Not Wanted: On Scharp’s Solution to the Liar
Abstract
Kevin Scharp argues that the concept of truth is defective, and is therefore unable to play its intended role in natural language truth-conditional semantics. As such, for this theoretical purpose, Scharp constructs two replacements: ascending truth and descending truth. Scharp applies the resultant theory, AD semantics, to the liar sentence, thereby obtaining a novel solution to the liar paradox. The aim of the present paper is fourfold. First, I show that, contrary to Scharp’s claims, AD semantics in fact yields an inconsistency when applied to standard liar sentences. Second, I diagnose the problem: AD semantics mishandles negation. I propose an alternative treatment, resulting in what I call AD* semantics. Third, I show that AD* semantics gives Scharp the resources required to respond to an alleged revenge paradox that has been raised against his view. Finally, I argue that, these consequences notwithstanding, it remains unclear whether AD* semantics provides an adequate account of alethic paradoxes more generally.
1 Introduction
Kevin Scharp believes that many of our concepts are defective.^{1} He tells us that, sometimes, a concept’s defect may prevent it from playing its intended explanatory role in some field of inquiry. In such a case, Scharp recommends replacing the defective concept with something better suited to playing the relevant explanatory role.
- (1)
Sentence (1) is not true,
If p, then X is ascending true.
If X is descending true, then p.
If X is safe, then: if X is ascending true, then p.
If X is safe, then: if p, then X is descending true.
The concepts ascending truth and descending truth are designed to play the theoretical role traditionally played by truth in truth-conditional, natural language semantics, the dominant approach to the empirical study of linguistic meaning in natural languages.^{4} Scharp develops a variant of the approach, called AD semantics. (To emphasise: AD semantics is an approach to the empirical study of meaning in natural languages; it is not an intended interpretation of the formal theory ADT).^{5} Whereas, for each sentence of the object language, truth-conditional semantic theories yield a biconditional that encodes a truth condition for that sentence, AD semantic theories yield both a biconditional encoding an ascending truth condition and a biconditional encoding a descending truth condition.
Developing AD semantics, Scharp provides an assessment-sensitive semantics for the English word “true”.^{6} The underlying idea is that, qua linguists, we can choose to interpret an ordinary utterance of “true” either as expressing ascending truth, or as expressing descending truth. Scharp claims that, either way, sentences such as (1) are found to be unsafe. In addition he claims that, whichever way we interpret an utterance of “true”, the reasoning involved in the liar paradox turns out to be invalid. Likewise for other alethic paradoxes, such as Curry’s paradox and Yablo’s paradox.^{7} These findings constitute Scharp’s solution to the liar and other alethic paradoxes.
Scharp argues that this solution has a major advantage over its competitors: it does not even face a revenge paradox.^{8} That is, Scharp argues that the usual strategies for constructing revenge sentences simply do not give rise to anything that the view cannot handle. Moreover, this is achieved without any expressive restrictions, without imposing an object/meta language distinction, and without endorsing a theory of truth that implies that its own axioms are not true. If right, this is an impressive result.
The aim of the present paper is fourfold. First, in Sect. 3, I show that, as it stands, AD semantics in fact does not yield the results Scharp claims: surprisingly, it yields an inconsistency even when applied to standard liar sentences. It follows that AD semantics is no advance on truth-conditional semantics, and directly undermines Scharp’s solution to the liar, AD semantics, and the principal theoretical motivation for ADT.
Second, in Sect. 4, on behalf of Scharp, I diagnose the problem. I argue that AD semantics mishandles negation, and show how to amend it accordingly. I call the resulting theory AD* semantics.
Third, in Sect. 5, I show that this result has an important consequence for Scharp: it gives him the resources required to respond to an alleged revenge paradox, which I raised in Pinder (2015). Using AD* semantics in place of AD semantics, the derivation of the revenge paradox is blocked, and we instead derive the intuitively correct result that the revenge sentence is unsafe.
Finally, in Sect. 6, I argue that, these consequences notwithstanding, it remains unclear whether AD* semantics provides an adequate account of alethic paradoxes more generally. AD* semantics, when combined with ADT, is stronger than ADT alone. The consequence is that, as things stand, there is no guarantee that AD* semantics can be consistently applied to ‘pathological’ sentences in general.
2 ADT and AD Semantics
ADT is a consistent, axiomatic theory expressed in first-order predicate logic (FOL).^{9} ADT consists of a list of twenty axiom schemas; I only introduce part of the theory herein.
