Abstract
Bounds consequence provides an interpretation of a multiple-conclusion consequence relation in which the derivability of a sequent is understood as the claim that it is conversationally out-of-bounds to take a position in which each member of Γ is asserted while each member of Δ is denied. Two of the foremost champions of bounds consequence—Greg Restall and David Ripley—have independently indicated that the shape of the bounds in question is determined by conversational practice. In this paper, I suggest that the standard treatments of bounds consequence have focused heavily on the matter of veridicality at the expense of ignoring other features by which conversational bounds are set, prime among them being the matter of content or subject-matter. Furthermore, I argue that the semantic behavior of propositions containing “monstrous” content—content whose introduction is inappropriate to a context independently of veridical considerations—leads to a weak Kleene account of bounds consequence.
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Notes
N.b. that the language is essentially two-sorted. The set of standard terms is provided by the signature and the distinguished terms \(\ulcorner \varphi \urcorner \) defined in parallel with the language. As a distinguished term will not appear in any formula in At, there is no need to define the language with a dual recursion.
We include a rule of contraction not present in Ripley’s formulation to make some proofs easier; as shown in [4], the addition of contraction makes no difference.
N.b. that it is a common alternative to require that t be a variable rather than an arbitrary “novel” term. The choice of presentation is intended to preserve maximal continuity with e.g. [4].
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Acknowledgments
Many parties—reviewing for this journal or participating in workshops at the University of Buenos Aires and the University of Connecticut—have generously offered helpful suggestions and criticisms. The work is much better for their insights and I appreciate their help.
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Appendices
Appendix A: Strict-Tolerant Logic ST
Afford the truth-values 1, \(\frac {1}{2}\), and 0 the standard Kleene interpretations of true, neither true nor false, and false, respectively. The strict-tolerant logic ST defined in [5] and [7] distinguishes between strict truth—cases in which a sentence affirmatively takes the value 1—and tolerant truth—cases of non-falsity taking either value 1 or \(\frac {1}{2}\).
To precisely review ST, we first describe a first-order language with a truth predicate. We take the signature of the language to be arbitrary, assuming only that there exists a set Var of variables and a set At that includes atomic formulae whose terms are elements of Var or members of a set of standard constants, each of which is constructed in the usual fashion. One of the most noteworthy features of ST is its ability to support a transparent truth predicate. The inclusion of the truth predicate \(\mathop {\boldsymbol {\textsf {T}}}\) demands that we include a term-forming operator \(\ulcorner \cdot \urcorner \) that produces a distinguished term \(\ulcorner \varphi \urcorner \) for every formula φ in the language.
Definition 7
\({\mathscr{L}}\) is a first-order language in a signature with ψ ∈At and x ∈Var.Footnote 1
Note that languages defined in this way are the same as the standard first-order language but for the inclusion of \(\mathop {\boldsymbol {\textsf {T}}}\).
We reproduce the sequent calculus for ST including transparent truth as described in [4].Footnote 2 In what follows, call a term t novel with respect to a sequent or if t appears in no side formulae.Footnote 3
Definition 8
The logic ST corresponds to the following sequent calculus. This includes structural rules:
Rules for propositional connectives:
Rules characterizing a transparent truth predicate:
And rules for the quantifiers:
In the Kleene-Kripke models of [4]—and the most frequently encountered approaches to Liar sentences, like those of Field or Priest—the semantic behavior of statements follows the strong Kleene matrices as described in Fig. 4.
