Quantified Orthologic Overview

• Quantified orthologic is a formal extension of minimal quantum logic defined over ortholattices and characterized by a rejection of distributivity and the absence of a standard implication connective.
• One approach develops first-order orthologic using natural deduction with optional Reductio ad Absurdum, while another interprets quantifiers as infinitary meets and joins in complete ortholattices.
• Research in this area also highlights decidable fragments and interpolation results, providing practical insights for applications in logic and quantum set theory.

Quantified orthologic is the quantified extension of orthologic, also called minimal quantum logic, the logic of ortholattices. In the recent literature it appears in two technically distinct but closely related forms. One line treats first-order orthologic in the implication-free signature with conjunction, disjunction, negation, and quantifiers, and obtains it from a weaker quantified introduction–elimination logic by adding Reductio ad Absurdum. Another line studies a quantified calculus in which quantifiers are interpreted directly as infinitary meets and joins in complete ortholattices. In both settings, orthologic remains a non-distributive weakening of classical logic, and implication is typically omitted from the core language because orthologic lacks a satisfactory implication connective in the standard signature (Holliday, 2022, Guilloud et al., 15 Jul 2025, Aguilera et al., 19 Mar 2025).

1. Scope and formal variants

Orthologic is the logic of ortholattices and is weaker than classical logic because distributivity fails. In the propositional setting, the standard signature is $\{\wedge,\vee,\neg\}$, often with $\top,\bot$ included as constants in formal developments. This implication-free stance carries over to quantified work: the first-order language isolated as the quantified base in the literature uses exactly conjunction, disjunction, negation, universal quantification, and existential quantification, and omits implication as primitive (Holliday, 2022).

Two quantified formalisms dominate the current landscape. In the first, first-order orthologic is obtained proof-theoretically from a weaker quantified introduction–elimination logic. In the second, quantified orthologic is presented algebraically as QOL, where quantifiers are written $\bigwedge x.\phi$ and $\bigvee x.\phi$ and interpreted as infinitary infima and suprema in complete ortholattices. These are not merely notational variants. The first is a first-order predicate logic without function symbols, constants, or identity, whereas the second is a quantified algebraic extension of orthologic in which formulas still denote ortholattice elements (Holliday, 2022, Guilloud et al., 15 Jul 2025).

This suggests that quantified orthologic is presently best understood as a family of tightly related formalisms rather than as a single universally fixed calculus. The common core is ortholattice semantics, involutive negation, and the refusal of distributivity; the main point of divergence is how quantification is added.

2. First-order orthologic from fundamental logic

A major proof-theoretic starting point is the quantified “fundamental” logic in the language

$$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$

with no function symbols, constants, or identity. Its natural deduction system contains only introduction and elimination rules for the logical constants. The paper presenting this framework argues that Reiteration and Reductio ad Absurdum are not introduction/elimination rules tied to the meanings of the connectives, and therefore should be treated as optional extensions rather than part of the basic logic (Holliday, 2022).

Within that framework, the central classification is exact. Base logic plus Reiteration yields intuitionistic logic in the same signature; base logic plus RAA yields orthologic; base logic plus both yields classical logic. In the propositional case the orthologic extension is equivalently described by adding double-negation elimination,

$\neg\neg\varphi \vdash \varphi.$

For the quantified case the paper states the same pattern explicitly: adding RAA to the quantified base logic yields first-order orthologic (Holliday, 2022).

The natural deduction rules for quantifiers are also fixed explicitly. Universal introduction has the side condition that the quantified variable is not free in the initial assumption; universal elimination is

$$\frac{\forall v\,\varphi}{\varphi^v_u}\;(\forall E)$$

when $u$ is substitutable for $v$. Existential introduction is

$$\frac{\varphi^v_u}{\exists v\,\varphi}\;(\exists I),$$

again under substitutability. Existential elimination is

$\top,\bot$0

with side condition $\top,\bot$1. The use of the side condition $\top,\bot$2, rather than a standard fresh-eigenvariable condition, is a distinctive feature of this system (Holliday, 2022).

