my ++ (x EX %A*)), m, w, L) Es). From (12) and (4) we get (13): (Ew)(w(x) = w(y) & tv((vx)(x E y ++ (x E x $:A *)), m, w, L) E s). From (13) and (5) we get (14): (Ew)(w(x) = w(x) & tv((x EY f) (x E x %A**)), m, w, L) E s). From (14)and(6)weget(15): (Ew)(tv((yEy*(yEylA*)),m,w,L)Es). From (15) and (7) we get (16): (Ew)(tv((y EY + (y EY GA>), m,w,L)Es&tv((yEy1-A*)~yEy),m,w,L)Es).From(l6)and(8) we get (17): (Ew)(tv((yEycA*),m, w, L)Es& tv(((yEy %:A*)


+y my), m, w, L) Es). From (17) and (9) we get (18): OW(tvK~ E Y %:A*) , m, w, L) E s & tv(y EY, m, w, L) E s). From (18) and repeated applications of (9) we get (19): (Ew)(tv(A*, m, w, L) Es). From (19) and (10) we get (20): (w)(tv(A*, m, w, L) Es). From (20) and repeated applications of (5) we get (21): (w)(tv(A, m, w, L) E s). And from (21) we get (22): tv(A, m, w, L) Es. Thus, from (1 l)-(22) we have (23): tv(COM(A*, n), m, w, L) E Des(L) > tv(A, m, w, L) Es. From (23) we get (24): (m)(w)(tv(COM(A*, n), m, w, L) E Des(L) > tv(A, m, w, L) Es).’ Then from (24) and (2) we get (25): s C Tru(L) & (m)(w)(tv(B, m, w, L) $ s) & (m)(w)(tv(COM(A *, n), m, w, L) E Des(L) 1 tv(A, m, w, L) Es). Then from (25) we have (26): CURY(2, n, L). Q.E.D.

Remark: If (THESIS(l, n, L) & (EB)(m)(w)(tv(B, m, w, L) $ Des(L))) is true, then THESIS(2, n, L) is true.

To prove that COM is inconsistent in a logic L, it suffices to show, then, there is an n such that either THESIS(l) n, L) or THESIS(2, n, L) is true. If either THESIS(l, n, L) or THESIS(2, n, L) is true, we say that COM generates a Curry Paradox of Order(n) in L. If either CURY (1, n, L) or CURY(2, n, L) is true, but neither THESIS(l, n, L) nor THESIS(2, n, L) is true we say that COM generates a Quasi Curry Paradox of Order(n) in L.

In subsequent sections of this paper I briefly describe a number of many- valued logics. The reader is requested to check the literature for further details, especially Rescher’s Many- Valued Logic. Except for the Boolean IV-valued logic(s), for all of the logics described in which COM is known to be inconsistent, there is an n such that COM generates a Curry Paradox of Order(n). I sketch a proof of this for the finite-valued Lukasiewiczian logics, and leave it to the reader to provide most of the details of the proofs for the other logics. COM is shown to generate a Quasi Curry Paradox of Order(l) in the Boolean W-valued logic(s). It is also easy to show, although I will not show it, that if COM generates a (Quasi) Curry Paradox of Order(j) in logic L, then COM generates a (Quasi) Curry Paradox of Order(n), n > j, in logic L. Moreover, for all of the logics L described in which COM is known to be inconsistent, except Sobocinski’s 3.valued logic and the finite-valued K’SEQ logics, there is an n such that CURY(2, n, L) is true. But if L is Sobocinski’s 3-valued logic or a finite-valued K2-SEQ logic, then only CURY (1, j, L), j > 1, is true. Actually, in all of the finite- valued logics L described in which COM is known to be inconsistent, except Sobocinski’s 3-valued logic and the finite-valued K2SEQ logics, there is an n such that both CURY(l, n, L) and CURY(2, n, L) are true.


IV. THE kUKASIEWICZIAN LOGICS

There are three types of -Lukasiewiczian logics (LUK lo&s): the finite n-valued (n > 2) logics, LUK(n); the infinite denumerable-valued logic, LUK(H,); and the indenumerable-valued logic, LUK(Hr).

Truth-values: Tru(LUK(n)) = (k/n - 110 < k < n - 11. Tru(LUK(H,)) = [0, l]#;Tru(LUK(Hr)) = [O,l]. For every LUK logic L, Des(L) = (1).

Truth-functions: For every LUK logic L, fu(i, L)(u) = 1 - u ; fu(A, L)(u, v) = min(u, v); fu(V, L)(u, v) = max(u, v); fu(+, L)(u, v) = min(1, 1 -u+v);fu(*,L)(u,v)= 1 -lu-vl;fu(g,L)(t)=lub(t); fu( tl, L)(t) = glb(t).

PROPOSITION 1: THESIS(2, n - 1, LUK(n)) is true.

To prove PROPOSITION 1 it suffices to prove that (THESIS(l, n - 1, LUK(n)) & (EB)(m)(w)(tv(B, m, w, LUK(n)) + Des(n))) is true. The proof of the second conjunct follows at once from the fact that (A)(m)(w)(tv((A A 1 A, m, w, LUK(n)) 6 Des(LUK(n))) is true. (I leave the details to the reader.)

In order to prove that THESIS( 1, n - 1, LUK(n)) is true, we have to prove that each of the conjuncts of THESIS(l) IZ - 1, LUK(n)) are true. Verification of Nof(Des(L), L), Var(Des(L), L), and Clo(Des(L), L), for every LUK logic L, are left to the reader. They are easily proved by mathematical induction on the length of wffs.

PROOF that Exis(Des(LUK(n)), Des(LUK(n)), LUK(n)) is true. Assume (1): tv((3y)A r, m, w, LUK(n)) Es. Given that m = (d, h), (1) then yields (2): bib {u I (Ewr)(Eb)(b Ed & vin(wr, m, LUK(n)) & wr(y) = b &u = tv(Ar, m, wl, LUK(n)) & fre(Ar,y, w, wr))] Es. From (2) and the finite- ness of Tru(LUK(n)) we then get (3): smax {u 1 (Ew,)(Eb)(b Ed & vin(wi, m, LUK(n)) &.vin(wr, m, LUK(n)) & w,(y) = b &u = tv(A1,m, wl, LUK(n))&fre(Ar,y, w, w,)))Es. From(3) and themeaning of ‘smax’ we get (4): (Ew)(tv(Ar, m, w, LUK(n)) Es. Then let A r be (Wx)(A c+ B), and s be Des(LUK(n)).