The strong Kleene interpretation of the quantifiers is as follows, where the functions are operations on sets of truth-values:
Bounds consequence, as understood by Ripley, mirrors these matrices. Hence, if the position is not out-of-bounds, , too, remains in-bounds. More formally, a Kleene-Kripke model is defined in [4] as follows:
Definition 9
A Kleene-Kripke model for a language \({\mathscr{L}}\) is a pair 〈D,I〉 where D is a domain of elements such that \({\mathscr{L}}\subseteq D\) and I is an interpretation such that:
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for a term t, I(t) ∈ D
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where \(\ulcorner \varphi \urcorner \) is a distinguished name for φ, \(I(\ulcorner \varphi \urcorner )=\varphi \)
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for an n-ary predicate R, I(R) maps n-tuples from Dn to \(\lbrace 0,\frac {1}{2},1\rbrace \)
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for atoms \(P(\vec {t})\), \(I(P(\vec {t}))=I(P)(I(\vec {t}))\)
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sentential connectives and quantifiers are given the strong Kleene interpretation
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\(I(\mathop {\boldsymbol {\textsf {T}}}\ulcorner \varphi \urcorner )=I(\varphi )\) for all formulae φ
Then we define strict-tolerant validity as follows:
Definition 10
An ST-counterexample to a position is a Kleene-Kripke model such that \(I[{\Gamma }]\cap \lbrace 0,\frac {1}{2}\rbrace =\varnothing \) and \(I[{\Delta }]\cap \lbrace \frac {1}{2},1\rbrace =\varnothing \). The position is ST-valid if there are no ST-counterexamples.
The sequent calculus described in [4] is sound and complete with respect to the consequence relation induced by the strong ST framework.
Appendix B: Formal Proofs
I provide detailed proofs of the formal results in this paper, including the soundness and completeness of the calculi iST1 and wST1 with respect to the provided model theory.
The sequent calculus iST1 has been defined by supplementing ST with principles concerning out-of-bounds positions that ST does not recognize. From an interpretative standpoint, this makes sense; there are two kinds of cases—veridical and content-theoretical—that can make the assertion of a disjunction out-of-bounds. From a formal perspective, the resulting sequent calculi’s possessing pairs of rules complicate the demonstration of meta-theoretic properties like completeness.
In order to make for a simpler proof, we initially will establish an equivalence between the calculi iST1 and wST1, on the one hand, and alternative sequent calculi iST2 and wST2, respectively.
Definition 11
The sequent calculus iST2 is defined by replacing the left disjunction rules ([L∨] and [L ∨w]) in iST1 with the following rule:
and replacing the right conjunction rules ([R∧] and [R ∧w]) with the single rule:
Definition 12
The sequent calculus wST2 is defined by replacing the disjunction and conjunction rules of wST1 in the above fashion while additionally replacing the left particular quantifier rules ([L∃] and [L∃w]) with the single rule:
(with the proviso that s is novel) and replacing the right universal quantifier rules ([R∀] and [R∀w]) with the single rule:
(with the proviso that s is novel).
Lemma 1
The calculi iST1 and iST2 are equivalent, as well as the calculi wST1 and wST2.
Proof
We prove this on induction of complexity of wST1 and wST2 proofs, i.e., we take as induction hypothesis that every subproof of a wST1 proof has a corresponding wST2 proof. As the two share axioms, the basis step is resolved. The intermediate case follows along analogous lines.
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To prove the interderivability between [L∨] and [L ∨w] on the one hand and [L ∨⋆], we first show that the two wST1 rules are emulable in wST2 and then show that the wST2 rule is emulable in wST1. First, suppose that one has wST1 proofs Ξ0 and Ξ1 of the sequents and . Then by induction hypothesis, we have proofs \({\Xi }_{0}^{\star }\) and \({\Xi }_{1}^{\star }\) in wST2 of these sequents. We construct a proof
Alternately, one may have wST1 proofs Ξ0 and Ξ1 of the sequents and ’ where ξ = φ or ξ = ψ. Without loss of generality, we consider the former case
Conversely, suppose that one has wST2 proofs of , , and ; call these Ξ0, Ξ1, and Ξ2, respectively. Then by induction hypothesis, we have proofs in wST1 \({\Xi }_{0}^{\star }\), \({\Xi }_{1}^{\star }\), and \({\Xi }_{2}^{\star }\) of these sequents. From these, we are able to construct the following wST1 proof of :
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The interderivability of the right conjunction rules follows a dual argument to that for left disjunction.
