ungroundedness in classical languages

TIMOTHY MCCARTHY

UNGROUNDEDNESS IN CLASSICAL LANGUAGES

I

The intuitive notion of groundedness for sentences in an interpreted language has received several characterizations in recent years. (See, for example, Kripke (1973, Gupta (1982) and Herzberger (1982).) These explanations tend to connect the property of ungroundedness with semantic closure: the ability of a language L to phrase a predicate which is true in L of precisely the sentences true in L (or of the Code1 numbers of these senten&). However, in an aside to his ‘Outline of a Theory of Truth’, Saul Kripke has called into question that ungroundedness is connected to semantic closure in an essential way:

One surprise to me was the fact that the orthodox approach by no means obviously guarantees groundcdness in the intuitive sense mentioned above. The concept of truth for C, sentences is itself Z, , and this fact may be used to construct sentences of the form (3) (viz., (3): (3) is true). Even if unrestricted truth definitions are in question, standard theorems easily allow us to construct a descending chain of first-order languages L,,, L,, 4, . . such that L, contains a truth predicate for L,., ,. 1 don’t know if such a chain can engender ungrounded sentences, or even quite how to state the problem here; some substantial technical questions in this area are yet to be solved. (p. 61).

This paper is about the issues raised in Kripke’s last sentence. In Sec- tion II, I shall construct a sequence L,, L,, . . . of classical first-order languages such that (i) for each i, Li contains a truth-predicate for Li+, and (ii) for each i, there is an ungrounded sentence in Li. This example is based upon a general result about the situation to which Kripke refers and will.be developed around the intuitive notion of grounding implicit in his discussion. In Sections III and IV, I will suggest a formal definition of groundedness for sentences in a classical language; this definition will explicate the notion of grounding underlying the construction of Section II. Finally, in Section V, I will consider some classes of ungrounded sentences.

Journal of Philosophical Logic 17 (1988) 61. 74. cc 1988 by D. Reidel Publishing C0mpan.v.

62 TIMOTHY MCCARTHY

II

By a classical language we mean a pair L = (L, 2I) where L is an un- interpreted first order language and ‘2l a structure for L. It will always be assumed that L is an extension of the language of Peano Arithmetic (PA) and that ?I satisfies PA. We assume a fixed Giidel numbering of the expressions of L; for any expression tl of L, ‘p will be its Giidel number. If S is a set of sentences from L, a one-variable formula ,4(x) in L will be said’ to be a truth predicate for S in L iff for each sentence # of S, d, is true in 2l iff A(T#l) holds in 2X. A truth predicate for a sublanguage L’ of L is a truth predicate for the set of all sentences of L’.

THEOREM 1. Let K he a consistent extension of PA. For any n, set L, = L, + T, + r,-, + . . . . Then there in a structure VI for Lo such that (II is a model of K and for each n, T, is a truth predicate for L”,,, in (L,, 9l).

Proof Fix a consistent extension K of PA. For any n, let W, be the set of all sentences of the form

T,(rA-‘) t* A

for A a sentence of L,,, , . Define

K* = u, W, v K.

It suffices to show that K* is consistent, for if ‘2I is any model of K*, for each i c is clearly a truth predicate for L,+, in (L,, 9l). For this, by compactness it suffices to show that any finite subtheory of K* is consistent. Thus, let (4,) . . . , &} be any finite subset of U, W, and set

K, = K u {4,, . . . , A>. Let T,,, . . . , T,., be the predicates q appearing in c#J,, . . . , c$,, where we assume

k, < k, < . . ’ < k,.

Let 2 be a model of K. Define an interpretation ‘!?I0 of Lq as follows. (u,(L, = 9. For any i, 1 < i < I, rU,,(T,,) is defined by induction on 1 - i. If 1 = i, ‘%,,(T,,) is the set of all standard integers in 9’ which are codes of sentences of L, true in 2’. For any p, 0 d p < 1, set L(p) = L, + Tk, + Tkl-, + . . . + Tk,-,. Now suppose that

UNGROUNDEDNESS IN CLASSICAL LANGUAGES 63

QIp(Tk,.-,) has been defined for some i, 0 < i < 1 - 1. Then flo(% (,+I) ) consists precisely of the standard integers in 2I which are Gijdel numbers of sentences which are true in L(i) on the induc- tively given interpretation of L(i). Thus for each i, 1 < i < I - 1, Tk,.-, is a truth predicate for L(i - 1) in ‘210. Now consider any sentence c$,~(O < s < n) of the form

for 0 < i < I and A a sentence of Lp,,. Then A belongs to L(i - 1) if i > 0 and to L, if i = 0. In either case we have ‘?I, k #,, and thus, since C#J, was arbitrary, that %,, + &. Thus K, is consistent, and we are done.’

