Conjectural Logics: Modeling Hypothetical Reasoning

• Conjectural logics are formal frameworks that treat tentative hypotheses as integral objects of reasoning while maintaining established facts.
• They employ diverse semantic architectures—including analogical mappings, Kripkean models, and RDF extensions—to address defeasibility and modal collapse.
• These logics extend to strategic settings and game-theoretic models, allowing agents to optimize their decisions based on learned conjectures.

Conjectural logics are logical and semantic frameworks in which hypothetical, tentative, or not-yet-settled content is treated as a first-class object of reasoning rather than as a mere absence of proof. In current work, the label covers several closely related programs: modal systems in which established facts are preserved across hypothetical layers by Axiom $C$, semantics in which conjectures are generated by analogy or encoded directly as special semantic objects, and inferential systems designed for defeasible, paracomplete, or paraconsistent reasoning about hypotheses (Vitali, 10 Aug 2025, Schlechta, 2019, Rolfini, 2021, Frittella et al., 2023). Related developments extend the topic to conditionals for team semantics, illocutionary operators for conjecturing, and strategic settings in which agents optimize against learned conjectures about others’ reactions (Barbero et al., 2 Mar 2026, Schumann, 2011, 0811.0048).

1. Conceptual scope

A central formulation treats conjectural logics as a third family of cognitive modal logics alongside doxastic and epistemic systems. On that view, doxastic logics represent possibly false or inconsistent beliefs, epistemic logics represent true beliefs through axiom $T$, and conjectural logics represent hypothetical reasoning grounded in truth: a reality world $R$ supplies established facts, and conjectural worlds extend that base with additional assumptions while preserving what is already settled. The characteristic axiom is

$\varphi \rightarrow \Box \varphi,$

which states that any established fact is preserved in all conjectural extensions (Vitali, 10 Aug 2025).

A second formulation starts from analogical reasoning. The basic schema uses a source domain $S$ and a target domain $T$: $S$ and $T$ are similar in certain known respects, $S$ has a further feature $Q$, and it is therefore plausible that $T$0 has $T$1 or an analogous $T$2. This is explicitly inductive and conjectural, because the conclusion is not guaranteed by the premises. The formal problem is to give the conjectural projection from source to target a logical and semantic shape (Schlechta, 2019).

A third formulation treats conjecture as an explicit semantic category. In RDF, conjectural triples are not asserted triples with annotations; they are distinct triples whose predicates belong to a separate conjectural property space. In illocutionary logic, conjecture is an illocutionary force $T$3, on a par with assertion, promise, or order, and the relevant semantic contrast is not only truth versus falsity but also successful versus unsuccessful performance (Rolfini, 2021, Schumann, 2011).

A further extension appears in strategic settings. In conjectural equilibrium and conjectural Stackelberg models, an agent does not optimize against a fully known reaction function. Instead it forms a conjecture about how its own action changes the state it experiences, or about how other players and a follower react, and then updates its strategy relative to that conjecture (0811.0048, Morri et al., 23 Jan 2025).

2. Semantic architectures

One semantic architecture is analogical. Let $T$4 be the source-language subalphabet and let

$T$5

be an injective, type-preserving map. For each source formula $T$6, the translated formula $T$7 is obtained by replacing each symbol with its image. Relative to a valuation $T$8, the source formulas split into positive support $T$9, negative support $R$0, and neutral or conjectural support $R$1: positive support contains formulas whose source and target truth values are both known and agree, negative support those whose truth values are both known and disagree, and neutral support those whose source value is known while the target value is not. The conjectural effect is exact: for $R$2, conjecture that $R$3. A set $R$4 of such analogy maps can then be ordered by a preference relation $R$5, with minimal elements interpreted as the best analogies, and semantic consequence is defined by truth across all $R$6-best analogies (Schlechta, 2019).

A second architecture is Kripkean and paracomplete. Each world $R$7 carries a partial valuation $R$8, the reality world $R$9 supplies the factual base, and the accessibility relation is definedness-preserving: $\varphi \rightarrow \Box \varphi,$0 Accessible worlds therefore extend the current valuation without contradicting what is already defined. Axiom $\varphi \rightarrow \Box \varphi,$1 is enforced as a propagation condition on valuations, not as an ordinary frame condition: truths at $\varphi \rightarrow \Box \varphi,$2 must remain true at all $\varphi \rightarrow \Box \varphi,$3 with $\varphi \rightarrow \Box \varphi,$4. On this basis the systems $\varphi \rightarrow \Box \varphi,$5, $\varphi \rightarrow \Box \varphi,$6, $\varphi \rightarrow \Box \varphi,$7, and $\varphi \rightarrow \Box \varphi,$8 are defined and semantically characterized by definedness-preserving, $\varphi \rightarrow \Box \varphi,$9-propagating frames, with seriality, transitivity, and Euclideanness added where required (Vitali, 10 Aug 2025).

