Reference and Disquotation

Reference and Disquotation ∗ Lavinia Picollo Work in progress

Abstract This investigation is a realization of Horwich’s minimalist project: the search for an encompassing criterion of selection of instances of the T-schema that could be employed in the construction of consistent axiomatic theories of disquotational truth. It is usually thought that pathological expressions can be identified by their underlying reference patterns. Given the lack of sensible notions of reference for formal languages in the literature, this thesis hasn’t been tested properly so far. I provide a sound and precise notion of reference for sentences of the language of arithmetic expanded with a truth predicate and employ it in the definition of several reference patterns. Since I take a proof-theoretic approach, these reference patterns are arithmetically definable, though not recursive. I use them to formulate three axiomatic theories of disquotational truth, which I show to be ω-consistent and proof-theoretically strong.

1 Introduction Deflationism is likely the most popular view on truth nowadays. It’s usually committed to the following two thesis: first, that the meaning of the truth predicate is exhausted by the T-schema, i.e. by all instances of T pϕq ↔ ϕ

(T-schema)

where ϕ is a sentence of the language and pϕq a quotational name for it; second, that the truth predicate is just a logico-linguistic device that exists in the language solely to allow us to express certain things, usually generalizations, we simply cannot express otherwise. Unfortunately, if the language is capable of self-reference and the underlying logic is classical, the T-schema leads to paradox. For we can formulate a (strengthened) liar sentence λ, that says ∗ This

is a cut of chapter 4 of my phd thesis, which is written in Spanish.

1

of itself that it’s not true, satisfying λ ↔ ¬T pλq,

(1)

which obviously contradicts the T-biconditional for λ. While most deflationists opt to abandon classical logic, others, like Horwich (1998), prefer to stay within its limits and restrict the T-schema to consistent instances. On the one hand, it would be desirable to include as many instances as possible. On the other, the criterion of selection we employ must be ‘simple’ enough. Though it isn’t obvious what is exactly meant by this, it’s clear that the criterion cannot be given in terms of complex notions that turn the truth predicate into a substantial property, something more than a logico-linguistic operator. Moreover, since the main goal of deflationism is to have a truth predicate that can be used to express and infer, like any other logical operator, the criterion must be of a low complexity, so it can be employed in the formulation of axiomatic theories. The search for a simple and encompassing policy for admissible T-biconditionals is known as minimalism. Many deflationists, following Tarski (1933), only admitted instances of the T-schema given by sentences not containing the truth predicate. Although this a very simple criterion, it’s too restrictive. Many innocuous instances are excluded, e.g. the one given by T p0 = 0q. Ideally, we find the cause of paradox, a simple property shared by all and only those expressions whose instances of the T-schema have unwanted consequences. An obvious candidate for a criterion is, precisely, consistency: paradoxes are those from which a contradiction can be derived. Unfortunately, maximality approaches fail badly. McGee (1992) has shown that there is not one but many different maximal consistent sets of T-biconditionals, all of which are highly complex, i.e. not arithmetically definable. For any sentence independent from the underlying base theory can be decided in either way by a certain T-biconditional. Moreover, each set makes arbitrary choices with respect to sentences containing T . Consider a liar cycle of length 2, this is, two sentences λ1 and λ2 such that the first one says of the second that it’s true, while the latter says of the former that it’s untrue: λ1 ↔ T pλ2q

(2)

λ2 ↔ ¬T pλ1q A contradiction can be obtained from the T-biconditionals for λ1 and λ2 taken together, but not separately. Thus, every maximal consistent set of T-biconditionals must contain exactly one of these, but there’s no way of deciding which, and none of the two options seems reasonable. Finding a sensible, encompassing and simple restriction on instances of the T-schema isn’t an easy task. Perhaps the criterion that fares best so far is Halbach (2009)’s T -positiveness: only sentences in which the truth predicate occurs positively are allowed in the T-schema. This is a

2

recursive restriction that results in an ω-consistent and powerful system when formulated over Peano arithmetic, called putb. putb can define the truth predicates of the Ramified Theory of Truth up to the ordinal 0 rt 1, Yx is untrue.

(Y1 )

... For all x > n, Yx is untrue.

(Yn )

...

Assume one of them, (Yn ), is true. Then, all the ones below must be untrue, i.e. (Yn+1 ) is untrue and so are the ones that come after (Yn+1 ). But this is precisely what (Yn+1 ) says, (Yn+1 ) is true after all. Since we started by assuming that an arbitrary sentence on the list was true and we reached a contradiction, we must conclude that none of them is. In that case, no sentence below (Y0 ) is true, so (Y0 ) it true, for that is what it says. Contradiction. Given the relevance of a non-self-referential paradox of truth, after its first appearance, Yablo’s list was carefully studied. First, it was formalized in LT , which seems the most natural choice. Then, three questions emerged: whether the existence of the list can be proved in pat as the liar’s; if so, whether a contradiction can be obtained from the sequence together with intuitive truth principles; and, third, whether there is really no trace of self-reference in the sentences that form the sequence. The latter started a specious debate, later truncated by Leitgeb (2002)’s remarks on the lack of a formally correct and materially adequate notion of reference to evaluate Yablo’s case as well as others, and on the hard obstacles any such notion must overcome. Next we introduce a formal version of Yablo’s paradox in LT , show how it’s existence can be obtained via diagonalization in pat, and make some short remarks on the paradoxicality of the list. Afterwards, in 3.2, we present the debate on the referential status of the list and its members. This will allow us to extract important conclusions on how a good notion of reference meant to evaluate semantic paradoxes and reference patterns underlying expressions involving truth should look like.

10

3.1 Formalization and paradoxicality of Yablo’s sequence The following is a natural formalization of Yablo’s list in LT : υ(0) = p∀x > 0¬T υ(x)q

(7)

υ(¯ 1) = p∀x > ¯1¬T υ(x)q ... υ(¯ n) = p∀x > n ¯ ¬T υ(x)q ... where υ(x) is a term of the language. To prove the existence of such term we can apply the strong diagonal lemma to ∀y > x¬T s. (z, y, ˙ pxq) over the variable z. This delivers a term t such that pat ` t = p∀y > x¬T s. (t, y, ˙ pxq)q Applying the function s. (y, x, ˙ pxq) to each side of the identity symbol, we get pat ` s. (t, x, ˙ pxq) = s. (p∀y > x¬T s. (t, y, ˙ pxq)q, x, ˙ pxq) Let υ(v) := s. (t, v, ˙ pxq). Then, pat ` υ(x) = p∀y > x¬T ˙ υ(y)q

(8)

or, equivalently, pat proves the Uniform Fixed-Point Yablo Principle:4 ∀x υ(x) = p∀y > x¬T ˙ υ(y)q

(UFPYP)

from which every sentence on the list follows as an instance. Yablo’s informal argument to a contradiction can be formalized as well: 1. 2. 3. 4. 5. 6. 7. 8. 9. 4 Following

T υ(n) ∀y > n¬T υ(y) ¬T υ(n + 1) ∀y > n + 1¬T υ(y) T υ(n + 1) ⊥ ¬T υ(n) ∀y¬T υ(y) ∀y > 0¬T υ(y)

assumption 1 2 2 4 3, 5 1-6 7 8

Ketland (2005)’s terminology.

11

10. 11.

T υ(0) ¬T υ(0)

9 8

It’s unclear what the justification of steps 2 and 8 is, though. For it’s not clear whether n is playing the role of a variable or the numeral of a given number. If the latter is the case, we’re not looking at a derivation but a derivation-scheme, with an instance for each n ∈ ω. Step 2 would then be given by (7) together with the Local Yablo Disquotation Principle: T p∀y > n ¯ ¬T υ(y)q ↔ ∀y > n ¯ ¬T υ(y)

(LYDP)

for each n ∈ ω, the T-schema restricted to the Yablo sentences. But step 8 is certainly not allowed, unless we worked with infinitary rules. Indeed, as Hardy (1995) and Ketland (2005) point out, pat + LYDP is consistent but ω-inconsistent. On the other hand, if n were a variable in the derivation, step 2 couldn’t be justified by (7) and (LYDP). Stronger principles are needed: the (UFPYP) and the Uniform Yablo Disquotation Principle: ∀tT p∀y > t. ¬T υ(y)q ↔ ∀y > t◦ ¬T υ(y)

(UYDP)

Then a contradiction follows, but the Yablo sentences are not responsible for it. It wasn’t them but the (UFPYP), a principle that Yablo didn’t even mention. Many authors concluded that Yablo’s isn’t a paradox but an ω-paradox. For the purposes of giving an ω-consistent theory of truth, this is bad enough. Yablo’s list is as problematic as the liar.

3.2 Self-reference in Yablo’s sequence Priest (1997) was the first to doubt Yablo’s counterexample to the orthodox view on paradoxes. He claimed that sentences like the liar in (1) are self-referential because they are fixed points of some predicate. Since Yablo’s list concerns a predicate υ(x) that is a fixed point of ∀y > x¬T s. (z, y, ˙ pxq), it must involve some kind of self-reference too.5 As Leitgeb (2002) pointed out, this isn’t what Yablo had in mind when he claimed his sentences are not self-referential, but the range of the quantifiers that occur in them—which range only over the sentences below in the list. There are at least two different notions of self-reference at play: a rather intuitive one, Yablo’s, and a rather technical one, Priest’s. The ‘intuitive’ notion of self-reference is based on an intuitive notion of reference, according to which a sentence can refer to an objet (that could be a sentence) in two ways: by mentioning 5 For

a discussion specifically over this point, see Sorensen (1998) and Beall (2001).