Theorem 1 enables us to construct examples of apparently ungrounded sentences.* For example, let L be the language of PA together with new constants co9 c, , c2, . . . . We assume a fixed Giidel numbering of the sentences of the language that results from L by adjoining the predicates T. K consists of PA together with all sentences

c,-I = ‘Tf(c,)-

for any i > 0. K is clearly consistent. Let 2I be the structure for Lo given by Theorem 1 for this choice of K. Then for each i, Y&(ci) is an ungrounded sentence of the language (Li, ?I). To see this for i = 0, let us try to evaluate the sentence

(1) To(co) using the Tarski schemata

r,,(‘Al) = A

and the semantics of L. We obtain a nonwellfounded chain of quivalences:

(1) is true in L, iff T, (c, ) is true in L, iff T?(Q) is true in L, iff , . .

The resulting chain of evaluations using the Tarski disquotational rule then fails to determine a truth value for (1) or, indeed. for any of the sentences T(ci). But it must do so if (1) is grounded in ‘QI. The present example seems quite analogous to a situation that can arise in

64 TlMOTHY MCCARTHY

a fixed point interpretation. Let L be the language of elementary number theory together with constants co, c,, . . . and enriched with a truth predicate 7’(x) in the manner of Kripke. In L each c, denotes the Giidel number of the sentence T(c,+,). Then for essentially the same reasons as above, the sentence T(c,) is ungrounded in L. In fact, it is easy to see that no sentence T(ci) receives a truth value in the minimal fixed point for L.

III

The suggested example involving the language f$ might seem puzzling in one respect. The structure ‘U for & delivered by Theorem 1 is a classical interpretation, and thus each primitive of L, has a totally defined extension in 91. In particular, each predicate r is totally defined in 2l. Thus, each of the sentences Ti(ci) must have a definite truth value in ‘$I. This fact may seem to conflict with the impression that these sentences are ungrounded with respect to that interpretation.

However, the intuitive property of groundedness which is in question here does not coincide with determinacy of truth value. Rather, the groundedness of a sentence consists in the occurrence of an outcome in an ideal process of evaluation involving the notion of truth (or other semantic properties). The process proceeds in stages. We attempt to determine the truth value of any sentence of the form

(2) rA1 is true.

by evaluating the sentence A. If A is determined to be true at some stage, (2) is counted as true at all later stages, and if A is determined to be false, (2) is counted as false at all later stages. The truth value of a first-order combination of length n is evaluated in terms of rules that determine how the truth value of such a sentence depends upon the truth values of sentences of length < n.

Let us consider the form of such rules. A disjunction A v B will count as true at a stage if either A or B has been evaluated as true at some earlier stage, and as false if both A and B have been evaluated as false. Otherwise it will be undertermined at that stage. A sentence - ,4 will count as true at a stage iff A has been evaluated as false at a previous stage and as false ifT A has been evaluated as true. A

UNGROUNDEDKESS IN CLASSICAL LANGUAGES 65

quantified statement (3x)4(x) is counted true at a stage iff some instance of A(X) has been evaluated as true at a previous stage, and as false iff each instance of A(x) has been evaluated as false. The truth values of atomic sentences not involving the notion of truth are determined directly at the initial stage from the interpretations of the relevant primitives. A sentence is grounded if it eventually receives a truth value in this process.3

Let us note several points about this informal conception. First, the notion of groundedness for a language is necessarily relative to a choice of predicates which are to be regarded as expressing semantic concepts for the language or for a fragment thereof. This is so because the process described above segregates the roles of semantic and nonsemantic notions: the truth values of atomic sentences involving nonsemantic notions are determined at the initial stage, whereas the truth values of sentences involving the concept of truth are determined cumulatively via the ‘disquotational’ rule by which (2) is replaced by the sentence represented by ‘A’.4 Secondly, the notion of truth involved in the process need not be univocal; the process may be ramified in such a way that various truth predicates are used which express concepts of truth for distinct fragments of the language in question (or even for the same fragment). Finally (and this is the point with which we began) there is nothing in the informal conception of groundedness that requires ungrounded sentences to lack truth value. A sentence may be true or false in the language in question and yet not be determinately so on the basis of the process of evaluation described above.