(A-In) | ϕ → A〈ϕ〉 |
(D-Out) | D〈ϕ〉 → ϕ |
(S) | S〈ϕ〉 ↔ (D〈ϕ〉 ∨ ~ A〈ϕ〉). |
(A-Out) | S〈ϕ〉 → (A〈ϕ〉 → ϕ) |
(D-In) | S〈ϕ〉 → (ϕ → D〈ϕ〉) |
T〈ϕ〉 ↔ ~ T〈~ϕ〉
T〈ϕ∧ψ〉 ↔ (T〈ϕ〉 ∧ T〈ψ〉)
(M) | D〈ϕ〉 ↔ ~A〈~ϕ〉 |
(∧-A) | (A〈ϕ〉 ∧ A〈ψ〉) → A〈ϕ∧ψ〉 |
(∧-D) | D〈ϕ∧ψ〉 → (D〈ϕ〉 ∧ D〈ψ〉) |
λ_{A} | ~Aλ_{A} |
λ_{D} | ~Dλ_{D} |
Sentence (1) is not true;
〈 w, t, d, s 〉
Individual constant: | λ. |
One-place predicates: | T; W. |
Logical connectives: | ~; ∧. |
For individual constant α and one-place predicate Γ, “Γα” is a sentence.
For sentences ϕ and ψ, “~ϕ” and “ϕ∧ψ” are sentences.
Thus, for example, our presemantic theory will represent “sentence (1) is written” as:Utterances in English- are represented in ε-, taking syntactic differences into account as appropriate, where: “λ” represents “sentence (1)”; “T” represents “is true”; “W” represents “is written”; “~” represents “not”; and “∧” represents “and”. When the assessor interprets “true” as an ascending truth predicate, the context of assessment is represented as “SA”; when she interprets “true” as a descending truth predicate, the context of assessment is represented as “SD”.
Wλ.
A core semantic theory is an axiomatic theory that yields semantic values for sentences of a formal language on the basis of their constituents and modes of composition. Our core semantic theory is formulated in a rich metalanguage of ε- that contains the language of set theory. As AD semantic theories ultimately yield both ascending truth conditions and descending truth conditions, we need to distinguish between what we might call ascending semantic values (at alethic standard s), represented by the function \( {}^{\text{A}} {\llbracket} {\cdot} {\rrbracket}_{s} \), and descending semantic values (at alethic standard s), represented by the function \( {}^{\text{D}} {\llbracket} {\cdot} {\rrbracket}_{s}. \)^{11} As usual, the semantic values of individual constants are objects, the semantic values of predicates are sets of objects, and the semantic values of sentences are 1 or 0.
(I) | \( {}^{V} \llbracket\uplambda \rrbracket_{s} \) = “sentence (1) is not true” |
(II) | \( {}^{\text{A}} \llbracket {\text{W}} \rrbracket_{s} \)= {o : it’s ascending true that o is written} |
(III) | \( {}^{\text{D}} \llbracket {\text{W}} \rrbracket_{s} \) = {o : it’s descending true that o is written} |
(IV) | \( {}^{\text{A}} \llbracket {\text{T}} \rrbracket_{\text{SA}} \) = {o : it’s ascending true that o_{A} is ascending true in English-A} |
(V) | \( {}^{\text{A}} \llbracket {\text{T}} \rrbracket_{\text{SD}} \) = {o : it’s ascending true that o_{D} is descending true in English-D} |
(VI) | \( {}^{\text{D}} \llbracket {\text{T}} \rrbracket_{\text{SA}} \) = {o : it’s descending true that o_{A} is ascending true in English-A} |
(VII) | \( {}^{\text{D}} \llbracket {\text{T}} \rrbracket_{\text{SD}} \) = {o : it’s descending true that o_{D} is descending true in English-D} |
(VIII) | \( {}^{V} \llbracket \varGamma \alpha \rrbracket_{s} \, = \,1 \) iff \( {}^{V} \llbracket \alpha \rrbracket_{s} \in \,^{V} \llbracket \varGamma \rrbracket_{s}.\) |
(IX) | \( {}^{V} \llbracket \sim\!\phi \rrbracket_{s} \, = \,1 \) iff \( {}^{V} \llbracket \phi \rrbracket_{s} \, = \,0 \). |
(X) | \( {}^{V} \llbracket \phi \wedge \psi \rrbracket_{s} \, = \,1 \) iff \( {}^{V} \llbracket \phi \rrbracket_{s} \, = \,1 \) and \( {}^{V} {\llbracket \psi \rrbracket}_{s} \, = \,1 \). |
\( {}^{\text{A}}\! \llbracket {\text{W}}\uplambda \rrbracket_{\text{SA}} \, = \,1 \) | iff \( {}^{\text{A}} \llbracket\uplambda \rrbracket_{\text{SA}} \, \in \,^{\text{A}} \llbracket {\text{W}} \rrbracket_{\text{SA}} \) | [By (VIII)] |
iff “sentence (1) is not true” ∈ {o : it’s a-true that o is written} | ||
[By (I) and (II)] |
An uttered sentence, p, is ascending true from a context of assessment iff: where ϕ is the representation of p and s is the representation of the context of assessment, \( {}^{\text{A}}\! \llbracket \phi \rrbracket_{s}\! \, = \,1. \)
An uttered sentence, p, is descending true from a context of assessment iff: where ϕ is the representation of p and s is the representation of the context of assessment, \( {}^{\text{D}} \llbracket \phi \rrbracket_{s} \, = \,1. \)
~Tλ.
