(p.152) Appendix I: What Can Be Said?
(p.152) Appendix I: What Can Be Said?
I.1
Towards the end of his paper on truth Kripke wrote:
If we think of the language as one thing, and the acts we perform in speaking the language as another, then Kripke is, I think, wrong. One way to say something is to deny a claim. What is to stop us from speaking the language Kripke studies, asserting some of its sentences, denying others? It is straightforward to extend the language in Section 2.5 so that it contains its own Kripkean truth predicate and validates denials of liar sentences. Speaking that language one can certainly say what the status of the liar is.It seems likely that many who have worked on the truth‐gap approach to the semantic paradoxes have hoped for a universal language, one in which everything that can be stated at all can be expressed. . . . the present approach certainly does not claim to give a universal language, and I doubt that such a goal can be achieved. . . . there are assertions we can make about the object language [studied in the paper] which we cannot make in the object language. For example, Liar sentences are not true in the object language . . . but we are precluded from saying this in the object language by our interpretation of negation and the truth predicate. . . . The ghost of the [Tarskian] hierarchy is still with us.^{1}
According to Kripke, ‘is [a] true [sentence of English]’ is only partially defined on the set of English sentences: there are sentences which are in neither the extension of ‘true’ nor its anti‐extension (the set of things of which it is false). Kripke takes the extension and anti‐extension of ‘true’ to be determined in stages as follows. Start with an interpreted propositional language—call the model interpreting it M—in which truth‐value gaps may arise; assume that the connectives are given the strong Kleene treatment. Now add a truth predicate ‘T’ to the language, as well as constants ‘a’, ‘b’, etc. and quotation names to the language; call the resulting language L. Extend M to a model M(0) as follows: the constants name sentences of the (extended) language; ‘S’ names its interior; the extension and anti‐extension of ‘T’ are empty. M(0) makes no ascriptions of truth or falsity true or false, though of course it makes many sentences true, others false.
Consider the model obtained from M(0) by putting the sentences M(0) makes true into ‘T”s extension, those M(0) makes false into its anti‐extension; call this M(1). What M(0) made true (false), M(1) makes so. But some ascriptions of truth that were without truth‐value in M(0) are now themselves true or false. Consider now the sequence of models one gets by iterating this procedure, using the rule:
(p.153) One can carry this sequence into the transfinite in the following way: After following the procedure infinitely many times (at a limit ordinal, as they say), construct a model by making the extension of ‘T’ the set of those sentences which have been put into the predicate's extension at some point in the preceding sequence of models; analogously for the predicate's anti‐extension. Then go back to using rule (R). In this sequence the extension of ‘T’ at each stage includes its previous extension; likewise for ‘T's anti‐extension. Thus, one must eventually reach a point where these sets stop growing. That is, one must eventually reach a point where the set of sentences which are true (i.e. are made true by the model) is the same as the set of sentences to which one can correctly ascribe truth. At this point—the so‐called fixed point—we have arrived at a model in which ‘T’ is a truth predicate for the language.
(R) M(n + 1) is the model one obtains from M(n) by making the extension of ‘T’ the set of sentences which M(n) makes true, the anti‐extension of ‘T’ the set of sentences which M(n) makes false.
We want to add a truth predicate to our language using Kripke's method. Before we do, however, we should address some issues about truth and force. If I am saying something when I deny that Jo is bald by uttering ‘Jo is not bald’, I can be correctly reported as having said something, presumably by uttering ‘Richard said that’ and then echoing the words I used. Adopt for present purposes a conventional view of attitude ascriptions, on which they involve a two‐place predicate (‘believes’, ‘hopes’, etc.) whose second argument place is typically filled by a ‘term’ formed from a complementizer, such as ‘that’, and a sentence. To regiment such ascriptions in a language like that of Section 2.5 we would introduce (besides names and ‘said’) a complementizer—let it be ‘that’—which combined with unforced sentences to yield terms for the regimentation of the likes of ‘Alfred believes that snow is white’. We would also allow ‘that’ to combine with forced sentences, so that our language allows for sentences along the lines of
which we would use to report a denial. A first approximation of the semantics of (1) has it true just in case whatever a names produced an utterance whose force (in the language he was then speaking) was the conventional force of ‘not Fb’.^{2} That is, (1) is true iff what a names uttered something which expresses the second‐order commitment conventionally associated with ‘not Fb’ in the language of (1).
(1) a said that not Fb.
It is not just after a verb of attitude that something of the form that S may occur; it can also occur after ‘It's true’. But things such as ‘not Fb’ do not make truth‐evaluable claims. The idea that things of the form that S pick out something not truth‐evaluable is, of course, quite against the grain of entrenched philosophical common sense. I have already made clear why I think we ought to go against the grain here, and won't pause to further defend this. What does call for some discussion is the combination of things like ‘that not Fb’ with a truth predicate. And this impacts what we say about the sentential truth predicate, at least given a desire that sentences of the form It is true that S and ‘S’ is true should run in tandem.