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Now consider the case of [L∃], [L∃w], and [L∃⋆]. If a wST1 proof of terminates with an application of [L∃w], then one has wST1 proofs Ξ0 and Ξ1 of the sequents and . By induction hypothesis, one has wST2 proofs of these sequents, whence we can construct a wST2 proof of .
In case one had applied [L∃], there exists a wST1 proof of the sequent where s does not appear in Γ ∪Δ. Then by induction hypothesis, we have a wST2 proof \({\Xi }_{0}^{\star }\) of this sequent. Then we are able to prove in wST2 as follows:
Now, suppose that one has a wST2 proof of terminating with an application of [L∃⋆]. One must therefore have wST2 proofs Ξ0 and Ξ1 of the sequents and . By induction hypothesis, there exist corresponding wST1 proofs \({\Xi }_{0}^{\star }\) and \({\Xi }_{1}^{\star }\), from which one can construct a wST1 derivation of :
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The case of the right universal quantifier rules follows a dual argument to that used for the left existential quantifier rules.
□
Lemma 2
The rule [L ∨⋆] preserves intermediate and weak ST validity.
Proof
We prove the contrapositive. Suppose that the sequent is not weak (intermediate, respectively) ST valid. Then there exists a three-valued valuation v mapping all elements of Γ ∪{φ ∨ ψ} to 1 and all elements of Δ to 0. Because v(φ ∨ ψ) = 1, there are three possibilities concerning the values v assigns to φ and ψ, and we examine these cases individually.
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If v(φ) = v(ψ) = 1, then v maps all elements of Γ ∪{φ,ψ} to 1 (while still mapping each formula in Δ to 0). Hence, v is a weak (intermediate) Kleene-Kripke counterexample to the sequent .
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If v(φ) = 1 while v(ψ) = 0, then v assigns each formula of Γ ∪{φ} the value 1 while assigning each formula of Δ ∪{ψ} the value 0, and therefore serves as a weak (intermediate) Kleene-Kripke counterexample to .
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By swapping φ and ψ in the previous case, that v(ψ) = 1 while v(ψ) = 0 entails that v is a weak (intermediate) Kleene-Kripke counterexample to .
These cases exhaust the possible cases for v. Hence, the weak ST invalidity of entails that at least one of the sequents , , and is also invalid. But this is the contrapositive of the rule. □
Lemma 3
The rule [R ∧⋆] preserves both weak and intermediate ST validity.
Proof
Analogous to the previous lemma. □
Lemma 4
The rule [L∃⋆] preserves weak and intermediate ST validity.
Proof
The strong Kleene interpretation of ∃ follows from standard ST, so we show that this holds for the weak Kleene quantifier. Suppose that is not weak ST valid and that v is a weak Kleene-Kripke counterexample for the sequent. Let X be the set of truth-values that v assigns to formulae φ(c). Then there are two cases: one in which X = {1} and one in which X = {0, 1}.
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In the first case, because for any term c whose interpretation is an element of the domain v(φ(c)) = 1, one can either find a term s not appearing in Γ ∪Δ—or introduce a new constant—such that v(φ(s)) = 1. One can also find a second term t for which this holds. In this case, v maps all elements of Γ ∪{φ(s),φ(t)} to 1 yet continues to map all elements of Δ to 0, making it a weak Kleene-Kripke counterexample to the sequent where s does not appear in Γ or Δ.
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In the second case, one loses the assurance that for any term c, v(φ(c)) = 1. However, one can nevertheless guarantee that for any term t—including terms appearing in Γ ∪Δ—either v(φ(t)) = 1 or v(φ(t)) = 0. Hence, for any term t, one of two things occurs:
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v maps all elements of Γ ∪{φ(t)} to 1 and all elements of Δ to 0, or
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v maps all elements of Γ to 1 and all elements of Δ ∪{φ(t)} to 0.