PROOF that Univ(Des(LUK(n), LUK(n)) is true. Let L be any LUK logic. Let s be {u I u E Tru(L) &u > v}. Clearly, if ZI = 1, then s = Des(L). Assume (1): tv((vx)A, m, w, L) Es. (1) implies (2): tv((tlx)d,m,w,L)>v.


Implicit in (2) is (3): m = (d, h) & vin(w, m, L) & (Eb)(b Ed & w(x) = b). Given that m = (d, h), (2) implies (4): glb{u 1 (Ew,)(Eb)(b Ed & vin(w,, m, L) & wr(x) = b &u = tv(A, m, wl, L) & fre(A,x, w, wr))}> z1. From (4) we get (5): (u)((EwJ(Eb)(vin(wr, m, L) &b Ed & wl(x) = b & u = tv(A;m, wl, L) &fre(A,x, w, wi)) >u 2~). From (5) we get (6): (u)(wr)((vin(wr, m, L) & (Eb)(b Ed & wl(x) = b) &u = tv(A, m, wl, L) & fre(A, x, w, wl)) 1 u > v). (6) implies (7): (vin(w, m, L) & (Eb)(b Ed & w(x) = b & fre(A,x, w, w) & tv(A, m, w, L) = tv(A, m, w, L)) > tv(A, m, w, L) > V. Since ‘fre(A, x, w, w)’ and ‘tv(A, m, w, L) = tv(A, m, w, L)’ are logical truths, (7) implies (8): (vin(w, m, L) & (Eb)(b Ed & w(x) = b)) > tv(A, m, w, L) > o. (8) and (3) then imply (9): tv(A, m, w, L) Es.

PROOF that Equi(Des(LUK(n), LUK(n)) is true. Let L be any LUK logic. Assume (1): tv((A *B), m, w, L) E Des(L). From (1) and the fact that Des(L) = (1) we get (2): 1 - I tv(A, m, w, L) - tv(B, m, w, L) I = 1. From (2) we get (3): tv(A, m, w, L) = tv(B, m, w, L). Then from (3) we get (4): tv((A +B), m, w, L) E Des(L) & tv((B +A), m, w, L) E Des(L).

In order to prove that Ab (n - 1, Des(LUK(n)), LUK(n)) is true, we need the following lemma:

LEMMA 1: For every LUK logic L and every integer n > 1, tv((A SB), m, w, L) = min(1, n(1 - tv(A, m, w, L)) + tv(B, m, w, L)).

The proof of LEMMA 1 is by mathematical induction on the integer over the arrow. Here is a guide to the proof, the details of which are left to the reader.

The Base Step is trivial. The Induction Step proceeds by cases. Assume the inductionhypothesis H: tv((A s:), m, w, L) = min(1, k(l - tv(A, m, w, L)) + tv(B, m, w, L)). For the first case, assume (i): tv(A, m, w, L) > tv((A SB) , m, w, L). Then show that (i) and H together imply (iii): tv((A s B), m, w, L) = min(1, (k + 1)(1 - tv(A, m, w, L)) + tv(B, m, w, L)). For the second case, assume (ii): tv(A, m, w, L) < tv((A 5 B), m, w, L). Then show that (ii) and H together imply (iii).

PROOF that Ab(n - 1, Des(LUK(n)), LUK(n)) is true. The proof is by reductio ad absurdum. Assume (1): tv((A 1: B), m, w, LUK(n)) E Des(LUK(n)). Also assume (2): tv((A SB), m, w, LUK(n)) B

PARADOXES AND MANY-VALUED SET THEORY 277 .4

Des(LUK(n)). Since Des(LUK(n)) = {l}, (1) is just (3): tv((A 3 B), m, W, LUK(n)) = 1. Similarly, (2) is just (4): tv((A SB) m, w, LUK(n)) < 1. From LEMMA 1 and (3) we get (5): min(1, n(1 - tv(A, m, w, LUK(n))) + tv(B, m, w, LUK(n))) = 1. And from LEMMA 1 and (4) we get (6): min(1, (n - l)(l - tv(A, m, w, LUK(n))) + tv(B, m, w, LUK(n)))< 1. From (5) we get (7): n(1 - tv(A, m, w, LUK(n))) + tv(B, m, w, LUK(n)) > 1. From (6) we get (8): (n - l)(l - tv(A, m, w, LUK(n))) + tv(B, m, w, LUK(n)) < 1. From (8) and the fact that 0 < tv(A, m, w, LUK(n)) < 1, we get (9): tv(B, m, w, LUK(n)) < 1; and we also get (10): (n - l)(l - tv(A, m, w, LUK(n))) < 1. But from (10) we get (11): tv(A, m, w, LUK(n)) > n - 2/n - 1. From (11) and the fact that Tru(LUK(n)) = (k/n - 1 IO < k <n - l} we get (12): tv(A, m, w, LUK(n)) = 1. Then from (12) and (7) we get (13): tv(f3, m, w, LUK(n)) > 1. But (13) contradicts (9). Thus, if (1) is true, then (2) is false.

PROOF that MP(Des(LUK(n)), LUK(n)) is true. Let L be any LUK logic. Assume (1): tv((A +I?), m, w, L)E Des(L) & tv(A, m, w, L) E Des(L). From (1) and the fact that Des(L) = {l}, we get (2): tv((A +B), m, w, L) = 1 & tv(A, m, w, L) = 1. From (2) we then have (3): min(1, tv(B, m, w, L)) = 1. From (3) we get (4): tv(B, m, w, L) > 1. Then from (4) and the fact that (u)(u E Tru(L) 1 u < 1) we get (5): ts@, m, w, L) = 1.