12

it or by quantifying over it. For instance, a sentence of the form T t refers to whatever t denotes. If t = p0 = 0q, T t isn’t self-referential. But if t = pT tq, like a truth-teller sentence, T t refers to itself. In general, a sentence ϕ refers to an object o by mention if and only if ϕ contains a term that denotes o. Also, sentences like ∀v(ϕ(v) → T v) intuitively refer to the ϕs. In general, sentences of the form ∀v(ϕ(v) → ψ(v)) refer by quantification to whatever satisfies the predicate ϕ. Therefore, according to this notion, the n-th sentence in Yablo’s list refers to all expressions whose code is greater than n. Since under a monotone coding it’s very likely that the n-th sentence’s code is greater than n, Yablo’s list might turn out to be self-referential. However, if we reformulate the list as follows: υ(¯ n) = p∀y∀z(y = υ(z) ∧ z > n ¯ → ¬T yq,

(9)

(which is also possible via diagonalization) each sentence refers via quantification just to sentences that come after it in the list. We might also want to consider the possibility of indirect reference. The sentences that form the liar 2-cycle in (2) are not directly but indirectly self-referential. Thus, reference simpliciter should be closed under transitivity. Even taking the transitive closure of reference by quantification, Yablo’s reformulated sentences do not refer to themselves. On the other hand, Priest’s ‘technical’ notion of self-reference is based on the presence of fixed points. It’s usually said that the diagonal lemmata are sources of self-reference, as in G¨odel’s first incompleteness result and Tarski’s undefinability of truth theorem. Accordingly, an LT -formula ϕ is self-referential if it’s equivalent in pat to a formula ψ containing a term that denotes ϕ, i.e. pat ` ϕ ↔ ψ(pϕq)

(10)

For instance, λ in (1), λs in (3) and κs in (4) are self-referential. By (8), Yablo’s predicate ∀y > x¬T ˙ υ(y) contains a term υ denoting ∀y > x¬T ˙ υ(y), so it’s self-referential. Since Yablo’s sentences are given in terms of υ, they do involve some kind of self-reference, contradicting Yablo’s claims. The intuitive and the technical notions seem to be irreconcilable. Nonetheless, as Leitgeb notices, they shouldn’t be taken so seriously for they are highly deficient. Regarding the intuitive notion, it’s incomplete: nothing has been said about the reference of quantified expressions that are not of the form ∀v(ϕ(v) → ψ(v)). According to Leitgeb, this should be done in such a way that what he calls equivalence condition is satisfied, i.e. that logically equivalent expressions should refer to the same objects. However, in that case, the notion would become trivial, both by mention and by quantification: every sentence ϕ is logically equivalent to ϕ ∧ (T t → T t), for every term t, and every expression of the form ∀v(ϕ(v) → ψ(v)) is logically equivalent to

13

∀v((χ(v) → χ(v)) → (ϕ(v) → ψ(v))), for every formula χ(v). With respect to the technical notion of self-reference, it already satisfies the equivalence condition, but it is also already trivial: take t in ϕ ∧ (T t → T t) to be a name for ϕ. It is not easy to avoid this problem. Requiring that ϕ is obtained by a diagonalization process could be an option, if a sound notion of diagonalization process is given first.6 Demanding that identities instead of equivalences are proved, this is, that the strong rather than the weak diagonal lemma is employed, would mean that languages without enough function symbols are not capable of self-reference.7 But this would give us a non-self-referential paradox for free, for λ in (1) can be obtained by weak diagonalization, and a contradiction derived from it along with the corresponding instance of the T-schema. Therefore, Leitgeb’s poses a challenge: to find a notion of self-reference that is formally correct and materially adequate, and—this might follow from material adequacy—satisfies the equivalence condition. Urbaniak (2009) was the only one so far who tried to meet the challenge, with a notion of aboutness based on the idea that terms and predicates must occur informatively in a sentence to be a source of reference. E.g. in ϕ ∧ (T t → T t) t does not occur informatively, so ϕ would not be about the denotation of t despite being logically equivalent to ϕ ∧ (T t → T t). Although the notion represents an improvement with respect to the few other concepts already available in the literature,8 Urbaniak’s aboutness fails to capture the most basic intuitions. The logical form of sentences is neglected. Sentences are about the objects they informatively mention or fall under the primitive predicates they informatively mention, e.g. T p0 = 0q is about 0 = 0 but also about whatever is in the extension of T (so under the most reasonable interpretations of LT T p0 = 0q is self-referential), and a Yablo sentence would be self-referential if it falls under T ’s extension. Also, sentences of the form ∀v(¬P (v) → Q(v)), where P and Q are primitive predicates, are about the P s and the Qs, but not about the non-P s. Thus, if ∀v(¬P (v) → ¬T (v)) is the only non-P , ∀v(¬P (v) → ¬T (v)) is a non-self-referential liar sentence. Finally, although the equivalence condition is satisfied, Urbaniak himself recognizes that his notion cannot account for the self-referentiality of all sentences obtained via weak diagonalization, for they are given not just by logical but arithmetical equivalence. Before becoming skeptical about finding sound and precise notions of self-reference and reference, let us cast some doubts on the legitimacy of the equivalence condition. Leitgeb gives three arguments in its favor. First, he argues that logically equivalent sentences are extensionally equivalent in every logically possible world, and thus indistinguishable from a logical view point. Second, that reference is a semantic notion, for it’s defined in terms of the denotation of terms, which is an obviously semantic relation. And, thirdly, that, if the equivalence condition 6 This

is the goal of a future project. and Visser (2014a,b) seem to favor this decision. 8 Cf. Ryle (1933), Putnam (1958), Goodman (1961). 7 Halbach

14

isn’t satisfied and, moreover, if two pat-equivalent sentences do not refer to the same objects, “[. . . ] no philosopher may any longer argue in the following way: “By G¨odel’s diagonalization lemma, we know that there is a sentence ϕ such that ϕ is equivalent to ‘¬T (pϕq)’ in arithmetic. Thus there is a self-referential sentence, that is, ϕ”.” (Leitgeb, 2002, p. 9) With respect to his first argument, logic cannot distinguish between two logically equivalent sentences because it doesn’t talk about them, it just uses them. As soon as sentences become the objects of discourse and their syntactic properties are considered (such as their logical form), they become distinguishable, for they have different syntactic properties.9 Like truth, reference isn’t a purely logical notion, it’s at most a logico-linguistic concept. As a consequence, denotation of terms isn’t the only concept at play in a notion of reference or aboutness. Since syntax matters, reference isn’t a purely semantic notion either, as Leitgeb wants. Regarding Leitgeb’s third argument, it’s not clear that every two pat-equivalent sentences must refer to the same objects for the diagonalization lemma to be a means of self-reference. A closer look at the sentences the weak diagonalization delivers shows that they are of the form ∀v2 (Diag(p∀v2 (Diag(v1 , v2 ) → ϕ(v2 , ~v ))q, v2 ) → ϕ(v2 , ~v )), and they satisfy their own antecedent. Thus, they turn out to be self-referential not by the equivalence condition but by one of the requisites already imposed on the intuitive notion. In what follows, we provide two different notions of reference. First, we briefly sketch a concept for sentences of Lpa that corresponds to the intuitive notion, and will be useful to assess the referential status of diagonal expressions. Secondly, we develop a notion especially designed to study the reference patterns that underlie semantic paradoxes in LT , a concept of alethic reference. It will become clear that it’s not possible to merge both concepts in one. None of the concepts we introduce satisfy the equivalence condition; they are intensional.

4 Reference in arithmetic In this section we focus on the reference of sentences of Lpa to other sentences of the same language via G¨ odel coding. Considering other languages, formulae instead of sentences, or reference to numbers, the primary objects of Lpa , are the objects of another paper. The goal here is to provide a definition of reference that completes and articulates the intuitive notion of reference at play in the debate on the referential status of Yablo’s paradox. We first introduce definitions of reference by mention and by quantification, and consider a few examples. We show that these two notions are enough to account for strong and weak 9 This

argument can already be found in Urbaniak (2009).

15

diagonalization as devices for self-reference. Then we define direct and indirect reference, and set some examples regarding the transitivity of reference. In the third subsection we give notions of self-reference and well-foundedness in terms of indirect reference, and study the reference patterns underlying G¨ odel sentences, cycles and chains. We close this section with a proof of the undefinablility of reference.

4.1 Reference by mention The following definition covers the cases of direct reference by mention. Definition 4.1 (Direct m-reference) Let ϕ, ψ ∈ Lpa . ϕ directly m-refers to ψ iff ϕ contains a term t such that N  t = pψq. Definition 4.1 is enough to regard as self-referential every expression obtained by the strong diagonalization lemma. Let ¬Bewpa (g) be pa’s strong G¨odel sentence, i.e. pa ` g = p¬Bewpa (g)q,

(11)

obtained by the strong diagonalization lemma. Since ¬Bewpa (g) contains a term g that denotes ¬Bewpa (g), it m-refers to itself. However, ¬Bewpa (g) m-refers to other sentences too. The proof of the strong diagonal lemma shows that g is a complex term that contains, as a subterm, a numeral, pϕ(s. (v1 , v1 , pv1q), ~v )q. Thus, g also contains as subterms the numerals of all numbers whose code is smaller than ϕ(s. (v1 , v1 , pv1q), ~v )’s. Therefore, ¬Bewpa (g) m-refers to all sentences whose code is smaller than ϕ(s. (v1 , v1 , pv1q), ~v )’s. Ignoring subterms for reference isn’t feasible. ¬Bewpa (v) isn’t a primitive predicate of Lpa but a complex formulae in which v might always occur as part or some complex term, like Sv, or ¬. v. In that case, g would be itself a subterm, so ¬Bewpa (g) wouldn’t be selfreferential after all. The overgeneration of definition 4.1 is an inevitable consequence of the lack of individual constants in Lpa for the objects in the domain of the standard model.