It would seem that if a can say that S, we can entertain the thought that it is true that S. Likewise, if we can use S to say something, it would seem that we can use ‘S’ is true to say something. So if (1) is in order, so are (p.154)
This doesn't mean that such sentences are possibly true. If ‘that not Fb’ in (2a) picks out the claim—i.e. second‐order commitment—made by uttering ‘Not Fb’, and that claim isn't the sort of thing which can be true or false, then (2a) itself is either false or without truth‐value. The same sort of thing applies to (2b). I will take sentences such as the (2)s to be without truth‐value. Since they are not even candidates for a truth‐value, I also take it to be always appropriate to deny them. These stipulations are a matter of convenience—matters seem to run most smoothly in what follows if one so proceeds. So far as I can see, nothing hangs on this decision.
(2a) It's true that not Fb.
(2b) ‘Not Fb’ is true.
Consider now (an interpreted version of) the propositional language of Section 2.5; add to it constants, quotation names, and a truth predicate ‘T’; stipulate that applying the predicate ‘T’ to any constant or quotation name results in an unforced sentence. A moment's reflection should make it clear that we can straightforwardly apply Kripke's procedure for defining truth to this language. The only novelty here is the presence of forced sentences. But since these are not assigned truth‐values by the semantics, they will sit on the sidelines as we progress through the sequence of models which define ‘T’. We can prove—we will sketch a proof below—that we reach a fixed point at which ‘T’ is a truth predicate for the language: in which, if S is an unforced sentence of the language which is true (false, undetermined), T(‘S’) is true (false, undetermined).
If we speak this language—call it P—we do not have to be embarrassed about being prepared to seriously utter absolutely any instance in our language of the equivalence schema
—S unforced—even when S is a sentence which, because of vagueness or pathology, is without truth‐value. For any such sentence is appropriate—indeed, any such sentence is a logical truth, since it will be appropriate in any model of the language.^{3} Modest extensions of the language allow us to express all sorts of things which are commonly held to be inexpressible. For example, a sentence such as
(E) ‘S’ is true iff S.
is a declarative sentence, even given that its ‘and’ and ‘not’ are force indicators. (It's not as if the sentence is, say, interrogative or imperative.) So
(3) This is a sentence of P, and this is not true in P.
with ‘and’ and ‘not’ construed as force indicators, is a predicate. Add it to the language and interpret it in the obvious way. It is now a non‐truth predicate for P, in the following sense: A sentence of the form
(3′) x is a sentence of P, and x is not true in P.
(p.155) is appropriate iff (what) t (names) is a sentence of P which is not true.^{4} That is, such a sentence has conventionally associated with it a second‐order commitment which is appropriate if and only if (what) t (names) is a P‐sentence either false (viz. one with a true negation) or without truth‐value.
(3″) t is a sentence of P, and t is not true in P.
This is not to say, of course, that such a sentence is true if and only if (what) t (names) is a P‐sentence which is not true; we cannot, for familiar reasons, have such a predicate in P, which after all has ideology enough to create paradoxical sentences. But there is really no reason we need such a predicate. Why, after all, think that we need such a predicate, save because we think that if it is a fact that a sentence of a language is false or without truth‐value, we should be able to say in the language that it is so? But we have in P a predicate which allows us to say this sort of thing: (3″).
Languages like that being sketched are not ones in which one can ‘give the semantics of the language in the language’, for the language does not even contain quantifiers, much less the syntactic ideology necessary for this. But it should be clear that by adding quantifiers and ideology we can at least start to do this sort of thing. Since we can incur second‐order commitments without committing to the truth of any proposition, it would seem there is a perfectly good sense of ‘give the semantics for a language in the language itself’ in which we can start doing this: not by making assertions, but by making the appropriate claims—i.e. incurring appropriate second‐order commitments—about the language. For example, suppose we extended the language with quantifiers and syntactic ideology, and added a vague predicate like ‘bald’, interpreting this last as a predicate defined only on some of the objects in the domain. We can surely correctly—i.e. appropriately—say in the language that
Here all of the connectives save boldface ‘not’ are understood as force operators; boldface ‘not’ is truth‐functional. An utterance of (J) quite precisely characterizes the semantic facts about sentences of the form n is bald; when we know that the commitment (J) expresses is appropriate, we know all there is to know about the conditions under which such sentences are true, not true, and otherwise.
(J) For any individual object u and singular term t of our language: if t names u in our language, then:
(1) the sentence t is bald is true (in our language) iff u is bald;
(2) the sentence t is bald is not true (in our language) iff u is not bald;
(3) otherwise, the sentence is not true (i.e. if (u is not bald and u is not not bald), then the sentence is not true).