For example, consider the sentences q(ci) in the interpretation ‘2I of L, constructed above. Each of these sentences receives a well defined truth value in 2l. But it is easy to see, assuming that the stages in the process are well ordered under the relation earlier than, that that process does not determine a truth value for any of these sentences. Suppose, for a contradiction, that some such sentence does receive a truth value. Let cx be the earliest stage at which any such sentence is valuated, and let Ti(ci) be a sentence which is valuated at a. Then 7;+, (c,,,) must be valuated at some stage preceeding CC This con- tradicts the minimality assumption about ~1. Thus no sentence 7;(ci) is ever valuated.

66 TIMOTHY MCCARTHY

Our next task is to make the intuitive picture just described precise for classical languages. By a frame for a classical language L = (L, ‘2l) we mean a function F mapping a collection of one-place predicates of L into a collection of sets of sentences of L such that, for any predicate P in the domain of F, P is a truth predicate for F(P) in L; i.e., for any sentence 4 in F(P), Pr&‘) holds in L iff 4 is true in L. If each predicate P for which F is defined is primitive, we say that F is a normal frame for L. We sey that the elements of the domain of Fare the truth constants for F (or for L with respect to F). A frame for L, then, is essentially a choice of predicates which are to be regarded as expressing truth concepts for finitely many fragments of L.

Let L = (L, 9I) be a classical language and F a normal frame for L. We shall characterize the sentences of L which are grounded over F; these will be the sentences which are eventually valuated by the process described above when the elements in the domain of F are treated as truth predicates for the corresponding sets of sentences. We expand L to a language I,+ by adding constants for the elements of IaI(, and apply the frame F to L’ . We define transfinite sequences A,(a), BF(a) (~1 an ordinal) as follows. Intuitively, for any ordinal a, AF(a) will consist of all sentences of L+ valuated as true at stage a, and B,(a) of all the sentences of L’ valuated as false at a. A,(O) consists of all true sentences in L+ of the forms (i) P(t,, . . . , I,) where t,, . . . , t, are closed terms of L+ and P is not in the domain of F, and (ii) P(t) where P belongs to the domain of F and t does not denote the Gbdel number of a sentence in F(P). BF(0) consists of all sentences in L+ of either of these forms which are false in Lt.

Now suppose that A&) and B&3) have been specified for each ordinal /l < a. Set

&(a) = B~wAF(BX &(a) = p~e&W.

Then AJa) consists of all sentences C#J satisfying any of the following: (i) 4 E AF.(a), (ii) C#J is of the form A v B where A E A,(a) or B E A,.(a), (iii) 4 is of the form A & B where A E AF.(a) and B E Ape(a), (iv) C#J is of the form (3x),4 where A(c/x) E AF(a) for some constant c, (v) 4 is of the form (Vx)A where A(c/x) belongs to A,.(a) for each constant c, (vi) 4 is of the form P(t) where P E dom (F) and t is a term for the


Code1 number of a sentence ij of F(P) such that tj belongs to AF.(a). A symmetrical set of conditions characterizes B,(a).

The basis case of this construction lays down all of the true and false atomic sentences of L+ other than those involving the semantic notions distinguished by F. The induction clauses evaluate complex sentences in terms of their parts, and, where P is a truth constant applicable to a sentence 4 of L, sentences that apply P to 4 are evaluated disquotationally, i.e., by evaluating 4. It is straightforward to show by transhnite induction on x that AF(u) contains only true sentences of L+ and B,(r) contains only false sentences of L+ .

The sequences (AF(a): z an ordinal), (B,-(a): a an ordinal) are clearly increasing. Thus there is an ordinal z such that

A,(a) = A,(a + 1)

&(a) = B,(r + I);

let a,, be the least such ordinal. The elements of A,(a,) in L are naturally interpreted as the sentences of L which are determined to be true on the basis of the evaluation scheme described above, and the elements of B,(a,) in L as those which are determined to be false. The set T(L, F) of grounded sentences of L in F may be identified with the collection of all sentences of either type. That is if, for each ordinal a, we set G(L, F, a) = AF(a) u BP(m), T(L, F) = {C#I E G(L, F, a,.): 4 a sentence of L}.

IV

Thus far we have considered only normal frames, i.e., frames in which each truth constant is a primitive predicate. In the general case, however, we shall wish to consider non-primitive predicates. For example, as Kripke notes, the concept of truth for C, sentences of number theory is Z, , and this circumstance allows us to construct examples of ungrounded sentences (see Section V). But the relevant truth predicate is properly C,, and is thus non-primitive.

