SYMPOSIUM

The Structural Collapse Approach Reconsidered

Ignacio Ojea

Universidad de Buenos aires - CONICET
ignacioojea@gmail.com

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Abstract

I will argue that Roy Cook's (forthcoming) reformulation of Yablo's Paradox in the infinitary system D is a genuinely non-circular paradox, but for different reasons than the ones he sustained. In fact, the first part of the job will be to show that his argument regarding the absence of fixed points in the construction is insufficient to prove the noncircularity of it; at much it proves its non-self referentiality. The second is to reconsider the structural collapse approach Cook rejects, and argue that a correct understanding of it leads us to the claim that the infinitary paradox is actually non-circular.

KEY WORDS: Paradox; Circularity; Yablo; Structural collapse.

Resumen

En este trabajo argumentaré que la reformulación que Roy Cook (forthcoming) hace de la paradoja de yablo en el sistema infinitario D es una genuina paradoja no circular, pero por motivos distintos a los defendidos por ese autor. La primera parte del trabajo consiste en mostrar que la ausencia de puntos fijos en la construcción es insuficiente para demostrar su no circularidad, a lo sumo prueba su no autorreferencialidad. La segunda parte consiste en volver a considerar el enfoque del colapso estructural que Cook rechaza, y argumentar que una correcta comprensión del mismo revela que la paradoja es genuinamente no circular.

PALABRAS CLAVE: Paradoja; Circularidad; Yablo; Colapso estructural.

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1. Fixed points

As previously mentioned, Cook (forthcoming) presents an infinitary system D for LP including infinitary rules of derivations, and shows that the (infinitary) Yablo's Paradox presented in Barrio's previous contribution leads to contradictions. furthermore, he proves THEOREM 2.4.3, where he claims that the absence of weak fixed points is enough to show the non-circularity of the construction. I believe this is false.
Consider the set of sentence's names {S₁ , S₂ , S₃ } and the following denotation function d:

d(S₁) = F(S₂).
d(S₂) = F(S₃) .
d(S₃) = F(S₁).

It is clear that the pair <{S₁ , S₂ , S₃ }, d> is closed and it is easy to check that we can derive a contradiction from those statements (in δD, the same for Yablo's paradox). Furthermore, it seems natural to claim that there is a circularity involved in {S₁ , S₂ , S₃}. after all, S₁ claims the falsity of S₂ , S₂ claims the falsity of S₃ and S₃ claims the falsity of S₁ . Nevertheless, neither S₁ , S₂ nor S₃ are weak fixed points of <{S₁ , S₂ , S₃ }, d>, at least with Cook's definitions for D. let us start with the notion of fixed point uner consideration:

Φ is a weak sentential fixed point of <{S_β}_β∈B , d> if and only if there is an a ∈ B, a formula Ψ and statement name Sγ occurring in Ψ; such that:

d(S_a ) = Φ, and both
Φ ⇒ Ψ[S_γ /S_a ] and
Ψ[S_γ /S_a ] ⇒ Φ are theorems in D.

Since D is sound over the assignment semantics previously presented, to prove that there are no fixed points in {S₁ , S₂ , S₃} I only need to give countermodels to the conditionals in the definition. This is in fact the very same strategy Cook uses to show THEOREM 2.4.3. I will only do it for S₁ , the proofs for S₂ and S₃ are completely analogous.
Suppose d(S₁) is a weak fixed point. Given that d(S₁ ) = F (S₂ ), the formation rules for the extended LP language used in the deductive system guarantee that Ψ[S_γ , S₁] must take one of two forms, T (S₁) or (Φ₁ ∧ Φ₂ ∧...F(S₁ )... ∧ Φ_n ∧ Φ_n+1), where T (x) works like a truth predicate. So we can only have:

a. F (S₂ ) ⇔ T (S₁), or
b. F (S₂ ) ⇔ (Φ₁ ∧ Φ₂ ∧ ... F (S₁ )... ∧ Φ_n ∧ Φ_n+1).

But now consider the following denotation function d and acceptable assignments σ_a and σ_b over it:

d'(S₁ ) = F(S₂) ∧ F(S₃), d'(S₂) = F(S₁) ∧ F(S₃) and d'(S₃) = F(S₂) ∧ F(S₁).
σa(S₁) = f , σ_a(S₂) = f and σ_a(S₃) = t.
σb(S₁) = t, σ_b(S₂) = f and σ_b(S₃) = f .

