1.1 Generalized schematic definitions

Our aim in offering a revision semantics is to enrich a classical ground language with circularly defined predicates.Footnote ⁴ For our ground language $\mathscr L$ we may take any first-order language with identity and the standard classical connectives.Footnote ⁵ We will call a classical model $M \ (= \langle D, I \rangle )$ for $\mathscr L$ a ground model. D is the domain of M. I, the interpretation function of M, assigns each n-place predicate a set of n-tuples from $D^n$ , each individual constant an element of D, and each n-place function symbol a function $D^n \to D$ . For a set of sentences $\Gamma $ , I’ll use $M \models \Gamma $ to say that $M \models A$ for all $A \in \Gamma $ .

We expand $\mathscr L$ to $\mathscr L^+$ by introducing a countable (finite or infinite) set of new unary predicates $G_1, G_2, \ldots $ , each defined by a corresponding definition $\mathscr D_i$ . Let $\mathscr D$ be the set of all such definitions $\mathscr D_i$ . We may assume that all definitions in $\mathscr D$ conform to the highly flexible form of generalized schematic definitions [Reference Gupta13, sec. 145].Footnote ⁶ A generalized schematic definition of G uses a (generalized) scheme $\phi $ , which is a function:

1. (i) that maps none, some, or all closed terms t of $\mathscr L$ to sentences $\phi _t$ of $\mathscr L^+$ (which may contain any of $G_1, G_2, \ldots $ ); and

2. (ii) that maps one variable x of $\mathscr L$ to a formula $\phi _x$ of $\mathscr L^+$ with only x free.

The scheme $\phi $ is a function in our metalanguage, and need not (though may) be represented in $\mathscr L^+$ . Given a scheme $\phi _i$ for one of our $G_i$ , we may write a generalized schematic definition as

(𝒟 _i ) $$\begin{align} G_i t =_{Df} \phi_{i,t}; \text{ otherwise, } G_i x =_{Df} \phi_{i,x}. \end{align}$$

The first clause is called the partial-definition clause of 𝒟 _i ; the second is the explicit definition clause. $G_i$ is the defined term. We say that $\phi _{i,t}$ [respectively, $\phi _{i,x}$ ] is the definiens for $G_i t$ [ $ \! G_i x$ ] in $\phi _i$ (and also in 𝒟 _i ). When the domain of $\phi _i$ contains finitely many closed terms and $\phi _{i,x} = x \neq x$ , we obtain the partial definition

(𝒟 _partial ) $$\begin{align} G_i t_0 =_{Df} A_0; \quad G_i t_1 =_{Df} A_1; \quad \ldots ; \; \quad G_i t_n =_{Df} A_n. \end{align}$$

Alternatively, we can let the domain of $\phi _i$ contain only x to obtain the explicit definition

(𝒟 _exp ) $$\begin{align} G_i x =_{Df} \phi_{i,x}. \end{align}$$

When $\phi _i$ contains terms other than x in its domain, the definition 𝒟 _i carries a presupposition, notated $Pr_{\mathscr D_i}$ , that distinct closed terms in the domain of $\phi _i$ refer to different things. We may identify $Pr_{\mathscr D_i}$ with the set of sentences of the form $t_1 \neq t_2$ , where $t_1$ and $t_2$ are distinct closed terms in the domain of $\phi _i$ . A ground model M is normal for 𝒟 _i iff $M \models A$ for all $A \in Pr_{\mathscr D_i}$ . According to the semantical system presented below, when a model is not normal for a definition 𝒟 _i of $G_i$ , $G_i$ will be categorically false of everything in the domain.

1.2 Categorical truth in $\mathbf {S}^{\#}$ and $\mathbf {S}^{\#N}$

We will define our systems’ key semantical notion of categorical truth by isolating a set of “optimal” hypotheses, or interpretations of the defined terms.

Let $\mathscr L$ be a ground language and $M \ (= \langle D, I \rangle )$ a ground model for $\mathscr L$ . A hypothesis for a set of defined predicates $G_1, G_2, \ldots $ is a function h that assigns a candidate extension $h(G_i) \subseteq D$ to each defined predicate $G_i$ . The set of hypotheses is the set of all such functions, which we will denote $Hyp$ . A ground model M may be supplemented with any $h \in Hyp$ to produce the model $M + h$ that is just like M except that each $G_i$ is assigned the extension $h(G_i)$ . When there is only one defined term, we’ll abuse notation by treating h simply as the extension of $G_1$ .

We can view circular definitions as instructions for refining hypotheses. Relative to a ground model M, the set of definitions $\mathscr D$ yields a revision rule $\delta _{\mathscr D, M} : Hyp \to Hyp $ , defined as follows. For all i, hypotheses h, and $d \in D$ : $d \in \delta _{\mathscr D, M}(h)(G_i)$ iff either

1. (i) M is normal for 𝒟 _i and there is a term t in the domain of $\phi _i$ such that $I(t) = d$ and $\phi _{i,t}$ is true in $M + h$ ; or

2. (ii) M is normal for 𝒟 _i , there is no such term t, and d satisfies $\phi _{i,x}$ in $M + h$ .

