Global Self-Applicative Truth Predicate

• Global self-applicative truth predicates are defined as predicates that apply to the names of sentences within their own language, ensuring a T-schema like biconditional without resorting to a metalanguage hierarchy.
• Several approaches use fixed-point semantics, transfinite recursion, or compositional axioms to secure consistency and sidestep liar-type paradoxes in self-referential truth theories.
• Alternative frameworks employing notions of meaningfulness, assertibility, and restricted nonclassical logics offer nuanced strategies to manage paradoxes while influencing proof-theoretic strength.

A global self-applicative truth predicate is a predicate $T$ formulated in a language that can name its own sentences and can apply $T$ to those names, including names of sentences that themselves contain $T$. In the literature represented here, this notion appears in several technically distinct forms: a Strong Kleene fixed-point semantics with a final classical valuation (Čulina, 2021); transfinite fixed-point constructions over fully interpreted base languages [(Heikkilä, 2013); (Heikkilä, 2015); (Heikkilä, 2017)]; compositional arithmetical truth theories whose added principles alter proof-theoretic strength (Enayat et al., 2018, Łełyk et al., 2017); intuitionistic systems that tie truth to meaningfulness and assertibility (Weaver, 11 Jul 2025, Weaver, 9 Oct 2025); and restricted or nonclassical logics that define their own truth or satisfaction predicates internally [(0811.0964); (Sikter, 2022)]. The common objective is to sustain some form of $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$, or an exact analogue, without collapse into liar-type contradiction.

1. Core notion and formal variants

The defining feature of self-application is that the truth predicate ranges over sentences of the very language in which the predicate occurs. In the Čulina–Kripke construction, the language $L^T$ is formed by adding unary predicates $S(x)$ and $T(x)$, together with the formation rule that if $\phi$ is any formula then $\phi$ is also a term, so that expressions such as $T(\ulcorner \phi\urcorner)$ are well formed (Čulina, 2021). In Heikkilä’s MTT, every sentence of the fixed-point language $T$0, including sentences mentioning $T$1 even self-referentially, is assigned truth and falsity by the fixed-point interpretation (Heikkilä, 2013). In EFPL, the internally defined satisfaction predicate $T$2 yields a special case $T$3, and EFPL proves $T$4 for every closed EFPL formula, including formulas that mention $T$5 itself (0811.0964).

The literature differs sharply over the exact form of the T-schema. Heikkilä presents unrestricted biconditionality for sentences in the fixed-point language: $T$6 This is taken as the operative biconditional schema in MTT, and substituting $T$7 for $T$8 yields explicit self-applicative closure (Heikkilä, 2013). Sikter’s TVL gives a closed-sentence truth schema of the same form,

$T$9

but only in a language lacking full negation, unbounded universal quantification, implication, and biconditional as primitives (Sikter, 2022). Weaver’s 2025 account is different again: truth is introduced only “subjunctively” and only under a meaningfulness hypothesis,

$T$0

so the biconditional is not available as a raw unconditional axiom (Weaver, 11 Jul 2025).

Compositionality also appears in distinct ways. In arithmetic truth theories such as $T$1, compositionality is axiomatized clause by clause for atomic formulas, connectives, and quantifiers (Enayat et al., 2018). In Weaver’s framework, compositionality is described as automatic once one reasons under the assertibility predicate $T$2 and then releases only in meaningful contexts (Weaver, 11 Jul 2025). In EFPL, the clauses defining $T$3 mirror the semantic clauses for atomic, Boolean, existential, and induction-assertion formulas, and the Tarski biconditionals are obtained from the least fixed-point definition itself (0811.0964).