(2) | (a) | \( {}^{\text{A}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1 \) |
(b) | \( {}^{\text{D}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,0. \)^{16} |
3 Inconsistency
There is a striking feature of Scharp’s discussion. Despite developing substantive formal machinery, he only reasons informally when explicitly addressing the liar paradox. In particular, Scharp does not prove (2a, b), merely relying on an informal comparison to descending liars. Unfortunately, the details of AD semantics do not bear out the result.
(3) | \( {}^{V} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1 \) | iff \( {}^{V} \llbracket {\text{T}}\uplambda \rrbracket_{\text{SD}} \, = 0 \) | [By (IX)] |
iff \( {}^{V} \llbracket\uplambda \rrbracket_{\text{SD}} \notin \,^{V} \llbracket {\text{T}} \rrbracket_{\text{SD}} \) | [By (VIII)] |
\( {}^{\text{D}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1 \) | iff \( {}^{\text{D}} \llbracket\uplambda \rrbracket_{\text{SD}} \notin \,^{\text{D}} \llbracket {\text{T}} \rrbracket_{\text{SD}} \) |
(4) | \( {}^{\text{D}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1 \) | iff “sentence (1) is not true” ∉ |
{o : it’s descending true that o_{D} is descending true in English-D} |
~Dλ_{D},
(5) | ~D〈Dλ_{D}〉 |
(i) | │ D〈Dλ_{D}〉 | [Hypothesis, for reductio] |
(ii) | │ Dλ_{D} | [By (i), (D-Out)] |
(iii) | │ D〈 ~Dλ_{D}〉 | [By (ii), substitution] |
(iv) | │ ~ Dλ_{D} | [By (iii), (D-Out)] |
(v) | │ Dλ_{D} ∧ ~ Dλ_{D} | [By (ii), (iv), conjunction introduction] |
(vi) | ~D〈Dλ_{D}〉 | [By (i)–(v), reductio ad absurdum] |
(6) | \( {}^{\text{D}} \llbracket \,\sim\!\!\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1. \) |
This result is troubling. It immediately follows that, from SD, (1) is safe—that the liar sentence is not ‘pathological’ after all. This immediately casts Scharp’s account into doubt.
\( {}^{\text{A}} \llbracket {\text{D}} \rrbracket_{s} \)= {o : it’s ascending true that o is descending true}
\( {}^{\text{D}} \llbracket {\text{D}} \rrbracket_{s} \)= {o : it’s descending true that o is descending true}
(XI) | \( {}^{\text{A}} \llbracket {\text{D}} \rrbracket_{\text{SA}} \) = {o : it’s ascending true that o_{A} is descending true in English-A} |
(XII) | \( {}^{\text{A}} \llbracket {\text{D}} \rrbracket_{\text{SD}} \) = {o : it’s ascending true that o_{D} is descending true in English-D} |
(XIII) | \( {}^{\text{D}} \llbracket {\text{D}} \rrbracket_{\text{SA}} \) = {o : it’s descending true that o_{A} is descending true in English-A} |
(XIV) | \( {}^{\text{D}} \llbracket {\text{D}} \rrbracket_{\text{SD}} \) = {o : it’s descending true that o_{D} is descending true in English-D} |
(7) | \( {}^{\text{A}} \llbracket \sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,0. \)^{17} |
Pulling the above results together, it follows that, from SD:
(1) is ascending false and (1) is descending true.
4 Negation
These results are deeply puzzling. If ADT is consistent, and AD semantics interprets “T” as expressing ascending truth or descending truth, then we might naturally expect AD semantics to be consistent. So what is going on?