The same work extends a negative translation $\top,\bot$3 to quantifiers by

$\top,\bot$4

That translation illuminates the relation between quantified orthologic and the weaker quantified base logic, but it does not amount to a separate completeness theory for quantified orthologic as an independent first-order system (Holliday, 2022).

3. Semantic frameworks

The first-order introduction–elimination approach is paired with algebraic and relational semantics for the base quantified logic. Algebraically, the relevant structures are bounded lattices with weak pseudocomplementation, that is, lattices equipped with a unary $\top,\bot$5 satisfying antitonicity, semicomplementation, and double-negation introduction: $\top,\bot$6 Orthologic is then recovered by strengthening weak pseudocomplementation to orthocomplementation, equivalently by adding double-negation elimination $\top,\bot$7 (Holliday, 2022).

The same paper also gives a quantified relational semantics over pseudosymmetric reflexive frames. A first-order frame has the form $\top,\bot$8, where $\top,\bot$9 is the space of states and $\bigwedge x.\phi$0 is the object domain. With assignments $\bigwedge x.\phi$1, the quantified forcing clauses are

$\bigwedge x.\phi$2

$\bigwedge x.\phi$3

The completeness theorem is for the base quantified logic: $\bigwedge x.\phi$4 where $\bigwedge x.\phi$5 is the class of pseudosymmetric reflexive first-order frames. The paper explicitly treats this as a semantic architecture beneath quantified orthologic rather than as a dedicated completeness theorem for quantified orthologic itself (Holliday, 2022).

A different semantic framework appears in QOL. There the model is a complete ortholattice $\bigwedge x.\phi$6 with an assignment $\bigwedge x.\phi$7, and quantifiers are interpreted directly by infinitary lattice operations: $\bigwedge x.\phi$8

$\bigwedge x.\phi$9

This is not the first-order theory of ortholattices; it is a quantified extension in which formulas themselves continue to denote ortholattice elements. The semantic need for complete ortholattices arises precisely because quantifiers are interpreted as arbitrary joins and meets (Guilloud et al., 15 Jul 2025).

4. QOL over complete ortholattices

The most direct current treatment of quantified orthologic is the system QOL. Its syntax extends propositional orthologic by allowing

$\bigvee x.\phi$0

Its proof theory is sequent-based and retains the characteristic orthologic restriction that a sequent is a set of at most two annotated formulas. The quantifier rules are the expected lattice-theoretic analogues of universal and existential reasoning: $\bigvee x.\phi$1 and

$\bigvee x.\phi$2

with the usual freshness restriction on $\bigvee x.\phi$3 (Guilloud et al., 15 Jul 2025).

The central metatheorem is a full semantic characterization: $\bigvee x.\phi$4 for every sequent $\bigvee x.\phi$5, where $\bigvee x.\phi$6 means validity in all complete ortholattices under all assignments. The completeness proof proceeds by forming the quotient of formulas modulo provable equivalence, showing that the required joins and meets for quantified formulas already exist in the resulting syntactic ortholattice, and then embedding that ortholattice into its MacNeille completion, which is again a complete ortholattice (Guilloud et al., 15 Jul 2025).

QOL does not admit quantifier elimination in general. The explicit counterexample is that there is no quantifier-free $\bigvee x.\phi$7 with

$\bigvee x.\phi$8

The failure is semantic rather than merely proof-theoretic: the quantified formula takes different values in different ortholattices in ways no quantifier-free one-variable orthologic formula can match (Guilloud et al., 15 Jul 2025).

Despite that failure, QOL has interpolation in the ortholattice-order sense. The paper shows that if $\bigvee x.\phi$9, then there exists $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$0 such that

$$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$1

and

$$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$2

The interpolant is extracted from a proof by an explicit algorithm whose running time is linear in the size of the given proof. The same work also shows that a stronger, refutation-based notion of interpolation fails in orthologic, so the lattice-theoretic formulation is the appropriate one (Guilloud et al., 15 Jul 2025).