I.2
I have spoken of second‐order commitments as being appropriate or otherwise. Appropriateness is not truth: it is appropriate to deny a liar sentence, but that appropriateness is not a reflection of the liar sentence's truth or falsity, and the denial itself is not truth‐valued. (p.156) Still, appropriateness is in some important sense a semantic notion, and one would want a semantic theory to characterize the conditions under which sentences (strictly: the associated second‐order commitments—SOCs, for short) are appropriate or otherwise. Indeed, if we are speaking a language whose claims we both assert and deny and want to do as much of the language's semantics within it as possible, we will want the language we speak to contain its own appropriateness predicate.
Such a predicate threatens to produce something very much like the liar. Suppose that ‘Ap’ is a predicate of the language, and Ap(a) is appropriate—i.e. has associated with it an SOC which is appropriate—iff a names a sentence of our language which has associated with it an SOC which is appropriate.^{5} Isn't this trouble? For presumably the status of the SOC associated with a sentence S will be that of the sentence Ap(‘S’). Thus, any instance of
will be c‐valid. But just as we can construct a sentence which says of itself that it is not true, given a truth predicate, a negation operator, and sufficient sentence‐christening power, so we can construct a sentence which ‘denies of itself that it is appropriate’, given an appropriateness predicate, a denial operator, and sentence‐naming technology of sufficient strength:
(4) Ap(‘S’) iff S.
Given the identity ‘(5) = “Not Ap(5)” ’ and this instance of 4:
(5) Not Ap(5).
identity elimination yields
(4′) Ap(‘Not Ap(5)’) iff Not Ap(5).
And (4″) is problematic. The problem is not a problem with truth: (4″) is not a candidate for truth to begin with, as its principle operator is a force connective. But (4″) c‐follows from premises which are apparently themselves appropriate, and so must be appropriate. But it is not.
(4″) Ap(5) iff Not Ap(5).
This argument does not show that a language cannot contain its own appropriateness predicate. It assumes that any instance of (4) is appropriate, given that S and Ap(‘S’) always have the same ‘appropriateness status’. But we know that this sort of inference is not generally valid. For example: S and the sentence ‘S’ is true invariably have the same ‘truth‐value status’. It doesn't follow that any instance of S ↔ S’ is true is itself true. In particular, when S (and thus an ascription of truth to S) is neither true nor false, the biconditional will also be without truth‐value. Analogously, if there can be ‘appropriateness gaps'—if it can be neither appropriate nor inappropriate to assume the second‐order commitment associated with a sentence—then instances of (4) themselves may be neither appropriate nor inappropriate. If, in particular, (5) suffers from an appropriateness gap, then (4′) should too. Thus, the argument just rehearsed does not show that the language cannot contain an appropriateness predicate for itself.
(p.157) Is it coherent to suppose that a sentence(’s SOC) might be neither appropriate nor inappropriate? One might well doubt that it is, given the way in which we introduced the notion of appropriateness. When we introduced the notion, we proceeded roughly as follows. Fix a (possibly partial) assignment of truth‐values to the ‘non‐semantic’ atomic sentences of the language. If a(n unforced) sentence S comes out true on that assignment, it's appropriate and not S is inappropriate; otherwise, S is inappropriate and not S is appropriate. Appropriateness is then assigned to (unforced and) forced compounds in such a way that when all the subsentences of a compound are appropriate or inappropriate, so is the compound. Since every atomic sentence starts out as appropriate or not, how could appropriateness gaps (AGs, for short) arise?
But the language of Section 2.5 does not contain an appropriateness predicate. For such a language, appropriateness is determined as above, and AGs can't arise.^{6} Once we introduce an appropriateness predicate, however, we should not be surprised that appropriateness gaps can occur, given the obvious analogies between truth and appropriateness. Just as the truth predicate is truth‐disquotational for sentences with truth‐values—i.e. S ↔ ‘S’ is true is true provided that S has a truth‐value (and so the sentences flanking the biconditional will have the same truth status if true or false)—so the appropriateness predicate is ‘appropriateness disquotational’: S iff ‘S’ is appropriate is appropriate, provided that S does not suffer from an AG (and so the flanking sentences are either both apt or both inapt). A language with a predicate, like ‘is true’, which is truth‐disquotational, an operator, like ‘‐’, which reverses truth‐value, and sufficient sentence‐naming ability can construct sentences, like the liar, the truth‐teller (b: ‘b is true’), and so forth. In such a language, truth‐value gaps must arise. A language with a predicate which is appropriateness‐disquotational, contains an operator, like ‘not’, which reverses appropriateness, and sufficient sentence‐naming ability can construct sentences like (5), the self‐appropriator (c: ‘c is appropriate’), and so forth. In such a language, appropriateness gaps must arise.
