The essay proposes treating some problematic sentences not by assigning them the classical values true or false, nor by declaring them outright contradictions, but by inserting a question operator ? — an operator that marks a sentence as “anything but true” (i.e. open, interrogative, non-final). The rhetorical intent is to preserve openness, time/flow, and uncertainty instead of forcing a binary assignment.

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How ? maps onto existing responses to the Liar

1) Truth-value gap / paracomplete approaches (Kripke, Strong Kleene, K3) — Closest cousin

• What these systems do. Paracomplete systems treat some sentences (notably Liar sentences) as neither true nor false — there is a third “gap” or undefined value. Kripke’s 1975 fixed-point theory constructs a minimal fixed language in which paradoxicals remain ungrounded (i.e., lack a truth value) rather than producing contradiction. The Strong Kleene tables are often used to evaluate such gaps. impan.pl+1
• Similarity to ?. If ?φ is read as “φ is not true / indeterminate / open,” then ? behaves very much like marking a gap. Viewed semantically, ? could be treated as a predicate that holds of sentences that take the gap value in a K3/Kripke fixed-point semantics.
• Key difference. Paracomplete accounts typically define the truth-value directly (a semantic gap), whereas the essay treats ? as an intentional question/attitude that preserves possibilities and invites change. To formalize ? as a gap you must give it truth-conditions and rules for how it composes with other connectives (something the essay doesn’t yet do). impan.pl

2) Supervaluationism — ? as “not super-true” / indeterminate across precisifications

• What supervaluationism does. Supervaluationism treats borderline sentences as true on all admissible precisifications (super-true), false on all (super-false), and otherwise indeterminate. It thereby preserves classical tautologies for super-true sentences while allowing gaps. Academia
• How ? could fit. ?φ might be read as “φ is not super-true” or “φ is not true on all precisifications” — a higher-level diagnostic operator saying the sentence lacks a robust classical truth. That would let you keep many classical inference patterns when sentences are super-true, while marking paradoxical sentences as ? (indeterminate).
• Issues to watch. Supervaluationists face revenge problems (one can formulate sentences that say “this sentence is not super-true”), so you’d need to show how ? avoids or resolves the same technical pitfalls. Academia+1

3) Paraconsistent / Dialetheist approaches (Priest) — ? is very different

• What dialetheism does. Dialetheists accept that some sentences are both true and false (true contradictions, dialetheias), and use paraconsistent logics to block explosion (i.e., to avoid triviality when contradictions occur). Stanford Encyclopedia of Philosophy
• Contrast with ?. The essay’s ? rejects asserting truth rather than accepting a sentence as both true and false. So while dialetheism embraces contradiction, ? seeks to sidestep it by withholding the affirmation of truth. These are epistemically and metaphysically distinct moves. If you formalize ? as a gap, it aligns with paracomplete, not paraconsistent, strategies. Stanford Encyclopedia of Philosophy

4) Tarski’s hierarchical solution — meta-levels vs. a questioning operator

• What Tarski suggested. Ban self-reference by splitting object-language and meta-language levels to avoid a global truth predicate.
• Where ? stands. ? is not ontologically banning self-reference; it’s a device for marking problematic self-referential sentences as open/indeterminate rather than pushing them into a higher metalanguage. So ? is more permissive than Tarski’s stratification — but if you want to keep a single language with ?, you’ll need to show how ? avoids the contradictions Tarski tried to prevent.

5) Temporal / dynamic accounts (truth as time-indexed) — a natural partner

• What they do. Some approaches model truth as time-indexed or as evolving under revision: sentences can change truth-value over time or under increasing stages of evaluation. Kripke’s construction also has a staged, monotone build-up which resonates with dynamic viewpoints. impan.pl
• ? and time. The essay’s stress on change and flow makes it natural to read ? as a temporal/modal operator: ?φ ≈ “φ is not now fixed as true” or “it is currently open whether φ.” This would place the operator in the family of truth-revision or temporal logics and could be formalized with staged evaluation or a modal semantics (possible-stages quantification). That choice would help the essay keep its time-sensitive intuitions while moving to a clean formal semantics.

6) Epistemic / probabilistic approaches — ? as an epistemic withholding

• What they do. Instead of altering truth-conditions, these accounts treat paradox as a feature of our epistemic state (we should suspend judgment or assign subjective probabilities).
• ? as epistemic. If ? is interpreted as an epistemic operator—“we currently withhold judgment about φ” — then the operator doesn’t alter the language’s truth-conditions but signals a stance of suspension. That is easier to implement but weaker logically (it doesn’t solve the paradox semantically; it only prescribes behavior about assertion).