It is now easy to check that σ_a falsifies the a. biconditional, and σ_b falsifies b. biconditional. Hence neither biconditional is a theorem of D. The same can be proved for S₂ and S₃.

In conclusion, the absence of weak fixed points can be -at much- evidence for the non self- referentiality of the construction, but not for its non circularity.

2. The structural collapse approach reconsidered

There is, nevertheless, a way to argue in favor of the noncircularity of Cook's construction: by using the structural collapse approach. I will avoid presenting it because it was fully explained previously. Cook offers two central arguments against it. The first one states, in a nutshell, that "the analogy -between statements and sets- can be, at most, suggestive (...) the identity of the characteristic sets does not imply the identity of the statements of which they are characteristic". Hence any information we might get by comparing characteristic sets may not be accurate regarding the statements of which they are characteristic. The second one runs like this: "even were the structural collapse account successful, it is not clear that it would provide what its defender presumably desires: an argument that the yablo paradox involves circularity of the sort that can be blamed for the paradoxes. in other words, even if the structural collapse account entails the circularity of the Yablo paradox, it does not entail that the yablo paradox suffers from a sort of circularity which can be blamed for the paradoxes".
I believe both arguments assume that the structural collapse approach fails in mapping sets of statements into (non-well-founded) sets in a way that the referential structure of the statements is isomorphic to the membership relation of the sets. To put it the other way around: if all the referential -hence semantic- properties of each (set of) statement(s) have a counterpart in membership properties of the characteristic set(s) and vice versa, and the characteristic set(s) of some (set of) statement(s) is clearly circular -under some understanding of this notion-, then it must be the case that the (set of) statement(s) is circular; this is exactly what the isomorphism guarantees. So the "stronger" the isomorphism is -in the sense that preserves all the referential and semantic properties over the mapping- the stronger the analogy is. in particular, if we can ensure that properties such as paradoxicality have some kind of counterpart in the non-well-founded sets, and the characteristic set of Yablo's paradox is both circular and paradoxical in that sense, then "the Yablo paradox suffers from a sort of circularity which can be blamed for paradox". But the reciprocal will also be the case: if the mapping genuinely preserves all semantic properties and the characteristic set of Yablo's paradox is both paradoxical and non-circular; then we have found a noncircular paradox. I will argue this is the case.
Let us start considering the mapping operation. It is a composition of two operations, a mapping from sentences to direct graphs and a mapping from graphs to (possibly non-well-founded) sets using some set theory. The connection between graph theory and Lp is straightforward. Let D_d (S_i) be the set of all sentence names that appear on the d-denotation of S_i . So for example if A is a set of indexes and d(S_i) = Λ{F (S_a) : a ∈ A}, then D_d (S_i) = {S_a : a ∈ A}. A pair with set of sentence names and a denotation function, < {S_β }_β∈B , d >, is closed iff for any a, β, if β ∈ B and S_a∈ D_d (S_β), then a ∈ B. All pairs considered here will be closed in this sense.
An assignment is a function σ : C → {t, f} and it is acceptable over a denotation function d if and only if for every β ∈ B: σ(S_β) = t iff, for all S_a∈ D_d (Sβ), σ(S_a) = f. Given a set of indexes B and <{S_β }_β∈B , d> closed, let Dep_d({S_β }_β∈B) = {<S_a , S_γ>: a, γ ∈ B; Sγ ∈ D_d (S_a)} be the overall dependence relation over all the sentences' names considered. Then the pair:

<{S_β }_β∈B , Dep_δ ({S_β }_β∈B)>

is a serial directed graph. it is also easy to check that given some sentence name S_a in the language, the subgraph of <{S_β }_β∈B , Depd ({S_β }_β∈B)> that starts with Sα is an accessible pointed directed serial graph.
Furthermore, we can represent assignments of truth values to sentences in Lp relative to a denotation function d with coloring of graphs. A coloring of a graph is nothing more than an assignment of a color to each node in the graph. We shall consider only two colors, turquoise and fuchsia, and we shall impose the following conditions on acceptable colorings:

Given a serial, directed graph , a coloring of <N, E > is acceptable if and only if, for any n ∈ N : n is colored turquoise if and only if, for any node m such that <n, m>∈ E, m is colored fuchsia.