One expects the extension of ‘appropriate’ to be determined in a way parallel to the way the truth predicate's extension is determined. If the truth predicate is Kripkean, a truth‐value cannot be assigned to a sentence of the form a is true or of the form a is not true before a truth‐value is assigned to the sentence named by a. This prevents liars, truth‐tellers, and the rest from receiving a truth‐value.^{7} If appropriateness is determined in an analogous fashion, then sentences of the form a is appropriate and a is not appropriate can be assigned the status appropriate or not only after the sentence named by a receives such a status. And this will prevent sentences such as (5) from being decided as appropriate or otherwise.
I.3
I'll sketch what seems the most natural way of introducing an appropriateness predicate.^{8} To keep things simple, we continue with the sentential case. The language contains (p.158) atomic letters, truth‐functional ‘−’ and ‘v’, the force operators ‘not’ and ‘or’, constants, quotation marks and the predicates ‘Tr’ and ‘Ap’. We stipulate

1. Constants are terms, as is any result of quoting a sentence.

2. Any atomic letter is an unforced sentence.

3. Applying ‘−’ to an unforced sentence, or ‘v’ to two unforced sentences, yields an unforced sentence.

4. Applying not to a sentence, or or to two sentences, yields a forced sentence.

5. Applying ‘Tr’ or ‘Ap’ to a term yields an unforced sentence.

6. Nothing is a term or sentence save in virtue of (1) through (5).
I.3.1
The hierarchy of models. We define a sequence of models for LTA, adapting the technique of Kripke (1975). The initial model, M(0), consists of a function D and a tuple M(0)_{E}=<M(0)_{T+}, M(0)_{T−}, M(0)_{A+}, M(0)_{A−}>. D maps the terms of LTA onto its sentences (mapping quote names, of course, to their interiors); it also assigns truth‐values to some or all of the atomic letters of the language. M(0)_{E}'s members are, respectively, the extension of ‘Tr’, its anti‐extension, the extension of ‘Ap’, and its anti‐extension. In M(0) all of them are empty.
The definitions of truth and appropriateness in a model (given below) determine relative to D and M(0)_{E} a tuple M(0)_{V}=<M(0)_{W+}, M(0)_{W−}, M(0)_{G+}, M(0)_{G−}>. These are, respectively, the wffs which are true, false, appropriate, and inappropriate in M(0). Analogously for models M(i) latter in the sequence to be defined. When convenient, we speak of this latter tuple as also being part of the model.
Given a model M(i), M(i + 1) is the model consisting of D along with the tuple M(i+1)_{E}=M(i)_{V}=<M(i)_{W+}, M(i)_{W−}, M(i)_{G+}, M(i)_{G−}>. That is: M(i + 1) is obtained by holding D constant and making the extensions (anti‐extensions) of ‘Tr’ and ‘Ap’ the sets of wffs which were made true (false) and appropriate (inappropriate) in M(i). When i is a limit ordinal, M(i) is the result of pairing D with the quartet of sets which result from taking unions of the extensions and anti‐extensions of ‘Tr’ and ‘Ap’ at levels below i. We call the first M(i) such that M(i)_{E}=M(i)_{V}—i.e. the first fixed point in the sequence—F(0).
We now wish to define a second sequence of models which will again culminate with a fixed point. To define the first such model, M*(0), we first define, for ϕ a sentence of our language, the family of ϕ as follows:
Say that a wff ϕ is safe for M*(0) when it is unforced and its family contains no occurrence of ‘Ap’. Let the model M*(0) be the result of pairing D with (p.159) <F(0)_{T+}, F(0)_{T−}, F(0)_{A+}, F(0)_{A−}∪S>, where S is the set of wffs which are safe for M*(0) and which are absent from the extensions and anti‐extensions of ‘Tr’ and ‘Ap’ in F(0). If an unforced wff is safe for M*(0) and is without a truth‐value in F(0), it will never receive a truth‐value, and thus its use is inappropriate. Adding these wffs—call them the initial wffs for M*(0)—to M*(0)_{A−} is recording that they will never be assigned a truth‐value, and are thus inappropriate.If ϕ is an atomic letter, the family of ϕ is {ϕ}.
If ϕ is Tr(α) or Ap(α), the family of ϕ is the set of what α names plus the members of its family.
If ϕ is −ψ or not ψ, the family of ϕ is the family of ψ plus ψ.
If ϕ is ψ v χ or ψ or χ, the family of ϕ is the union of the families of ψ and χ along with ψ and χ.
We now proceed as above, obtaining M*(i + 1) from M*(i) by making the extension of ‘Tr’ in M*(i + 1) the set of wffs made true in M*(i) (and so on, for the predicate's anti‐extension and for ‘Ap”s extension and anti‐extension); we take unions of extensions and anti‐extensions from lower levels at limit ordinals to form models at the limit. We call the first fixed point in this new sequence F(1).