(IX_{A}) | \( {}^{\text{A}} \llbracket \sim\!\!\phi \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{A}} \llbracket \phi \rrbracket_{s} \, = \,0 \) . |
(IX_{D}) | \( {}^{\text{D}} {\llbracket \sim\phi \rrbracket}_{s} = 1\,\,\text{iff}\,\, {}^{\text{D}} \llbracket \phi \rrbracket_{s} =0\) . |
(M) | D〈ϕ〉 ↔ ~A〈~ϕ〉 |
(IX_{A}*) | \( {}^{\text{A}} \llbracket \sim\!\phi \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{D}} \llbracket \phi \rrbracket_{s} \, = \,0 \) . |
(IX_{D}*) | \( {}^{\text{D}} \llbracket \sim\!\phi \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{A}} \llbracket \phi \rrbracket_{s} \, = \,0 \) . |
(M*) | ~D〈~ϕ〉 ↔ A〈ϕ〉. |
The negated sentence is ascending/descending true from s iff the un-negated sentence is not descending/ascending true from s.
A〈~ϕ〉 ↔ ~ D〈ϕ〉
D〈~ϕ〉 ↔ ~ A〈ϕ〉
The second reason to prefer (IX_{A}*) and (IX_{D}*) is philosophical. Suppose that 〈ϕ〉 is unsafe: it is ascending true and descending false. Then, what should we think of 〈~ϕ〉? At face value, we might expect the negation of an unsafe sentence to also be unsafe: negating a ‘pathological’ sentence does not negate its ‘pathological’ nature. This line of thought is borne out by (IX_{A}*) and (IX_{D}*): if 〈ϕ〉 is unsafe (i.e. \( {}^{A} \llbracket \phi \rrbracket_{s} \, = \,1 \) and \( {}^{D} \llbracket \phi \rrbracket_{s} \, = \,0 \), then they entail that 〈~ϕ〉 is also unsafe (i.e. \( {}^{\text{A}} \llbracket \sim\!\!\phi \rrbracket_{s} \, = \,1 \) and \( {}^{\text{D}} \llbracket \sim\!\!\phi \rrbracket_{s} \, = \,0 \)).
This is in stark contrast to (IX_{A}) and (IX_{D}). These axioms entail that, if 〈ϕ〉 is unsafe, then 〈~ϕ〉 is ascending false and descending true (i.e. \( {}^{\text{A}} \llbracket \sim\!\!\phi \rrbracket_{s} \, = \,0 \) and \( {}^{\text{D}} \llbracket \sim\!\!\phi \rrbracket_{s} \, = \,1 \))—an impossibility. This is precisely the problem that arose when we applied AD semantics to (1): the sentence “Tλ” is unsafe, so the rules for negation entail that “~Tλ” is ascending false and descending true.
(8) | (a) | \( {}^{\text{A}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1\;\;\text{iff}\;\; {}^{\text{D}} \llbracket \,{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,0 \) | [By (IX_{A}*)] |
(b) | \( {}^{\text{D}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1\,\,\text{iff}\,\, {}^{\text{A}} \llbracket \,{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,0 \) | [By (IX_{D}*)] |
(9) | (a) | \( {}^{\text{A}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1\;\;\text{iff}\;\; {}^{\text{D}} \llbracket\uplambda \rrbracket_{\text{SD}} \notin \;^{\text{D}} \llbracket {\text{T}} \rrbracket_{\text{SD}} \) | [By (8a) and (VIII)] |
(b) | \( {}^{\text{D}} \llbracket \,\sim\!{\text{T}}\uplambda \rrbracket_{\text{SD}} \, = \,1\;\;\text{iff}\;\; {}^{\text{A}} \llbracket\uplambda \rrbracket_{\text{SD}} \notin \;^{\text{A}} \llbracket {\text{T}} \rrbracket_{\text{SD}} \) | [By (8b) and (VIII)] |
~A〈Dλ_{D}〉,
From SD, then, it can be shown using AD* semantics that (1) is ascending true and descending false, and that (1) is thus unsafe. We obtain the same result from SA. When applied to the liar sentence, AD* semantics recovers the intuitively right result.
5 Revenge
AD* semantics also gives Scharp the resources to respond to a revenge paradox, which I introduced in Pinder (2015). I begin by reconstructing the revenge paradox.
(10) | Sentence (10) is not true-from-some-context. |
~Cρ.
