Logics of Formal Undeterminedness (LFUs)

• Logics of Formal Undeterminedness (LFUs) are paracomplete systems that use special unary connectives to signal undeterminedness and locally recover classical behavior.
• They employ varied semantic frameworks—such as De Morgan algebras, topological Boolean algebras, twist structures, and probabilistic models—to rigorously capture missing evidence.
• Recent advances in LFU research include analytic calculi, quantified extensions, and duality studies with LFIs, which pave the way for refined nonclassical logic applications.

Logics of Formal Undeterminedness (LFUs) are paracomplete logics in which the failure of excluded middle is not merely tolerated semantically but is also tracked inside the object language by special unary connectives expressing determinedness or undeterminedness. In the standard framework associated with João Marcos and later work by Carnielli–Coniglio–Rodrigues, an LFU weakens $A \lor \neg A$ while validating a controlled recovery principle, so that classical behavior can be restored formula by formula rather than imposed globally. Recent work places LFUs in several exact settings: De Morgan and involutive Stone algebra semantics, topological Boolean algebras, twist-structure semantics, prelinear quasi-Nelson algebras, and probability-theoretic frameworks that interpret undeterminedness as missing evidence (Gomes et al., 2021, Fuenmayor, 2021, Flaminio et al., 30 Jun 2026, Basu et al., 6 Jul 2025).

1. Conceptual profile and relation to paracompleteness

LFUs are characterized by the failure of excluded middle together with an internal recovery device. The topological treatment states the paracomplete side explicitly as

$\top \nvdash A \lor \neg A,$

and treats LFUs as the dual counterpart of LFIs, where the latter weaken explosion instead (Fuenmayor, 2021). In the probabilistic LFU $mb$, paracompleteness is presented by the condition that there exists some $\alpha$ such that

$\not\vdash \alpha\lor\neg\alpha,$

while the logic also validates an “included middle”

$\vdash \alpha\lor\neg\alpha\lor \alpha,$

where the last occurrence denotes the undeterminedness formula for $\alpha$ in the paper’s notation (Basu et al., 6 Jul 2025).

This family resemblance does not reduce LFUs to one fixed calculus. One line of work develops semantics for “some families of paracomplete Logics of Formal Undeterminedness” rather than axiomatizing a single named system (Fuenmayor, 2021). Another studies concrete algebraically motivated LFUs such as $PP^{\le}$, the order-preserving logic of perfect paradefinite algebras, and presents it as simultaneously paradefinite, an LFI, and an LFU (Gomes et al., 2021). A further strand treats four-valued Belnap–Dunn expansions and argues that BD2 is not merely a four-valued logic with classical negation but a logic in which formal inconsistency and formal undeterminedness are both expressible (Coniglio et al., 2022).

The resulting picture is that LFUs are best understood as a class of recovery-operator logics rather than as a single semantic tradition. Some are naturally many-valued, some are order-preserving, some are topological, and some are residuated. What remains stable is the logical profile: paracompleteness plus an internal mechanism for restoring determinedness in controlled contexts (Fuenmayor, 2021, Flaminio et al., 30 Jun 2026).

2. Recovery operators and the formalization of determinedness

The central technical device of an LFU is a unary connective that marks formulas whose behavior is sufficiently classical. The notation varies across frameworks. In the topological setting the historical notation is a determinedness connective $\delta A$, read as “$A$ is determined,” and the recovery principle is given as

$\top \nvdash A \lor \neg A,$0

In the same framework, determinedness is reconstructed algebraically as a fixed-point operator: $\top \nvdash A \lor \neg A,$1 equivalently as the complement of border,

$\top \nvdash A \lor \neg A,$2

so that $\top \nvdash A \lor \neg A,$3 is determined exactly when it is closed, $\top \nvdash A \lor \neg A,$4 (Fuenmayor, 2021).