I.3.2
Definitions of truth and appropriateness. Truth in a model M(i) is defined much as it is in Kripke (1975), using a strong Kleene treatment of the truth‐functional connectives. Forced wffs, since they are not candidates for truth or falsity, are never assigned a truth‐value. Making things explicit: As noted above, we use M(i)_{W+} to name the set of wffs true in M(i), M(i)_{W−} to name the set of wffs false in M(i), etc. Then, where M(i) is any of the models described in the previous section (including those in the sequence from M*(0) through F(1)):
1. If p is an atom of LTA,
p is in M(i)_{W+} iff D(p) = t
p is in M(i)_{W−} iff D(p) = f
Otherwise, p does not have a truth‐value in M(i).
2. If p is Tr(a), and a names q,
p is in M(i)_{W+} iff q is in M(i)_{T+}
p is in M(i)_{W−} iff q is in M(i)_{T−}
Otherwise, p does not have a truth‐value in M(i).
3. If p is Ap(a), and a names q,
p is in M(i)_{W+} iff q is in M(i)_{A+}
p is in M(i)_{W−} iff q is in M(i)_{A−}
Otherwise, p does not have a truth‐value in M(i).
4. If p is −q,
p is in M(i)_{W+} iff q is in M(i)_{W−}
p is in M(i)_{W−} iff q is in M(i)_{W+}
Otherwise, p does not have a truth‐value in M(i).
5. If p is q v r,
p is in M(i)_{W+} iff (q is in M(i)_{W+} or r is in M(i)_{W+})
p is in M(i)_{W−} iff (q is in M(i)_{W−} and r is in M(i)_{W−})
Otherwise, p does not have a truth‐value in M(i).
6. If p is a forced wff,
p does not have a truth‐value in M(i).
The wffs which are initial for M*(0) are, in effect, stipulated to be inappropriate when M*(0) is constructed. Thus, the following definition of p is appropriate in M(i) begins with a clause which establishes this stipulation:
If M(i) is M*(j) for some j and p is initial for M*(0), then p is inappropriate (i.e. is in M(i)_{G−}) in M(i). Otherwise, whether p is appropriate or inappropriate in M(i) is determined by the following rules: (p.160)
1. If p is atomic,
p is in M(i)_{G+} iff D(p) = t
p is in M(i)_{G−} iff D(p) is not t.
2. If p is Tr(a) and a names q,
p is in M(i)_{G−} if q is forced. Otherwise:
p is in M(i)_{G+} iff q is in M(i)_{T+}
p is in M(i)_{G−} iff q is in M(i)_{T−}
Otherwise, p is neither app. nor inapp. in M(i).
3. If p is Ap(a) and a names q,
p is in M(i)_{G+} iff q is in M(i)_{A+}
p is in M(i)_{G−} iff q is in M(i)_{A−}
Otherwise, p is neither app. nor inapp. in M(i).
4. If p is −q,
p is in M(i)_{G+} iff q is in M(i)_{W−}
p is in M(i)_{G−} iff (q is in M(i)_{W+} or q is in M(i)_{G+})
Otherwise, p is neither app. nor inapp. in M(i).
5. If p is q v r,
p is in M(i)_{G+} iff (q is in M(i)_{W+} or r is in M(i)_{W+})
p is in M(i)_{G−} iff both q and r are in either M(i)_{W−}
or M(i)_{G−}
Otherwise, p is neither app. nor inapp. in M(i).
6. If p is not q,
p is in M(i)_{G+} iff q is in M(i)_{G−}
p is in M(i)_{G−} iff q is in M(i)_{G+}
Otherwise, p is neither app. nor inapp. in M(i).
7. If p is q or r,
p is in M(i)_{G+} iff at least one of q, r is in M(i)_{G+}
P is in M(i)_{G−} iff both q and r are in M(i)_{G−}
Otherwise, p is neither app. nor inapp. in M(i).
I.3.3
Fixed points. Let M and M′ be two models in the hierarchy. M′_{E} extends M_{E} provided that each member of the first is, pointwise, a superset of each member of the second; M′_{V} extends M_{V} if the same relation holds between them; M′ extends M when M′_{E} extends M_{E} and M′_{V} extends M_{V}. Notice that by the construction of M(0) and M*(0), M(1)_{E} extends M(0)_{E}, and M*(1)_{E} extends M*(0)_{E}. Given this, from
it follows that
(I) For any models M and M′, if M′_{E} extends M_{E}, then M′_{V} extends M_{V}.