It follows that when M is not normal for 𝒟 _i the extension $\delta _{\mathscr D, M}(h)(G_i)$ is empty. Informally speaking, given a hypothesis h, the revision rule produces revised extensions $\delta _{\mathscr D, M}(h)(G_1)$ , $\delta _{\mathscr D, M}(h)(G_2), \ldots $ comprising the members of the domain that would satisfy the definientia according to h.

We can distinguish optimal hypotheses using revision sequences. Let $On$ be the class of all ordinals, let $\mathscr S$ be an $On$ -long sequence of elements of $Hyp$ , and let $\mathscr S_{\beta }$ be the $\beta $ th member of $\mathscr S$ .Footnote ^{7 ,} Footnote ⁸ If $\alpha $ is a limit ordinal, then an element $d \in D$ is stably $G_i$ [stably not $G_i$ ] in $\mathscr S$ at $\alpha $ iff

$$ \begin{align*} \exists \gamma < \alpha \ \forall \beta \ (\text{if } \gamma \leq \beta < \alpha \text{ then } d \in [\notin] \mathscr S_{\beta}(G_i)). \end{align*} $$

We say that d is stable for $G_i$ in $\mathscr S$ at $\alpha $ iff d is either stably $G_i$ or stably not $G_i$ in $\mathscr S$ at $\alpha $ ; otherwise, d is unstable for $G_i$ in $\mathscr S$ at $\alpha $ . A hypothesis h coheres with $\mathscr S$ at a limit ordinal $\alpha $ iff for all $G_i$ :

1. (i) if $d \in D$ is stably $G_i$ in $\mathscr S$ at $\alpha $ then $d \in h(G_i)$ , and

2. (ii) if $d \in D$ is stably not $G_i$ in $\mathscr S$ at $\alpha $ then $d \notin h(G_i)$ .

Finally, $\mathscr S$ is a revision sequence for a revision rule $\rho $ iff $\mathscr S$ is an $On$ -long sequence of hypotheses such that, for all ordinals $\alpha $ and $\beta $ ,

1. (i) if $ \alpha = \beta +1$ then $\mathscr S_{\alpha } = \rho (\mathscr S_{\beta })$ , and

2. (ii) if $\alpha $ is limit then $\mathscr S_{\alpha }$ coheres with $\mathscr S$ at $\alpha $ .

A hypothesis h is cofinal in a revision sequence $\mathscr S$ iff for all $\alpha $ some $\beta \geq \alpha $ is such that $\mathscr S_{\beta } = h$ . Cardinality considerations imply that all revision sequences contain cofinal hypotheses (RTT, Theorem 5C.7). A hypothesis h is recurring for a revision rule $\rho $ iff h is cofinal in some revision sequence for $\rho $ . Since the revision process never eliminates recurring hypotheses, we can reasonably consider these optimal for $\rho $ .

The definitions of categorical truth in $\mathbf {S}^{\#}$ and $\mathbf {S}^{\#N}$ are most transparent when set alongside the corresponding definition of a weaker system, $\mathbf {S}^*$ , presented in RTT. This system is the most well-known $\omega $ -consistent alternative to $\mathbf {S}^{\#}$ . It defines categorical truth as truth on all recurring hypotheses.

Definition 1.1 (Categorical truth in $\mathbf {S}^*$ )

Let A be a sentence of $\mathscr L^+$ and M a ground model. A is categorically true, or simply categorical,Footnote ⁹ in M relative to $\mathscr D$ in the system $\mathbf {S}^*$ (notation: $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\:\! *}}\,\,} A$ ) iff

$$ \begin{align*} \forall h \text{ recurring for } \delta_{\mathscr D, M}, M + \delta_{\mathscr D, M} (h) \models A. \end{align*} $$

In $\mathbf {S}^*$ , as well as the systems presented below, a sentence is categorically false iff its negation is categorically true, and pathological iff it is neither categorically true nor categorically false.Footnote ¹⁰

Unfortunately, $\mathbf {S}^*$ ’s definition of categorical truth is unsatisfactory, for many of the recurring hypotheses contain disruptive artifacts that creep in at limit stages (see Example 6.3). We can clean the recurring hypotheses up by revising them finitely many more times. This motivates Definition 1.2. In this definition, $\rho ^n (h)$ denotes the result of applying the revision rule $\rho\ n$ times to h, with $\rho ^0(h) = h$ .

Definition 1.2 (Categorical truth in $\mathbf {S}^{\#}$ )

Let A be a sentence of $\mathscr L^+$ and M a ground model. A is categorically true, or categorical, in M relative to $\mathscr D$ in the system $\mathbf {S}^{\#}$ (notation: $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\#}}\,\,} A$ ) iff

(1) $$ \begin{align} \exists n \forall h \text{ recurring for } \delta_{\mathscr D, M}, M + \delta_{\mathscr D, M}^n (h) \models A. \end{align} $$

Again, a sentence is categorically false iff its negation is categorically true.Footnote ¹¹

My new systems $\mathbf {S}^{\#N}$ offer a different way to improve upon $\mathbf {S}^*$ : rather than revising the recurring hypotheses an arbitrary number of times, as in Definition 1.2, we revise them only up to some fixed natural number N. We may call the parameter N a revision strength.Footnote ¹² I’ll postpone further philosophical discussion of revision strengths to §3. For now, we have the following definition.