(XV) | \( {}^{V} \llbracket\uprho \rrbracket_{s} \) = “sentence (10) is not true-from-some-context”. |
(XVI) | \( {}^{\text{A}} \llbracket {\text{C}} \rrbracket_{s} \) = {o : it’s ascending true that o_{A} is ascending true from SA, or it’s ascending true that o_{D} is descending true from SD} |
(XVII) | \( {}^{\text{D}} \llbracket {\text{C}} \rrbracket_{s} \) = {o : it’s descending true that o_{A} is ascending true from SA, or it’s descending true that o_{D} is descending true from SD} |
(11) | (a) | \( {}^{\text{A}} \llbracket \,\sim\!{\text{C}}\uprho \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{A}} \llbracket\uprho \rrbracket_{s} \notin \,^{\text{A}} \llbracket {\text{C}} \rrbracket_{s} \) |
(b) | \( {}^{\text{D}} \llbracket \,\sim\!{\text{C}}\uprho \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{D}} \llbracket\uprho \rrbracket_{s} \notin \,^{\text{D}} \llbracket {\text{C}} \rrbracket_{s} \) |
- \( {}^{\text{A}} \llbracket \sim\!{\text{C}}\uprho \rrbracket_{s} \, = \,1 \)
iff it’s not ascending true that “sentence (10) is not true-from-some-context” is ascending true from SA, and it’s not ascending true that “sentence (10) is not true-from-some-context” is descending true from SD
- \( {}^{\text{D}} \llbracket \sim\!{\text{C}}\uprho \rrbracket_{s} \, = \,1 \)
iff it’s not descending true that “sentence (10) is not true-from-some-context” is ascending true from SA, and it’s not descending true that “sentence (10) is not true-from-some-context” is descending true from SD
“Sentence (10) is not true-from-some-context” is ascending true from s iff
it’s not ascending true that “sentence (10) is not true-from-some-context” is ascending true from SA, and it’s not ascending true that “sentence (10) is not true- from-some-context” is descending true from SD.
“Sentence (10) is not true-from-some-context” is descending true from s iff
it’s not descending true that “sentence (10) is not true-from-some-context” is ascending true from SA, and it’s not descending true that “sentence (10) is not true- from-some-context” is descending true from SD.
A〈 ~Cρ〉_{[·]} ↔ ~A〈A〈 ~ Cρ〉〉_{[SA]} ∧ ~A〈D〈 ~Cρ〉〉_{[SD]}
D〈 ~Cρ〉_{[·]} ↔ ~D〈A〈 ~ Cρ〉〉_{[SA]} ∧ ~D〈D〈 ~Cρ〉〉_{[SD]}
(12) | (a) | Aρ_{[·]} ↔ ~A〈Aρ〉_{[SA]} ∧ ~A〈Dρ〉_{[SD]} |
(b) | Dρ_{[·]} ↔ ~D〈Aρ〉_{[SA]} ∧ ~D〈Dρ〉_{[SD]} |
(13) | (a) | A〈Cρ〉_{[·]} ↔ A〈Aρ〉_{[SA]} ∨ A〈Dρ〉_{[SD]} |
(b) | D〈Cρ〉_{[·]} ↔ D〈Aρ〉_{[SA]} ∨ D〈Dρ〉〉_{[SD]}. |
(14) | (a) | Aρ_{[SA]} ↔ Aρ_{[SD]} |
(b) | Dρ_{[SA]} ↔ Dρ_{[SD]} |
(i) | │ Aρ_{[SA]} | [Hypothesis, for reductio] |
(ii) | │ ~A〈Aρ〉_{[SA]} | [By (i), (12a →), ∧-elimination) |
(iii) | │ ~Aρ_{[SA]} | [By (ii), (A-In), modus tollens) |
(iv) | ~Aρ_{[SA]} | [By (i)–(iii), reductio ad absurdum] |
(v) | ~Dρ_{[SA]} | [By (iv), (A-In), (D-Out), modus tollens] |
(vi) | ~Dρ_{[SD]} | [By (v), (14b)] |
(vii) | ~Aρ_{[SA]} ∧ ~ Dρ_{[SD]} | [By (iv), (vi), ∧-introduction] |
(viii) | ~(Aρ_{[SA]} ∨ Dρ_{[SD]}) |
(ix) | ~ A〈 ~Cρ〉_{[SA]} | [By (iv), substitution] |
(x) | D〈Cρ〉_{[SA]} | [By (ix), (M)] |
(xi) | D〈Aρ〉_{[SA]} ∨ D〈Dρ〉_{[SD]} | [By (x), (13b →)] |
(xii) | Aρ_{[SA]} ∨ Dρ_{[SD]} | [By (xi), (D-Out), disjunctive reasoning] |
And, finally: | ||
(xiii) | ⊥ | [By (viii) and (xii)] |
