COLIN McLARTY

ANTI-FOUNDATION AND SELF-REFERENCE'

ABSTRACT. This note argues against'Barwise and Etchemendy's claim that theirsemantics for self~reference requires use of Aczel's anti-foundational set theory, AFA,and that any alternative "would involve us in complexities of considerable magnitude,ones irrelevant to the task at hand" (The Liar, p. 35).

Switching from ZF to AFA neither adds nor precludes any isomorphism types ofsets. So it makes no difference to ordinary mathematics. I argue against the author'sclaim that a certain kind of 'naturalness' nevertheless makes AFA preferable to ZF fortheir purposes. I cast their semantics in a natural, isomorphism invariant form withself-reference as a fixed point property for propositional operators. Independent of theparticulars of any set theory, this form is somewhat simpler than theirs and easier toadapt to other theories of self-reference.

Barwise and Etchemedy give an elegant semantics for self-reference inThe Liar1 They also claim this semantics requires a new set theory,Aczel's anti-foundational set theory, AFA, and that any alternative"would involve us in complexities of considerable magnitude, onesirrelevant to the task at hand" (p. 35). Gupta has agreed, saying themain obstacle to semantics of self-reference has been "the difficulty ofconstructing, within the confines of standard mathematics (that is, settheory), a natural account of circular propositions" (parenthesis isGupta's).' This note will show the foundations of set theory havenothing to do with it.

I do not object to AFA. In particular I do not say 2ormelo­Fraenkel set theory, ZF, has any "clearer and more compelling" amotivation than Aczel has given for AFA' To me, that claim merelyexpresses familiarity with ZF. I do say the difference between AFAand ZF makes no difference where normal working methods inmathematics are concerned, and those methods suffice as well forBarwise and Etchemendy's semantics.

First I point out that extending a ZF universe of sets to an AFAuniverse adds no new isomorphism types of sets, so it adds no struc­tures at all np to isomorphism.' I briefly argue on general grounds

Journal of Philosophical Logic 22: 19-28, 1993.© 1993 Kluwer Academic,Publishers. Printed in the Netherlands.

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against Barwise and Etchemendy's claim that a certain considerationof 'naturalness' in semantics nevertheless makes AFA preferable toZF for their purposes. Then I give a natural form of Barwise andEtchemendy's semantics which can be defined in any set theory. Areader who only wants to see the semantics can go directly to Section 3.

1. AFA AND ISOMORPHISM

The theory AFA includes all the ZFaxioms except well-founding sothe well-founded sets of any model of AFA form a model of ZF, thefamiliar inner model of the well-founding axiom. Every ordinal iswell-founded. Since AFA includes the axiom of choice every AFA setis isomorphic to an ordinal and thus to a set in the model of ZF. Soevery structure existing in an AFA universe is isomorphic to one in aZF universe. Conversely, Aczel's relative consistency proof showsevery ZF model embeds into an AFA model so every structure in aZF model exists in an AFA model (see The Liar, Ch. 3). Passing fromZF to AFA neither adds nor precludes any isomorphism types of setsor of other structures, and so neither adds nor removes any theoremsstated in isomorphism invariant form.

This is not the sense in which the authors admit "AFA does notgive rise to any new mathematical structures" (p. 47). They mean anystatement about AFA sets can be interpreted as a statement aboutequivalence classes of graphs in ZF. Via this interpretation the theo­rems on sets in AFA appear as theorems on equivalence classes ofgraphs in ZF. The authors justly compare this to interpreting state­ments about the complex numbers as statements about pairs of realnumbers: It obscures the natural sense of the structure (AFA sets, orcomplex numbers, as the case may be). If that were the only sense inwhich AFA gave no new structures I would agree with them on usingAFA. My point is that so long as you keep to isomorphism invariantstatements, as is usual in mathematics, the theorems and proofs ofAFA already are theorems and proofs of ZF and vice versa.