Definition 1.3 (Categorical truth in the systems $\mathbf {S}^{\#N}$ )

Let A be a sentence of $\mathscr L^+$ , M a ground model, and $N \in \mathbb {N}$ . A is categorically true, or categorical, in M relative to $\mathscr D$ in the system $\mathbf {S}^{\#N}$ (notation: $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{ \#N}}\,\,} A$ ) iff

$$ \begin{align*} \forall h \text{ recurring for } \delta_{\mathscr D, M}, M + \delta_{\mathscr D, M}^N (h) \models A. \end{align*} $$

I wishFootnote ¹³ to stress that, since N is a parameter that may vary, Definition 1.3 defines a sequence of revision systems $\mathbf {S}^{\#0}$ , $\mathbf {S}^{\#1}$ , etc.

The term “revision strength” is motivated by the fact that increasing N results in more and more sentences becoming categorical in $\mathbf {S}^{\#N}$ . $\mathbf {S}^{\#0}$ is just $\mathbf {S}^*$ ; increasing N results in progressively stronger systems, with $\mathbf {S}^{\#}$ the limit of this sequence. Relatedly, we have the following proposition, which will be important for my subsequent discussion.

6.1 The calculus $\mathbf {C}$

The characteristic feature of deductions in $\mathbf {C}$ is that to each formula is attached an index $j \in \mathbb {Z}$ , written as a superscript; e.g., $A^j$ .Footnote ²⁵ One can think of the indices as distinguishing between revision stages: a move from index j to $j+1$ indicates a move to the next stage in a revision sequence.

The inference rules for definitions illustrate the index policy. If $G_i$ is defined by the explicit definition

(𝒟 _i ) $$\begin{align} G_i x =_{Df} \phi_{i,x} \end{align}$$

we have the following Definiendum Introduction and Elimination rules, for all indices j Footnote ²⁶ :

1. (DfI _r ) $\phi _{i,x}[t/x]^{j}$ ; therefore $G_{i} t^{j+1}$ .

2. (DfE _r ) $G_i t^{j+1}$ ; therefore $\phi _{i,x} [t/x]^j$ .

Here $\phi _{i,x} [t/x]$ is the result of uniformly substituting a term t for all free occurrences of x in $\phi _{i,x}$ . One can notice a tight correspondence between the definition rules and the definition’s rule of revision.

All deductions of classical logic are legitimate in $\mathbf {C}$ , so long as the premises and conclusion bear the same index. Moreover, one may use versions of Reductio and Universal Generalization that relax the restrictions on indices. The rule supplying classical logic is:

1. Classical Logic (CL) (i) If the rules of classical first-order logic with identity allow the derivation of B from $A_1, \ldots , A_n$ , then $B^j$ may be derived from $A_1^j, \ldots , A_n^j$ , for any integer j.

2. (ii) If $[B \wedge \neg B]^{j_{n+2}}$ is derivable from $A_1^{j_1}, \ldots , A_{n+1}^{j_{n+1}}$ , then $[\neg A_{n+1}]^{j_{n+1}}$ is derivable from $A_1^{j_1}, \ldots , A_{n}^{j_n}$ .

3. (iii) If $[A_{n+1}(z)]^{j_{n+1}}$ is derivable from $A_1^{j_1}, \ldots , A_{n}^{j_n}$ and the variable z is not free in $A_1, \ldots , A_n$ , then $[\forall z A_{n+1}(z)]^{j_{n+1}}$ is derivable from $A_1^{j_1}, \ldots , A_{n}^{j_n}$ .

Finally, we may shift the index of any formula not containing defined terms:

1. Index Shift (IS) If A does not contain any defined term, then, for any integers $j_1$ and $j_2$ , $A^{j_2}$ may be deduced from $A^{j_1}$ .

We now define $\mathbf {C}$ as the proof system consisting of DfI _r , DfE _r , CL, and IS. To define derivability in the non-indexed language, say that a sentence B is derivable in $\mathbf {C}$ from sentences $A_1, A_2, \ldots $ on the basis of definition $\mathscr D$ (notation: $A_1, A_2, \ldots {\vdash \!\!\!\!^{^{\mathscr D}}_{_{}}\,\,} B$ ) iff, for some integer j, $B^j$ can be derived from $A_1^j, A_2^j, \ldots $ using the rules of $\mathbf {C}$ (notation: $A_1^j, A_2^j, \ldots {\vdash \!\!\!\!^{^{\mathscr D}}_{_{}}\,\,} B^j)$ . It is straightforward to check that $\mathbf {C}$ is sound for $\mathbf {S}^{\#}$ for explicit definitions.Footnote ²⁷

For an example of a deduction in $\mathbf {C}$ , we can use definition 𝒟₀ from Example 2.1 to prove the theorem $Ha \to Hb$ :

Long description

The table consists of three columns: step number, logical formula, and justification.

* Step 1: Vertical bar followed by open bracket H a wedge neg H b close bracket super 0. Justification: Ass.

* Step 2: Vertical bar followed by H a super 0. Justification: 1; C L.