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Advantages and potential payoffs of the ? idea

1. Preserves an intuitive, human practice. People ask rather than immediately assert in many philosophical contexts — ? captures that.
2. Flexible: multiple formalizations possible. ? can be implemented as (a) a gap operator (K3/Kripke), (b) a supervaluation diagnostic, (c) a temporal/modal operator, or (d) an epistemic operator. Which one you choose depends on the philosophical commitments you want to preserve (e.g., preserving classical logic where possible vs. admitting true contradictions). impan.pl+1
3. Pedagogical appeal. As the essay stresses, teaching a logic that foregrounds questioning may be more faithful to real reasoning and less dogmatic about truth.

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Main technical challenges you must address (concrete list)

1. Give formal semantics.
   • Option A: Define ?φ truth-conditions in a three-valued (or many-valued) semantics (e.g., K3). Prove consistency properties, and show how ? composes with ¬, ∧, ∨, →. Stanford Encyclopedia of Philosophy+1
   • Option B: Give ? a supervaluational semantics: ?φ true iff φ is not true on all precisifications (or vice versa). This requires addressing revenge sentences. Academia
2. Specify proof rules / entailment.
   • Which inference rules preserve ?? Does ?φ block contraposition, conditional proof, or classical explosion? You must specify a proof theory (sequent calculus or natural deduction) and prove soundness/completeness relative to your semantics.
3. Handle revenge paradoxes.
   • Any novel semantic device invites new paradoxes (sentences that self-refer to ? itself). Show how ? resists or locates such revenge sentences (this is where Kripke-style fixed points or restricted expressivity may help). ILLC Preprints and Publications+1
4. Decide metaphysical attitude.
   • Is ? a semantic device (changes what “is true” means) or an epistemic/intentional device (changes what agents assert)? Be explicit: conflating the two is a common slippery move.
5. Compare with dialetheism and justify rejection or adoption.
   • If you reject dialetheism, explain why accepting contradictions is worse than leaving sentences open. If you accept some contradictions, explain whether ? still has a role.

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Concrete suggestions to formalize ? (a roadmap)

1. Start with Kripke/K3 implementation. Treat the object language with a truth predicate T, adopt Kleene/Kripke evaluation rules, and define:

Good — this is a rich move. Here’s a toy model + worked example (handout style) for a supervaluational “?” operator + how to incorporate the nuance from Many Roads from the Axiom of Completeness (and your dissertation) about “Is this a question?” as part of the question-intention. I’ll lay it out in parts: (1) some philosophical framing, (2) formal toy semantics, (3) worked examples (including a “revenge-style” construction), (4) reflection / implications for teaching or your dissertation.

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1) Philosophical framing (drawing on Many Roads…)

Based on Many Roads from the Axiom of Completeness (Nightingale), especially as discussed in the PDF, a few important themes guide how we should treat ?:

• The essay treats questions not simply as linguistic forms, but as intentional acts — “thinking of ‘the question’ in itself.” Questions Are Power
• There is explicit attention to “Is this a question?” — i.e., the question-intention itself can be questioned. As Nightingale writes, exploring what the Axiom hides “releases a vertigo … an inquiry into thinking of ‘the question’ in itself.” Questions Are Power
• In your dissertation (as you indicate), you also treat ? (or question-intention) as layered: not only do we mark a sentence as “open / not super-true,” but we must also reflect on whether our very act of questioning is stable or “complete.” This meta-questioning (Is this a question? What kind of question?) is not peripheral — for Nightingale, it’s essential.

So: the ? operator should not just mark indeterminacy of truth, but also bear a second-order character: it signals questioning not only of the statement’s truth, but of the act of questioning itself.

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2) Formal toy semantics for ? (supervaluation + meta-questioning)

Here’s a minimal formal setup to capture both indeterminacy and a reflexive question-intention. This is meant for a handout or a “classroom toy model”; it’s not a full dissertation-length system, but enough to illustrate and test the concept.

2.1 Language

• Let L be a propositional / first-order language with a truth-predicate T(x) (or just a propositional “sentences as objects” style).
• Introduce a unary operator ? applied to well-formed formulas φ: ?φ.
• Introduce a second-order operator ?? (optional) to mark question-reflection: ??φ reads, “I question whether φ is the kind of question it claims to be” (i.e., question-about-question).