For, since M(i + 1)_{E} is M(i)_{V}, I guarantees that if M(i)_{E} extends M(i − 1)_{E}, then M(i + 1)_{V} extends M(i)_{V}. So M(j) must extend M(i) when j > i. This monotonicity entails (II).
(II) Both the fixed points F(0) and F(1) exist.
We prove (I) by an induction on the complexity of a wff p, showing that if M′_{E} extends M_{E}, then if p is in a given member of M_{V}, it is in the corresponding member of M′_{V}. The proof is somewhat tedious, because there are four cases to consider for each possible form of a wff. Since the denotation of terms does not shift across models, in what follows we write such things as a is in M(i) _{T+}, instead of the somewhat more cumbersome D(a) is in M(i) _{T+}.
(p.161) Suppose that M′_{E} extends M_{E}; call this assumption E. Basis. When p is a propositional letter, p's status with respect to whether it is true or false, and to whether it is appropriate or inappropriate, is fixed by whether it is true or otherwise, which is in turn fixed by D, which does not vary across models. So, trivially, if p is in M_{W+} (M_{W−}, . . .), it is in M′_{W+} (M′_{W−}, . . .). Suppose that p is of the form Tr(a). Then
Suppose that p is of the form Ap(a). ThenIf p is in M_{W+}, then a is in M_{T+} (truth definition=TD), so a is in M_{T+} (by E), and so p is in M′_{W+} (definition of truth=TD).
If p is in M_{W−}, then a is in M_{T−} (TD), so a is in M′_{T−} (E), so p is in M′_{W−} (TD).
If p is in M_{G+}, then a is in M_{T+} (definition of appropriateness=AD), so a is in M=_{T+} (E), so p is in M=_{G+} (definition of appropriateness=AD).
Suppose p is in M_{G−}. If a is forced, p must be in M′_{G−} (TD). If a is unforced, then a is in M_{T−} (AD), so a is in M′_{T−} (E), so p is in M′_{G−} (AD).
If p is in M_{W+}, then a is in M_{A+} (TD), so a is in M′_{A+} (E), so p is in M′_{W+} (TD).
If p is in M_{W−}, then a is in M_{A−} (TD), so a is in M′_{A−} (E), so p is in M′_{W−} (TD).
If p is in M_{G+}, then a is in M_{A+} (AD), so a is in M′_{A+} (E), so p is in M′_{G+} (AD).
If p is in M_{G−}, then a is in M_{A−} (AD), so a is in M′_{A−} (E), so p is in M′_{G−} (AD).
Use both force operators and truth functors to determine logical complexity. Induction step. The induction hypothesis (IH) is that if M′_{E} extends M_{E}, then for any sentence S of logical complexity less than n, if S is in M_{V}, then it is in M′_{V}. Again assume that M′_{E} extends M_{E} and call the assumption E. Suppose that p is of length n. If p is −q, then
The argument for the case in which p is q v r is similar enough to the last that it can be left as an exercise. Suppose p is not q. Here, p cannot be in M_{W+} or M_{W−}, as it is forced.If p is in M_{W+}, then q is in M_{W−} (TD), so by IH q is in M′_{W−}, so p is in M′_{W+} (TD). Analogously, if p is in M_{W−}.
If p is in M_{G+}, then q is in M_{W−} (AD), so q is in M′_{W−} (IH), so p is in M′_{G+}. Analogously, if p is in M_{G−}.
The case where p is q or r is routine.If p is in M_{G+}, then q is in M_{G−} (AD), so by IH q is in M′_{G−}, so p is in M′_{G+} (AD). Analogously, for the case in which p is in M_{G−}.
I.3.4
Consistency. It is not altogether obvious that in F(1) no wff has incompatible properties—that, for example, there is no p that is true and inappropriate, or false and appropriate. So it is worth proving:
Consider first the sequence from M(0) to F(0), and the status therein of those wffs whose family does not contain a sentence in which ‘Tr’ or ‘Ap’ occurs. A routine induction (p.162) on logical complexity shows that (III) holds for these wffs in these models. Using that result to prove the basis, one can then show by induction on the number of occurrences of ‘Tr’ and ‘Ap’ in a wff that the theorem holds for all wffs in the set of models. Given this, the theorem holds at M′(0), and thus another induction aping the two inductions just mentioned shows that the theorem holds in the sequence from M′(0) to F(1).
(III) In each model M, no wff in M_{W+} is in M_{W−} or M_{G−}; no wff in M_{G+} is in M_{W−} or M_{G−}.