* Step 3: Vertical bar followed by open bracket open parenthesis a = a vee a = b close parenthesis wedge neg H a wedge neg H b close bracket super -1. Justification: 2; D f E sub r.

* Step 4: Vertical bar followed by open bracket open parenthesis b = a vee b = b close parenthesis wedge neg H a wedge neg H b close bracket super -1. Justification: 3; C L.

* Step 5: Vertical bar followed by H b super 0. Justification: 4; D f I sub r.

* Step 6: Open bracket neg open parenthesis H a wedge neg H b close parenthesis close bracket super 0. Justification: 1-5; C L.

* Step 7: Open bracket H a arrow H b close bracket super 0. Justification: 6; C L.

6.2 The calculi $\mathbf {C}^{N}$

For reasoning with explicit definitions in the system $\mathbf {S}^{\#N}$ , we can use the calculus $\mathbf {C}^N$ , a weakening of $\mathbf {C}$ . As before, an index j is attached to each formula in a $\mathbf {C}^N$ -deduction. Now, however, no negative indices are allowed; indices are always natural numbers. The rules for deductions in $\mathbf {C}^N$ are otherwise exactly as in $\mathbf {C}$ . The revision strength N determines how liberal our definition of derivability will be. We say that the sentence B of $\mathscr L^+$ is derivable in $\mathbf {C}^N$ from $A_1, A_2, \ldots $ on the basis of definition $\mathscr D$ (notation: $A_1, A_2, \ldots\ {\vdash \!\!\!\!^{^{\mathscr D}}_{_{N}}}\ B$ ) iff, for some natural number $j \leq N$ , $B^j$ can be derived using the rules of $\mathbf {C}^N$ from $A_1^j, A_2^j, \ldots $ (notation: $A_1^{j}, A_2^{j}, \ldots\ {\vdash \!\!\!\!^{^{\mathscr D}}_{_{N}}}\ B^j$ ). It is again straightforward to check that $\mathbf {C}^N$ is sound for $\mathbf {S}^{\#N}$ for explicit definitions.

If we wished to carry out the above sample derivation in $\mathbf {C}^N$ , we would only need to increment each index by $1$ . (We assume $N \geq 1$ ; $N = 0$ is just classical logic.)

6.3 A test case: Finite natural implication

Gupta’s desideratum of preserving finite natural implication concerns the specially well-behaved class of finite definitions. Our main example of a finite definition will be 𝒟₀ from Example 2.1, which is in fact the only finite definition appearing in this paper. We characterize these definitions as follows. A hypothesis h is finitely reflexive for a revision rule $\rho $ iff there is $n> 0$ (called h’s period) such that $\rho ^n(h) = h$ . In the model M of Example 2.1, the hypotheses $\varnothing $ and $\{0,1\}$ (and only these) are finitely reflexive, with period $2$ . A definition $\mathscr D$ is finite iff, for all ground models M there is a number n such that, for all hypotheses h, $\delta _{\mathscr D, M}^n(h)$ is finitely reflexive. Note that all partial definitions (§1.1) are finite.

An attractive feature of finite definitions is that we can eliminate all “suboptimal” hypotheses after finitely many revision stages. In view of this fact, finite definitions can be given a natural semantics: we declare a sentence A categorically true in M relative to a finite definition $\mathscr D$ iff it is true on all hypotheses finitely reflexive for $\delta _{\mathscr D, M}$ . Finite natural implication is the corresponding implication relation restricted to finite definitions. That is, where $\mathscr D$ is finite, B is a finite natural implication of $\Gamma $ (notation: $\Gamma {\models \!\!\!\!\!^{^{\mathscr D}}_{_{f}}\,\,} B$ ) iff for all ground models M,

$$ \begin{align*} \forall h \left(\text{if } h \text{is finitely reflexive for } \delta_{\mathscr D, M}, \text{ then } M + h \models \Gamma \text{ implies } M + h \models B \right). \end{align*} $$

As I understand it, Gupta’s desideratum for a proof system $\mathbf {C}^{\circ }$ is:

1. (FNI) $\mathbf {C}^{\circ }$ preserves (i.e., is complete for) finite natural implication.

The next proposition, proved in the Appendix, compares $\mathbf {C}$ and the calculi $\mathbf {C}^{N}$ with respect to FNI.

I’ll unpack the significance of these three claims in turn. Gupta’s argument for $\omega $ -inconsistency from FNI proceeds by observing (i) and that FNI is not satisfied by any known proof system sound for an $\omega $ -consistent semantics. To illustrate the point, consider Example 2.1 again. The sentence $Ha \leftrightarrow Hb$ is a finite natural implication of the empty premise set, yet Table 1 implies that no proof system for $\mathbf {S}^*$ can preserve this entailment. Elsewhere Gupta [Reference Gupta13, sec. 176] refers to this weakness of $\mathbf {S}^*$ as a “fatal flaw.”

As stated, it is unclear whether the calculi $\mathbf {C}^N$ satisfy FNI: individually, they don’t (claim (ii)), but, collectively, they do (claim (iii)). I suggest that the situation is ultimately favorable for these calculi: given my proposed understanding of revision strengths, claim (iii) is all we should desire. If this is true, finite natural implication does not in fact favor $\mathbf {C}$ over the calculi $\mathbf {C}^N$ .