Any assignment σ that is acceptable on <{S_β }_β∈B , δ> induces a coloring on the graph <{S_β}_β∈B , Dep_d({S_β}_β∈B > (and vice versa) in which (a) S_β is colored turquoise if and only if σ(S_β ) = t, and (b) S_β is colored fuchsia if and only if σ(S_β) = f .
So there is a one to one correspondence between sets of sentences in Lp and graphs that preserves semantic properties. If a set of sentences is paradoxical, it does not admit an acceptable assignment; so its associate graph does not admit an acceptable coloring. In this sense, we are able to characterize a graph as paradoxical whenever it does not admit an acceptable coloring.

Cook observes there are mainly four ways of mapping graphs into non-well founded sets. In order to take a closer look to them, some definitions are required. Say U is the universe of non-well-founded sets and <N, E, p> is an accessible pointed directed graph -APG- (where N is the set of nodes, E ⊆ N² and p is the origin of the graph), then:

a function f : N → U is a decoration of the APG <n, e, p> iff for any n₁ , n₂ ∈ N : f (n₁) ∈ f (n₂) iff 1 , n₂>∈E.
A decoration f on an APG <N, E, p> is a picture of S ∈ U iff f (p) = S.
A decoration f on an APG is exact iff, for any distinct n₁, n₂ ∈ n , f (n₁) ≠ f (n₂).
An APG <N, E, p> is an exact picture of S iff there is a decoration f such that (a) f is a picture of S, and (b) f is exact.

Each set can have at most one exact picture (up to isomorphism), and each graph can be the exact picture of at most one set. further, all four candidates for anti-foundation axiom agree that any APG is a picture of a set. Where there they disagree is in terms of which graphs provide exact pictures of sets. The strongest candidate for an anti-foundation axioms (strongest in the sense that it provides the weakest criterion for a graph providing an exact picture, and thus allows for the greatest variety of distinct non-well-founded sets) is the Boffa anti-foundation axiom (or BAFA) BAFA states, in essence, that every extensional APG has is an exact picture, where extensional is defined as follows:

An APG <N, E, p> is extensional iff, for any n₁ , n₂ ∈ N : If for all m ∈ N , < n₁ , m> ∈ E iff <n₂ , m> ∈ E, then n₁ = n₂ .

In this case, the 'Liar' set Ω and 'Yablo's' under Cook's infinitary reformulation are distinct sets, since they correspond to distinct extensional graphs. Thus, if BAFA is the correct, or best, account of nonwell-founded set theory, then the structural collapse account of circularity does not entail the circularity of the Yablo paradox.
The second anti-foundation axiom is FAFA (whose addition to ZFC-foundation results in Finzler-Aczel set theory). FAFA states that an APG <N, E, p> must not only be extensional, but must also be isomorphism-extensional, if it is to be an exact picture of a set:

Given an accessible pointed graph <N, E, p>, the sub-APG induced by m (where m ∈ N) is <N*, E*, m> where (a) N* = {q : there is a path from m to q} ∪ {m}, and (b) E* = E ∩ (N* x N*).

An accessible pointed directed graph < N, E, p > is isomorphism-extensional iff, for any n₁, n₂ ∈ N , if the sub-APG induced by n₁ is isomorphic to the sub-APG induced by n₂ , then n₁ = n₂ .

Intuitively, an APG is an exact picture of a set if no two sub-APGs are isomorphic. In particular, if we look at the APG induced by the infinitary version of Yablo's paradox, the subgraphs induced by each of the nodes are isomorphic. So that graph is not an exact picture of a set. On the other hand, the graph is a picture of the liar set, since we can decorate each node with the liar set Ω. As a result, in Finzler-Aczel set theory the characteristic set(s) corresponding to the Yablo paradox are not distinct from the characteristic set corresponding to the Liar paradox.
The two other theories, known as SAFA and AFA -for Scott and Aczel-, will not be treated here, since we have the following result. Letting GΦ be the class of APGs that are exact pictures of sets relative to a particular (non-well-founded) set theory whose anti-foundation axiom is Φ:

G_{AF A} ⊆ G_{SAF A} ⊆ G_{FAF A} ⊆ G_{BAF A}

Hence, the APG corresponding to Yablo's Paradox will not be an exact picture neither in AFA nor SAFA; and in both theories it will be a picture of the Liar set Ω.
In conclusion, it seems that in order to decide whether the paradox is circular we need to choose between non-well-founded set theories, since AFA, SAFA and FAFA regard it as structurally equivalent to the liar and BAFA does not. But we do have a reason for choosing BAFA: it is the only theory that preserves semantic properties such as paradoxicality.
Consider the following sentences in Lp :

d(S₁) = F (S₂)
d(S₂) = F (S₁)