\( {}^{\text{A}} \llbracket \,\sim\!{\text{C}}\uprho \rrbracket_{s} \, = \,1\,\,\text{iff}\,\,{}^{\text{D}} \llbracket\uprho \rrbracket_{s} \notin \,^{\text{D}} \llbracket {\text{C}} \rrbracket_{s} \)
\( {}^{\text{D}} \llbracket \,\sim\!{\text{C}}\uprho \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{A}} \llbracket\uprho \rrbracket_{s} \notin \,^{\text{A}} \llbracket {\text{C}} \rrbracket_{s} \)
(15) | (a) | Aρ_{[·]} ↔ ~D〈Aρ〉_{[SA]} ∧ ~D〈Dρ〉_{[SD]} |
(b) | Dρ_{[·]} ↔ ~A〈Aρ〉_{[SA]} ∧ ~A〈Dρ〉_{[SD]} |
(i) | │ ~Aρ_{[SA]} | [Hypothesis, for reductio] |
(ii) | │ D〈Aρ〉_{[SA]} ∨ D〈Dρ〉_{[SD]} | [By (i), (15a)] |
(iii) | ││ D〈Aρ〉_{[SA]} | [Hypothesis, for reductio] |
(iv) | ││ Aρ_{[SA]} | [By (iii), (D-Out)] |
(v) | ││ Aρ_{[SA]} ∧ ~Aρ_{[SA]} | [By (i), (iv)] |
(vi) | │ ~D〈Aρ〉_{[SA]} | [By (iii)–(vi), reductio ad absurdum] |
(vii) | │ D〈Dρ〉_{[SD]} | [By (ii), (vi), disjunctive syllogism] |
(viii) | │ Dρ_{[SD]} | [By (vii), (D-Out)] |
(ix) | │ ~A〈Aρ〉_{[SA]} ∧ ~A〈Dρ〉_{[SD]} | [By (viii), (15b)] |
(x) | │ ~A〈Dρ〉_{[SD]} | [By (ix)] |
(xi) | │ ~Dρ_{[SD]} | [By (x), (A-In), modus tollens] |
(xii) | │ Dρ_{[SD]} ∧ ~Dρ_{[SD]} | [By (viii), (xi)] |
(xiii) | Aρ_{[SA]} | [By (i)–(xii), reductio ad absurdum] |
(xiv) | │ Dρ_{[SA]} | [Hypothesis, for reductio] |
(xv) | │ ~A〈Aρ〉_{[SA]} ∧ ~A〈Dρ〉_{[SD]} | [By (xiv), (15b)] |
(xvi) | │ ~A〈Aρ〉_{[SA]} | [By (xv)] |
(xvii) | │ ~Aρ_{[SA]} | [By (xvi), (A-In), modus tollens] |
(xviii) | │ Dρ_{[SD]} ∧ ~ Dρ_{[SD]} | [By (xvii), and (i)–(xii) above] |
(xix) | ~Dρ_{[SA]} | [By (xiv)–(xviii), reductio ad absurdum] |
All of this suggests that the revenge paradox presented in Pinder (2015) is, in fact, another symptom of AD semantics’ mishandling of negation. When we shift to AD* semantics, however, the problem appears to be resolved. As things stand, no revenge paradox has even been raised against AD* semantics.
6 Adequacy
Is AD* semantics consistent? Given that AD* semantics is an approach to natural language semantics, partially constituted by informal theories, the question is not well-posed. The threat of inconsistency only arises when we take AD* semantics to have consequences that we can reformulate in \( {\mathcal{L}} \), the language of ADT.
The basic threat derives from the axioms of the core semantic theory. On minimal assumptions about the presemantic and postsemantic theories, the core semantic theory generates substantive results that we can reformulate in \( {\mathcal{L}} \). When we do this, there are two important threats to distinguish. First, we might find that the recursive clauses governing the connectives lead to results that are inconsistent with the axioms of ADT. This is essentially the problem that I highlighted in Sect. 4: the recursive clause (IX), when applied to unsafe sentences, yields results that, formulated in \( {\mathcal{L}} \), are inconsistent with (A-In) and (D-Out). Second, we might invent a new term that extends natural language such that, when we formulate corresponding axioms in the core semantic theory, we can obtain a theorem that, formulated in \( {\mathcal{L}} \), is inconsistent with ADT. This is essentially the problem I highlighted in Sect. 5: in Pinder (2015), I introduced the term “true-from-some-context”, motivating the axioms (XV)–(XVII), leading to ascending/descending truth conditions that, reformulated in \( {\mathcal{L}} \), lead to contradiction.