Whether this is plausible depends on the considerations motivating FNI. First, Gupta claims that a proof system should preserve finite natural implications because they are “intuitively compelling” [Reference Gupta12, p. 190]. (Witness the straightforward proof of $Ha \to Hb$ on p. 17.) I agree that it would be a red flag if the entire apparatus of proof calculi $\mathbf {C}^N$ couldn’t capture some finite natural implication. But when we have multiple calculi at our disposal, I see no problem with providing, for each finite definition, the revision strength appropriate to capture its finite natural implications.

Gupta’s second motivation for FNI points out that finite definitions are needed for applications of the Revision Theory, such as to rational choice, and the applications require that we can reason with these definitions.Footnote ²⁸ But when stipulating a set $\mathscr D$ of finite definitions for the purposes of some application, we can always stipulate a revision strength N for which $\mathbf {C}^N$ preserves finite natural implication. (We just need to say, “Let the revision strength of our dialect be the minimum strength for which finite natural implication is preserved.”) Moreover, the finite definitions employed in applications can be easily transformed into the especially manageable form mentioned by Gupta [Reference Gupta, Chapuis and Gupta10]. His observations there show that from the syntactic form of such definitions one can read off an explicit revision strength that preserves finite natural implications. Even when users of the definition don’t stipulate a strength, Agreement implies that, on the default strength N, $\mathbf {S}^{\#N}$ will agree with $\mathbf {S}^{\#}$ on categorical truth for sentences in use. From this fact one can show that finite natural implications on sentences used in the application are guaranteed to preserve categorical truth.Footnote ²⁹ In summary: the calculi $\mathbf {C}^{N}$ provide for each job a tool that can handle it. We need not wish further for a single “supertool” that can perform every job.

7.1 Compositional truth principles

Compositional truth principles say that a conjunction is true iff its conjuncts are; that the negation of a sentence is true iff the sentence is not true; and so on. These principles are key features of how the concept of truth functions in semantics. Unlike other leading classical truth theories, $\mathbf {S}^{\#}$ and $\mathbf {S}^{\#N}$ both validate all the compositional principles.Footnote ³⁰ To make this claim precise, suppose $\langle M, \ulcorner \ \urcorner \rangle $ is a full sentential frame for $\mathscr L^+$ , which contains a truth predicate. Further suppose that the ground language $\mathscr L$ contains the predicates $Neg(x,y)$ , $Conj(x,y,z)$ , $Ui(x,y)$ , and $Ref(x,y)$ , which M interprets as follows. For $d_1, d_2, d_3 \in D$ ,

• • $\langle d_1, d_2 \rangle \in I(Neg)$ iff $d_1, d_2 \in \mathscr L^+$ and $d_1$ is the negation of $d_2$ ;

• • $\langle d_1, d_2, d_3 \rangle \in I(Conj)$ iff $d_1, d_2, d_3 \in \mathscr L^+$ and $d_1$ is a conjunction of $d_2$ and $d_3$ ;

• • $\langle d_1, d_2 \rangle \in I(Ui)$ iff $d_1, d_2 \in \mathscr L^+$ and $d_1$ is a universal quantification of $d_2$ ;

• • $\langle d_1, d_2 \rangle \in I(Ref)$ iff $d_1$ is a closed term of $\mathscr L^+$ and $d_1$ refers to $d_2$ in M.

Then we have the desideratum:

1. (Compositional Principles) Where $Neg$ , $Conj$ , $Ui$ , and $Ref$ are interpreted as above in a full sentential frame $\langle M, \ulcorner \ \urcorner \rangle $ , the following principles are categorical:

(C¬) $$ \begin{align} &\forall x \forall y [Neg(x,y) \to (Tx \leftrightarrow \neg Ty)] \end{align} $$

(C∧) $$ \begin{align} &\forall x \forall y \forall z [Conj(x,y,z) \to (Tx \leftrightarrow Ty \wedge Tz)] \end{align} $$

(C∀) $$ \begin{align} &\forall y \exists x Ref(x,y) \to \forall x [\exists y Ui(x,y) \to (Tx \leftrightarrow \forall y (Ui(x,y) \to Ty))]. \end{align} $$

Both $\mathbf {S}^{\#}$ and $\mathbf {S}^{\#N}$ satisfy Compositional Principles—provided, in the latter case, that the language’s revision strength is at least $1$ . In a language where $\neg $ , $\wedge $ , and $\forall $ are the primitive logical constants (apart from identity), (C¬)-(C∀) are the core compositional principles. If we take some other combination of logical constants as primitive, both theories will render compositional principles for them categorical, too.

7.2 Blind truth ascriptions

As many philosophers have stressed, the truth predicate serves important expressive functions. One of these is to allow us to make so-called blind truth ascriptions, in which one ascribes truth to some sentence one isn’t in a position to explicitly assert. As Horwich [Reference Horwich18, pp. 2–3] writes:

What Horwich says about propositions also holds for sentences, the bearers of truth considered in this section. For now let us focus on the case of blind ascriptions of truth to single sentences; we will consider the case of generalizations in the next subsection.