4.2 Reference by quantification Regarding reference by quantification, as Leitgeb (2002) pointed out, we’d like that sentences of the form ∀v(ϕ(v) → ψ(v)) referred to the ϕs. However, this condition doesn’t always exhaust the reference the quantifiers provide. Consider the sentence ∀x(x < g → ¬Bewpa (¬. x))

16

(12)

This expression can be understood as saying that the negation of every sentence whose code is greater than pa’s G¨ odel sentence’s is not provable in pa. Thus, it seems to refer, not only to the sentences satisfying the predicate x < g, but also to their negations. Consider this other example: ∀x(x = g → ∀y(y = ¬. x → ϕ(y)))

(13)

This expression seems to say that the negation of pa’s G¨odel sentence is a ϕ; it doesn’t only refer to ¬Bewpa (g) but also to ¬¬Bewpa (g). The simplest solution in both cases is to establish, recursively, that sentences of the form ∀v(ϕ → ψ) refer by quantification both to sentences satisfying ϕ and to sentences to which (ϕ → ψ)[¯ n/v] refers, where n ∈ ω satisfies ϕ. If the variable of the quantifier isn’t free in the antecedent or the consequent, the clause applies all the same: ∀x(SentLpa (g) → ϕ) would refer to everything, while ∀x(SentLpa (px = xq) → ϕ) to nothing by quantification. We still have to make decisions with respect to sentences given by quantifiers followed by atomic or negated formulae (recall ¬, → and ∀ are the only logical operators in the language). Since ∀v(ϕ → ψ) intuitively refers to the ϕs (and other things), ∀vψ, where ψ isn’t itself of conditional form, must refer to everything, for the universal quantifier occurring in it isn’t relativized to any formula. The same applies if v isn’t free in ψ. The only way to restrict reference via quantification is by a conditional expression. If the language contained existential quantifiers as well as conjunctions, restricted reference could be extended to formulae of the form ∃v(ϕ ∧ ψ) to the ϕs, in harmonious way. Extending these ideas to sentences of the form ∀v1 . . . vn (ϕ → ψ), it seems we should say that they refer to the tuples of sentences satisfying ϕ. However, since we want to keep reference as a relation between sentences, we disassemble these tuples and take reference to be a relation to their components. Thus, ∀v1 . . . vn (ϕ → ψ) will refer to all sentences χ such that, for some 1 ≤ i ≤ n, χ satisfies ∃v1 . . . vi−1 vi+1 vn ϕ, and as before to all sentences to which ∀v1 . . . vi−1 vi+1 vn (ϕ → ¯ i ] refers, for some 1 ≤ i ≤ n and k ∈ ω. Finally, ∀v1 . . . vn ϕ, where ϕ is an atomic or ψ)[k/v negated formula, refers to everything. Definition 4.2 (Direct q-reference) If ϕ, ψ ∈ Lpa , ϕ directly q-refers to ψ if and only if one of the following conditions is satisfied: 1. ϕ := ∀v1 . . . vn χ and (a) χ is an atomic or a negated expression, or (b) χ := δ → γ and, for some 1 ≤ i ≤ n, N  ∃v1 . . . vi−1 vi+1 . . . vn δ[pψq/vi ]

17

or exists a k ∈ ω such that ¯ i] N  ∃v1 . . . vi−1 vi+1 . . . vn δ[k/v ¯ i ] directly q-refers to ψ or contains an occurrence and ∀v1 . . . vi−1 vi+1 . . . vn (δ → γ)[k/v of a term t that doesn’t occur in δ → γ such that N  t = pψq. 2. ϕ := ¬χ and χ directly q-refers to ψ. 3. ϕ := χ → δ and either χ or δ directly q-refer to ψ. To the end of clause 1.(b) we must specify that t should be a ‘new’ term that doesn’t occur already in δ → γ for otherwise cases of m-reference would be mistaken as cases of q-reference. Definition 4.2 is enough to account for the self-referential status of sentences that obtain from weak diagonalization. As we mentioned before, those sentences are of the form ∀v2 (Diag(p∀v2 (Diag(v1 , v2 ) → ϕ(v2 , ~v ))q, v2 ) → ϕ(v2 , ~v )), and satisfy the predicate Diag(p∀v2 (Diag(v1 , v2 ) → ϕ(v2 , ~v ))q, v2 ), whatever ϕ is. Thus, e.g. G¨ odel sentences obtained this way are self-referential. Contrary to Leitgeb (2002)’s beliefs, it isn’t the equivalence between a sentence ϕ and ψ(pϕq) what makes ϕ self-referential but the special way that equivalence has been obtained. Thus, definitions 4.1 and 4.2 taken together can accommodate Priest (1997)’s ‘technical’ notion of self-reference.

4.3 Direct and indirect reference Definition 4.3 (Direct reference) If ϕ, ψ ∈ Lpa , ϕ refers directly to ψ if and only if ϕ directly m-refers to ψ or directly q-refers to ψ. As Leitgeb pointed out, sentences can also refer to others indirectly. By proposition 2.4, we can prove that there are cycles of sentences in pa, such as this 3-G¨odel cycle: g1 = pBewpa (g2 )q

(14)

g2 = pBewpa (g3 )q g3 = p¬Bewpa (g1 )q where the first two sentences say of the one immediately below that it’s provable in pa and the third one says of the first one that it isn’t provable. Actually, (14) can be used to prove G¨odel’s first incompleteness theorem as well as (11), pa’s traditional G¨odel sentence. Thus, we’d like to say these sentences refer to themselves. 18

In the same fashion, proposition 2.5 allows us to formulae chains of sentences like g0 = p¬Bewpa (g1 )q

(15)

g1 = p¬Bewpa (g2 )q g2 = p¬Bewpa (g3 )q ... for which we’d like to be able to say that each sentence in the sequence refers indirectly to all those that come below. Chains of q-reference are also possible. Via the strong diagonal lemma, we can formulate Yablo-like sequences in pa, such as t(¯ n) = p∀w∀u > n ¯ (w = t(u) ˙ → ¬Bewpa (w))q,

(16)

with which G¨ odel’s first incompleteness result can also be proved.10 Definition 4.4 (Chains of direct reference) A sequence of sentences χ0 , . . . , χn , . . . of Lpa of length α is a chain of direct reference if and only if, for each ordinal β < α, χβ directly refers to χβ+1 . Definition 4.5 (Reference) If ϕ, ψ ∈ Lpa , ϕ refers to ψ if and only if exists a finite chain of direct reference χ0 , . . . , χn such that χ0 is ϕ and χn is ψ. This definition covers cases of indirect reference, such as (14), (15) and (16). However, transitivity isn’t always a desirable feature for sentences of Lpa . Consider the following: SentLpa (pU (t, s, p0 = 0q)q)

(17)

This expression says that t, that consists of a string of quantifiers over the variables that are the arguments of s followed by 0 = 0, is a sentence. It directly m-refers to U (t, s, p0 = 0q), which in turn directly m-refers to 0 = 0. By definition 4.5, we must conclude that (17) also refers to 0 = 0. This might be a bit counterintuitive, for (17) says something about the syntax of U (t, s, p0 = 0q) that, contrary to truth or theoremhood predications, cannot translate into an assertion about the objects the latter refers to, i.e. 0 = 0. Distinguishing between predicates for which the transitivity of reference is plausible and those for which it isn’t is not an easy task. But one can always consider indirect reference just when it’s convenient. 10 See

Cie´sli´ nski and Urbaniak (2013).

19

4.4 Self-reference and well-foundedness The last two definitions allow us to introduce the following concepts. Definition 4.6 (Self-reference) If ϕ ∈ Lpa , ϕ is self-referential if and only if it refers to itself.

Definition 4.7 (Well-foundedness) If ϕ ∈ Lpa , ϕ is well-founded if and only if there is no infinite chain of direct reference starting with ϕ. Thus, self-referential sentences as ¬Bewpa (g) and the ones that form the 3-G¨odel cycle are non-well-founded, as well as the expressions that are part of the chains in (15) and (16). Sadly, given how terms in proposition 2.5 are obtained, every sentence in a chain is self-referential. Each tn is such that tn = pϕ(tn+1 )q = pϕ(g. (num(pϕ(tn )q)))q. Thus, ϕ(tn+1 ) (= ϕ(g. (num(pϕ(tn )q)))) directly m-refers to ϕ(tn ), which in turn directly m-refers to ϕ(tn+1 ). Something similar affects the Yablo-like sequences like (16). The code of each sentence ∀w∀u > n ¯ (w = t(u) ˙ → ϕ(w)) (= t(¯ n)) in the sequence is greater than n, and each sentence directly q-refers to every sentence whose code is greater than n, by clause 1.(b) of definition 4.2. For t(u) ˙ represents a function ˙ ¯ ¯ and, therefore, for each k > n we have that N  ∃w(k > n ¯ ∧ w = t(k)) and, a fortiori, N  ∃w(t(¯ n) > n ¯ ∧ w = t. (num(t(¯ n)))). Again, we can blame this overgeneration on the austerity of Lpa . Clearly, the concept of well-foundedness we introduced in definition 4.7 has little in common with the notion of grounding of Kripke (1975) or Leitgeb (2005). According to the latter, every sentence in Lpa is grounded because it expresses a non-semantical fact, even if it’s possible to have an infinite chain of direct reference, like in the case of (15).