2. 'NATURALNESS' IN SEMANTICS

Barwise and Etchemendy know every non-well-founded set is isomor­phic to a well-founded One but they claim something more than

ANTI-FOUNDATION AND SELF-REFERENCE 21

description up to isomorphism is needed for semantics. They say "Byfar the most natural way to model a proposition about a given objectis to use some set-theoretic construct containing that object (or itsrepresentative) as a constituent, that is, where the object appears inthe construct's hereditary membership relation" (p. 34). So theyrequire that the relation "x is an object that the proposition y isabout" must be modeled not by just any set theoretic relation withthe right structure, but specifically by some fragment of the hereditarymembership relation. Of course this is not isomorphism invariant.Isomorphisms of sets do not preserve membership. And ZF can notmodel a self-referential proposition 'naturally' in this sense since nowell-founded set is in its own hereditary membership relation, i.e., inits own transitive closure.

The general set-theoretic point of this note is to show that theauthor's concern with membership is an unnecessary complication. Ifsuch concern with the specific members of sets was ever current inmathematics, as perhaps under logicist influence, it was chased out byBourbaki. Bourbaki realized that what matters in mathematics is thestructural relations between sets, not the particulars of their members,and so he describes structures axiomatically and thus up to isomor­phism without specifying the members. Axiomatic descriptions needconsistency proofs, and for that Bourbaki uses models in a set theoryrather like ZF and says, in principle, exactly what are the elements ofeach set in the model. But the particular model is not only glossedover after that, it is entirely irrelevant to proofs from the axioms, andthere is no use worrying whether it happened to interpret some givenrelation in the axioms by membership.

Of course the authors know the difference between a theory and amodel and they prefer to work with a model. Barwise explains hispreference saying "set-theoretic modelling has proven so extra­ordinarily powerful a method in logic".' But in ordinary practice thepowerful uses of set theoretic models are not ones where you takea single model as the object of study, much less ones where youworry about the foundational details of constructing the model. Theyare ones where you construct and compare various models in respectof their model-theoretic"structure, which is invariant under isomor­phism of models. From this point of view we want not one model of

22 COLIN McLARTY

Russellian (or Austinian) propositions but a general theory of models.The account of algebras of prepropositions below is such a theory,with Barwise-Etchemendy algebras as the isomorphism invariant des­cription of the one model the authors give.

I will point out that Lawvere furthered Bourbakiste struCturalismby axiomatizing set theory itself so that sets are, quite rigorously,described only up to isomorphism. There is no membership relationbetween sets in his elementary theory of the category of sets, ETCS6 Ithought in terms of ETCS in writing this, but I will not spell it out orargue for it. The point is that the semantics described here can beformalized in the routine way in ZF, AFA, ETCS or any reasonableset theory.

3. SEMANTICS OF SELF-REFERENCE

I treat Barwise and Etchemendy's Russellian theory of propositionsbut the technique applies as well to Austinian propositions. I assumefamiliarity with The Liar.

Our version of the semantics makes self-reference a fixed pointproperty. For example, we define [FaJ as a function taking proposi­tions to propositions. Then a liar proposition is a fixed point for[FaJ, a solution to p = [Fap]. Barwise and Etchemendy's concernthat self-reference leads to set-theoretic problems is completelyimplausible when you look at it this way. No one expects a real num­ber x to be in the transitive closure of its sine, sin (x). No one sees aset theoretic problem in the equation 0 = sin (0). So we do notexpect p to be in the transitive closure of [Fap] nor see any settheoretic problem in self-referential propositions.

Define an algebra ofprepropositions to be a class PrePROP with:

I. Elements [aH c] and [aH c] where a is Claire or Max and cisa card.

2. A function LBelJ from {Claire, Max} x PrePROP to Pre­PROP and a similar function LBelJ

3. Functions [Tr J and [Tr J both from PrePROP to PrePROP.We use [FaJ to abbreviate [Tr J.

4. Functions [/\ Xl and [v X] taking subsets X of PrePROP tomembers of PrePROP.

ANTI-FOUNDATION AND SELF-REFERENCE 23

The definition so far is not equivalent to Barwise and Etchemendy's.We can build an algebra of prepropositions by starting with the pre­propositions of clause I and inductively applying clauses 2-4, addingnew members to PrePROP for each new application of a function toarguments. Barwise and Etchemendy would say we had "built Pre­PROP up from below, by a standard inductive characterization"(p. 63). We can do this in any reasonable set theory, such as ZF orAFA or ETCS, to get an algebra with no circular reference. In thisalgebra a preproposition can only refer to prepropositions formedbefore it was. Call this the free algebra.