In order for my blind ascription ‘What Oscar said is true’ to serve its intended purpose, it must be in some sense equivalent to the sentence Oscar said. A very weak form of equivalence is co-affirmability: we require that a blind ascription of truth to B is affirmable iff B itself is affirmable (cf. [Reference Gupta and Standefer15]). In the Revision Theory, it is plausible to say that a sentence is affirmable iff it is categorically true. Thus the following informal condition should be a desideratum for a revision theory:

1. (Blind Ascription) Where $t_B$ is a singular term referring to the sentence B, any use of $Tt_B$ is categorically true iff B is.

Blind Ascription corresponds to a formal constraint on revision systems. Let $\mathscr L^+$ be a language with a set of definitions $\mathscr D$ that includes the truth definition, and which is interpreted via a full sentential frame $\langle M, \ulcorner \ \urcorner \rangle $ . Using ${\models \!\!\!\!\!^{^{\mathscr D}}_{_{\!\;\circ }}\,\,} $ to indicate categorical truth in an unspecified revision system, we have

1. (T⊧) $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\!\;\circ }}\,\,} B$ iff $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\!\;\circ }}\,\,} Tt_B$ , where $t_B$ is any singular term referring to B in M.

$\mathbf {S}^{\#}$ satisfies ( ${T}$ ⊧) for all sentences B (RTT, Theorem 6C.1). Once again, the situation is more complicated with $\mathbf {S}^{\#N}$ , but I will argue that it comes out in $\mathbf {S}^{\#N}$ ’s favor. The systems $\mathbf {S}^{\#N}$ do not satisfy $({T}$ ⊧) in full generality. For a counterexample, we can use the language, ground model, and definitions from Example 2.3. As recorded in (3) and (4), $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\#N}}\,\,} G\overline {(N-2)}$ even though $M \not {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\#N}}\,\,} T\ulcorner G\overline {(N-2)} \urcorner $ , whenever $N \geq 2$ . Nevertheless, $\mathbf {S}^{\#N}$ can salvage a restricted principle. If a proponent of $\mathbf {S}^{\#N}$ accepts Agreement, she may claim that, in a language with default strength N, the system $\mathbf {S}^{\#N}$ satisfies $({T}$ ⊧) for all truth ascriptions $Tt_B$ that are ever used. (This fact follows from the more general Proposition 7.1, below.) As I formulated Blind Ascription, this is all the desideratum does and should demand. The expressive functions of truth only require $({T}$ ⊧) for the sentences that are in use: any further instances don’t help anyone express anything. An analogy is helpful: a refrigerator light has the function of making the contents of the refrigerator visible. But that function only requires the light to be on when the door is open. No fridge is defective because its light doesn’t stay on 24/7.

As with refrigerator lights, economy in performing functions proves valuable in the case of truth: cutting down on excess instances of $({T}$ ⊧) allows $\mathbf {S}^{\#N}$ to dodge Gupta’s second argument for $\omega $ -inconsistency. McGee’s Theorem states that any classical truth theory satisfying $({T}$ ⊧) and (C󀂠)-(C∀) is $\omega $ -inconsistent when syntax is arithmetized [Reference McGee20].Footnote ³¹ Gupta [Reference Gupta13] claims that all of these principles are needed for truth to fully serve its expressive functions; hence, truth gives rise to $\omega $ -inconsistencies. But, as we have seen, the full principle $({T}$ ⊧) is far stronger than needed for truth’s expressive functions. In my view, $\mathbf {S}^{\#N}$ achieves the optimal response to McGee’s Theorem: we maintain classical logic, $\omega $ -consistency, the semantical laws, and every instance of $({T}$ ⊧) needed for truth’s expressive functions.Footnote ³²

7.3 The generalization function

As already noted, truth’s expressive functions allow us not just to blindly affirm single sentences, but to affirm infinitely many sentences in a single breath. If I want to affirm all the axioms of Peano Arithmetic, it’s clearly impossible for me to do so by individually asserting each of the infinitely many axioms. I have no option but to use a generalizing device. The obvious solution is to say “all the axioms of Peano Arithmetic are true.”

We face the same expressive problem in trying to affirm any infinite set of sentences. It is plausible that the truth predicate provides an all-purpose solution. By all appearances, we can use the truth predicate to affirm any infinite set of sentences, provided some way of specifying those sentences. This expressive capacity is one aspect of what is often called truth’s generalization function.Footnote ³³

Suppose $\Gamma $ is a set of sentences that we can specify with some predicate $\gamma (x)$ true of all and only the sentences in $\Gamma $ . If truth performs the generalization function, saying

(G) $$ \begin{align} \text{Every sentence with the property } \gamma \text{ is true.} \end{align} $$

should be in some sense equivalent to individually affirming each sentence of $\Gamma $ . Again, we can let co-affirmability be our guide in formulating a desideratum, yielding:

1. (Generalization) Where $\gamma $ is a non-circular predicate true of precisely the sentences $\Gamma $ , any use of (G) is categorically true iff all the sentences $\Gamma $ are categorically true.