4.5 Undefinability of reference To know whether a sentence of the form ∀v(ϕ → ψ) directly q-refers to another sentence χ, we first need to know if ϕ(pχq) is true in N. Direct q-reference is a semantic notion and cannot be defined in Lpa . If it could, it would satisfy the following principle for every sentence ϕ ∈ Lpa , where Rqd (v1 , v2 ) ∈ Lpa expresses direct q-reference: ∀x(Rqd (p∀v1 . . . vn (ϕ → ψ)q, x) ↔ SentLpa (x) ∧ n _

(18)

(∃v1 . . . vi−1 vi+1 . . . vn ϕ[x/vi ] ∨ ∃v1 . . . vn (ϕ ∧ Rqd (p∀v1 . . . vi−1 vi+1 . . . vn (ϕ → ψ)[v˙ i /vi ]q, x)))).

i=1

20

Proposition 4.8 (Undefinability of direct q-reference) If L ⊇ Lpa contains a predicate Rqd and th ⊇ pa is formulated in L, th + (18) is inconsistent. Proof Applying the strong diagonalization lemma to ¬Rqd (∀. pyq(x→ . pyq= . pyq), p0 = 0q), we get a term t such that th ` t = p¬Rqd (∀. pyq(t→ . pyq= . pyq), p0 = 0q)q

(19)

i.e. a sentence that says of itself that it isn’t satisfied by 0 = 0, some kind of liar sentence. Instantiating (18) with ∀y(¬Rqd (∀. pyq(t→ . pyq= . pyq), p0 = 0q) → y = y), since the antecedent contains no free variables, we get in th + (18) that d Rqd (p∀y(¬Rqd (∀. pyq(t→ . pyq= . pyq), p0 = 0q) → y = y)q, x) ↔ ¬Rq (∀. pyq(t→ . pyq= . pyq), p0 = 0q) d Rqd (p∀y(¬Rqd (∀. pyq(t→ . pyq= . pyq), p0 = 0q) → y = y)q, p0 = 0q) ↔ ¬Rq (∀. pyq(t→ . pyq= . pyq), p0 = 0q) d Rqd (∀. pyq(t→ . pyq= . pyq), p0 = 0q) ↔ ¬Rq (∀. pyq(t→ . pyq= . pyq), p0 = 0q)

a

by (19).

Therefore, the notions of reference we introduced in this section, though intuitively compelling, are far too complex to be part of an axiomatic theory. We cannot extend them to LT hoping to find a simple restriction on the T-schema we can use to provide disquotational theories of truth. In the next section I introduce analogous notions for sentences of the language of LT , replacing the semantic concepts in the definition of direct q-reference by proof-theoretic ones.

5 Alethic reference In this section we provide a notion of reference designed to study the reference patterns underlying semantic paradoxes and other pathological expressions. We call it alethic reference. Since the final goal is to use these patterns to restrict the T-schema in an axiomatic way, the notion we provide is proof-theoretic. Instead of talking about Lpa ’s standard model as in the previous section, we turn to provability in a theory. The notion of alethic reference cannot be obtained just by replacing semantic predicates with proof-theoretic ones in the definitions in the last section, and extending them to LT , for two reasons. First, because alethic reference is meant as a bridge to truth. We don’t know anything about T yet, we have neither a standard model for LT in mind, nor an axiomatic system beyond pat. Thus, clauses like 1.(b) in definition 4.2 make little sense for sentences 21

of the form ∀v(T v → ϕ), for we would need to know whether a sentence is (provably) in the extension of T to know the reference of ∀v(T v → ϕ). The notions introduced in the last section are just to be applied to languages with a standard interpretation. Secondly, because we are now interested just in reference via the truth predicate, and the notion of reference defined in 4.5 allows a sentence to refer even when the truth predicate doesn’t occur in it. Only sentences containing T can refer alethically, but there are further differences. For instance, pϕq = pψq ∧ T pχq would m-refer to ϕ, ψ and χ, but only to the latter via the truth predicate. ∀x(x = g → T ¬. x) would q-refer to pa’s strong G¨odel sentence and its negation, but only to the latter via the truth predicate, for it says that ¬¬Bewpa (g) is true, but nothing about the truth of ¬Bewpa (g) itself. In this section we first introduce notions of alethic reference by mention and quantification, and show they can account for the self-referentiality of certain paradoxical expressions that can be obtained via the strong and the weak diagonalization lemmata, respectively. We then give notions of direct and indirect alethic reference, as well as alethic self-reference and well-foundedness. We study cycles, chains and Yablo-like sequences and classify them according to these concepts, in a sound and illuminating way. In 5.5 we show that the notions we provide in this section are all definable in Lpa and some of them representable or weakly representable in pa. 5.6 notices that alethic reference in pa suffers from incompleteness and puts forward some axioms to complete it, that give rise to a hierarchy of theories. This systems are proven to be ω-consistent. Let th be an axiomatizable theory in LT that extends pat (possibly) with axioms in Lpa that are true in N.

5.1 Alethic reference by mention Since there is only one predicate that generates reference now, and this predicate is a primitive symbol of the language, complex terms are no longer a problem. We can consider just whole terms, occurrences of terms that don’t occur as proper subterms of other terms. Definition 5.1 (Direct alethic m-reference) If ϕ, ψ ∈ LT , ϕ directly alethically m-refers to ψ if and only if T t is a subformula of ϕ and pat ` t = pψq. Since pa proves all true identities, this definition covers all reference by mention cases, there’s no need to resort to the standard model. Definition 5.1 also covers the cases of many pathological sentence obtained by the strong diagonal lemma. Any sentence that results from applying theorem 2.1 to a formula ϕ(v) ∈ LT , where T v occurs in ϕ, alethically m-refers to itself. E.g.

22

the strong diagonal lemma applied to ¬T x delivers a term l such that l = p¬T lq

(20)

¬T l is a strong (strengthened) liar sentence, and it alethically m-refers to itself. The diagonal lemma also implies the existence of a term c such that c = pT c → 0 6= 0q

(21)

T c → 0 6= 0 says of itself that it implies a falsehood, it’s a strong (strengthened) Curry sentence, and it alethically m-refers to itself too. However, alternative versions of the liar and Curry’s sentences, also obtained by the strong diagonal lemma, such as l0 = pT ¬. l0q

(22)

c0 = pT (c0 → . p0 6= 0q)q,

(23)

and

do not alethically m-refer to themselves. For T ¬. l0 doesn’t say anything about itself but it says about it’s negation that it’s true. Something similar applies to T (c→ . p0 6= 0q). One might find this a bit counterintuitive but, as we will see later, it will not represent a problem at all.

5.2 Alethic reference by quantification As we have anticipated, some changes with respect to definition 4.2 must be done to provide a notion of alethic q-reference, besides switching to a proof-theoretic notion. The only thing that will remain unchanged is that reference is closed under negation and conditionals. To begin with, superfluous quantifiers won’t be a source of q-reference anymore, for this reference isn’t via the truth predicate. If v isn’t free in ϕ, ∀vϕ and ϕ alethically q-refer to the same sentences. Go further to require that v is under the scope of T in ϕ would give the wrong verdict in cases like ∀x(x = l → ∀y(y = x → ¬T y)), where x is doing the job so the sentence refers to the strong liar, but isn’t under the scope of T . Second, sentences of the form ∀v1 . . . vn T t, where v1 , . . . , vn are free in t, will no longer refer to every LT -sentence but just to those that fall under the scope of T , this is, to every sentence denoted by a term t[k¯1 /v1 ] . . . [k¯n /vn ], for k1 , . . . kn ∈ ω. ∀xT x, for example, alethically q-refers to all sentences, but ∀xT ¬. x only to negated ones. For, just as in Yablo’s list reformulation (3.2), expressions like ∀xT ¬. x can be thought as restricting quantification, as ∀x(∃y x = ¬. y → T x). Something similar applies to sentences of the form ∀v1 . . . vn ¬δ, but with a little twist. 23

They will q-refer to all sentences denoted by terms t such that there is an occurrence of T t in ¬δ[k¯1 /v1 ] . . . [k¯n /vn ] that wasn’t in δ. It would not be sound to say that ∀x(¬. x = ¬. x ∧ T (x→ . x) ∧ T l) alethically q-refers to the liar and all negations but just to conditionals. Also, ¯ ¬δ[k1 /v1 ] . . . [k¯n /vn ] could happen to alethically refer to a sentence, not by mention but by quantification. For instance, in ∀x¬∀y(T (x→ . y)), ¬∀y(T (pϕq→ . y)) doesn’t m-refer to ϕ or any other sentence; it q-refers to every conditional expression whose antecedent is ϕ. Thus, we adopt a recursive clause for expressions of the form ∀v1 . . . vn ¬δ. Finally, expressions of the form ∀v1 . . . vn (δ → γ) should be treated carefully. Since we can only know if a sentence or number (provably) falls under a predicate in case it doesn’t contain T (or isn’t logically contingent, but this would just introduce more complications), to avoid risks, if T occurs in both δ and γ, we treat ∀v1 . . . vn (δ → γ) like ∀v1 . . . vn ¬(δ → γ): it alethically q-refers to everything (δ → γ)[k¯1 /v1 ] . . . [k¯n /vn ], with k1 , . . . , kn ∈ ω, alethically m or q-refers to. It’s better to have an overgenerating criterion than an undergenerating one, because the notions we introduce here are meant to built consistent restrictions on the T-schema. But if T occurs only in γ or in δ, restricted quantification is possible. If T occurs only in γ, prima facie ∀v1 . . . vn (δ → γ) q-refers just to the—sentences which we can prove to be—δs, and vice versa, if T occurs only in δ, ∀v1 . . . vn (δ → γ) q-refers just to the non-γs. This works fine for sentences of the form ∀v(δ → T v), where T doesn’t occur in δ and applies directly to v. For expressions like ∀v(δ → T f (v)) or ∀v1 (δ → ∀v2 T f 0 (v1 , v2 )), where f and f 0 are function symbols of the language, we should rather say, like in definition 4.2, that ¯ and ∀v1 (δ → ∀v2 T f 0 (v1 , v2 )) to ∀v(δ → T f (v)) q-refers also to all sentences denoted by f (k), ¯ v2 ) alethically refers to, where k ∈ ω is a δ. Since we’re only interested whatever each ∀v2 T f 0 (k, in reference via the truth predicate, we will stay just with the latter; it doesn’t matter if a sentence satisfies δ. Definition 5.2 (Direct alethic q-reference) If ϕ, ψ ∈ LT , ϕ directly alethically q-refers to ψ in th if and only if T occurs in ϕ and one of the following conditions is satisfied: 1. ϕ := ∀v1 . . . vn χ, v1 , . . . , vn are free in χ and (a) χ := T t or χ := ¬δ and, for some k1 , . . . , kn ∈ ω, χ[k¯1 /v1 ] . . . [k¯n /vn ] directly alethically q-refers to ψ or contains an occurrence of T s that doesn’t occur in χ such that th ` s = pψq, or (b) χ := δ → γ and i. δ and γ contain T and, for some k1 , . . . , kn ∈ ω, χ[k¯1 /v1 ] . . . [k¯n /vn ] directly alethically q-refers to ψ or contains an occurrence of T t that doesn’t occur in χ such that th ` t = pψq, or