5. Decidable fragments and constructive bases

A different quantified development isolates a decidable fragment, effectively propositional orthologic, which is the orthologic analogue of classical EPR or Bernays–Schönfinkel–Ramsey. Here formulas are quantifier-free combinations of predicate atoms by $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$3, variables are understood through closure under instantiation, and there are no function symbols. For a set $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$4 of axioms, the expansion $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$5 is the union of all instances of the axioms, and the decision problem asks whether the expanded instance is a valid orthologic deduction problem (Guilloud et al., 2023).

The main quantitative result is that for an instance $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$6 of size $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$7 and degree $$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$8,

$$\varphi ::= P(v_1,\dots,v_n)\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid (\varphi\vee\varphi)\mid \forall v\,\varphi\mid \exists v\,\varphi,$$9

with the more precise bound

$\neg\neg\varphi \vdash \varphi.$0

As a consequence, the fragment is decidable, and it is fixed-parameter tractable for a bounded maximal number of variables in each axiom. The same paper gives an instantiation rule and unification-based proof rules, and notes that Datalog programs can be evaluated using orthologic, yielding a generalization of Datalog with negation and disjunction (Guilloud et al., 2023).

Quantified orthologic also has a significant propositional boundary condition. The propositional paper on constructive quantum logics does not treat quantifiers, but it identifies the exact implication-free common fragment of orthologic and intuitionistic logic by adding a single axiom $\neg\neg\varphi \vdash \varphi.$1 to Holliday’s fundamental logic. In particular,

$\neg\neg\varphi \vdash \varphi.$2

and the paper argues that this is the sharpest currently available propositional base for future quantified orthologic that aims to combine orthologic with constructive principles. It also states explicitly that any quantified extension should preserve these propositional results at the quantifier-free level, while providing no first-order syntax, no quantified proof system, and no quantified version of $\neg\neg\varphi \vdash \varphi.$3 (Aguilera et al., 19 Mar 2025).

6. Related systems and terminological boundaries

Quantified orthologic sits close to, but should not be identified with, several adjacent research programs. A prominent neighboring development is quantified orthomodular logic. A natural deduction system for orthomodular logic uses ordered antecedent sequents, interprets implication as the Sasaki arrow, and extends to two quantified predicate systems, one sound for Takeuti’s quantum set theory and one sound for a Weaver-style quantum logic. Those systems are technically rich and contain explicit quantifier rules, but they are not systems for orthologic proper: their semantics and completeness arguments are tied to orthomodularity rather than mere ortholattice structure, and the paper explicitly states that it does not present a quantified system for orthologic as the logic of all ortholattices (Kornell, 2021).

Other nearby directions are modal rather than quantified. The orthologic of epistemic modals is based on ortholattices and possibility semantics with compatibility, but it explicitly sets aside quantification and remains propositional. This suggests that ortholattice-based modalization and ortholattice-based quantification are complementary extensions rather than interchangeable ones (Holliday et al., 2022).

A final terminological boundary is important because “orthologic” also has a geometric meaning. In geometry, orthologic triangles are triangles for which perpendiculars from the vertices of one to the sides of the other are concurrent, and orthologic pairs generate a one-parameter linear family in which any two triangles are orthologic. Likewise, orthologic tetrahedra are tetrahedra whose corresponding normal lines are concurrent, with a stronger orthosecting theory. These geometric notions are unrelated to orthologic as a non-distributive logic of ortholattices, despite the shared term (Bakaev et al., 2023, Schröcker, 2009).

Taken together, the current literature yields a precise but still incomplete picture. Quantified orthologic now has a sound and complete sequent calculus for complete ortholattices, a first-order introduction–elimination route to first-order orthologic via RAA, and a decidable effectively propositional fragment. At the same time, quantifier elimination fails in general, the constructive extension remains only propositional, and the richest quantified systems with implication currently belong to orthomodular rather than orthologic logic. This suggests that the main open task is not to define quantified orthologic from nothing, but to reconcile these existing quantified formalisms into a more unified first-order theory.