We could take the free algebra and add a new member fto Pre­PROP, specify [Fail = fthen build up a new algebra by inductivelyapplying clauses 2-4. This algebra has the one liar f and preprop­ositions built from it but no other circular reference among pre­propositions.

Another variant takes the free algebra and adds two new membersfandf', specifying [Fail = fand [Faf'] = f'. Then build up a newPrePROP by inductively ilPplying clauses 2-4. This gives an algebrawith two distinct liars, f andf'. There can not be distinct liars onBarwise and Etchemendy's approach. Their Theorem;j (p. 72) provesthere is a unique solution p to the equation p = [Fap].

Someone might claim the Russellian account in The Liar is basic­ally cOITect, except that each new utterance of "This proposition isfalse" states a new proposition: My indexical 'this' can not refer tothe same proposition as uttered by any of innumerable other peopleon occasions unknown to me. We can formalize that position as astep in considering the argument, using semantics in an algebra ofprepropositions with infinitely many distinct liars. For Barwise andEtchemendy to formalize the position in a 'natural' way by their stan­dard, they would first need to invent another set theory.'

We expand on this last claim since it is central to the author's"most natural way to model a proposition" and to their argumentagainst using ZF for their semantics. There are many ways to code

, propositions, as Barwise and Etchemendy stress several times. Anynumber of sets p could solve the equation p = [Fap] depending onthe coding. But once a coding is chosen we have a unique equationp = [Fap]. The author's argument against ZF is that no 'natural'

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coding gives this equation any solutions in ZF. To get one solutionwith a 'natural' coding in their sense they use AFA, but then there isprovably only one. The proof of Barwise and Etchemendy's Theorem3 (p. 70) would be unaltered by supposing a proposition shouldinclude some "representation of its context or its speaker or whatever.As long as "This proposition is false." translates into a specific settheoretic equation of the kind used in the author's solution lemma ithas a unique solution in AFA. If you want more than one solutionyou must use some other set theory - or abandon the authors'sallegedly natural link between the representation of propositions andset theoretic membership.

We get Barwise and Etchemendy's particular semantics in our formby applying the idea of anti-foundation directly to prepropositions.Define the immediate constituents of a preproposition as follows. Theimmediate constituents of[aHc] are: a, H, and c. Those of[aHc] are:Not, a, H, and c. Those of[aBelp] are:.a, Bel, and p. For [aBelp]add Not. The immediate constituents of [Tr p] and its negation areobvious. For [A X] they are A and the prepropositions in X. Similarlyfor [v X]. A preproposition in this simple language is uniquely deter­ntined by its immediate constituents. Define an isomorphism ofalgebras as an isomorphism between the classes which preserves thedistinguished elements and the immediate constituent relation.

Define a tagged graph as in The Liar (p. 39) and define a preprop­ositionally tagged graph as a tagged graph in which every node is ofone of these kinds:

I. A card node is a node with three children, each childless andtagged by: one of Claire or Max, H, and a card. A negatedcard node is the same but with the addition of a childlesschild tagged by Not.

2. A belief node is a node with three children, one with childrenof its own, one tagged by Bel and one tagged by Claire orMax. A negated belief node has an additional childless childtagged by Not. J

3. A truth node is a node with one child with children andone childless child tagged by Tr. Negated truth nodes areobvious.

I '

ANTI-FOUNDATION AND SELF-REFERENCE 25

4. A conjunction node has any set of children with children, andone childless child tagged by /\. For a disjunction node use v

instead of /\.

A decoration in an algebra PrePROP for a tagged graph is a func­tion D defined on the nodes of the graph such that for each node xwith no children D(x) = tag (x), whereas if x has children then D(x)is the element of PrePROP whose immediate constituents are all andonly the D(y) for y a child of x.