A third architecture extends RDF Simple Interpretation. In addition to the usual sets $S$0, $S$1, $S$2, $S$3, and $S$4, the semantics introduces a disjoint set $S$5 of conjectural properties, an injective mapping

$S$6

and a mapping $S$7. A conjectural triple $S$8 is true when $S$9 is a conjectural property, $T$0 lies in its conjectural extension, and $T$1 is linked by $T$2 to some ordinary predicate $T$3. Collapse to reality is then represented semantically by adding the ordinary triple $T$4 together with an explicit collapse relation between the asserted and conjectural forms (Rolfini, 2021).

These frameworks differ in ontology—analogies, worlds, and triples—but converge on one point: conjecture is modeled as an organized extension mechanism, not as arbitrary guesswork.

3. Inferential profile: defeasibility, partiality, and inconsistency

Analogical conjecture is explicitly nonmonotonic. A formula supported by the best current analogies may cease to be supported when new negative analogies are discovered or when the ordering on analogies changes. The preferential clause

$T$5

therefore behaves like a defeasible consequence relation. By importing the machinery of preferential structures, the framework also imports conditions such as smoothness and rankedness, together with the associated representation results and cumulative-style properties (Schlechta, 2019).

In modal conjectural logic, the crucial inferential issue is modal collapse. If one combines $T$6 with $T$7 in a classical bivalent setting, one gets $T$8, collapsing the distinction between factual and conjectural layers. The proposed response is to reject $T$9 and adopt a paracomplete base logic, such as Weak Kleene logic or Description Logic with open-world semantics. Undefined propositions can then coexist with modal assertions, so $S$0 may be true even when $S$1 is undefined at the base world, and the layering between fact and conjecture is preserved (Vitali, 10 Aug 2025).

Paraconsistent work pushes the same theme in a different direction. Using Belnap–Dunn logic expanded with a Baaz Delta operator, case models allow presumptive reasoning from inconsistent premises without explosion. An argument $S$2 can be negatively, positively, or strongly coherent, conclusive, or presumptively valid, depending on whether some preferred case supports $S$3, merely fails to contradict $S$4, or supports exactly the “true only” status $S$5. This yields a finer taxonomy of conjectural status than classical validity allows, because a conclusion may be supported, unsupported, both supported and contradicted, or neither (Frittella et al., 2023).

Across these approaches, conjectural consequence is stable only relative to a semantic discipline: best analogies, $S$6-propagating extensions, or preferred paraconsistent cases.

4. Conditionals, counterfactual structure, and illocutionary force

A major question is whether a logic of conjecture can internalize consequence by means of a well-behaved conditional. For team semantics, the answer depends sharply on closure properties. Downward closed logics admit intuitionistic implication

$S$7

which preserves downward closure and supports Modus Ponens and a Deduction Theorem. Upward closed logics admit the dual conditional

$S$8

which plays the same role for upward closed formulas. By contrast, union closed, convex, and intersection closed logics do not in general admit any binary connective that simultaneously preserves the relevant closure property and validates both Modus Ponens and Deduction Theorem; the no-go arguments are Hardegree-style impossibility results driven by failures of distributive principles. The same work also studies weaker operators, including maximal and minimal implications, linear implication, relevant conditional, and epistemic conditionals satisfying reduced requirements (Barbero et al., 2 Mar 2026).

Illocutionary logic gives conjecture a different internal form. A simple illocutionary act has the form $S$9, where $T$0 is an illocutionary force such as assertion, conjecture, or promise. In the basic many-valued matrix, propositional contents take values $T$1 or $T$2, while performed acts take $T$3 for successful performance and $T$4 for unsuccessful performance. In the more general non-Archimedean semantics, each force $T$5 receives a nonstandard value in a Boolean ultrapower $T$6, and complex illocutionary acts are evaluated by the induced operations. Formulas such as $T$7, $T$8, and $T$9 are listed as illocutionary tautologies in the framework (Schumann, 2011).

A cognitive pre-semantics for counterfactuals supplies yet another perspective. Instead of evaluating hypothetical reasoning over explicit tables of worlds and distances, it posits a layer of pictures, neuron groups, positive and negative connections, and attentional selection. On this account, counterfactual and similar logics are conjectural because they operate by retrieving and recomposing non-actual scenarios, while similarity is functionally approximated by accessibility, connection strength, and interference rather than by an explicitly represented metric (Schlechta, 2016).

5. Dynamics of conjecture, settlement, and collapse

Some conjectural logics are explicitly dynamic. In modal conjectural logic, the operation

$S$0

models the transition from conjecture to accepted fact. If $S$1 is undefined in the current reality $S$2, a settlement assigns it a definite value in an updated reality $S$3, increases the domain of the valuation, preserves $S$4 or $S$5 under suitable consistency conditions, and leaves modal depth unchanged. Independent settlements commute exactly when the settled formulas are syntactically independent. The intended interpretation is that reality becomes more defined while conjectural worlds are reclassified relative to the updated base (Vitali, 10 Aug 2025).