Having now shown that COM is inconsistent in each finite-valued LUK logic, what can be said about the consistency status of COM in the infmite- valued LUK logics? Although the answer to this question is not known, the evidence does seem to weigh in favor of COM being consistent. First of all, it has never been shown that COM is inconsistent in the infinite-valued LUK logics.

Secondly, if L = LUK(He) or L = LUK(Hr), then Ab(n, Des(L), L) is false for every n > 1. For suppose(l): tv(A, m, w, L) = n/n + 1; and (2): tv(B, m, w, L) = 0. Then from (1) and LEMMA 1 we infer (3): tv((A SB), m, w, L) = 1. Similarly, from (2) and LEMMA 1 we infer (4): tv((A 5 B), m, w, L) = n/n + 1. Thus, from (3) and (4) we get (5): tv((A *B), m, w, L) E Des(L) & tv((A SB), m, w, L) r$ Des(L). While significant, all that this proves of course is that the very same method for


proving that COM is inconsistent in the finite-valued LUK logics is not applicable to proving that COM is inconsistent in the infinite-valued logics.

The third and perhaps most significant reason for believing that COM is consistent in the infinite-valued LUK logics relates to the fact that certain so-called partial consistency proofs are available. By restricting what can count as the wff F(x, z 1, . . ., z,) of COM, various very general subsets of COM, the union of which equals COM, can be proved to be consistent in LUK(Hi). In particular, we can show that (3y)(Vx)(x EY * G(x)), where G(x) is a wff in which only x is free, is consistent in LUK(Kr). (See Maydole’s Many- Valued Logic as a Basis for Set Theory, especially Chapter III.) The upshot of this is that CURY (1, n, LUK(H r)) and CURY(2, n, LUK(H,)) are false for all n > 1, and hence that COM generates neither a Curry Paradox nor a Quasi Curry Paradox of any Order in LUK(Hr).

V. THE BOCHVARIAN LOGICS

There are three types of Bochvarian logics (BOC logics): the finite n-valued (n > 2) logics, BOG(n); the infinite denumerable-valued logic BOC(He); and the indenumerable-valued logic, BOC(Hi).

Truth-values: Tru(BOC(n)) = (k/n - 110 < k <n - 11; Tru(BOC(H,)) = [0, l]#; Tru(BOC(tSi)) = [0, 11. For every BOC logic B, Des(B) = (1).

Truth-finctions: For every BOC logic B, fu(l, B)(u) = 1 -u; fu( 3, B)(t) = lub(t); fu(V, B)(t) = glb(t). If u E (0, l> and VE (0, 11, then for every BOC logic B, fu(A, B)(u, v) = uv; fu(V, B)(u, v) = min(1, u + v); fu(+, B)(u, v) = min(1, 1 -u + v); fu(+*, B)(u, v) = 1 - (u + v) + 224~. If either u $ (0, l} or v B (0, l}, then for every BOC logic B, fu(A, B)(u, v) = fu(v, B)(u, v) = fu(+, B)(u, v) = fu(*, B)(u, v) = Z; where Z = n - 1/2(n - 2) if B is BOG(n), and Z = l/2 if B is infinite- valued.

PROPOSITION 2: THESIS(2,1, B) is true for every BOC logic B.

PROPOSITION 2 can be proved by showing that (THESIS( 1, 1, B) & WXm Wttvt~ 9 m, w, B) B Des(B))) is true for every BOC logic B. In proving this it is important to note two things. First,


(A)(m)(w)(tv(A P\ 1 A), m, w, B) B Des(B)) is true for every BOC logic B. Second, in the infinite-valued BOC logics every wff containing occurrences of +, ff, A, or v has a truth-values equal to 1, l/2 or 0 on any model. As indicated, I leave the details of the proof to the reader.

VI. THE GC)DELIAN LOGICS

There are three types of Godelian logics (GOL logics): the finite n-valued (n > 2) logics, GOL(n); the infinite denumerable-valued logic, GOL(H,); and the indenumerable-valued logic, GOL(H r).

Truth-ualues: Tru(GOL(n)) = {k/n - 110 Q k < n - 11; Tru(GOL(H,,)) = [0, l]#; Tru(GOL(Hr)) = [0, 11. For every GOL logic G, Des(G) = (1).

i’lurh-functions: For every GOL logic G, fu(i, G)(u) = 1, if u = 0; fu(l, G)(u) = 0, if u # 0; fu(A, G)(u, v) = min(u, v); fu(V, G)(u, v) = max(u,v);fu(+,G)(u,v)= l,ifu<v;fu(+,G)(u,v)=v,ifu>v; fu(++, G)(u, v) = 1, if u = v; fu(++, G)(u, v) = min(u, v), if u #v; fu( 3, G)(r) = lub(t); fu(V, G) = glb(t).

PROPOSITION 3: THESIS(2, 1, G) is true for every GOL logic G.

Let G be any GOL logic. Let A be any wff. Let s = (u 1 u E Tru((G) & u > 01. Clearly, (m)(w)(tv((A A 1 A), m, w, G) = 0). Thus, to prove that THESIS(2,1, G) is true it suffices to prove that Exis(Des(G), s, G), Nof(s, G), Univ(s, G), Var(s, G), Equi(s, G), Ab(1, s, G), MP(s, G) and Clo(s, G) are true.

VII. THE STANDARD SEQUENCE LOGICS

There are three different families of Standard Sequence logics (SEQ logics): the R-SEQ logics, the K-SEQ logics, and the C-SEQ logics. In addition, there is a single SEQ logic that is not a member of any of these families: the T-SEQ logic. In each family of SEQ logics S (other than T-SEQ), there are five types of logics: the finite n-valued (n > 2) lo&s, S(n); the infinite denumerable-valued logic, S(He); the infinite denumerable-valued gap logic, S(cC,, 1); the indenumerable-valued logic, S(Nr); and the indenumerable- valued gap logic, S(H r , 1).