An algebra of prepropositions PrePROP is called a Barwise­Etchernendy algebra, or B-E algebra, if and only if: Every preprop­ositionally tagged graph can be decorated in PrePROP in exactly oneway, and every member of PrePROP occurs in SOme decoration. Itis easy to see the class of prepropositions defined by Barwise andEtchemendy, with the obvious definition of functions LBelJ and[Tr J and so on, is a B-E algebra in AFA.

Take the reasoning Aczel used to show AFA is consistent if ZF is.(See The Liar, Ch. 3.) A simplification of this proves the B-E algebraaxioms are consistent if ZF is. We take Claire, Max, H, the fifty twocards, Bel, Not, Tr, /\, and v as urelements. Wherever he says agraph represents a set we say it represents a preproposition. Where hesays one AFA set is a member of another we say one prepropositionis an immediate constituent of another. We look not at all taggedgraphs but only at prepropositionally tagged ones, and we verify notthe AFA axioms but the B-E algebra axioms. This simplifies a goodbit as the B-E algebra axioms are much simpler than those for AFA.This proof, like Aczel's relative consistency proof for AFA, can begiven in any set theory strong enough to describe the same graphs asZF.

The proof of the solution lemma for AFA sets has an analogue forprepropositions in a B-E algebra. Given any prepropositional algebraPrePROP and a set of indeterminates X define PrePROP[X] byextending clause I in the definition of a prepropositional algebra toinclude members of X along with the elements [aH c] and [a He],and apply the other clauses inductively on this new base. In effectPrePROP[X] consists of PrePROP plus expressions like those in Pre­PROP but with indeterminates in place of some prepropositions.

26 COLIN McLARTY

Define a prepropositional equation in X to be an expression of theform

x = a

where x E X and a E PrePROP[X]. The analogy to The Liar, p. 49,should be clear. Our solution lemma says that if PrePROP is a B-Ealgebra then for every set of indetenninates X every set of preprop­ositional equations in X has a unique solution. (The reader may provethe converse as an exercise: If every such family has a unique solutionthen PrePROP is a B-E algebra.)

The plausibility argument that Barwise and Etchemendy give fortheir solution lemma works just as well for ours. The only differenceis that we have several ways to form prepropositions out of otherswhile they have only set formation. Each equation in a family gives agraph, and the graphs are then to be hooked up with one another.Aczel's formal proof of the lemma for AFA sets also yields a prooffor prepropositions just by substituting prepropositions for sets,immediate constituents for members, and prepropositionally taggedgraphs for all tagged graphs.

The same change of wording turns any proof that there is a uniqueclass meeting Barwise and Etchemendy's definition of PrePROP into aproof that all B-E algebras are isomorphic. We follow the authors insaying the result "follows easily from the general considerationsdiscussed in Chapter 3", (p. 62). Since we apply those considerationsto the constituent relation in any B-E algebra they are sound inany set theory, while the authors apply them only to the membershiprelation in AFA.

To prove a B-E algebra contains a particular prepropositioIJ, suchas a uniquefsuch thatf = [Fafl, we can use decorations of taggedgraphs. Specifically, f decorates the one untagged node in the taggedgraph consisting of: one childless node tagged by Not, one childlesstagged by Tr and one more node with each of those and itself aschildren. This parallels the author's use of tagged graphs to show par­ticular sets exist in AFA (p. 4Iff). Or we could use the solutionlemma, interpreted as applying to prepropositions in a B-E algebra asabove, the same way that they do.

ANTI-FOUNDATION AND SELF-REFERENCE 27

All the rest of Chapters 4-7 of The Liar can be stated for any B-Ealgebra and all the results proved in essentially the same way thatBarwise and Etchemendy do. The only difference is that we applytheir theorems and proofs not to membership in AFA but to the con­stituent relation in any B-E algebra.