In perfect paradefinite algebras the relevant operator is $\top \nvdash A \lor \neg A,$5, called a perfection operator. Its intended role is to express when an element behaves classically with respect to negation, and the paper derives

$\top \nvdash A \lor \neg A,$6

and

$\top \nvdash A \lor \neg A,$7

These equations show that $\top \nvdash A \lor \neg A,$8 internalizes both determinedness and consistency: under $\top \nvdash A \lor \neg A,$9, excluded middle and non-contradiction are recovered for $mb$0 (Gomes et al., 2021).

In the four-valued logic BD2 the primitive operator is a weak consistency connective $mb$1, but the system is analyzed as an LFU because a dual operator $mb$2 for formal undeterminedness is definable and validates

$mb$3

The same paper states that $mb$4 is the consistency operator and $mb$5 is the operator of undeterminedness, making BD2 a logic of formal inconsistency and formal undeterminedness in one framework (Coniglio et al., 2022).

A major qualification emerges in non-involutive settings. In prelinear quasi-Nelson logic, consistency and undeterminedness operators are “no longer duals of one another,” because Boolean elements and explosive elements no longer coincide in general. The paper formulates this algebraically as

$mb$6

with equality only in the involutive case, and accordingly distinguishes $mb$7-, $mb$8-, and $mb$9-undeterminedness operators instead of collapsing them into one notion (Flaminio et al., 30 Jun 2026). This marks a decisive shift from the involutive LFU literature: determinedness is not always obtainable by a simple negation-dual of consistency.

3. Semantic and algebraic frameworks

One major semantic environment for LFUs is the category of De Morgan-based algebras. Perfect paradefinite algebras are De Morgan algebras expanded by $\alpha$0 in the signature

$\alpha$1

subject to equations including

$\alpha$2

A central result shows that the resulting variety $\alpha$3 is term-equivalent to the variety of involutive Stone algebras via

$\alpha$4

and that

$\alpha$5

The associated order-preserving logic $\alpha$6 is characterized by a single six-valued matrix (Gomes et al., 2021).

A second framework is topological Boolean algebra semantics. Here the base is a Boolean algebra with topological operators such as closure $\alpha$7, interior $\alpha$8, exterior $\alpha$9, and border $\not\vdash \alpha\lor\neg\alpha,$0. The LFU-negation is the paracomplete negative modality

$\not\vdash \alpha\lor\neg\alpha,$1

and recovery of excluded middle is governed by closure fixed points: $\not\vdash \alpha\lor\neg\alpha,$2 The same work organizes closure, interior, exterior, border, and fixed-point transformations into a “topological cube of opposition,” and extends the semantics uniformly to propositional, first-order, and higher-order settings inside Isabelle/HOL (Fuenmayor, 2021).

A third framework is four-valued twist semantics. BD2 uses the Belnap–Dunn values $\not\vdash \alpha\lor\neg\alpha,$3, where $\not\vdash \alpha\lor\neg\alpha,$4 corresponds to both true and false and $\not\vdash \alpha\lor\neg\alpha,$5 to neither true nor false. The paper identifies these values with pairs in $\not\vdash \alpha\lor\neg\alpha,$6,

$\not\vdash \alpha\lor\neg\alpha,$7

and defines the connectives by twist-structure operations, including

$\not\vdash \alpha\lor\neg\alpha,$8

This semantics underwrites the claim that BD2 is a genuine LFIU rather than merely a logic with an additional negation (Coniglio et al., 2022).

A fourth line studies logics of upsets of De Morgan lattices. These papers do not use the LFU label directly, but their semantic apparatus is immediately relevant: the four-element De Morgan lattice $\not\vdash \alpha\lor\neg\alpha,$9 contains the value

$\vdash \alpha\lor\neg\alpha\lor \alpha,$0

and the framework analyzes upward closed designated sets, prime upsets, and $\vdash \alpha\lor\neg\alpha\lor \alpha,$1-filters. Shramko’s logic of “anything but falsehood,” based on the matrix

$\vdash \alpha\lor\neg\alpha\lor \alpha,$2

is especially close to LFU concerns because the undetermined value $\vdash \alpha\lor\neg\alpha\lor \alpha,$3 is designated as non-false (Přenosil, 2021).