Neither threat of inconsistency can be ruled out here. Firstly, it is unclear that the possibility of a revenge paradox could ever be ruled out. One might show that AD* semantics, when applied to some fragments of language, only yields results compatible with ADT. But, as AD* semantics is an approach to the empirical study of meaning in natural languages, and natural languages can always be extended with novel terms and concepts, there always remains the possibility of revenge. If we invent or discover a novel concept (such as truth-from-some-context), we are justified in formulating new axioms in our core semantic theory to govern a term that expresses that concept. If we can reverse engineer a concept to justify axioms that yield results that, when formulated in \( {\mathcal{L}} \), are inconsistent with ADT, we will find a revenge paradox. I do not know how to prove that such a project must be unsuccessful.
(X_{A}) | \( {}^{\text{A}} \llbracket \phi \wedge \psi \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{A}} \llbracket \phi \rrbracket_{s} \, = \,1\,\,\text{and}\,\, {}^{\text{A}} \llbracket \psi \rrbracket_{s} \, = \,1.\) |
(X_{D}) | \( {}^{\text{D}} \llbracket \phi \wedge \psi \rrbracket_{s} \, = \,1\,\,\text{iff}\,\, {}^{\text{D}} \llbracket \phi \rrbracket_{s} \, = \,1 \,\,\text{and}\,\, {}^{\text{D}} \llbracket \psi \rrbracket_{s} \, = \,1.\) |
(∧-A) | (A〈ϕ〉 ∧ A〈ψ〉) → A〈ϕ∧ψ〉 |
(∧-D) | D〈ϕ∧ψ〉 → (D〈ϕ〉 ∧ D〈ψ〉) |
\( {}^{\text{V}} \llbracket \phi \to \psi \rrbracket_{s} \, = \,1 \) iff if \( {}^{\text{V}} \llbracket \phi \rrbracket_{s} \, = \,1 \) then \( {}^{\text{V}} \llbracket \psi \rrbracket_{s}=1 \)
What this shows is that AD* semantics, when combined with ADT, is much stronger than ADT alone. The immediate consequence is that, even putting revenge paradoxes aside, the consistency of ADT does not suffice to protect AD* semantics from the threat of inconsistency. Although AD* semantics seems to deal adequately with liar sentences such as (1), it remains possible that there are alethic paradoxes that AD* semantics cannot handle.
This raises an important difficulty for Scharp. The concepts ascending truth and descending truth have been designed to play the central theoretical role in AD* semantics. What the present discussion shows is that those concepts are not, as they stand, adequate for their intended task. Their role in AD* semantics demands more than is supplied by ADT. So Scharp needs either to strengthen ADT, or to weaken AD* semantics. The former option requires care, and there is no guarantee of success. As Bacon (2019: 383–385) has discussed, there are several ways that Scharp might seek to strengthen ADT, but the options are not all jointly compatible. It remains an open question whether Scharp can strengthen ADT sufficiently to support its role in AD* semantics, without leading to inconsistency. In contrast, it is unclear whether the latter option is even viable. For example, the additional strength of AD* semantics derives in part from the fact that, where ADT has conditionals, AD* semantics has biconditionals. But having biconditionals is non-negotiable: if AD* semantics is to be a viable approach to natural language semantics, it must yield conditions necessary and sufficient for the ascending/descending truth of each sentence in the relevant fragment of natural language. Such conditions are expressed with biconditionals.
The upshot is this. Even if AD* semantics can deal adequately with the liar paradox and my earlier revenge paradox, it remains unclear whether AD* semantics deals adequately with alethic paradoxes more generally. The underlying problem is that the concepts ascending truth and descending truth are not, as they stand, adequate to play the central theoretical role in AD* semantics—and that it is unclear whether those concepts can be further developed to make them adequate for this theoretical role. As things stand, the consistency of ascending truth and descending truth does not guarantee that AD* semantics can be consistently applied to ‘pathological’ sentences in general.^{28}
Footnotes
- 1.
See e.g. Scharp (2013a: 290).
- 2.
- 3.
Throughout, I say that a sentence is ascending false just in case it is not ascending true, and descending false just in case it is not descending true.
- 4.