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But this is just to say that for any t, v is either a weak Kleene-Kripke counterexample to or a counterexample to .
Put together, then, that is not weak ST valid entails that either is not valid for an s not appearing in Γ or Δ, or that for an arbitrary term t, either is invalid or is invalid. This proves the contraposition of the rule. □
Lemma 5
The rule [R∀⋆] preserves weak and intermediate ST validity.
Proof
Analogous to the previous lemma. □
Theorem 1 (Soundness)
iST1 and wST1 are sound with respect to the proposed semantics.
Proof
All axioms are valid, and it is well-established that all rules for these calculi preserve validity besides [L ∨⋆], [R ∧⋆], [L∃⋆], and [R∀⋆]. Lemmas 2–5 establish, then, that iST2 and wST2 are sound; Lemma 1 establishes that this result carries over to iST1 and wST1, respectively. □
Theorem 2 (Completeness)
iST1 and wST1 are complete with respect to the proposed semantics.
Proof Proof Theorem 2:
We follow closely the completeness proof in [4], which is relatively standard. Let be an unprovable sequent. We can construct a tree iteratively by extending each node sequent with nodes above it whose sequents have the property that both \({\Gamma }\subseteq {\Gamma }^{\prime }\) and \({\Delta }\subseteq {\Delta }^{\prime }\). We say that a node is closed if \({\Gamma }\cap {\Delta }\neq \varnothing \) and open otherwise.
At each stage α + 1, we apply reduction steps to the formulae carried over in a sequent from stage α.
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A negation on the right is reduced in a node by extending the branch by the sequent
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A negation on the left is reduced in a node by extending the branch by the sequent
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A disjunction on the right is reduced in a node by extending the branch by the sequent
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A disjunction on the left is reduced in a node by splitting the branch into three branches by the nodes , , and
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An existentially quantified formula on the right is reduced in a node by the sequent where t has not yet been used in a reduction of this formula in this position
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A formula \(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner )\) on the right is reduced in a node by extending the branch with
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A formula \(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner )\) on the left is reduced in a node by extending the branch with
In the case of iST2, assuming that we have an enumeration of terms of our language, we add:
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An existentially quantified formula on the left is reduced in by the sequent where s is the least term not appearing in Γ, φ(x), or Δ
while in the case of wST2, we add instead:
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An existentially quantified formula on the left is reduced in by splitting the branch into two branches with the sequents and where s is the least term not appearing in Γ, φ(x), or Δ and t has not yet been used in the reduction of the formula ∃xφ(x) in that position
If all reduction steps are exhausted, because is not provable, then there must remain an open branch.
Define to be the union of all the sequents found on this open branch. I.e., let Γω be the union of all antecedents appearing in some sequent on the branch and let Δω be the union of all succedents appearing in some sequent on the branch.
From a model 〈Dω,Iω〉 may be defined as follows. Dω is the set of terms appearing in some formula in Γω ∪Δω and the interpretation is defined so that:
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For all non-distinguished terms t, Iω(t) = t
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For all distinguished terms \(\ulcorner \varphi \urcorner \), \(I_{\omega }(\ulcorner \varphi \urcorner )=\varphi \)
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For all R, \(I_{\omega }(R)(I_{\omega }(t_{0}),...,I_{\omega }(t_{n-1}))=\begin {cases} 1 & \text {if }R(t_{0},...,t_{n-1})\in {\Gamma }_{\omega } \\ \frac {1}{2} & \text { if } R(t_{0},...,t_{n-1})\notin {\Gamma }_{\omega }\cup {\Delta }_{\omega }\\ 0 & \text { if }R(t_{0},...,t_{n-1})\in {\Delta }_{\omega }\end {cases}\)
□
At this point, we pause the proof of Theorem 2 to establish several facts about iST, wST and the interpretation Iω.
Lemma 6
Both intermediate and weak Kleene-Kripke interpretations are monotonic.