I.3.5
Sample applications. Consider the wffs
Neither of these wffs is made true or false in any model through F(0), and thus neither is made appropriate or inappropriate either. Thus, each is added to A−, the anti‐extension of ‘Ap’ in M*(0), and each is stipulated to be inappropriate (i.e. to be in M*(0)_{G}). Thus, in M*(0) the wffs
(a) −Ta
(b) Ta
are appropriate. Furthermore, in M*(0), since a and b are in A−, the wffs
(c) not −Ta
(d) not Ta
will be false, since the relevant sentences are in the anti‐extension of ‘Ap’. Thus,
(e) Ap(a)
(f) Ap(b)
are both true in M*(0) and F(1). Thus, in M*(0) and F(1) we can say truly what the status of a liar sentence such as (a) is: we can say, truly, that it is inappropriate. Finally, note that in any model in which ‘a’ names ‘−Tr(a)’, c and g will both be appropriate. Thus, the sentence
(g) −Ap(a)
(h) −Ap(b)
will be a logical truth, in the sense of being c‐valid.^{9} Thus (speaking a little loosely), when we truly say that a liar sentence such as a is inappropriate, we say something logically equivalent to what we say when we deny that the liar is true.If a = ‘−Ta’, then (not −Tr(a) iff −Ap(a)).
What prevents paradox from occurring in the language? Let us look at a simple attempt to construct a paradoxical sentence:
If we try to ape canonical derivations of a contradiction from the strengthened liar using (i), we will have to employ both (p.163)
(i) −Ap(i)
One might think that these should both be appropriate (so that ‘not Ap(i) iff Ap(i)’ would be): (j), one might think, should hold because, quite generally, a sentence is inappropriate iff it is not true; (k) because −Ap(i) is true iff it is appropriate. But in fact neither (j) nor (k) is appropriate. The principle that a sentence is inappropriate iff not true holds only for sentences whose family is ‘Ap’‐free; sentences such as (k) will be neither appropriate nor inappropriate when their flanking formulas suffer from an appropriateness gap.
(j) −Ap(i) iff not Ap(i)
(k) −Ap(i) iff Ap(‘−Ap(i)’)
Finally, consider the wff
What is its status in the final fixed point for the language? It should be clear that it is neither appropriate nor inappropriate (and neither true nor false). For (l) is not safe for the model based on the first fixed point (as it contains ‘Ap’), and thus will not be declared inappropriate therein. (And obviously it can't become so before this model.) Given this, its appropriateness will continue to be undecided through the second fixed point.
(l) not Ap(l)
Will (l) lead to a paradox‐like situation? It would if the schema
were c‐valid. And this schema would be valid, if every wff was declared appropriate or inappropriate in the second fixed point. But this does not happen. So far as I can see, (l) leads to no paradox‐like problems.
(m) Ap(‘S’) iff S
To generalize the preceding to a first‐order language requires generalizing the notion of the family of a wff, in order to adequately characterize the class of sentences declared inappropriate in the construction of the model M*(0). The way to do this, I think, is to make the family of a wff a class of (atomic) predicates: The family of an atom contains the predicate from which it is constructed; the family of Tr(α) contains the family of (what is named by) α as well as (what is named by) α itself; etc. In the construction of M*(0), we put a sentence in the anti‐extension of ‘Ap’ if its family is ‘Ap’‐free and the sentence's truth‐value has not been decided in the fixed point on which M*(0) is based. I leave an investigation of the details to anyone who has gotten this far.
I.4
Perhaps you feel like echoing Kripke:
Well, no and yes.there are claims we can make about the object language which we cannot make in the object language. For example, sentences such as a = ‘not Ap(a)’ are not appropriate in the object language . . . but we are precluded from saying this in the object language by our interpretation of denial and the appropriateness predicate. . . . The ghost of the hierarchy is still with us.
Surely we can say in the language of sentence a that it is not appropriate. Why shouldn't there be a kind of speech act—call it rejection—not definable in terms of assertion, denial, or the other speech acts introduced in Chapter 2, which is sometimes voiced with the (p.164) idioms of negation? After all, if it is possible to introduce an appropriateness predicate, sentence (a) is a perfectly good sentence, and we can see that the claim that it makes isn't right. In saying that this claim ‘isn't right’ we aren't asserting anything—if we were, contradiction would ensue.^{10} In saying that a isn't appropriate, we can't be denying anything either. So we must be doing something else.
There is no reason that rejection must be ‘lexicalized’—either with a construction conventionally marked to express it, or via a predicate for classifying utterances which achieve it—any more than assertion or denial need be. (Which came first, warnings or ‘warning’?) Neither need we have a well‐developed concept of rejection—or of denial as something distinct from the assertion of negation—in order to reject a denial. That a particular utterance is a denial, as opposed to the assertion of a negation, or is a rejection, as opposed to an assertion or denial—this is surely often a matter of what interpretation of the utterance makes the most sense of it.