These frameworks agree on a common theme: undeterminedness is represented structurally rather than metalinguistically. What differs is the locus of structure—lattice order, topological closure, twist coordinates, or designated upsets.

4. Canonical systems and proof theory

The order-preserving logic $\vdash \alpha\lor\neg\alpha\lor \alpha,$4 is one of the best-developed algebraic LFUs. It is determined by a single six-valued matrix, is simultaneously an LFI and an LFU, and has an analytic finite Hilbert-style symmetrical calculus. The paper further states that the order-preserving logic associated with PP-algebras is fully self-extensional, non-protoalgebraic, and therefore non-algebraizable, even though it is semantically neat and proof-theoretically well behaved (Gomes et al., 2021).

The same line was extended by adding implication. The key negative result is that there is no expansion of $\vdash \alpha\lor\neg\alpha\lor \alpha,$5 with a classic-like implication that preserves self-extensionality. The positive result is to define implication as the relative pseudo-complement

$\vdash \alpha\lor\neg\alpha\lor \alpha,$6

expand $\vdash \alpha\lor\neg\alpha\lor \alpha,$7 to $\vdash \alpha\lor\neg\alpha\lor \alpha,$8, and study the resulting order-preserving and assertional logics. The order-preserving Set-Set calculus remains analytic, the top-assertional companion is algebraizable with equivalent algebraic semantics $\vdash \alpha\lor\neg\alpha\lor \alpha,$9, and the paper connects the new variety with symmetric Heyting algebras and Moisil’s symmetric modal logic (Greati et al., 2023).

BD2 supplies a second canonical family. The propositional logic is given a Hilbert-style characterization, and its first-order extension QBD2 uses partial structures with four-valued predicate interpretations. In QBD2, the quantifiers are interdefinable by means of the paracomplete and paraconsistent negation: $\alpha$0 but not by means of the classical negation $\alpha$1. The system comes with soundness and completeness theorems, and compactness is also established (Coniglio et al., 2022).

In prelinear quasi-Nelson logic, LFU proof theory takes a different shape. The logics $\alpha$2, $\alpha$3, and $\alpha$4 are Hilbert-style algebraizable expansions by an undeterminedness connective $\alpha$5, with core principles such as

$\alpha$6

Only the Boolean-minimum version remains semilinear as a variety, and the semilinearized LFU logics collapse: $\alpha$7 This gives a chain-complete LFU family in a non-involutive residuated setting (Flaminio et al., 30 Jun 2026).

A recurring metatheoretic pattern is the Derivability Adjustment Theorem. For perfect paradefinite algebras,

$\alpha$8

while in the prelinear quasi-Nelson case classical reasoning is recovered by adding the dual determinedness assumptions $\alpha$9 (Gomes et al., 2021, Flaminio et al., 30 Jun 2026). The general significance is that LFUs do not reject classical reasoning outright; they recast it as a locally recoverable regime.

5. Quantification, probability, and adjacent information-sensitive frameworks

LFUs are not confined to the propositional level. The topological semantics based on closure and interior is explicitly uniform across propositional, first-order, and higher-order logic, and also treats unrestricted, constant-domain, and varying-domain quantification in HOL (Fuenmayor, 2021). QBD2 provides a distinct first-order route through partial structures and four-valued quantifier semantics, with

$PP^{\le}$0

These developments show that formal undeterminedness can be integrated with nonclassical quantification rather than remaining a propositional anomaly (Coniglio et al., 2022).