24

ii. T occurs only in γ and exist k ∈ ω and 1 ≤ i ≤ n such that ¯ i] th ` ∃v1 . . . vi−1 vi+1 . . . vn δ[k/v ¯ i ] directly alethically q-refers to ψ or conand ∀v1 . . . vi−1 vi+1 . . . vn (δ → γ)[k/v tains an occurrence of T t that doesn’t occur in γ and th ` t = pψq, or iii. T occurs only in δ and exist k ∈ ω and 1 ≤ i ≤ n such that ¯ i] th ` ∃v1 . . . vi−1 vi+1 . . . vn ¬γ[k/v ¯ i ] q-directly alethically q-refers to ψ or conand ∀v1 . . . vi−1 vi+1 . . . vn (δ → γ)[k/v tains an occurrence of T t that doesn’t occur in δ and th ` t = pψq. 2. ϕ := ∀v1 . . . vn χ, vi , with 1 ≤ i ≤ n, isn’t free in χ and ∀v1 . . . vi−1 vi+1 . . . vn χ directly alethically q-refers to ψ. 3. ϕ := ¬χ y χ directly alethically q-refers to ψ. 4. ϕ := χ → δ and χ or δ directly alethically q-refer to ψ. For instance, ∀x(x = l → ∀y(y = ¬. x → T (y→ . y)) alethically q-refers to everything that ∀y(y = ¬. l → T (y→ . y) m or q-refers to, i.e. to ¬T l → ¬T l. Like before, every sentence that obtains from a weak diagonalization process applied to a formula in which T v occurs will q-refer to itself. E.g. the weak liar sentence λ in (1) is actually ∀v2 (Diag(p∀v2 (Diag(v1 , v2 ) → ¬T v2 )q, v2 ) → ¬T v2 ) and is obviously self-referential by clause 1.(b)ii. of definition 5.2. But if v occurs in ϕ as a subterm in the scope of T there’s no guarantee that the weak diagonalization of ϕ is self-referential, for similar reasons as (22)’s. Alethic q-reference isn’t a semantic notion anymore. Now we don’t need to know which formulae are satisfied by which numbers or sentences in the standard model to find out the referential status of an expression. We do however need to know if our base theory th proves something or not. That is why alethic q-reference is always relative to some theory or other, as well as other concepts defined in terms of it. We explore the consequences of this later in this section.

25

5.3 Direct and indirect alethic reference Sentences of LT can directly alethically refer to others by mention and by quantification at the same time, like T c ∧ ∀x(x = l → ¬T x). Definition 5.3 (Direct alethic reference) If ϕ, ψ ∈ LT , ϕ directly alethically refers to ψ in th if and only if ϕ directly alethically m-refers or q-refers to ψ in th. Observation 5.4 Let ϕ, ψ, χ ∈ LT . 1. If ϕ ∈ Lpa , ϕ doesn’t directly alethically refer to ψ. 2. ϕ directly alethically refers to ψ if and only if ¬ϕ directly alethically refers to ψ. 3. ϕ → χ directly alethically refers to ψ if and only if ϕ or χ directly alethically refers to ψ.

Unlike reference in Lpa , transitivity is a fully desirable property for alethic reference. Definition 5.5 (Chains of direct alethic reference) A sequence of sentences χ0 , . . . , χn , . . . of LT of length α is a chain of direct alethic reference in th if and only if, for each ordinal β < α, χβ directly alethically refers to χβ+1 in th. Definition 5.6 (Alethic reference) If ϕ, ψ ∈ LT , ϕ alethically refers to ψ in th if and only if exists a finite direct alethic reference chain χ0 , . . . , χn in th such that χ0 is ϕ and χn is ψ. Applying proposition 2.4 to formulae containing occurrences of T v we can obtain cycles of sentences that do not directly refer to themselves but only indirectly. For example, we can get in pat the strong truth-teller cycle given by pat ` t1 = pT (t2 )q pat ` t2 = pT (t1 )q

5.4 Alethic self-reference and well-foundedness Now we can define notions of alethic self-reference and alethic well-foundedness. Definition 5.7 (Alethic self-reference) If ϕ ∈ LT , ϕ is alethically self-referential in th if and only if ϕ alethically refers to itself in th.

26

Definition 5.8 (Alethic well-foundedness) If ϕ ∈ LT , ϕ is alethically well-founded in th if and only if there is no infinite chain of direct alethic reference in th starting with ϕ. Let’s see some examples. Yablo’s sentences, both in their original and in their alternative formulations ((7) and (9)), are alethically non-well-founded and non-self-referential in pat. We take a look at the former. Each term υ(¯ n) denotes ∀x(x > n ¯ → ¬T υ(x)) ¯ where pat ` k¯ > n that doesn’t m-refer but q-refers to every sentence denoted by υ(k), ¯ . Each sentence in the list directly refers just to sentences coming after it. Thus, they are not selfreferential. But, since every infinite subsequence of Yablo’s list is a chain of direct alethic reference in pat, all its members are non-well-founded. Yablo’s dual version, which can also be obtained in pat by strong diagonalization, receives the same verdict:11 ¬∀x(x > 0 → T κ(x))

(24)

¬∀x(x > ¯1 → T κ(x)) ¬∀x(x > ¯2 → T κ(x)) ... where each κ(¯ n) with n ∈ ω denotes the n-th expression in the sequence. And further alternative lists with different reference patterns can also be formulated: ∀(x > 0 → ¬∀y(y > x → T η(y)))

(25)

∀(x > ¯1 → ¬∀y(y > x → T η(y))) ∀(x > ¯2 → ¬∀y(y > x → T η(y))) ... where each η(¯ n) with n ∈ ω denotes the n-th expression on the list. These refer directly to what each ¬∀y(y > k¯ → T η(y)) with k¯ > n ¯ refers to, this is, to every sentence T η(m) ¯ where m ¯ > k¯ > n ¯ . Then, every sentence refers to all sentences that occur two or more positions ahead in the sequence, but not to the one that follows immediately. Thus, they are all non-self-referential and non-well-founded. By proposition 2.5, we can also generate ω-chains of sentences containing T such that each of them directly alethically m-refers just to the one that comes next. For instance, the following 11 Cf.

Cook (2014).

27

chain of truth-tellers in pat: t0 = pT t1q

(26)

t1 = pT t2q t2 = pT t3q ... Clearly, every sentence in the chain is non-well-founded. Unlike the notion of reference introduced in the previous section, sentences in chains are not alethically self-referential, because subterms are no longer taken into consideration. Finally, alethic reference for sentences like ∀xT f (x, p0 = 0q), where f (0, pϕq) := pϕq and f (S(x), pϕq) := pT num(f (x, pϕq))◦q (f writes T x times before ϕ), and McGee’s corresponds to our intuitions. ∀xT f (x, p0 = 0q) says that all iterations of T over 0 = 0 are true, and alethically ¯ p0 = 0q) with k ∈ ω, i.e. to 0 = 0 (denoted by refers to every sentence denoted by a f (k, f (0, p0 = 0q)), T p0 = 0q (by f (¯ 1, p0 = 0q)), T pT p0 = 0qq (by f (¯2, p0 = 0q)), etc. Since each of these sentences refers just to a finite number of other sentences, there’s no infinite chain of direct reference; ∀xT f (x, p0 = 0q) is well-founded. McGee’s sentence ¬∀xT f (x, m) obtains by applying the strong diagonal lemma to ¬∀xT f (x, y). pat ` m = p¬∀xT f (x, m)q

(27)

¬∀xT f (x, m) says of itself that not every iteration of T over itself is true. It’s usually called an ω-liar. A contradiction can be derived form it together with some compositionality principles for ¯ m) for each k ∈ ω. Thus, it refers to truth.12 ¬∀xT f (x, m) refers to sentences denoted by f (k, itself (k = 0), and it’s both self-referential and non-well-founded. Alethic well-foundedness is closer to Kripke (1975)’s or Leitgeb (2002)’s notions of grounding than the concept of well-foundedness introduced by definition 4.7, but they still don’t coincide. For instance, ∀x(T x → T x) alethically refers to every sentence, while it’s grounded in Leitgeb’s sense, for it’s true in every expansion of N to LT . If t = pt = t ∨ T tq obtains in pat diagonalizing x = x ∨ T x, t = t ∨ T t alethically refers to itself and is, therefore, non-well-founded, but it’s grounded both according to Kripke and Leitgeb. In general, unlike many other approaches, alethically self-referential expressions are alethically non-well-founded, because we can produce infinite chains of direct reference starting with them. Self-reference is, from this view point, a special kind of non-well-foundedness. 12 See

McGee (1985).