To arrive at a formal condition, let $\mathscr L$ be a language that is expanded with a truth predicate T (and perhaps other new predicates) to $\mathscr L^+$ , and let $\langle M, \ulcorner \ \urcorner \rangle $ be a full sentential frame for $\mathscr L^+$ . Then the generalization function requires of a revision system ${\models \!\!\!\!\!^{^{\mathscr D}}_{_{\!\;\circ }}} $ :

1. (Generalization (Formal)) If the $\mathscr L$ -formula $\gamma (x)$ is true of precisely the set of $\mathscr L^+$ -sentences $\Gamma $ in M, then $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\!\;\circ }}\,\,} \Gamma $ iff $M {\models \!\!\!\!\!^{^{\mathscr D}}_{_{\!\;\circ }}\,\,} \forall x (\gamma (x) \to Tx)$ .

Truth in $\mathbf {S}^{\#}$ does not always satisfy Generalization (Formal), on account of $\omega $ -inconsistencies. To see this we can again use the language $\mathscr L^+$ , ground model M, and definitions $\mathscr D_1$ of Example 2.3. Someone who uses $\mathscr L^+$ might notice that all the sentences in the set $\Gamma = \{G0, G0', \ldots \}$ are categorically true. She might then reasonably wish to affirm all of $\Gamma $ . Of course, to do so she needs some generalizing device. If the truth predicate T performs the generalization function, then she can affirm all these sentences by asserting

(GG) $$ \begin{align} \forall x (\exists j \ x = g(j) \to Tx). \end{align} $$

Colloquially, we’d say, “the G-predication of every number is true.” Since the formula $\exists j \ x = g(j)$ is true of precisely the sentences in $\Gamma $ , (GG) is a way of saying that the sentences in $\Gamma $ are true. However, (GG) is categorically false according to $\mathbf {S}^{\#}$ ; this is a failure of Generalization (Formal).

It is easy to see that the systems $\mathbf {S}^{\#N}$ , too, don’t always satisfy Generalization (Formal). Whenever $N \geq 2$ , all the sentences $G0, G0', \ldots , G\overline {(N-2)}$ are categorically true (see (3)); and the formula $\exists y \ (y \leq \overline {(N-2)} \ \wedge \ x = g(y))$ is true in M of precisely these sentences. But the generalization

$$ \begin{align*} \forall x [\exists y (y \leq \overline{(N-2)} \wedge x = g(y)) \to Tx] \end{align*} $$

(“all G-predications of numbers less than or equal to $N-2$ are true”) is pathological according to $\mathbf {S}^{\#N}$ . The crucial advantage of the theory $\mathbf {S}^{\#N}$ over $\mathbf {S}^{\#}$ , though, is that the informal requirement Generalization only ought to apply to sentences that are used by speakers to make generalizations. By appealing to Agreement, $\mathbf {S}^{\#N}$ can once again salvage a restricted principle and save the generalization function. If finitely many sentences of $\mathscr L^+$ are used and Agreement yields default strength N, then $\mathbf {S}^{\#N}$ satisfies Generalization (Formal) for every generalization that is ever used. Proposition 7.1 makes this claim precise.

Proposition 7.1. Let $\mathscr L^+$ be a ground language expanded with any set $\mathscr D$ of definitions that includes the truth definition 𝒯, and let $\langle M, \ulcorner \ \urcorner \rangle $ be a full sentential frame for $\mathscr L^+$ . Suppose the $\mathscr L$ -formula $\gamma (x)$ defines the set $\Gamma $ of $\mathscr L^+$ -sentences in M, and suppose N is such that $\mathbf {S}^{\#N}$ and $\mathbf {S}^{\#}$ agree on whether $\forall x (\gamma (x) \to Tx)$ is categorically true in M relative to $\mathscr D$ . Then also

$$ \begin{align*} M {\models\!\!\!\!\!^{^{\mathscr D}}_{_{\#N}}\,\,} \Gamma \text{ iff } M {\models\!\!\!\!\!^{^{\mathscr D}}_{_{\#N}}\,\,} \forall x (\gamma (x) \to Tx). \end{align*} $$

At first one might have thought that, even if Agreement ensures that truth in $\mathbf {S}^{\#N}$ performs well on the sentences actually in use, once we start generalizing over unused sentences the theory will run into problems. But Proposition 7.1 shows that exactly the opposite is true: it is precisely on generalizations like (GG) that $\mathbf {S}^{\#N}$ performs well and $\mathbf {S}^{\#}$ does not.

8.1 A generalization of Agreement

The previous section’s argument involved the assumption that linguistic usage is finite. In this section I briefly discuss default strengths and the expressive functions of truth for languages whose usage is infinite.

First, I’ll show that these languages present no particular problem for setting default strengths, by suitably generalizing Agreement. To settle on a principle, I’ll begin with some examples. The most obvious test case is the language $ \mathscr L^+$ from Example 2.3 with the defined predicates G and T. Imagine now that its speakers use all the sentences $\exists i \neg Gi, G0, G0', \ldots \space $ . If these are the only sentences used, the most natural choice of a revision strength seems simply to be $0$ . Trying to maximize the number of sentences whose semantical status agrees with $\mathbf {S}^{\#}$ no longer singles out any revision strength. In the absence of good reasons to pick one revision strength over any other, we break the tie by choosing the least candidate.