Probability has also entered the LFU literature. The logic $PP^{\le}$1 is presented as an LFU in which undeterminedness is interpreted as missing evidence, and a probability function $PP^{\le}$2 is defined relative to the logic. The paper proves a theorem of total paracomplete probability, a paracomplete Bayes rule, and a completeness result for the probabilistic semantics: $PP^{\le}$3 It also introduces a paracomplete probability space

$PP^{\le}$4

based on a $PP^{\le}$5-algebra rather than an ordinary $PP^{\le}$6-algebra, reflecting the fact that ordinary set complement does not match paracomplete negation (Basu et al., 6 Jul 2025).

Two neighboring frameworks clarify the boundaries of LFU research. “First-order logic with incomplete information” evaluates formulas over model sets with positive and negative satisfaction relations $PP^{\le}$7 and $PP^{\le}$8, so that informational undeterminedness appears when neither relation holds. The paper is explicit that this yields a strong adjacent framework rather than a standard LFU proper (Kuusisto, 2017). Likewise, “A Logic of Uncertain Interpretation” models uncertainty not about truth-values but about which event a sentence means at a world. Its key connective

$PP^{\le}$9

captures meaning entailment, and the induced evidential support is exactly a Dempster–Shafer belief function. That work is LFU-adjacent rather than an LFU in the usual many-valued or recovery-operator sense (Bjorndahl, 15 Mar 2025).

A common misconception is therefore that every formal treatment of “undeterminedness” belongs to LFU theory proper. The current literature distinguishes at least three cases: object-level LFUs with recovery operators, information-state semantics where undeterminedness emerges from incomplete information, and interpretation-uncertainty frameworks where the indeterminacy concerns meaning rather than truth.

6. Dualities, limits, and current directions

The strongest organizing idea in the LFU literature is duality with LFIs. The topological program makes this explicit: paraconsistent negation is

$\delta A$0

paracomplete negation is

$\delta A$1

recovery on the LFI side is tied to openness $\delta A$2, and recovery on the LFU side to closedness $\delta A$3 (Fuenmayor, 2021). Perfect paradefinite algebras sharpen the same point by showing that one operator, $\delta A$4, can serve simultaneously as a consistency connective and a determinedness connective (Gomes et al., 2021).

That duality is nevertheless not exact in all settings. The topological paper remarks that “exact mirror symmetry breaks” once one examines border behavior and quantifier interaction, especially with varying domains (Fuenmayor, 2021). The quasi-Nelson paper reaches the same conclusion algebraically: without involutivity, consistency and undeterminedness “are no longer duals of one another,” and the operator taxonomy must be refined (Flaminio et al., 30 Jun 2026). A plausible implication is that the LFI/LFU duality is robust as a methodological guide but unstable as a universal definitional equivalence.

Several open directions are stated explicitly. The perfect paradefinite program proposes weakening the PP-equations, studying settings with two negations and perhaps separate recovery operators for consistency and determinedness, and adding implication via relative pseudocomplementation (Gomes et al., 2021). The topological approach notes that it is primarily semantic and does not yet provide complete calculi for the resulting LFUs (Fuenmayor, 2021). The probabilistic approach leaves open whether the probability-on-sentences and probability-on-sets approaches are equivalent in the paracomplete case (Basu et al., 6 Jul 2025). The implicative extension of PP-logics isolates a sharp design tension: classical-style implication conflicts with self-extensionality, whereas residuated implication preserves the algebraic character of the LFU (Greati et al., 2023).

The cumulative trajectory of the field is therefore twofold. On one side, LFUs now possess well-developed algebraic semantics, finite matrices, analytic calculi, chain semantics, quantified extensions, and probabilistic enrichments. On the other, the literature increasingly emphasizes that “formal undeterminedness” is not a single mechanism. It may be modeled by perfection operators in De Morgan settings, fixed-point operators in topological Boolean algebras, underdeterminedness connectives in four-valued or residuated logics, or evidence-sensitive formalisms that remain adjacent to, rather than identical with, LFU theory proper (Coniglio et al., 2022, Greati et al., 2023, Flaminio et al., 30 Jun 2026).