28

5.5 Definability of alethic reference We show that Lpa ⊆ LT contains formulae defining all the notions we introduced in the preceding subsection. We can do so because no semantic concepts are involved in them. The following Lpa -formula defines direct alethic m-reference of a sentence x to a sentence y of LT : ◦ SentLT (x→ . y) ∧ ∃t < x(Subf orm(x, pT t.q) ∧ t = y)

(28)

d d We write Rm (x, y). Since it’s a ∆0 -expression, Rm (x, y) strongly represents direct alethic m-

reference in pat. To show direct alethic q-reference is also definable in Lpa we describe a machine that takes two arguments, x and y, and answers ‘yes’ when both x and y are sentences of LT and x directly alethically q-refers to y, and doesn’t stop or says ‘no’ otherwise. This machine only uses recursive procedures. As a consequence, direct alethic q-reference is semi-recursive and, thus, definable in Lpa by a Σ1 -formula we note Rqd (x, y). Moreover, Rqd (x, y) weakly represents direct alethic q-reference in pat. Our machine takes as inputs two natural numbers, x and y, and first checks whether they both codify LT -sentences, and an open term occurs under the scope of T . If not, it answers ‘no’ and stops. If so, it removes superfluous quantifiers in each propositional component in x in which an open term occurs under the scope of T and assigns a label [0], [1], . . . to the resulting formulae, according to their order of appearance in x. The machine operates from now on with the following three phases, starting by phase 1: Phase 1: The machine checks sentence labeled [j]—where, [j − 1] is the label of the formula checked immediately before, if any; otherwise, j = 0. If [j] is of the form ∀v1 . . . vn γ, where γ is an atomic formulae, a negation or a conditional where T occurrs both in the antecedent and in the consequent, the machine goes to phase 2 with γ, the first n-tuple hk1 , . . . , kn i ∈ ω n that hasn’t been used yet for this label according to some fixed coding of tuples with natural numbers, variables v1 , . . . , vn and sentence y. If [j] is of the form ∀v1 . . . vn (δ → γ) where T occurs only in γ (δ), the machine checks if the first pair hn, mi that has not yet been checked for this label according to some fixed ordering is such that n is the code of a proof of ∃v1 . . . vi−1 vi+1 vn δ[m/v ¯ i ] (∃v1 . . . vi−1 vi+1 vn ¬γ[m/v ¯ i ]) in th. If not, it moves to label [j + 1], if there’s any, and to [0] otherwise. If so, it goes to phase 2 with ∀v1 . . . vi−1 vi+1 . . . vn (δ → γ), m, vi and y. Phase 2: The machine receives as inputs a formula ϕ, numbers k1 , . . . , kn , variables v1 , . . . , vn and a sentence ψ, and checks whether ϕ[k¯1 /v1 ] . . . [k¯n /vn ] contains a new occurrence of T t such that t = pψq. If not, it moves to phase 3 with ϕ[k¯1 /v1 ] . . . [k¯n /vn ] and ψ as inputs. If

29

so, it answers ‘yes’ and stops. Phase 3: The machine takes two formulae ϕ and ψ, removes all superfluous quantifiers from each propositional component of ϕ in which an open term occurs under the scope of T . It labels the ones that are still universal assertions with [n], [n + 1], . . . according to their ordering of appearance in ϕ, starting with the least number not already used as a label and returns to phase 1. Thus, direct alethic reference can be defined in Lpa and weakly represented in pat by the Σ1 d formula Rm (x, y) ∨ Rqd (x, y), which we abbreviate Rd (x, y). The following Σ1 -expression defines

and weakly represents in pat the set of finite chains of direct alethic reference: Seq(x) ∧ ∀i < lg(x)Rd ((x)i , (x)i+1 )

(29)

which we note Cref (x). Therefore, reference can also be defined and weakly represented in pat by the Σ1 -formula ∃z(Cref (z) ∧ x = (z)0 ∧ y = (z)lg(z) )

(30)

which we abbreviate R(x, y). Turning to this predicate, we can define alethic self-reference and well-foundedness in Lpa by SR(x) := R(x, x) and SentLT (x) ∧ ∀y(Cref (y) ∧ (y)0 = x →

(31)

∃z > lg(y)∀w(Seq(w) ∧ lg(w) = z ∧ ∀i ≤ lg(y)(y)i = (w)i → ¬Cref (w))) We write Bf (x). According to this formula, for every reference chain y starting with x there must be a number z such that y cannot be extended to a reference chain of length z. Thus, every reference chain must be finite. While self-reference is semi-recursive, as SR(x) is Σ1 , well-foundedness is more complex.

5.6 The incompleteness of alethic reference and hierarchies Since alethic reference can be expressed by a Σ1 -formula in Lpa , by pa’s Σ1 -completeness, pat can already prove all positive cases of alethic reference. However, some negative cases will not be provable. For instance, th cannot prove that ∀x(Bewth (x) → T x) doesn’t refer in th to 0 6= 0, because, by G¨ odel’s second incompleteness result, th cannot prove ¬Bewth (p0 = 0q), i.e. its own consistency, on pain of triviality. Since we want to include as many instances of the T-schema as possible in our theories, and these will be selected according to their reference patters, the more we know about them, 30

the better. In particular, self-reference and non-well-foundedness will turn out to be dangerous. Thus, we would like to know, e.g. that ∀x(x = p0 = 0q → T x) doesn’t refer to itself or to other sentences besides 0 = 0, that it isn’t dangerous. The obvious solution is to add axioms that help us know more about the non-reference relations between LT -sentences. Definition 5.9 qr(th) is the theory in LT that extends th with the following axiom: qr(th) ∀x(SentLT (x) → (Bewth (¬. x) → ¬Bewth (x))) qr(th) proves the consistency of th, as expected. It also allows us to know, for instance, that ∀x(Bewth (x) → T x) doesn’t refer to 0 6= 0, because th proves that Bewth (p¬0 6= 0q), and by qr(th) we now know that ¬Bewth (p0 6= 0q). We also know in qr(th) that ∀x(x = p0 = 0q → T x) doesn’t refer to itself or to anything different than 0 = 0. As a third example, we cannot know in th that Yablo’s sentences aren’t self-referential in th in th but only in qr(th). Naturally, qr(th) isn’t enough to negation-complete th, because it’s an axiomatic theory extending pa and therefore subject to G¨odel’s incompleteness results. For example, qr(th) isn’t enough to conclude that ∀x(Bewqr(th) (x) → T x) doesn’t refer to 0 6= 0 in th. Observation 5.10 qr(th) is ω-consistent. Proof For axiom qr(th) is true in N, as well as th.

a

qr(th) generates certain stratification. Consider the sentence ∀x(¬Rq (p∀x(x = 0 → T x)q, x) → T x)

(32)

Intuitively, (32) q-refers to every sentence to which ∀x(x = 0 → T x) doesn’t q-refer. Since qr(th) implies that this sentence doesn’t q-refer to, e.g. ¬T l, (32) seems to refer to the strong liar in th. But it doesn’t. Because it’s qr(th) the one that proves that ¬T l satisfies the antecedent of (32), not th. This is not a case of reference in th that th cannot prove; reference is Σ1 . It’s simply false that (32) refers to the liar in th. It does in qr(th), though. Thus, we have a new notion of reference, relative to qr(th), replacing th with qr(th), for which we might want to add a new axiom like qr, generating a new theory with a corresponding notion of reference, etc. We work with a hierarchy of theories qr0 , qr1 , . . . over pat and their corresponding alethic reference notions, definable in Lpa by the Σ1 -formulae R0 , R1 , . . . We can even extend this hierarchy to transfinite levels. This will give us the possibility to know more about the reference relations of sentences in LT but, most importantly, about the non-relations of reference among them. 31

Definition 5.11 Let qr0 := pat and, if 0 < α ≤ 0 , qrα extends pat with the following axiom-scheme for each ordinal β < α: qrα ∀x(Bewqrβ (¬. x) → ¬Bewqrβ (x)) Unlike alethic m-reference, which is independent of the theory we are working on, for each ordinal α < 0 there are different notions of alethic direct q-reference, direct reference, reference, self-reference and well-foundedness in qrα , definable and weakly representable in pat by the formulae Rqα (x, y), Rdα (x, y), Rα (x, y), SRα (x) and Bf α (x), correspondingly. Observation 5.12 For each α ≤ 0 , qraα is ω-consistent. Observation 5.13 Let α ≤ 0 , and t and s be two terms. If qraα ` Rqα (t, s), then, for all β > α, qraβ ` Rqβ (t, s). This observation notices that, if a sentence alethically refers to another sentence at some point in the hierarchy, it will refer too at every higher point, which means that reference is cumulative. For certain sentences, but not for all of them, we can also show that if they don’t refer to other sentences at some point in the hierarchy, they will not refer either at any higher point. First we need some auxiliary definitions and results. Definition 5.14 (Decidable formulae) If ϕ ∈ LT with v1 , . . . , vn as its only free variables, ϕ is decidable in th if and only if, for any k1 , . . . , kn ∈ ω, th ` ϕ(k¯1 , . . . , k¯n ) or th ` ¬ϕ(k¯1 , . . . , k¯n ). Decidable formulae in th are represent the set they define in th. Unfortunately, this notion isn’t even semi-recursive. The least complex formula that defines it in Lpa is Π2 : F ormLT (x) ∧ ∀y(Seq(y) ∧ lg(y) = f ree(x) → Bewth (~s. (x, y)) ∨ Bewth (¬. ~s. (x, y))) We abbreviate it with Decth (x). However complex, we can prove for many formulae that they are decidable in pat. For example, every formula that is equivalent in th to a ∆1 -formula is provably decidable in th. Proposition 5.15 If ϕ ∈ LT is th-equivalent to a ∆1 -formula, th ` Decth (pϕq). Proof Let ϕ(v1 , . . . , vn ), ψ(v1 , . . . , vn ) ∈ LT be such that th ` ϕ ↔ ψ and ψ ∈ ∆1 . Then, exist ψΣ (v1 , . . . , vn ) ∈ Σ1 and ψΠ (v1 , . . . , vn ) ∈ Π1 , both equivalent to ψ and, therefore, to ϕ in th. Note that ¬ψΠ is also Σ1 . Since pa knows about its own Σ1 -completeness and, moreover, about