280 ROBERTE.MAYDOLE

Zkuth-values: Tru(T-SEQ) = [0, 11. For every SEQ logic S other than T-SEQ, Tru(S(n)) = {k/n - 1 IO <k G n - 11; Tru(S(H,)) = [0, I]#; True%, !I>> = 1% ll#-- (l/2}; T~(S(W) = 10, 11; TNS(b, 4)) = 10, 11 - {l/2). F or every SEQ logic S other than T-SEQ, Des(S) = {u I u E Tru(S) &U > l/2}. Des(T-SEQ) = (1).

Truth-functions: For every SEQ logic S, fu(l, S)(U) = 1 -u; fu(V, S)(u, v) = max(u, v); fu(A, S)(u, v) = min(u, v); fu(3, S)(t) = lub(t); fu(tl, S)(t) = glb(t). F or every R-SEQ logic R, fu(+, R)(u, v) = 1, if u<v;fu(+,R)(u,v)=O,ifu>v;fu(*,R)(u,v)=i,ifu=v; fu(++, R)(u, v) = 0, if u # v. For every K-SEQ logic K, fu(+, K)(u, v) = max(1 -u,v); fu(+t, K)@,v)=min(max(l -u,v),max(l -v,u)). For every C-SEQ logic C, fu(+, C)(u, v) = 1, if u G v; fu(+, C)(U, v) = v, if u>v;fu(~,C)(~,v)=l,ifu=v;fu(*,C)(u,v)=min(u,v),ifu#v. fu(~,T-SEQXu,v)=l,ifu#lorv=l.fu(~,T-SEQ)(u,v)=O,ifu=l and v # 1. fu(*, T-SEQ)(u, v) = min(fu(+, T-SEQ)(u, v), fu (--‘, T-SEQ)(ZJ, u)).

PROPOSITION 4: THESIS(2,1, S) is true for every SEQ logic S.

Since (A)(m)(w)(tv((A A 1 A), m, w, S) $ Des(S)) is true for every SEQ logic S, PROPOSITION 4 can be proved by showing that THESIS( 1, 1, S) is also true for every SEQ logic S. Save for the proof that Exis(Des(S), Des(S), S) is true for every infinite-valued SEQ logic S, the proof that THESIS( 1, 1, S) is true for every SEQ logic S is fairly straight-forward. In proving that Exis(Des(S), Des(S), S) is true if S is T-SEQ or S is an infmite- valued R-SEQ logic, it is important to note that every wff containing occurrences of either + or *has a truth-value equal to 1 or 0 on every model of S. If S is an infinite-valued K-SEQ logic or S is an infinite-valued C-SEQ logic, then in proving that Exis(Des(S), Des(S), S) is true it is important to note that if lub(t) = u and u E Des(S), then (Ek)(k > 0 & (24 - l/k) > l/2).

Readers of Rescher’s Many- VaZued Logic might think that my claim that COM generates a Curry Paradox of Order( 1) in the SEQ logics depends on a somewhat arbitrary choice of the designated truth-values for these logics. Except for T-SEQ, Rescher appears to leave the designations of the SEQ logics ‘open’. But according to the view of first order logics I am supporting

PARADOXESANDMANY-VALUEDSETTHEORY 281

in this paper, a first order logic has a ‘futed’ set of designated truth-values. It might then be asked whether COM would generate a Curry Paradox of some Order in other so-called Standard Sequence logics that could be formulated, say, in accordance with Rescher’s range of ‘open’ designation possibilities.

Let 0 < v < 1. Let R.-SEQ be like R-SEQ except that Des(R) = (tcluETru(R)& u , v > }f or every R,-SEQ logic R. Then both THESIS(2, 1, R) and THESIS( 1 , 1, R) are true for every R.-SEQ logic R. In particular, for v = 1, Des(R) = { 1 > and both THESIS(2, 1, R) and THESIS( 1, 1, R) are true for every Rr-SEQ logic R.

Let 0 < ZI =G 1. Let R’,-SEQ be like R-SEQ except that Des(R) = {uluETru(R)&u> }f v or every R’,-SEQ logic R. Then both THESIS(2, 1, R) and THESIS(1, 1, R) are true for every R’,-SEQ logic R.

Let l/2 <v < 1. Let K,-SEQ be like K-SEQ except that Des(K) = {uluETru(K)&u>v)f or every K,-SEQ logic K. Then THESIS(2, 1, K) is true for every K,-SEQ logic K. In particular, for v = 1, Des(K) = {l}, and THESIS(2, 1, K) is true for every Ki-SEQ logic K.

Let l/2 d v G 1. Let KL-SEQ be like K-SEQ except that Des(K) = (uluETru(K)&u> }f v or every K’,-SEQ logic K. Then THESIS(2,1, K) and THESIS(1, 1, K) are true for every KL-SEQ logic K.

Let KZ-SEQ be like K-SEQ except that Des(K) = {U I u E Tru(K) & II > l/2) for every K2-SEQ logic K. Then THESIS(1, 1, K) is true for every finite-valued and every infinite-valued gap K2-SEQ logic K. Note that K2-SEQ(He, 1) = K-SEQ(&, 4) and K2-SEQ(Kr, f) = K-SEQ(Hr, 4). So THESIS(2, 1, K2-SEQ(&, 1)) and THESIS(2, 1, K2-SEQ(tEr, 1)) are also true. It appears to be the case, however, that for all n > 1, COM does not generate a Curry Paradox of Order(n) in K2-SEQ(&) or K2-SEQ(Hr).

Let 0 < v < 1. Let C,-SEQ be like C-SEQ except that Des(C) = {uluETru(C)&u>v)f or every C,-SEQ logic C. Then THESIS(2, 1, C) is true for every C,-SEQ logic C. In proving this it is valuable to note that (A)(m)(w)(tv(l(A +A), m, w, C) = 0) is true for every C,-SEQ logic C. In particular, for v = 1, Des(C) = (1 }, and THESIS(2, 1, C) is true for every Cr-SEQ logic C.

Let 0 < v < 1. Let C$SEQ be like C-SEQ except that Des(C) = {UlUETru(C)&u>v}f or every Ch-SEQ logic C. Then both THESIS(2, 1, C) and THESIS( 1, 1, C) are true for every Cb-SEQ logic C.