The suggested definition of groundedness cannot be directly applied to frames with non-primitive truth constants. The difficulty is that the

68 TIMOTHY MCCARTHY

inductive process described above will settle the extensions of the noncomplex expressions appearing in such a constant at the initial stage, and thus its extension will be completely defined at all stages exceeding its first-order complexity. The idea behind the above construction was to “mask” the extensions of the truth constants of a frame for L in the semantic evaluation of sentences in L. If F is a frame for L which is defined for a predicate P, the evaluation of a sentence of the form P(‘@) for 4 E F(P)dependsupon a prior evaluation of 4, and does not make use of the extension of P. But if P is non-primitive, the extension of P will eventually be constructed by reference to those of its atomic subformulas; thus all sentences of the form P(‘c$l) will eventually receive truth values.

The simplest way out of this difficulty is to replace a frame in which non-primitive predicates are identified as truth constants with one in which these constants are represented by primitive predicates. To each non-normal frame F for L, we will associate a frame F*, the evaluator of F. F* is a frame for a language L* extending L. L* arises from L by the addition of primitive predicates (Q,, . . . , Q,} corresponding to the truth constants r,, . . . , T, in the domain of F. (It is assumed that the Qis do not occur in L and that distinct lJs are associated with distinct Qs.) To each sentence C/J of L we associate a translate c$* in L*; 4* results from C#J by replacing each occurrence of a predicate q with the corresponding Qi. It is assumed that the Gddel numbering is arranged in such a way that the Giidel number of any sentence in L coincides with that of its translate in L*. The extension of each predicate Qi in L* is then just that of T in L. The domain of F* is the set {Q,, . . . , Q,}, and for any i, 1 < i < n, we have

F*(Qi) = (d*: 4 E F(Qt>}. F* is thus a normal frame for L*. The above construction then determines the grounded sentences of L* over F*. The grounded sentences of L over Fare now those sentences of L whose translates are ungrounded in L* over F*. The grounded sentences in L over Fare thus roughly those which are evaluated as grounded by the original construction when the constants in the domain of Fare treated as (noncomplex) truth predicates.


The present definition may seem odd in one respect. For it seems to imply that, for example, a quantified statement may be ungrounded even though its instances are grounded. Similarly, if - B(X) is a formula of number theory which is treated as a truth predicate, a sentence of the form B(a) may count as grounded but - B(a) as ungrounded.

However, it is necessary to be careful here. What is defined above is not a property, c$ is grounded, but a relation, C$I is grounded over the frame F, where F is essentially a choice of formulas which are to be treated as truth predicates. These may include subformulas of the formula C#J. There is a clear sense in which the formula - B(a) above is grounded; like every sentence of a classical language, it is grounded over each frame for the language in question that does not treat any of its subformulas as truth predicates, Suppose, however, that - B(x) is treated as a truth predicate, and that there is a term t in the language designating the Gijdel number of the sentence - B(t). This sentence therefore says of itself, under the relevant numerical coding, that it is true, and is thus intuitively an ungrounded sentence. It will be ungrounded, according to the suggested criterion, over any frame for which - B(x) is a truth predicate for a set containing the sentence - B(t). The sentence B(t) may nonetheless count as grounded over such a frame, unless B(x) itself is treated as expressing a semantic concept (say,falsity). If the predicate B(x) is not treated as expressing a semantic concept, the sentence B(t) simply expresses a number theoretic fact, and so construed it is clearly grounded.

The general point illustrated by such cases is that the question of the groundedness of a sentence makes sense only relative to a choice of predicates which are taken as expressing truth concepts for fragments of the language in question. Such a choice is described by a frame. The present example shows also that groundedmw relative to a frame is a strongly intensional notion. In ‘fact, we can use the example of the predicate -B(x) ,above to show that this property is not even generally preserved by substitution of logically equivalent formulas. If N B(x) is treated as a truth predicate for - B(t), this sentence is ungrounded over the corresponding frame. Thus the sentence - - B(t) is also ungrounded over that frame. But the (logially equivalent) sentence B(t) is grounded over that frame.

70 TIMOTHY MCCARTHY

V

I will conclude with some applications of a simple criterion of groundedness. Let L = (L,, ‘$I) be a classical language, F a frame for L, and I a set of sentences in L. Say that I is regular over F iff each sentence in I is of the form Q(l) for some predicate Q E dom (F) and some term t denoting the Godel number of a sentence in I n J(Q).