32

iΣ1 ’s Σ1 -completeness, if χ(v1 , . . . , vn ) ∈ Σ1 ,13 pa ` χ(v1 , . . . , vn ) → BewiΣ1 (pχ(v˙ 1 , . . . , v˙ n )q)

(33)

Also, by the third condition of Hilbert-Bernays on Bewth , pa ` Bewth (px→ . yq) → (Bewth (x) → Bewth (y))

(34)

Then, reasoning in th we get that, on the one hand, ψΣ (v1 , . . . , vn ) → BewiΣ1 (pψΣ (v˙ 1 , . . . , v˙ n )q)

by (33)

→ Bewth (pψΣ (v˙ 1 , . . . , v˙ n )q) → Bewth (pϕ(v˙ 1 , . . . , v˙ n )q)

by (34)

and, on the other, ¬ψΠ (v1 , . . . , vn ) → BewiΣ1 (p¬ψΠ (v˙ 1 , . . . , v˙ n )q)

by (33)

→ Bewth (p¬ψΠ (v˙ 1 , . . . , v˙ n )q) → Bewth (p¬ϕ(v˙ 1 , . . . , v˙ n )q)

by (34)

Since ϕ ∨ ¬ϕ implies ψΣ ∨ ¬ψΠ , th ` Bewth (pϕ(v˙ 1 , . . . , v˙ n )q) ∨ Bewth (p¬ϕ(v˙ 1 , . . . , v˙ n )q) or, equivalently, th ` ∀v1 . . . vn (Bewth (pϕ(v˙ 1 , . . . , v˙ n )q) ∨ Bewth (p¬ϕ(v˙ 1 , . . . , v˙ n )q)). a Note that, although every formula that is logically equivalent to a ∆1 -formula is itself ∆1 , this is not necessarily the case for other expressions that are not logically equivalent but pa or th-equivalent to a ∆1 -formula. Proposition 5.15 entails more decidable formulae than the ones in ∆1 . Definition 5.16 (Rd-decidable sentences) ϕ ∈ LT is rd-decidable in th if and only if all the subformulae of ϕ of the form ψ → χ in which ψ contains T if and only if χ doesn’t are such 13 See Smorinsky (1985, p. 61, theorem 6.22) and Simpson (2009, p. 370-371, definition IX.3.4, lemma IX.3.5 and theorem IX.3.6). The proof of (33) can be easily adapted to pa, extending the canonical interpretation of the language of arithmetic without additional function symbols in pra’s language to Lpa , because the latter contains terms for each p.r. function.

33

that the one not containing T is decidable. Intuitively, rd-decidable sentences in th are such that their direct reference is completely determined by th. It’s impossible that in a consistent stronger system the sentence directly refers to more sentences than in th. For instance, ∀x(x = l → T x) is rd-decidable in th, for x = l is decidable in th. ∀x(x = l → ∀y(y > x → T y)) is also rd-decidable in th, because y > x is decidable. But ∀x(Bewth (x) → T x) isn’t rd-decidable in th, for Bewth (x) isn’t decidable in th. We define rd-decidability in Lpa in terms of Decth (x) by the formula RdDecth (x). Since Decth (x) ∈ Π2 and Decth (x) doesn’t occur under the scope of a negation in definition 5.16, RdDecth (x) ∈ Π2 too. Definition 5.17 (R-decidable sentences) ϕ ∈ LT is r-decidable in th if and only if it alethically refers just to rd-decidable sentences. For r-decidable sentences, alethic reference simpliciter is fixed for stronger (sound) theories. The set of r-decidable sentences in th is definible in Lpa in terms of RdDecth via the formula RdDecth (x) ∧ ∀y(Ra (x, y) → RdDecth (y)), which we abbreviate RDecth (x). Since Ra is Σ1 , RDecth is Π2 . Observation 5.18 Let t and s be terms such that t denotes an rd-decidable LT -sentence and let α ≤ 0 . If qraα ` ¬Rqα (t, s), then, for all β > α, qraβ ` ¬Rqβ (t, s).

5.7 A minimalist theory of truth In the last section we have introduced appropriate notions to study the reference patterns that underlie paradoxical expressions in LT . The next step is to identify the common notes these patterns share and use them to restrict the set of LT -sentences that will generate legitimate instances of the T-schema. Naturally, every inference we make over reference patterns common to the paradoxes will be informal, for we only have a limited sample. On the one hand, self-referential sentences such as the liar, Curry’s, McGee’s, and their corresponding cycles. On the other, non-well-founded Yablo-like lists. Since self-reference is a form of non-well-foundedness, we might want to restrict the T-schema to well-founded expressions, getting close to Leitgeb (2005) and Horwich (2005). A closer look reveals, however, that the expressions causing paradox and ω-paradox that are not self-referential always directly refer to an infinite number of other sentences, such as (7), (24) y (25). Thus, a second criterion, more fine-grained but riskier, could consist in including 34

al well-founded sentences plus all the non-well-founded ones that aren’t self-referential and only refer directly to a finite number of sentences. However, this criterion seems very permissive. For instance, T ¬. l0 in (22) and T l in (20) would both generate instances of the T-schema, which means that the criterion would give rise to truth theories that are incompatible with certain compositional axioms like the one that states that the truth predicate commutes with ¬ and the one that closes truth under conditionals, just like putb. Thus, we might want to consider a slightly more restrictive criterion that allows sentences in T-schema only if they are well-founded or directly refer to a finite number of sentences, none of which is self-referential. With little modifications, each of these three criteria will generate an ω-consistent theory of truth. Which theory to choose depends on the purposes for which it will be used. Of course, it might also be possible to refine the given restrictions to obtain better theories based on the notion of alethic reference. In the rest of the paper we first introduce Leitgeb (2005)’s notion of dependence, which we will later use to provide consistency results for our theories. Then we provide some bridge results, between Leitgeb’s dependence and alethic reference, that will allow us to translate the theorems for dependence into theorems of alethic reference. In 5.10 we give three disquotational systems, which we show to be ω-consistent, and briefly discuss their properties. Finally, we show that these systems are as strong as putb and rt 0∃y(y > x ∧ ¬T y).16 For this sentence to be true there must be a code from which on no sentence belongs to the extension of T ; it doesn’t matter how large the smallest such code is. Thus, the sentence depends on ω ⊇ {1, 2, . . . } ⊇ {2, 3, . . . } ⊇ . . . ; there’s no smallest set on which it depends. Observation 5.24 (Leitgeb) Let ϕ, ψ be LT -formulae and Γ, ∆, Γi ⊆ LT , with i ∈ ω. 1. If ϕ ∈ Lpa , ϕ essentially depends on ∅. 2. If ϕ := T t and N  t = pψq for some ψ ∈ LT , ϕ essentially depends on {ψ}. 3. If ϕ depends on Γ, ¬ϕ also depends on Γ. 4. If ϕ depends on Γ and ψ on ∆, ϕ → ψ depends on Γ ∪ ∆. 5. If, for each n ∈ ω, ϕ[¯ n/v] depends on Γn , ∀vϕ depends on

S

i∈ω

Γi .17

Thus, dependence obeys some kind of compositionality. We can resort to it to define notions of self-reference and grounding. Definition 5.25 (D-self-reference) ϕ ∈ LT is d-self-referential if and only if there are ϕ1 , . . . , ϕn ∈ LT such that, for every Γ on which ϕ depends, ϕ1 ∈ Γ; if 1 ≤ i < n, for each Γ on which ϕi depends, ϕi+1 ∈ Γ; and, for every Γ on which ϕn depends, ϕ ∈ Γ.18 This definition covers cases such as the liar sentences in (1), (3), (20), and (22), Curry sentences, truth-tellers, etc., but also cycles such as the liar’s in (2). 16 Cf.

Leitgeb (2005, p. 164, list of examples 1). Leitgeb (2005, p. 162, list of examples 1 and p. 165, lemma 5). 18 Leitgeb (2005) only defines direct self-reference. We extend his notion in a straightforward way to prove some satisfiability results in this and the next section. 17 Cf.