VIII. THE ULK LOGICS (VARIANTS ON THE LUK LOGICS)

There are three types of ULK logics: the finite n-valued (n > 2) logics, ULK(n); the infinite denumerable-valued logic, ULK&); and the indenumerable-valued logic ULK( H r ).

Truth-values: if n is odd, Tru(ULK(n)) = (u I (u = (n - 2k - l)/ n-l@u=(2k+1-~~)~n-l)&k~O&1~2k+l~n};ifniseven, Tru(ULK(n)) = (u I (u = (n -2k)/n8u=(2k-n)/n)&k>O& 0<2k<n};Tru(ULK&,))= [-l,l]#;Tru(ULK(N,))= [-l,l].For every ULK logic U, Des(U) = (u I u E Tru(U) & u > 01.

Truth-jimctions: For every ULK logic U, fu(l, U)(u) = -u; fu(A, U)(u, v) = min(u, e)); fu(V, II)+, v) = max(u, v); fu(+, U)(u, v) = 1, ifu--vO;fu(~,U)(u,v)=O,ifO<u--vl;fu(-*,U)(u,v)=-l,if u - 17 > 1; fu(*, U)(u, v) = min(fu(+, U)(u, v), fu(+, U)(v, u)); fu(3, U)(t) = lub(t); fu(V, U)(t) = glb(t).

PROPOSITION 5: THESIS(2,2, U) is true for every ULK logic U.

Since (A)(m)(w)(tv(i(A +A), m, w, U) = - 1) is true for every ULK logic U, PROPOSITION 5 can be proved by showing that THESIS( 1,2, U) is also true for every ULK logic U. While the proof of the latter is straightforward, the proof that Ab(2, Des(U), U) is true for an (arbitrary) ULK logic U is slightly longer and more complicated than the proofs that the other conjuncts of THESIS( 1,2, U) are true. Note that while Ab(2, Des(U), U) is true for every ULK logic U, Ab (1, Des(U), II) is true only if U is ULK(2) or ULK(3).

As a variation on the ULK logics, consider the C-ULK logics. A C-ULK logic is like an ULK logic except that fu(-+, C)(u, V) = min(- u, V) for every C-ULK logic C. It is easy to show that THESIS( 1, 1, C) and THESIS(2, 1, C) are true for every C-ULK logic C. Note that (A)(m)(w)(tv((A A 1 A), m, w, C) B: Des(C)) is true for every C-ULK logic C.

IX. THE FINITE-VALUED POSTIAN LOGICS

For each positive integer n > 2, there are n - 1 n-valued Postian logics: POS(n,j),l<jGn-1.


Ihrth-values: Tru(POS(n, j)) = (k I 1 < k Gn}. Des(POS(n, j)) = {uIuETru(POS(n,j)&u<j}.

Truth-findions: Let P be POS(n, j). Then fu(l, P)(u) = u + 1, if u # n fu(i, P)(u) = 1, if u = n; fu(V, P)(u, v) = min(u, v); fu(A, P)(u, v) = max(u, v); fu(+, P)(u, v) = v, if u < v and u <j; fu(+, P)(u, v) = max(l,l -u +v),ifu>voru>j;fu(*,P)(u,v)= max(fu(+, P)(u, v), fu(+, P)(v, u)); fu( 3, P)(t) = smin(t); fu(V, P)(t) = smax(t).

PROPOSITION 6: THESIS(2, n - 1, POS(n, j)), 1 <j Q n, is true.

Let P be be POS(n, j). Since (A)(m)(w)(tv(A A 7 A A ii A A i “-l .?YW 7 A, m, w, P) = n) is true, and n B Des(P), PROPOSITION 6 can be proved by showing that THESIS( 1, n - 1, P) is true. Except for the proof that Ab (n - 1, Des(P), P) is true when n > 3, the proof that THESIS( 1, n - 1, P) is true is quite straightforward. In order to prove that Ab(n - 1, Des(P), P) is true when n > 3, the following lemmas are needed:

LEMMA 2: For every k > 1, (tv(A, m, w, P) > j @ tv(A, m, w, P) > tv((A % B), m, w, P)) 1 (tv(A, m, w, P) > j @ tv(A, m, w, P) > tv((A s B), m, w, P)).

LEMMA 3: For every k > 1, (tv(A, m, w, P) > j 8 tv(A, m, w, P) > tv((A s B):m, w, P)) 1 tv((A % B), m, w, P) = max(1, (k + l)(l - tv(A, m, w, P)) + tv(B, m, w, P)).

The proof of LEMMA 2 is easy. It involves the use of a few simple tauto- logies of our two-valued metalogic together with the valuation of fu(+, P),

The proof of LEMMA 3 is slightly complicated. Mathematical induction on the integer over the arrow should be used. The induction step should be divided into two cases: the case where tv(A, m, w, P) < max(1, (k + l)(l - tv(A, m, w, P)) + tv(B, m, w, P)); and the case where tv(A, m, w, P) > max(1, (k + l)(l - tv(.4, m, w, P)) + tv(B, m, w, P)). The details are left to the reader.


X.THE INFINITE-VALUED POSTIAN LOGICS:POS(N,)andPOS(H,)

Truth-values: Tru(POS(H,)) = (1, l/2, l/4, l/8, . . . . (1/2)n, . . . . 01. Tru(POS(N,)) = [0, 11. Des(POS(&)) = Des(POS(Hr)) = (1).

Truth-finctions: Let P be either POS(He) or POS(H,). Then fu(l, P)(u) = u/2, if u # 0; fu(l, P)(u) = 1, if u = 0; the P truth-functions associated with V, A, +, ++, 3, and V are the same, respectively, as those of the LUK logics.

For reasons similar to those offered for believing that COM is consistent in the infinite-valued LUK logics, it appears to be the case that COM is consistent in the infinite-valued Postian logics.