It is easy to define an acceptable assignment for {S₁ , S₂} over d. We could use σ_a(S₁) = t and σ_a(S₂) = f or σ_b(S₁) = f and σb(S₂) = t. Hence that set of sentences is not paradoxical, admits acceptable assignments. analogously, we could construct two acceptable colorings for the associate graph of S₁ (and S₂), so the graph is also non-paradoxical.
Let us look now what happens with the different non-well-founded set theories. Note that the associate graph of S₁ is extensional: no two distinct nodes have the same children. Then in BAFA it is an exact picture of set distinct from the characteristic set of the liar. This is what we need, otherwise GAFA would identify a non-paradoxical sentence with a paradoxical one.
On the contrary, in FAFA fails at this point. In the graph of S₁, the sub-APG induced by S₁ is isomorphic to the sub-APG induced by S₂, but S₁ ≠ S₂. Hence the graph is not an exact picture of a set. But it can be decorated with the liar's set Ω. As a result, in Finzler-Aczel set theory the characteristic set corresponding to S₁ is not distinct from the characteristic set corresponding to the liar paradox. Using the previously mentioned fact that G_AFA ⊆ G_SAFA ⊆ G_FAFA ⊆ G_BAFA, we note that the graph of S₁ is not an exact picture in AFA and SAFA. And for the same reason as before, the characteristic set of S₁ is also Ω in those theories. In conclusion, FAFA, SAFA and AFA do not preserve the semantic properties of the graphs (sentences); since they identify as structurally isomorphic a paradoxical sentence and a non-paradoxical sentence.
It may be argued that we do not need to preserve paradoxicality in order to guarantee that the mapping preserves circularity. That is true, but I believe it would undermine the whole structural collapse approach. Circularity is not problematic per se, it is only problematic when it leads to inconsistency. What the structural collapse must guarantee is for as to have some way to identify the sort of circularity that leads to paradox. No set theory that dissipates the difference between sentences that admit acceptable assignments and sentences will be useful in doing so.
So if my point is conceded, we should take BAFA to be the adequate theory for the structural approach account. Then the characteristic set of Yablo's paradox is distinct from the characteristic set of the Liar. Now: have we shown that Cook's reconstruction of Yablo's paradox is not circular? Certainly not. At this point we can only claim that it lacks of some well known sort of circularity, the one present in the Liar. But we did not show that is non-circular simpliciter. That task would require a characterization of the notion of circularity for non-wellfounded sets, and it is beyond the scope of the present work. Nevertheless, some rough idea can be shortly presented.
Let us say that an APG <N, E, p> is circular whenever there is at least one node n ∈ N such that there is a finite path from n to n. And let us say a non-well-founded set is circular whenever its exact picture is circular. Analogously, a (set of) sentence(s) is circular if its characteristic set(s) is circular. Of course, this intuitive idea of circularity may be problematic in hard cases, but I take it to be simple and clear enough for the present purposes. Under the present definition, Yablo's paradox is not circular at all: in the graph associated with that set of sentences there is no node connected to itself by a finite path.

References

1. Barwise, J. and Etchemendy, J. (1987), The Liar: An Essay on Truth and Circularity, Oxford, Oxford University Press.         [ Links ]

2. Cook, R. (2004), "Patterns of Paradox", Journal of Symbolic Logic, 69, pp. 767-774.         [ Links ]

3. Cook, R. (2006), "There are non-circular Paradoxes (But Yablos' isn't one of Them)", The Monist, 89, pp. 118-149.         [ Links ]

4. Cook, R. (forthcoming), The Yablo Paradox: An Essay on Circularity, Oxford, Oxford University Press.         [ Links ]

5. Yablo, S. (1982), "Grounding, dependence, and paradox", Journal of Philoshical Logic, 11, pp. 117-137.         [ Links ]

6. Yablo, S. (1993), "Paradox without self-reference", Analysis, 53 (4), pp. 251- 252.         [ Links ]

7. Yablo, S. (2004), "Circularity and paradox", in Bolander, H., Hendricks, V. and Pedersen, S. A. (eds.) (2006), Self-Reference, Stanford, CSLI Publications, pp. 165-183.         [ Links ]