THEOREM 2. Let L be a classical language, F a frame for L, and r a set ofsentences which is reg&ar over F. Then any element of r is ungrounded in L over F.

Proof. It clearly suffices to consider only normal frames. We employ the notation of Section III. Let L = (L, 2l) and F be given and let I be regular over F. Suppose that I contains a sentence which is grounded in L over F. Let a be the least ordinal such that AF(~) u B,(a) contains some sentence C$ of L. By regularity such a sentence must be of the form Q(t) for Q E dom (F) and ‘%(t) = i-+7 for some sentence ij of I- n F(Q). Then by the evaluation rule for Q(t) J/ must appear in A&?) u B&I) for some ordinal /I < a, a contradiction.

Let us apply this result to the initial example of this paper. Let To, T,, T,, . . . be a fixed enumeration of distinct one place predicates. For any language L not containing any q and any n, we define L, = L + T,, + T,,, + _ _ _ . The canonical frame for L, is the frame defined for (T,,, T,+,, T,,+z, . . .} and mapping each T, onto the set of all sentences’ of Li+ , .

THEOREM 3. Let K be a consistent extension of PA. Then there D an extension by constants K+ of K and a structure ‘3 for the language (L, , ),, such that the foliowing conditions are satisfied for each n:

(4

(W

Set Li = (L,,.)i and Li = (Li, 2ljL,). Then T, is a truth predicate jbr L,+ , in L,. There is a sentence c#+, which is ungrounded in L, over the canonical.frame for L,.

Proof. Fix K. Let L+ = L, + c,,, c,, . . . , where the cis are constants not appearing in L,. Fix a Godel numbering for Lf . Let K+ be K together with all sentences c,-, = ‘T(q)1 for each i > 0. K+ is


clearly consistent. By Theorem 1, let ‘?l be a structure for L, satisfying (a) for each n. For each n, the set {K(ci): i = n, n + 1, . . .> is then regular over the canonical frame for L,. Thus by Theorem 2 if we take 4, to be T,(c,) condition (b) holds for each n as well.

It is roughly correct to say that the sentences which are grounded over a normal frame for a language are those which are forced to be true or false by the interpretations of expressions of the language which are not identified as truth constants by the frame, in conjuction with the rule that a sentence of the form Q(r~l), where Q is a truth constant, is assigned the truth value of I,&. This suggests that the grounded sentences of a language L = (L, 2t) over a normal frame F are those sentences which receive the same truth value in any structure for L that agrees with M on all expressions other than truth constants (relative to F), and in which those constants are truth predicates for the relevant fragments of L. More precisely, let us say that a structure Y for L is admissible for L over F iff

(a) IV = 14

(W 5X(t) = S?(s) holds for each primitive 5 6 dom (F)

(4 For each Q E dom (F) and each sentence 4 4 F(Q)

aQ(‘4’) = yQCrl)

Cd) For each Q E dom (F), Q is a truth predicate for F(Q) in 9.

It is then natural to conjecture that a sentence is grounded in L over F if and only if it assigned the same truth value by each structure for L which is admissible for L over F.

One half of this conjecture is readily verified:

THEOREM 4. Let L be a language and F a normal frame for L. Then a sentence of L is grounded in L over F only ifit has the same truth value in each structure which is admissible for L over F.

Proof. This property may be verified for each sentence of G(L, F, r) by transfinite induction on 01.

72 TIMOTHY MCCARTHY

However, it is interesting to note that the converse fails: not every sentence in L that receives the same truth value in each structure admissible for L over F need be grounded in L over F. For example, let L = (L,, VI), where L is L,, together with additional predicates T,, T, and additional constants cl, c2. ‘$I assigns the standard interpretations to the arithmetical primitives. Suppose that 9I(c,) = rT,(c,)l and Yl(c,) = ‘T,(c,)l. Define a frame F for L by taking as the truth constants for F the predicates T, and T, and setting F(T,) = fT2(cI)) and F( T,) = {T, (c, )}. Then the set {T, (c, ), T2(c2)} is regular in L over F. Thus by Theorem 2 the sentences T,(c,) and T?(q) are both ungrounded in L over F, whence the sentence

is ungrounded in L over F. But this sentence is clearly false in any structure which is admissible for L over F.