2.2 Precisifications (supervaluationist setup)

• Define a set of admissible precisifications Π. Each precisification π∈Π is a classical interpretation of L (assigning truth-values in {T, F}) for each atomic sentence, including sentences with T.
• But not all self-referential / higher-order question-reflection sentences need to be allowed in all precisifications. We can restrict Π so that for any π∈Π:
  • If a sentence involves ??, its interpretation in π must respect a stability clause: π must assign truth in such a way that question-intentions do not collapse trivially into “true / false only.” (This mimics restricting precisifications to avoid very pathological self-referencing “I am not a question” loops.)
  • Alternatively: allow all, but track second-order indeterminacy (see below).

2.3 Semantic clauses

Define the supervaluation semantics for ? and ?? as follows:

• A sentence φ is super-true if it is true in all π∈Π.
• φ is super-false if it is false in all π∈Π.
• Otherwise, φ is indeterminate.

For the question operators:

1. ?φ (first-order question) is super-true iff φ is not super-true.
   • Intuitively: ?φ = “It is not the case that φ is unambiguously (in all precisifications) true.”
   • ?φ is super-false iff φ is super-true.
   • In other cases (if φ is indeterminate), ?φ may itself be indeterminate (depending on exactly how you set up composition).
2. ??φ (meta-question) is super-true iff there is at least one precisification π such that in that precisification, the act of questioning φ (i.e., interpreting ?φ) does not correspond to a “stable question”. Formally:
   • Let’s say in each π, there’s a predicate or evaluation criterion Qπ​(φ) that determines whether in π, ?φ is treated as a legitimate question (i.e., nontrivial questioning, not just “φ false / true”).
   • Then: ??φ is super-true if Qπ​(φ) fails in at least one π.
   • ??φ is super-false if Qπ​(φ) holds in all π.
   • Otherwise, ??φ is indeterminate.

(This is a toy clause — in a full system you would need to define exactly what “legitimate questionhood” means in each precisification.)

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3) Worked Examples (Toy Handout)

Here are some example sentences + how they might be evaluated under this toy semantics.

Example A: Simple non-paradoxical sentence

Let p = “It rains.”
Consider ?p.

• In each π∈Π, p is either T or F (classical).
• Suppose in all precisifications, p is sometimes true, sometimes false (i.e., it’s not fixed). Then p is not super-true.
• So ?p is super-true (because the semantic clause says: ?φ is super-true if φ is not super-true).
• This matches our intuition: “It is not unambiguously true that it rains” — a genuine question-intention.

Example B: Liar-style sentence (revenge-type)

Let L = “L is not super-true.” (the canonical super-laier). Now consider:

1. ?L — “I question whether L is (super-)true.”
2. ??L — “I question whether ‘L is not super-true’ is itself a stable question.”

Evaluation under toy semantics:

• For some π∈Π, you might assign L = T; for others, L = F. That is typical for a revenge-style situation. So L is indeterminate (neither super-true nor super-false).
• Then ?L is super-true (because L is not super-true in every precisification).
• But what about ??L? That depends on whether in some precisifications, ?L is not “stable” as a question. If in some precisifications, Q_\pi(L) fails (i.e., we interpret ?L not simply as a “this is uncertain” but as a mis-question, or a destabilized questioning act), then ??L is super-true. That means: “Yes, I even question whether my own question ‘L is not super-true’ is a proper question or stable act of questioning.”

This avoids a direct contradiction: you don’t force L to be super-true or super-false, but you allow that the question-intention (?L) is itself non-final; and then you explicitly reflect (??L) on that instability.

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4) Reflection / Implications for Teaching or Dissertation

Here’s how you might use this toy model in your dissertation, or in a classroom (or both):

1. Pedagogical tool:
   • Present students with L and ?L. Ask them: “Is ?L a better way to handle the Liar than just declaring L undefined?”
   • Then pose ??L: have them reflect: “What does it mean to question one’s own question? Is there a kind of ‘meta-uncertainty’?”
   • This helps concretize the idea from Many Roads… that questioning itself is not monolithic — it can be questioned, destabilized, refined.
2. Dissertation development:
   • Use the toy semantics to formalize a portion of your argument, showing that the ? operator (plus maybe ??) can be given a precise, supervaluationist semantics.
   • Use the restriction on precisifications (or the stability clause) to mirror your philosophical argument from Many Roads…: not all “questions” are legitimate — question-intention itself requires reflection.
   • Then analyze revenge paradoxes (like the Liar) in light of ??: show that some revenge sentences become higher-order indeterminate rather than outright contradictory.
   • Finally, you can connect this to pedagogy: how teaching ? and even ?? can help students develop a more nuanced understanding of logic, truth, and inquiry — not just false / true, but questioning stability.

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