XI. THE 3-VALUED T-SPLIT LOGIC: SPL

lluth-values: Tru(SPL) = (1, l/2, O}. Des(SPL) = (1, l/2). Truth-functions: Let S be SPL. Then fu(i, S)(u) = 0, if u > 0;

fu(i, S)(u) = 1, if u = 0; fu(A, S)(u, v) = min(u, v); fu(V, S)(u, v) = max(u. v); fu(+, S)(u, v) = 0, if u > 0; fu(+, S)(u, v) = 1, if u = 0; fu(++, S)(zf, v) = min(fu(+, S)(u, v), fu(+, S)(v, u)); fu( 3, S)(t) = smax(t); fu(V, S)(t) = smin(t).

PROPOSITION 7: THESIS(2, 1, SPL) is true.

Since (A)(m)(w)(tv(A A 1 A, m, w, SPL) = 0) is true, PROPOSITION 7 can be proved by showing that THESIS( 1, 1, SPL) is true, which is quite straightforward and easy.

XII. SOBOCINSKI’S 3-VALUED LOGIC OF 1952: SOB

Zluth-values: Tru(SOB) = {l, l/2, O}. Des(SOB) = (1, l/2}. Truth-finctions: Let S be SOB. Then fu(i, S)(u) = 1 -u;

fu( 3, S)(t) = 0, if t = (0, l/2}; fu( 3, S)(t) = smax(t), if t # (0, l/2}; fu(V, S)(t) = 1, if t = {l/2, l}; fu(tl, S)(t) = smin(t), if t # {l/2, l};

u 1 1 1 112 112 l/2 0 0 0

il(“, S)(u, v) 1 1 l/2 1 0 1 1 1 l/2 l/2 0 0 1 1 0 l/2 0 0 fu(h, S)(u, v) 1 1 0 1 l/2 0 0 0 0 fu(+, S)(u, Y) 1 0 0 1 l/2 0 1 1 1 fil(*, S)(u, v) 1 0 0 0 l/2 0 0 0 1


PROPOSITION 8: THESIS( 1 , 1, SOB) is true.

Note that THESIS(2, j, SOB) is false for every j > 1. This is because no wff of SOB is uniformly non-designated on every model of SOB. Nevertheless, by virtue of PROPOSITION 8, COM still generates a Curry Paradox of Order( 1) in SOB, and is therefore inconsistent in SOB.

XIII. REICHENBACH’S 3-VALUED QUANTUM LOGIC: QUN

The language of QUN is like the language of the previously described first order logics, but with the following additions: the monadic sentential connectives, 7, -; the binary sentential connectives *, +f, *, and *.

Truth-values: Tru(QUN) = (1, l/2, O}. Des(QUN) = (1). Truth-fin&ions: Let Q be QUN. Then fu(i, Q)(u) = 1 -u;

fu(‘l,Q)(u)=u-1/2,ifu#O;fu(=‘l,Q)(u)= l,ifu=O; fu(-, Q)(u) = l/2, if 11 = 1; fu(-, Q)(u) = 1, if 1( # 1; fu(V, Q)(u, v) = max(u, v); fu(A, Q)(u, v) = min(u, v); fu(-+, Q)(u, v) = min(l,l-n++);fu(*,Q)(u,v)=O,ifu=landv#l; fu(*,Q)(u,v)= 1,ifufl orv= l;fu(*,Q)(u,v)=v,ifu= 1; fu(*, Q)(u, v) = l/2, if u # 1; fu(*, Q)(u, v) = 1 - 1 u -v I; fu(+, Q)(u, v) = 1, if u = v; fu(*, Q)(u, v) = 0, if u f v; fu(@, Q)(u, v) = min(fu(*, QXu, v), fu(*, QXv, ~1); fu( 3, Q)(t) = smax(t); fu(V, Q)(f) = smin(t).

PROPOSITION 9: THESIS(2,2, QUN) is true.

Since (A)(m)(w)(tv(A A 1 A, m, w, QUN) = l/2) is true, PROPOSITION 9 can easily be proved by showing that THESIS( 1,2, QUN) is also true.

There are three forms of QUN equivalence: * (standard equivalence), * (alternative equivalence), and tl, (quasi-equivalence). Might not the Principle of Comprehension be expressed in terms of alternative equivalence or quasi-equivalence, instead of in terms of standard equivalence, as in COM? Consider the following two ways of possibly expressing the Principle of Comprehension:

COMl: (V-z,) . . . (Vz,)(3y)(Vx)(x EY *F(x, ZI, . . . . z,)); COMz: (Vz,) . . . (Vz,)(3y)(Vx)(x EY +++ F(x, zl, . . . . z,));


whereF(x,zr, . . . . z,)isanywffofQUNwithatmostx,z, . . ..~~asfree variables. Whether the Principle of Comprehension is best expressed as COM, COMr or COMz need not concern us here; for COMr and COMz are also inconsistent in QUN.

Let THESIS@, n, QUIU), be like THESIS&, n, QUN), but with * in place of + and * in place of t+. Let THESIS(k, n, QUN), be like THESIS(k, n, QUN), but with * in place of + and 4, in place of ++. Clearly, if either THESIS(l, 1, QUN)r.or THESIS(2, 1, QUN), is true, then COMr is inconsistent in QUN. Similarly, if either THESIS( 1, 1, QUN)2 or THESIS(2, 1, QUN)? is true, then COMa is inconsistent in QUN.

PROPOSITION 10: THESIS(1, 1, QUN), and THESIS(2, 1, QUN), are true.

PROPOSITION 11: THESIS( 1, 1, QUN), and THESIS(2, 1, QUN), are true.

I leave the rather easy proofs of PROPOSITIONS 10 and 11 to the reader.

XIV. THE BOOLEAN W-VALUED LOGIC(S): BOLO

Truth-values: Let IV be a set of possible worlds. Tru(BOL(W)) = {s I s C W>. Des(BOL(W)) = W.

Truth-fin&ions: Let L be BOL(W). Then fu(l, L)(s) = F; fu(v, L)(s, t) = s u t; fu(A, L)(s, t) = s n t; fu(-+, L)(s, t) = F u t;

fu(*,L)(s,t)=(sUf)n(ius);fu(3,L)(t)= Ut;fu(V,L)(t)=flt.