Proof
The monotonicity of weak Kleene-Kripke models (and strong Kleene-Kripke models) is well-known and acknowledged in e.g. [11, 15], and [30]. Consequently the respective systems’ individual truth-functions interpreting connectives and quantifiers guarantee monotonicity. This holds a fortiori for the truth-functions used to extend intermediate Kleene-Kripke interpretations, namely the truth-functions of the weak Kleene connectives and strong Kleene quantifiers. Thus, intermediate Kleene-Kripke models are monotonic as well. □
Lemma 7
In the model 〈Dω,Iω〉,
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Iω(φ) = 1 if φ ∈Γω
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Iω(φ) = 0 if φ ∈Δω
Proof
As in the case of Ripley’s [4], induction on complexity of formulae establishes this fact. As an illustration, consider a case unique to the weak Kleene matrices in which φ = ψ ∨ ξ. In this case, φ ∈Γω only if a reduction step had been applied to some sequent on the branch. There are three possible outputs of this reduction step, one of which must have been added to the branch.
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if was added, by hypothesis, Iω(ψ) = 1 and Iω(ξ) = 1
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if was added, then Iω(ψ) = 1 and Iω(ξ) = 0
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if was added, Iω(ψ) = 0 and Iω(ξ) = 1
In each of these cases, Iω(ψ ∨ ξ) = 1, i.e., Iω(φ) = 1. □
Although these observations are sufficient to yield a model, 〈Dω,Iω〉 may not respect the clauses for T. An additional step is necessary to construct a provable counterexample to . The critical ingredient relies on Kremer’s notion of fixability.
Definition 13
A three-valued interpretation I is fixable if \(I(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner ))=I(\varphi )\) whenever \(I(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner ))\neq \frac {1}{2}\).
Lemma 8
The model 〈Dω,Iω〉 is fixable.
Proof
If \(I_{\omega }(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner ))\neq \frac {1}{2}\) then suppose without loss of generality that \(I_{\omega }(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner ))=1\). As an atomic formula, the definition of Iω ensures that \(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner )\in {\Gamma }_{\omega }\). Its appearance in Γω means that a sequent appears at some stage in the branch to which a reduction step was applied yielding the sequent . Thus φ ∈Γω and by Lemma 7, Iω(φ) = 1. □
In [30], Kremer proves the following improvement of Kripke’s Minimal Fixed Point Theorem from [15]. While Kripke’s result guarantees the existence of minimal fixed points if the initial interpretation of T maps all distinguished terms to \(\frac {1}{2}\), the model 〈Dω,Iω〉 may have already assigned values of 0 or 1 to some formulae \(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner )\).
Theorem 3 (Kripke/Kremer)
For a three-valued logic with monotonic valuations, if a model 〈D,I〉 is fixable then it can be extended to a fixed-point model \(\langle D,I^{\prime }\rangle \).
This gives us all the necessary tools to complete the proof of Theorem 2:
Proof Proof Theorem 2, Continued:
By Lemma 6 the applicable logic has monotonic valuations and by Lemma 8 the model 〈Dω,Iω〉 is fixable. Hence, by Theorem 3, Iω can be extended to a fixed-point model \(\langle D_{\omega },I^{\prime }_{\omega }\rangle \) in which \(I^{\prime }_{\omega }(\boldsymbol {\mathsf {T}}(\ulcorner \varphi \urcorner ))=I^{\prime }_{\omega }(\varphi )\) for all φ.
Thus, \(\langle D_{\omega },I^{\prime }_{\omega }\rangle \) satisfies the clause for the interpretation of T and is in fact a model of appropriate type for . It is a fortiori a counterexample to , establishing completeness for iST2 and wST2 with respect to the intended semantics. By Lemma 1, this extends to iST1 and wST1. □
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Ferguson, T.M. Monstrous Content and the Bounds of Discourse. J Philos Logic 52, 111–143 (2023). https://doi.org/10.1007/s10992-022-09666-4
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DOI: https://doi.org/10.1007/s10992-022-09666-4