Rejection must be like assertion and denial insofar as we can speak of a rejection as ‘getting it right’ or ‘getting it wrong’. Getting it right when one rejects something can't be a matter of truth or appropriateness; it must be a matter of the thing being rejected having yet some other property, being fit we might call it. And so it does look like we are still haunted by the hierarchy. For one suspects this process will never stop. Fitness is as much a semantic property as appropriateness; so we have to introduce a fitness predicate, if we want to say everything there is about semantics. But then there will be a sentence that says of itself that it's not fit. In evaluating it, we will be driven again to yet a higher level, and so on and on. If one tells the story I am telling, one must be at peace with the idea of an endless sequence of sui generis semantic properties evaluating an endless sequence of kinds of denial. Since each property in the sequence is sui generis with respect to the earlier ones, no language can express all the facts, semantic or otherwise.
Is this problematic? What seems most problematic about the hierarchy of truth predicates associated with Tarski's work on truth is the idea that there is a property—that expressed by the use of ‘true sentence of English’—about which it is impossible to adequately theorize. On the standard way of applying Tarski's ideas about truth to natural languages, we understand ‘true [in English]’ as (at least potentially) infinitely ambiguous, since a Tarskian truth predicate cannot be true of a sentence in which it occurs.^{11} There is, on this view, no understanding of ‘true English sentence’ on which it is true of . . . well, every true English sentence.^{12} Do we encounter this sort of problem on the current account?
Well, it doesn't seem that we do. Truth in L, on the Kripkean approach we've hijacked here, is of course defined in L. And so is appropriateness in L. There's no reason to think that the same thing won't be true of fitness in L or any of the other residents of the semantic stable. Neither does the charge, that we are unable to adequately theorize about the property of truth (or about the others) seem to be correct. The point here is perhaps seen most clearly if we consider the language discussed in Section I.1 above, which extends the (semantic‐vocabulary‐free) language of Section 2.5 simply by the addition of a truth (p.165) predicate. In this language, we are not only able to define truth, we are able to do such things as say that its liars are not [denial] true, and (using, of course, forced sentences) to give an accurate account of the conditions under which (unforced) sentences are true and false. We can, in this language, theorize about truth in the language.
It seems—though I have nothing like a proof for this—that something cognate is true of the language of the last section and the properties of both truth in that language and appropriateness in that language. Speaking that language and performing acts such as the above‐posited act of rejection, we should be able to adequately theorize about both truth and appropriateness in the language. The language, of course, isn't ‘semantically closed’. Its sentences will be fit or unfit, but—lacking a fitness predicate—one isn't able to ascribe fitness to sentences. But the language does seem able to ‘state all the facts’ about the truth, falsity, appropriateness, and inappropriateness of its sentences.
So far as I can see this sort of thing will continue as we rise in the hierarchy: For any finite set of the properties in the sequence that begins ‘truth, aptness, fitness, . . . ’ a language can contain predicates which characterize each of the properties (ones which have the properties' extensions and anti‐extensions), and its speakers can, by performing the appropriate speech acts (whose performance may or may not be associated with lexical items) indicate exactly how these properties are distributed to the sentences of the language. The apparent infinitude of semantic properties makes semantic closure impossible. But it would seem that it is possible—if the sort of thing just indicated does indeed continue as we ascend the hierarchy—to say as much as there is to say about the distribution of any finite collection of semantic properties to a language's sentences. Since in some fairly clear sense we can focus on only finitely many semantic properties anyway, we can, that is, say all there is to say about all of the semantic properties that we can or could actually get a fix on. And perhaps that's all one could ever want or need, in the way of semantic effability.
Notes:
(2) (I) completely ignores subtleties connected with the hyperintensionality of contexts like ‘says that . . . ’.
(3) We are at the moment operating with the definition of appropriateness given in Section 2.5. That definition undergoes modification below, when we add an appropriateness predicate to the language.
(4) If we think of the words of 3″ as retaining their English senses, then it is fair to say that 3″ literally says that t is a sentence of P that is not true.
(5) Lexicalizing force—introducing operators whose utterance encodes force—simplifies things by making it possible to make both the truth and the appropriateness predicate apply to the same sorts of things.
(6) This is perhaps too facile. Perhaps atomic sentences without semantic vocabulary can suffer from appropriateness gaps. I will just ignore this possibility.
(7) In the minimal fixed point. We can, of course, stipulate that some of these sentences—in particular, the truth‐teller—are true.
(8) Anyone familiar with Soames's elegant discussion (1999) of Kripke (1975) will recognize that the presentation in this section is modeled on Soames's.
(9) We assume: ‘iff’ is introduced so that p iff q abbreviates not(p or q) or not(not p or not q); analogously for ‘if . . . then’; ‘=’ means equality.
(10) I assume, of course, that an assertion is apt iff what's asserted is true.
(11) This is an upshot of Tarski's procedure of only defining true sentence of language L in a (‘richer’) language than L, so that such a predicate is never a part of the language of whose sentences it is potentially true.