PROPOSITION 12: CURY (1 , 1, BOL(W)) and CURY (2,1, BOL(W)) are true.

Let L be BOLOY). Since tv(A A-I A, m, w, L) = 8, PROPOSITION 12 can be proved by showing that CURY (1, 1, L) is true. In proving the latter the following eleven metatheorems are needed;

Ml: (A)(B)((m)(w)(tv(A + B, m, w, L) = W) 3 (mXw)MW)A m, w, L) = W 1 tv((V)B, m, w, L) = WI).

M2: (A)(m)(w)(tv((Vx)A + A, m, w, L) = W).


M3: t-WN((mW(t44 m,w,L)=W)&II<(A,B,x,y))> (mXw)tW% m, w, L) = WI.

M4: (A)(B)(m)(w)(tv((A c* B) + ((A + B) A (B +A)), m, w, L) = W). M5: (A)(B)(m)(w)(tv((A + (A + B)) + (A + II), m, w, L) = W). 1146: (-NW dt~d(mWt(tv(A + (B AA I), m, w, L> = W &

tv(B +B1, m, w, L) = W) 1 tv(A + (A, AB,), m, w, L) = W). M7: (A)(B)(A,)(m)(w)(tv((A r\B)+A1,m, w, L) = WI

tv((AAB)+(BAA,),m,w,L)= W). M8: (A)(B)(m)(w)(tv(((A + B) A A) + B, m, w, L) = W). M9: (A)(B)(A,)(m)(w)((tv(A + B, m, w, L) = W &

tv(B+A1),m,w,L)~W)3tv(A+A,,m,w,L)=W). MlO: (A)(m)(w)((cl(A) & tv(( 3y)A, m, w, L) = W) 1 tv(A, m, w, L) =

w. Ml 1: (A)(B)(m)(w)((tv(A + B, m, w, L) = W & tv(A, m, w, L) = W) 3

tv(B, m, w, L) = W). PROOF that CURY (1, 1, BOL(W)) is true. Again let L be BOL(W).

Assume(l): tv(COM(A*, l), m, w, L) = W. As an instance of M2 we have (2): (m)(w)(tv((Vx)(x E y * (x E x + A*)) + (xEy+(xEx+A*)),m,w,L)= W).(2)andM3imply(3): tv((Vx)(xEy++(xEx+A*))+(yEy*(yEy+A*)),m,w,i)= W. As an instance of M4 we have (4): tv((y E y ++ (y E y + A*)) + t(Y~y~(y~y~A*))A((y~y-tA*)~y~Y)),m,w,L)=W.As~ instanceofM5 wehave(5): tv((yEy+(yEy+A*))+ (y E y -+ A*), m, w, L) = W. (4), (5) and M6 imply (6): WY EY ++(Y EY -+A*))-‘(((Y EY +A*)+Y EY)A (y EY + A*)), m, w, L) = W. As a instance of M8 we have (7): tv((((y~y-,A*)~y~y)A(y~y~A*))~y~y,m,w,L)= W.(7) and M7 imply (8): tv((((yEy+A*)+yEy)A(yEy+A*))+ ((y EY + A*) Ay E y), m, w, L) = W. As an instance of M8 we have (9): tv(((y E y + A*) Ay E y) + A*, m, w, L) = W. (3) (6), (8) (9) and repeated application of M9 imply (10): tv((Vx)(x EY ++ (x EX -+A*))+A*,m, w, L)= W. (l),(lO)andMl imply(ll): tv(@y)A*,m, w, L)= W. (11) andM10 imply (12): tv(A*, m, w, L) = W. (12) and repeated use of M2 and Ml1 imply (13): tv(A, m, w, L) = W. Since W = Des(L), CURY(l, 1, L) is proved.

Remark: Note that since tv(( 3y)(B A -I B), m, w, L) = 0, steps (l)- (11) of the above proof directly demonstrate the inconsistency of COM in L if (II A -I B) is substituted uniformly for A*.


XV. GLIMPSES BACK AND BEYOND

It has now been shown that COM is inconsistent in a host of different finite and infinite-valued first order logics. That in itself should suffice to dispel the wide-spread and long standing misconception that the paradoxes of NST can be dodged by merely rejecting the Principle of Bivalence or the Law of Excluded Middle. It .is possible of course that certain set-theoretic comprehension principles weaker than COM might prove consistent in those many-valued logics in which COM proves consistent. But what would be the point of investigating the possibility of a many-vahred set theory with comprehension axioms weaker than COM? It could not be for the sole purpose of dodging the paradoxes of NST; for we know of various classical two- valued set theories with comprehension axioms weaker than COM that adequately dodge these paradoxes. There seems to be no point of both weakening the axioms of a theory and mutilating its logic in order to dodge the paradoxes of the theory, if those paradoxes can be dodged by just weakening the axioms. Clearly, a methodological variation of Occam’s Razor is in force here: mutilate your familiar systems only as much as is necessary. True, a many-valued set theory with comprehension axioms weaker than COM might prove very suitable for purposes other than the dodging of set-theoretic paradoxes, such as: the systematic grounding of assertions about so-called fuzzy sets; or the systematic grounding of assertions involving vague predicates; or providing a general mathematical framework for assertions designed to explain physical anomalies. But then a modified version of two-valued set theory might also prove suitable to these other purposes.

Of all the many-valued logics dealt with in the previous sections of this paper only LUK(&), LUK(H& POS(H,), POS(H& K2-SEQ&) and K2-SEQ(H1) appear to hold up under the weight of COM. There are of course sundry many-valued and non-classical logics that have not been dealt with above. Indeed, you could generate many-valued and non-classical logics forever. But in those non-classical and many-valued logics not dealt with above that come readily to mind, COM also proves inconsistent. COM can be shown, for example, to lead to a contradiction in Intuitionistic Quantifi: cation Theory; and in the quantificational extensions of the modal logics Sl-SS, even when the left-most biconditional of COM is construed as strict equivalence. Similarly, COM can be shown to be inconsistent in the first

PARADOXESANDMANY-VALUEDSE;TTHEORY 289

order generalization of the quasi truth-functional Probability Logic. And finally, it is easy to show that if COM is inconsistent in the logics Ll and L, then COM is also inconsistent in the product logic L1 X Lz.