Now imagine that $ \mathscr L^+$ contains, in addition to G and T, a new predicate H, which is given definition $ \mathscr D_0$ from Example 2.1. (This time use ‘ $0$ ’ and ‘ $\overline {1}$ ’ for ‘a’ and ‘b’, respectively.) The speakers of $ \mathscr L^+$ use all the sentences $\exists i \neg Gi, G0, G0', \ldots $ , plus the sentence $H0 \leftrightarrow H\overline {1}$ . Now it seems that $ \mathscr L^+$ ’s revision strength should be at least 1, so as to yield optimal results both for $\exists i \neg Gi$ and $H0 \leftrightarrow H\overline {1}$ . Beyond strength 1, however, there is complete symmetry between the other candidate strengths; we break the tie by choosing the least candidate, 1.

We can generalize the idea as follows. Define the stabilization point $st(B)$ of a sentence B as the least N for which B is either categorically true or categorically false in $\mathbf {S}^{\#N}$ , if such an N exists; and $0$ otherwise. Even in cases of infinite usage, speakers presumably have a tendency to use sentences with low stabilization points. Our guiding thought is that all sentences reflecting this tendency should stabilize. There may be several ways to cash this idea out. Here is one: we set the revision strength as the largest number at which “especially many” sentences stabilize. More precisely, we stipulate that the strength is the largest n such that, for all $m> n$ , more of the used sentences stabilize at n than at m; and it is $0$ if there is no such n. Formally, this is:

1. (Infinite Agreement) Let $\Gamma $ be the totality of $ \mathscr L^+$ -sentences used by its speakers, and let $ \Gamma _n = \{B \in \Gamma : st(B) = n\}$ . Then the default revision strength for $ \mathscr L^+$ is $\max (\{0\} \cup \{n \in \mathbb {N} : \forall m> n \ |\Gamma _n| > |\Gamma _m|\})$ .

Infinite Agreement entails both Agreement and the strengths suggested in the previous two scenarios. Alternative or more sophisticated versions of Infinite Agreement may be desirable, but for now we at least have a promising proof of concept.

8.2 Truth in languages with infinite usage

When we consider infinite cases, it is less plausible that $\mathbf {S}^{\#N}$ performs truth’s functions strictly better than $\mathbf {S}^{\#}$ . One can easily construct cases where $\mathbf {S}^{\#}$ but not $\mathbf {S}^{\#N}$ satisfies the expressive functions for all truth ascriptions in use. Consider again $ \mathscr L^+$ with defined predicates G and T, and suppose the only sentences used are $G0, G0', \ldots $ and $T\ulcorner G0 \urcorner , T\ulcorner G0' \urcorner , \ldots \space $ . $\mathbf {S}^{\#}$ satisfies Blind Ascription for all the truth predications, but for no choice of N can the system $\mathbf {S}^{\#N}$ match this feat. This fact somewhat dampens §7’s argument.

Nevertheless, there is a reasonable case that $\mathbf {S}^{\#}$ does not pull ahead of $\mathbf {S}^{\#N}$ even when we consider infinite usage. Failures of the expressive functions in the two theories go hand in hand, in a certain sense. The next proposition shows that, whenever $\mathbf {S}^{\#N}$ inevitably fails to satisfy Blind Ascription on some infinite set of sentences $\Delta $ , there is a natural generalization about $\Delta $ for which $\mathbf {S}^{\#}$ fails Generalization. The upshot is that, if we were to devise some way of “counting” failures of expressive functions, there’s no reason to think that $\mathbf {S}^{\#N}$ would have the higher count.

Proposition 8.1. Let $\langle M, \ulcorner \ \urcorner \rangle $ be a full sentential frame for a language $\mathscr L$ . Expand $\mathscr L$ to $\mathscr L^+$ with predicates defined by the set of definitions $\mathscr D$ ; in particular, $\mathscr D$ defines the predicate T by the truth definition $\mathscr T$ . Suppose $\Delta $ is a set of $\mathscr L^+$ -sentences such that, for all $N \in \mathbb {N}$ , it’s not the case that for all $B \in \Delta $ ,

$$ \begin{align*} M {\models\!\!\!\!\!^{^{\mathscr D}}_{_{\#N}}\,\,} B \text{ iff } M {\models\!\!\!\!\!^{^{\mathscr D}}_{_{\#N}}\,\,} T\ulcorner B \urcorner. \end{align*} $$

Then there is a set $\Gamma \subseteq \Delta $ such that, if an $\mathscr L$ -formula $\gamma (x)$ defines $\Gamma $ in M, then it’s not the case that

$$ \begin{align*} M {\models\!\!\!\!\!^{^{\mathscr D}}_{_{\#}}\,\,} \Gamma \text{ iff } M {\models\!\!\!\!\!^{^{\mathscr D}}_{_{\#}}\,\,} \forall x (\gamma(x) \to Tx). \end{align*} $$

In the end, I don’t think infinite cases significantly alter my verdict about truth. If one is inclined to place more weight on the case of finite usage, one should still judge $\mathbf {S}^{\#N}$ ’s truth theory superior to $\mathbf {S}^{\#}$ ’s. Even if not, one should probably judge the theories to be on a par.