Finally, let us consider some examples of ungrounded sentences over frames for elementary number theory. bA is the language of Peano arithmetic and N the standard interpretation thereof. L will be (L,,, A’“). In the following results, it will be convenient to assume that L,, has been supplemented with a primitive recursive set of terms, one for each primitive recursive function. Thse are the primitive recursive terms of L,,. A set of formulas in L, is said to be primitive recursively closed if it is primitive recursive and is closed under substitution of primitive recursive terms.

THEOREM 5. Let r be a primitive recursively closed set of formulus in L, and let T(x) E I- be a truth predicate in L for the set of all sentences in I-. Let F be a jkme for L,, such that F(T(x)) is defined and contains each sentence of r. Then there exists a sentence in I- which is ungrounded in L over F.

Proof: We apply a familiar self-referential argument. Let g be a primitive recursive function enumerating the Gddel numbers of one- variable formulas of F, where we set g(n) = ‘,4,(x)1. Let Sb be a primitive recursive function such that for any one-variable formula A(x) of L,, .Sb(n, ‘A(x)‘) = rA(ii/x)l holds for each n. For any n, set

h(n) = Sb(n, g(n)) = rA,(ii)‘.


Let b(x) be a primitive recursive term for It. Then since I- is primitive recursively closed, T(lr(x)) E I-. Let T(h(x)) be A,(x). Then

4,) = SW,, I&,)) = rT(h(fi,))l.

Thus the set (T(h(&,))} is regular over the frame F. Theorem 5 now follows from Theorem 2.

Theorem 5 furnishes an explicit procedure for constructing statements T(s) such that s denotes V(s)l; i.e., T(s) “says of itself” that it is true. We can use this result to construct many examples of ungrounded sentences over frames for LP,,. For example, call a prefix class of formulas in LpA homogeneous if it is of the form C,O or II: for a fixed n. It is easy to see that each homogeneous prefix class in L,, contains its own truth predicate.’ Let K be a homogeneous prefix class in L,,. Let F, be the frame for LpA which is defined solely for a fixed truth predicate T for K such that T E K, where we take F,&T) to be just the set of sentences in K. K is primitive recursively closed. Thus by Theorem 5 there exists a sentence in K which is ungrounded over F K.

NOTES

’ 1 would like to thank an anonymous referee for this Journal for drawing my atten- tion to a construction essentially similar to that of Theorem I in Albert Visser’s ‘Semantics and the Liar Paradox’ (Section 3.3.2), to appear in the Handbook of Philosophical Log,gic IV. Visscr also observes that the theory K* has no standard models. ’ Gupta (1982). Section 3, also gives examples of classical languages with ungrounded sentences under the assumption of restricted self-reference. ’ An elegant statement of this general view of groundedness may be found in the open- ing pages of Kripkc (1975). 4 A similar relativity appears in Kripke’s discussion (1975). Kripke suggests that a scn- tence is grounded in a Kleene three valued language if and only if it is valuated in the minimal fixed point for the language. However, the construction of the minimal fixed point requires us to distinguish the predicate that expresses the concepts of truth for the language from the other predicates of the language. The extensions and anti- extensions of the latter are given in advance; for the purpose of assessing groundedness, an interpretation of the truth predicate T(x) is constructed iteratively, by a method analogous to the process that generates the A,(z) below. However. T(x) might be associated in advance with an interpretation that determines truth values for certain Sentences which are not vatuatcd in the minimal fixed point (e.g., its interpretation

74 TIMOTHY MCCARTHY

could be given by a non-minimal fixed point). If T(x) is not distinguished as a truth predicate, indeed as a truth predicate of a quite special sort, these sentences should count as grounded. ’ For example, let A(u) be the formula

vx 3y vz S(x, y, z, u).

where S(I, m, n, k) holds iff k is the Godel number of a H, sentence and the triple (I, m, n) satisfies its matrix. Then A(u) is a truth predicate for IT, sentences of I+.,,, but since S is itself H, A(u) is H,. Obviously this technique can be extended to any homogeneous prefix class.

REFERENCES

[I] Anil Gupta, ‘Truth and PdKidOX', Journal of Philosophical Logic XI, 1 (February 1982): 1 -60.

[2] Hans Herzberger, ‘Notes on Naive Semantics’, ibid., 61-102. [3] Saul Kripke, ‘Outline of a Theory of Truth’, J~rnu/ of Philosophy, IXXII, 19

(November 6, 1975): 690-715.

Department of Philosophy, University of Illinois at Urbana-Champaign, 105 Gregory Hall, 810 South Wright St., Urbana. IL 61801, U.S.A.

ungroundedness in classical languages

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