Scharp emphasises this as the principal motivation for developing ascending truth and descending truth in his 2019.
- 5.
For the intended interpretation of ADT, see Scharp (2013a: 178–187)
- 6.
- 7.
See Scharp (2013a: 255f).
- 8.
See Scharp (2013a, especially 271–273).
- 9.
- 10.
- 11.
This is my terminology, not Scharp’s. Rather than distinguishing ascending semantic values and descending semantic values, Scharp recursively defines truth-in-a-model relative to an alethic value parameter which can take ascending truth or descending truth as its value (2013a: 214–215). Formally, the approaches are mere notational variants, so long as we treat \( {}^{A} \llbracket \cdot \rrbracket_{s} \) and \( {}^{D} \llbracket \cdot \rrbracket_{s} \) as functions obtained by setting the value of parameter V in the function \( {}^{V} \llbracket \cdot \rrbracket_{s} \).
- 12.
For Scharp’s discussion of predication, see his (2013a: 218–221).
- 13.
See Scharp (2013a: 249).
- 14.
See Scharp (2013a: 170).
- 15.
See Scharp (2013a: 247–256).
- 16.
See the top of Scharp (2013a: 252)
- 17.
So far as I am aware, there is no direct proof of this, analogous to the argument from (4) to (6) given above. Hence this roundabout way of establishing (7). Thanks to Dave Ripley for pointing me towards this result.
- 18.
It can also be shown that, if 〈ϕ〉 is ascending false, then 〈∼ϕ〉 is ascending true. But, if 〈ϕ〉 is ascending true, then we can again only infer that 〈∼ϕ〉 is ascending false if 〈∼ϕ〉 is safe.
- 19.
Bacon’s model of ADT straightforwardly satisfies (M*) as a direct consequence of Bacon’s definition of “A”. So we can consistently add (M*) to ADT. See Bacon (2019: 381f).
- 20.An anonymous reviewer notes that they were initially mislead by the following derivation:But, the reviewer points out, (*) is not valid according to ADT: just let ϕ be Dλ_{D} and let ψ by 1 = 0. Even if we require ϕ and ψ to be logically equivalent, it is unclear that the resulting principle would hold in ADT: Scharp writes that it is an “essential feature of ADT” that “logically interdeducible sentences might have different descending truth values or different ascending truth values” (2013a: 173).
(i)
D〈~ϕ〉 ↔ ~ A〈 ~ ~ϕ〉
[by (M)]
(ii)
D〈~ϕ〉 ↔ ~ A〈ϕ〉
[by *]
(iii)
~D〈~ϕ〉 ↔ A〈ϕ〉
[logic]
However, as they note, the step from (i) to (ii) appears to be invalid in ADT. The following principle would justify the step:
(*)
(ϕ ↔ ψ) ↔ (A〈ϕ〉 ↔ A〈ψ〉).
- 21.
I change the notation from Pinder (2015), bringing it in line with the notation herein.
- 22.
I assume here that, given Scharp’s comments about his semantics for “true”, these subscripts should apply to whole terms, and should not iterate. Thus, using (D-Out), we can infer “Dρ_{[SD]}” from “D〈Dρ〉_{[SD]}”; and sentences such as “D〈Dρ_{[SA]}〉_{[SD]}” are not well-formed.
- 23.
This substitution is explicitly permitted. See Scharp (2013a: 154).
- 24.
I have simplified the derivation from Pinder (2015) slightly.
- 25.
In Bacon’s model of ADT, the interpretation of “D” is not closed under conjunction introduction. So we obtain a counterexample to the converse of (∧-D) by letting ϕ and ψ be different axioms of ADT; and to the converse of (∧-A) by letting ϕ and ψ be the negations of different axioms of ADT. See Bacon (2019: 381–382, 386).
- 26.
See Scharp (2013a: 154, 248).
- 27.As it stands, this axiom does lead to results that are inconsistent with ADT. This can be demonstrated by considering a sentence of the form:(I omit the details.) As Scharp does not discuss conditionals, however, I assume that this constitutes future research, and so I will not push the point here.
- (16)
If sentence (16) is descending true, then sentence (16) is not descending true.
- 28.
An earlier version of this paper was presented at the Semantic Paradox and Revenge Workshop at the University of Salzburg, and the paper improved immeasurably as a result of the subsequent discussion. I thank Dave Ripley and Andrew Bacon in particular for their comments and suggestions. I am also grateful to two anonymous referees for this journal, whose insightful comments pushed me to develop and improve the paper even further.
Notes
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