RDF conjecture semantics has a parallel dynamic, but articulated through collapse rather than modal update. A conjecture triple $S$6 can be collapsed into an asserted triple $S$7 together with a conj:collapses relation that records the transition. The framework extends this to conjectural graphs, blank nodes in conjectural contexts, nested conjectures, conjectures about collapses, and cascading collapses. In cascading collapse, making an effective conj:collapses triple true forces the conjectured object graph itself to be collapsed, so conjectural and asserted content can coexist while preserving provenance (Rolfini, 2021).

These two traditions use different vocabularies, but they share a structural feature: conjectural content is not merely discarded when resolved. It is converted, reclassified, or linked to its effective form through explicit semantic machinery.

6. Strategic conjecture and learning

In game-theoretic settings, a conjecture is a belief function about reactions. In the water-filling game for frequency-selective interference channels, the state relevant to a user is the interference it experiences. A conjectural equilibrium consists of belief functions $S$8 and an action profile $S$9 such that each user optimizes against its own believed state mapping and, at the realized action, the conjectured state equals the actual state. For a single foresighted user, the paper adopts a linear local belief about stationary interference in each frequency bin,

$Q$0

learns $Q$1 and $Q$2 from observations, and proves that both Nash equilibrium and Stackelberg equilibrium are special cases of conjectural equilibrium. Under additional conditions, the set of conjectural equilibria is infinite, and the conjecture-based rate-maximization algorithm improves the foresighted user’s rate over iterative water-filling while often improving the rates of other users as well (0811.0048).

Conjectural Stackelberg games generalize the idea to multiple leaders and a follower. Each leader $Q$3 carries conjectures

$Q$4

representing beliefs about other leaders’ and the follower’s reactions to $Q$5. These conjectures may be affine, polynomial, or neural-network based, provided they are twice differentiable. A Conjectural Stackelberg Equilibrium is a profile in which each leader minimizes its conjectured objective

$Q$6

while the follower plays its true best response. Consistency is imposed locally through derivative matching, yielding Consistent Conjectural Stackelberg Equilibrium. Stackelberg equilibrium is then a refinement of this notion, obtained when conjectures coincide with actual best responses. The two-stage COSTAL procedure first learns conjectures from noisy response data and then performs decentralized gradient updates on the conjectured objectives; under standard stochastic-approximation assumptions, the iterates converge almost surely to a local CSE (Morri et al., 23 Jan 2025).

These models suggest a strategic reading of conjectural logic in which the semantic object is neither a world nor a sentence but a learned response map from own action to experienced consequence.

7. Metatheory, limitations, and boundaries

The strongest metatheoretic results occur in the modal and preferential strands. The modal systems $Q$7, $Q$8, $Q$9, and $T$00 are stated to be sound, complete, decidable, and robust under partial knowledge for the intended classes of definedness-preserving, $T$01-propagating frames (Vitali, 10 Aug 2025). In the analogical framework, the representation theory of preferential structures supplies abstract tools—minimality, smoothness, rankedness, and preservation under restriction—for classifying conjectural consequence relations, but the proposal is explicitly a sketch rather than a full axiomatic calculus (Schlechta, 2019).

Other strands remain mainly semantic. The RDF treatment develops a detailed model theory for conjectural triples, graphs, collapse, and nesting, but it does not provide a proof system, soundness/completeness theorems for such a calculus, or a complexity analysis of reasoning with nested and cascading structures (Rolfini, 2021). Team-semantics work reaches the opposite conclusion: for several important closure classes, the obstacle is not the absence of a calculus but the absence of any well-behaved conditional satisfying the desired preservation and inferential constraints (Barbero et al., 2 Mar 2026).

Several open directions recur across the literature. The modal framework is single-agent and studies only one dynamic operation, $T$02, leaving multi-agent conjectural reasoning and richer dynamic epistemic machinery open (Vitali, 10 Aug 2025). The analogical framework invites explicit syntactic operators, proof systems, or labeled calculi (Schlechta, 2019). The RDF framework leaves interaction with richer RDF(S)/OWL semantics largely undeveloped (Rolfini, 2021). The game-theoretic literature leaves the existence and global computation of stronger consistency notions open, since current learning procedures converge only to local equilibria (Morri et al., 23 Jan 2025).

A distinct but terminologically related line appears in descriptive complexity. There, the question is not how to reason conjecturally, but whether there exists a logic capturing classes such as $T$03 or $T$04 in Gurevich’s sense. Relativization results show that there is an oracle relative to which $T$05 has a logic while $T$06, and for higher intersection classes $T$07 the existence of a logic is equivalent to the existence of complete problems (Dahan et al., 2020). This suggests a useful terminological boundary: not every discussion of “conjectures” and “logics” concerns conjectural logics in the sense of hypothesis-preserving or tentative reasoning.

Taken together, these traditions present conjectural logics as a family rather than a single formalism. Their common task is to regiment reasoning in which content is projected, hypothesized, provisionally maintained, or strategically postulated under explicit semantic constraints, and their main differences concern the ontology of conjecture—world, analogy, triple, speech act, team, or response function—and the structural conditions under which conjectural consequence remains logically well behaved.