I stress that the proofs for B-E algebras differ from those for AFAvery little and are often simpler. We do not take Barwise and Etch­emendy's proofs and eliminate non-well-founded sets by substitutinglengthy considerations of equivalence classes of decorated graphs. Wedo not eliminate non-well-founded sets at all. We simply bypass issuesof set membership by dropping the demand that representatives ofconstituents of a preproposition be in the transitive closure of the rep­resentative of the proposition. The constituent relation between pre­propositions in a B-E algebra, which is after all the real point ofinterest for the semantics, is not well-founded no matter what settheory we use.

4. CONCLUSION

The point is not to oppose variant set theories. It is that immediateconstituents of propositions are not especially similar to members ofsets. Even if a semantics and a set theory can be paired so the consti­tuent relation and the membership relation have similar formal proper­ties, as Barwise and Etchemendy have done, the formal parallel addsnothing to the semantics.

Barwise and Etchemendy themselves insist most details of the repre­sentation of propositions are irrelevant:

We use [a He] for that set theoretic object which represents the proposition that a hasC, ... Exactly how these are represented by sets doesn't really matter, and burdeningour definitions with the inessential detail would just obscure matters. We will assume asa general feature of our coding, though, that the objects referred to in naming the settheoretic representative are members of its transitive closure. Thus we assume that a isin the transitive closure of [a He]. (p. 62)

Their coding is defined up to 'isomorphism plus a condition on tran­sitive closures', which latter condition serves no purpose but to pre­clude solving p = [Fap] and other equations for circular reference inZF' It is a mere distraction. The authors show the graph-theoretic

28 COLIN McLARTY

ideas Aczel used to motivate AFA are valuable in analyzing circularreference. These graphs may serve other ends besides, but so do lotsof structures and we would do as well not to worry about enshriningthem all in the membership relations of set theories. B-E algebrasapply the graph theory directly to semantics.

NOTES

* I thank John Mayberry for discussions of axiomatics which inspired this paper. JonBarwise's criticism of an earlier draft improved this one as did an anonymous referee.I J. Barwise and J. Etchemendy, The Liar: An Essay on Truth and Circularity (NewYork: Oxford University Press, 1987). Hereafter page numbers after quotes of Barwiseand Etchemendy refer to this book.2 "Jon Barwise and John Etchemendy's The Liar", Philosophy of Science 56 (1989):697-709. The quote is on p. 697.3 L. Moss takes this position in his review of The Liar, in Bull. Amer. Math. Soc. 20,no. 2 (1989): 216-225. The quote is from p. 222.4 When I say two sets are isomorphic I mean there is an invertible function betweenthem, that is a one-one unto function. When I say two structures are isomorphic Imean there is an invertible functio'n between their underlying sets, preserving the oper­ations and relations of the structure, whose inverse also preserves them.5 J. Barwise, The Situation in Logic, CSLI Lecture Notes no. 17 (Stanford: CSLI. 1989)p. 188.6 See F. W. Lawvere "An Elementary Theory of the Category of Sets", Proc. Nat.Acad. Sci. USA. 52 (1964): 1506-1511 or W. S. Hatcher, The Logical Foundations ofMathematics (New York: Pergamon Press, 1982).7 In fact Aczel Non-Well-Founded Sets, CSLI Lecture Notes No. 14 (Stanford: CSLI.1988), gives a number of variants on AFA. Each has a corresponding 'natural' variantaccount of self-reference in Barwise and Etchemendy's sense. You could look to seewhether one of these variants is philosophically useful but that would be the hard wayto do semantics. All these variants are algebras of prepropositions and we have amuch simpler and more flexible classification theorem for such algebras than weever will for the set theories. Algebras of prepropositions are models of an infinitelypresented infinitary algebraic theory (treat 1\ and v as classes of finitary and infinitaryfunctions), so they are precisely the quotients of free algebras on some number ofgenerators.8 Gupta misses the importance of the clause on transitive closures when he seeks toagree with the authors by saying all that matters in a model is "the kind of informationcoded by the modeling" (op. cit., p. 699, n. 4). B-E algebras are only one of manycodings for the same information which violate the condition on transitive closures anddefeat the argument against ZF.

Department of Philosophy,Case Western Reserve University,Cleveland, OH 44106, U.S.A.