Let us now momentarily focus on LUK(&), POS(H,J, and K2-SEQ(N,). It would seem as though these logics are not suitable for basing a set theory, even if it can be shown that COM is consistent in them. The reason is that there are models of these logics on which certain wffs have undefined truth- values. This is particularly true of certain wffs beginning with an existential or universal quantifier. For example, if it happens that lub(t) B Tru(LUK(tE,)), where t = (g 1 (Ew,)(Eb)(b Ed & vin(w,, m, LUK(H,))&w,(x)=b &g= tv(A, m, wl,LUK(Ho))& fre(A,x, w,wl))}, then tv(@x)A, m, w, LUK(&)), m =(d,h),will be undefined; and similarly for POS(H,-,) and K2-SEQ(H,). And this could happen, because the least upper bound of a set of rational numbers is not always a rational number.

Even if COM is consistent in LUK(H i), POS(H r), and K2-SEQ(H r), what is the significance? What it really comes to is a mere invitation to further pursue the investigation of a many-valued set theory based on these logics. To show that COM is consistent in either LUK(HI), POS(H,), or K2-SEQ(H1) is really only the beginning of the program of developing a viable many-valued set theory, and , perhaps, one of the easier problems to be encountered.

Related to the question of whether COM is consistent in LUK(Nl), POS(H r), and K2-SEQ(H r) is the question of whether the Generalized Principle of Comprehension, G-COM, is consistent in these logics. G-COM is expressed thus: (Vz i) . . . (Vz,)@y)(Vx)(x E y cf F(x, y, z r , . . ., z,)); where F(x, y, z 1, . . ., z,) is any wff with at most x,-v, zl, . . ., z, as free variables. Clearly, if G-COM is consistent in a logic L, then COM is consistent in L.

Some of the other problems that loom large in developing a full-fledged many-valued set theory based on LUK(HI), POS(Hr), or K2-SEQ&), and which are still very open ended are reflected in the questions: (1) How should we axiomatize LUK(H1), POS(HI) and K2-SEQ(&)? (2) Besides COM or G-COM what other set-theoretic axioms should we adopt? (3) How should we construe set-theoretic identity? (4) What truths of classical mathematics can be deduced? (5) What kind of deduction theorems hold? (6) How should the more than two truth-values be interpreted? (7) What


should we take as a standard, intended, or intuitive model of such a theory? (8) What kind of completeness (incompleteness) theorems are provable?

There are those who may be prone to infer a victory for two-valued logic from the results reflected in this paper. You might hear them arguing like this:

After all, three or four truth-values - well that’s one thing; but having to accept an indenumerable number of truth-values in order to save COM (GCOM) - well, that’s something else again. And then there are all the knotty, open ended questions you mentioned in the last paragraph. The choice between saving COM (GCOM) while accepting an indenumerable-valued logic and rejecting COM (G-COM) while accepting two-valued logic is clear: reject COM (G-COM) and stick with two-valued logic.

The sentiments expressed in this argument run deep. But the inference to an out and out victory for two-valued logic is not all that clear. Many-valued set theory is still in the embryonic stage. Let us be careful that we are not condemning something just because it is new and different.

APPENDIX

Listed below are those expressions of our metalanguage which require a brief word of explanation. Other expressions of this metalanguage are either explicitely defined in the text of the paper, clear in context, or commonly used in the literature of logic and mathematics. Sentential connectives, quantifiers, and punctuation marks of our first order object language(s) serve in this metalanguage as names of themselves, with the distinction in usage being clear in context. Note, however, that we do have special sentential connectives and quantifiers in this metalanguage.

1. x,y,z,xl,yl,zl . . . . forindividualvariables. 2. A,B,A1,B1,A2,B2 . . . . forwffs. 3. d,dl,dz . . . . for semantical domains. 4. a,b,c,a,...: for members of semantical domains. 5. m, m r, mp . . .: for models. 6. w, wl, w2 . . . . for v-interpretations. 7. j, k, n, jr . . .: for integers. 8. U, V, ur, zll . . .: for real numbers. 9. s, t, sl, Cl . . . . for sets.

10. [u, V] : the set of real numbers in the closed interval between u and V.


11. [u,v]#: therationalsin [u,v]. 12. smin(t): the minimum of the set t. 13. smax (t): the maximum of the set t. 14. N - - -: not - - -. 15. ---&-:---and-. 16. - - - o -: - - - or -. 17. --- 2-5 if---then-, 18. ---z-: - - - if and only if -. 19. (E , . .) - - -: there is a . . . such that - - -. 20. (...)---: forevery . . . . ---. 21. vin(w, m, L): w is a vinterpretation on the model m of the logic L. 22. tv(A, m, w, L): the truth-value of the wff A of logic L with respect

to, model m and *interpretation w. 23. fr(x, A): x is free in the wff A. 24. fre(A,x, w, wr): (y)((fr(y,A) &- (y isx)) > wr(y) = w(y)). 25. cl(A): wff A does not contain any free variables. 26. lk(A, B, x, y): wff B is like wff A except for containing free occur-

rences of either x or y wherever A contains free occurrences of x.

Davidson College Davidson, North Carolina

BIBLIOGRAPHY

Chang, Chen Chung: ‘The Axiom of Comprehension in Infinite Valued Logic’, M&he- matica Scandinavica 13 (1963), 9-30.

Chang, Chen Chung: ‘Infinite Valued Logic as a Basis for Set Theory’, Logic, Methodo- logy and Philosophy of Science, ed. by Yehoshua Bar-Hillel, North-HoLland Pub. Co, Amsterdam, 1965, pp. 93-100.

Maydole, Robert E.: Many-Valued Logic as a Basis for Set Theory; Ph.D. Thesis, Boston University, Boston, Mass., 1972.

Rescher, Nicholas: Many-Valued Logic, McGraw Hill Pub. Co., New York, 1969.

Paradoxes and many-valued set theory

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