37

The set of grounded sentences is the result of a fixed-point construction. It consists, roughly, of sentences that depend on the empty set, plus sentences that depend on sets of sentences that depend on the empty set, plus sentences that depend on sets of sentences that depend on sets of sentences that depend on the empty set, etc. Definition 5.26 (∆f p ) Let ∆0 := ∅, ∆α+1 := {ϕ ∈ LT : ϕ depends on ∆α }, for each ordinal S α, and, if α is a limit ordinal, ∆α := β n + 1, which is impossible, and ∆∗ϕ ∩ Γ = ∅.

a

Theorem 5.29 If Γ = {ϕ1 , ϕ2 , . . . } ⊆ LT is an infinite set of sentences that essentially depend on a finite set and are not d-self-referential, there is a Θ ⊆ LT such that hN, Θi  T  (∆f p ∪Γ).20 Proof We assume Γ ∩ ∆f p = ∅. For each n ∈ ω, let ∆n be ϕn ’s essential dependence, Γn := {ϕ1 , . . . , ϕn } and Ξn := {Θ ⊆ ω : Θf p ⊆ Θ ⊆ Θf p ∪

[

(∆i ∪ {ϕi }) and hN, Θi  T pϕq ↔ ϕ, if ϕ ∈ Γn }

i≤n

Each Ξn is finite, because

S

i≤n (∆i

∪ {ϕi }) is so. We show that for every n ∈ ω there are

Θ1 , . . . , Θn such that Θ1 ⊆ Θ2 ⊆ · · · ⊆ Θn , where Θi ∈ Ξi for each 1 ≤ i ≤ n. First, by theorem 5.28, for each n ∈ ω there is a model hN, Θi such that Θf p ⊆ Θ in which T  (∆f p ∪ Γn ) es true. S For 1 ≤ i ≤ n, let Θi := Θ ∩ (Θf p ∪ k≤i (∆k ∪ {ϕk })). Then, for each i ≤ n, Θi ⊆ Θi+1 . S Since, by observation 5.22, each ϕ ∈ Γi depends on Θf p ∪ k≤i (∆k ∪ {ϕk }), hN, Θi i  T  Γi , and so, Θi ∈ Ξi . We recursively define a tree A as a set of sequences of members of {∅} ∪ 20 This

result was proved in collaboration with Thomas Schindler.

39

S

n∈ω

Ξn as follows:

1. h∅i ∈ A. 2. If hΘ0 , . . . , Θn i ∈ A and Θn ⊆ Θn+1 ∈ Ξn+1 , then hΘ0 , . . . , Θn , Θn+1 i ∈ A. 3. Nothing else is in A. Notice that, for each n ∈ ω, A contains a sequence hΘ0 , Θ1 , . . . , Θn i of length n. For any such sequence, ∅ = Θ0 ⊆ Θ1 ⊆ · · · ⊆ Θn and Θi ∈ Ξi for every 0 < i ≤ n. Thus, any two members of any two sequences are connected via ∅. And, since Ξi is finite but contains at least one element, A is an infinite finitely branching tree—i.e. each node belongs only to a finite number of sequences. Then, by K¨ onig’s lemma, A must contain an infinite sequence Θ0 ⊆ Θ1 ⊆ · · · ⊆ Θn ⊆ . . . with Θi ∈ Ξi for every i > 0.21 S Let Θ := n∈ω Θn . We show that hN, Θi  T  (∆f p ∪ Γ). Since, for each n ∈ ω, ∆f p ∩ Θn = Θf p , ∆f p ∩ Θ = Θf p , and, by observation 5.21, hN, Θi  T  ∆f p . For each ϕi ∈ Γ let m > i by the least natural number such that, for every k ≥ m, Θk doesn’t contain members of ∆i that are not already in Θm . m must exist because ∆i is always finite and sequences of Θj with j ∈ ω are monotonic. Since i < m and Θm ∈ Ξm , hN, Θm i  T pϕiq ↔ ϕi . Given the selection process for m, observation 5.21 implies that hN, Θi  T pϕiq ↔ ϕi .

a

To know whether a sentence can be consistently inserted into the T-schema, we can appeal to the notion of dependence and invoke theorems 5.27, 5.28, and 5.29. However, these results cannot be—and are not mean to be—employed in the construction of axiomatic theories of truth. For to know if a sentence depends on a set we first need to know its truth value under several extensions of N. Leitgeb’s notion is highly complex, not even arithmetical. It’s conjectured to be Π11 , i.e. it can only be defined by a formula consisting of a second-order universal quantifier followed by a first order expression, or even more complex formulae. Consequently, to provide a disquotational system over pat turning to dependence we would first need an axiomatization of a dependence predicate taken as a primitive.22 Thus, truth would be given in terms of a nonarithmetical, rather semantical notion, introducing inflationary means in our notion of truth. Leitgeb’s dependence incompatible with the minimalist project.

5.9 Bridge results In this subsection we prove several results that connect alethic reference with Leitgeb’s dependence. They will allow us to import theorems from the latter to the former and establish ω-consistency results for the truth systems we provide below. 21 K¨ onig’s lemma establishes that, if A is a connected graph (for any two nodes there is a connecting path) with infinitely many nodes, each of which belongs to finitely many sequences, then A contains an infinite sequence. 22 See Schindler (2014).

40

Theorem 5.30 If ϕ, ψ ∈ LT and ϕ is rd-decidable in th, if ϕ doesn’t alethically refer to ψ in th, there’s a Γ such that ϕ depends on Γ and ψ ∈ / Γ. Proof By induction over ϕ’s logical complexity. We assume ϕ ∈ / Lpa ; otherwise the result is trivial. If ϕ is atomic, it must be of the form T t, where th ` t 6= pψq. Thus, by clause 2 of observation 5.24, ϕ depends on a Γ such that ψ ∈ / Γ. Suppose that every sentence that is less complex than ϕ, rd-decidable in th and doesn’t alethically refer to ψ in th depends on a set Γ such that ψ ∈ / Γ. If ϕ := ¬χ, since ϕ doesn’t alethically refer to ψ, by clause 2 of observation 5.4, χ doesn’t refer to ψ either. By inductive hypothesis, there is a set Γ on which χ depends such that ψ ∈ / Γ. By clause 3 of observation 5.24, ϕ also depends on Γ. The case for conditional expressions is analogous. Let ϕ := ∀v1 . . . vn χ. If for some 1 ≤ i ≤ n vi isn’t free in χ, ϕ refers to the same sentences as ∀v1 . . . vi−1 vi+1 . . . vn χ, which, therefore, doesn’t refer to ψ. By inductive hypothesis, exists a Γ on which this sentence depends such that ψ ∈ / Γ and, by observation 5.22, ϕ also depends on Γ. Assume that for every 1 ≤ i ≤ n vi is free in χ. If χ is an atomic formula, a negation or a conditional expression where both the antecedent and the consequent contain T , there are no ¯ i ] refers to ψ. By inductive hypothesis, k ∈ ω and 1 ≤ i ≤ n such that ∀v1 . . . vi−1 vi+1 . . . vn χ[k/v ¯ i ] depends such for any k ∈ ω and 1 ≤ i ≤ n there is a set Γ on which ∀v1 . . . vi−1 vi+1 . . . vn χ[k/v that ψ ∈ / Γ. Then, by clause 5 of observation 5.24, there exists a Γ on which ∀v1 . . . vn χ depends such that ψ ∈ / Γ. Finally, let ϕ := ∀v1 . . . vn (δ → γ), where γ contains T and δ doesn’t. The converse case is analogous. Since ϕ doesn’t refer to ψ, for all k ∈ ω and all 1 ≤ i ≤ n, if ∀v1 . . . vi−1 vi+1 . . . vn (δ → ¯ i ] refers to ψ, th 0 ∃v1 . . . vi−1 vi+1 . . . vn δ[k/v ¯ i ]. As ϕ is rd-decidable, by definition 5.16, δ γ)[k/v ¯ i ]. Since th is true in every extension must be decidable, this is, th ` ¬∃v1 . . . vi−1 vi+1 . . . vn δ[k/v ¯ of N to LT , N 2 ∃v1 . . . vi−1 vi+1 . . . vn δ[k/vi ]. Thus, for every k ∈ ω and every 1 ≤ i ≤ n, if ¯ i ], ∀v1 . . . vi−1 vi+1 . . . vn (δ → γ)[k/v ¯ i ] doesn’t refer to ψ. By N  ∃v1 . . . vi−1 vi+1 . . . vn δ[k/v inductive hypothesis, for each of these instances there is a Γ on which it depends and ψ ∈ / Γ. By clause 5 of observation 5.24, there is a Γ on which ∀v1 . . . vn (δ → γ) depends such that ψ ∈ / Γ. a Ideally, we wouldn’t restrict this result to rd-decidable formulae. Unfortunately, if we remove this restriction, the statement is false. Consider the sentence ∀x(¬Bewth (x) → T x). th’s G¨odel sentence belongs to every set ∀x(¬Bewth (x) → T x) depends on but this sentence doesn’t refer in th to th’s G¨ odel sentence, but in qr(th). That a sentence doesn’t refer directly to another sentence in a theory doesn’t mean it cannot depend on it: only if it’s rd-decidable. Corollary 5.31 If ϕ ∈ LT is r-decidable and alethically well-founded in th, ϕ ∈ ∆f p . Proof By definition 5.8, every chain of direct alethic reference in th starting with ϕ must be of 41

a finite length n. Let mϕ := max{n : there is a chain of direct alethic reference in th starting with ϕ of length n}. mϕ > 0 can be a transfinite ordinal, as e.g. for ∀xT x p0 = 0q. We show that ϕ ∈ ∆mϕ , where ∆i is as in definition 5.26, by transfinite induction over mϕ > 0. If mϕ = 1, by definition 5.5, ϕ doesn’t alethically refer to anything in th. Thus, by theorem 5.30, it depends on ∅ and, therefore, ϕ ∈ ∆1 . Assume that every r-decidable sentence ψ that is alethically well-founded in th and mψ < mϕ , is such that ψ ∈ ∆mψ . Since ϕ is r-decidable and alethically well-founded in th, it refers only to rd-decidables ψ in th such that mψ < mϕ wich, by inductive hypothesis, belong to ∆mψ . Then, S by theorem 5.30, ϕ depends on i