2. Fixed-point and recursion-based constructions

One major family of approaches builds a global self-applicative truth predicate from a fixed-point semantics. Čulina’s construction begins with an ordinary interpreted first-order language $T$4, expands it to $T$5, and defines a three-valued “primary” semantics

$T$6

with Strong Kleene clauses for $T$7, together with the fixed-point condition

$T$8

The resulting partial classical valuation $T$9 is characterized as the unique maximal intrinsic fixed point. A second valuation $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$0 is then defined by reading off the $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$1-cases of the primary semantics and extending classically to compounds; Proposition 4.2 states that $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$2 is a total two-valued classical truth valuation on $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$3 and extends the primary semantics where that semantics is defined (Čulina, 2021). The worked liar example is treated by assigning the liar sentence the value $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$4 in the primary semantics and then describing a consistent two-valued final semantics for it.

Heikkilä’s MTT proceeds by a different fixed-point mechanism. Starting from a mathematically agreeable language $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$5, one enlarges the language by a unary predicate $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$6, fixes Gödel numbering, and defines operators $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$7 and $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$8 on sets of Gödel numbers by recursion on sentence complexity. A subset $T(\ulcorner \phi\urcorner)\leftrightarrow \phi$9 is consistent when it contains no pair $L^T$0, and Theorem 3.1 yields the smallest consistent fixed point $L^T$1 with

$L^T$2

The language $L^T$3 consists of those sentences whose Gödel numbers lie in $L^T$4, and truth is interpreted by membership in $L^T$5. Lemma 4.1 states T-biconditionality for every $L^T$6, and Lemma 4.2 states preservation of base-language truth (Heikkilä, 2013).

The 2015 and 2017 Heikkilä constructions develop the same general pattern in greater recursive detail. They define $L^T$7, successive closure sets $L^T$8, and finally

$L^T$9

then iterate $S(x)$0 with unions at limit stages until stabilization at a least consistent fixed point $S(x)$1 (Heikkilä, 2015). In the 2017 formulation, this yields a fully interpreted extension $S(x)$2 of the original language with the fixed-point property

$S(x)$3

from which the biconditional $S(x)$4 is obtained for every sentence $S(x)$5 of $S(x)$6 (Heikkilä, 2017).

Taken together, these fixed-point constructions replace an explicit Tarskian hierarchy with recursion, partiality, or transfinite stabilization. In MTT this is stated directly as “no hierarchy,” while in the 2015 construction the fixed point is obtained without allowing both a sentence and its negation into the designated set, and in the Čulina construction the three-valued primary semantics absorbs paradoxical cases before the final classical valuation is read off [(Heikkilä, 2013); (Heikkilä, 2015); (Čulina, 2021)].

3. Compositional truth in arithmetic and proof-theoretic strength

A different line of work studies truth predicates over arithmetic by compositional axioms rather than by semantic fixed-point definitions. In $S(x)$7, the language is $S(x)$8, where $S(x)$9 is the usual language of first-order arithmetic. The theory adds atomic correctness, compositional clauses for negation and disjunction, and quantifier clauses such as

$T(x)$0

The additional schema $T(x)$1 requires that $T(x)$2 commute with disjunctions of arbitrary finite size. The principal result is that

$T(x)$3

where $T(x)$4 is $T(x)$5 plus $T(x)$6-induction in the expanded language, and this strengthened theory proves $T(x)$7 (Enayat et al., 2018).

The proof route described in that paper proceeds through “Inductive Correctness”: $T(x)$8 To obtain this, Enayat and Pakhomov develop a two-sorted theory $T(x)$9 of iterated truth biconditionals and prove a new general form of Visser’s theorem: no theory extending $\phi$0 plus a chain of truth-biconditionals can have an infinite descending chain of indices (Enayat et al., 2018). The stated significance is that the seemingly weak axiom of disjunctive correctness already destroys conservativity over $\phi$1, and the paper describes the boundary between conservative truth theory and reflection-rich principles as “fragile.”

Wcisło and Łełyk study a related strengthening phenomenon for a modified compositional truth theory. Their language $\phi$2 extends arithmetic by $\phi$3, and $\phi$4 consists of $\phi$5 plus $\phi$6-induction for formulas mentioning $\phi$7. The extension $\phi$8 adds generalized regularity $\phi$9, stating that substituting co-denoting term sequences into any formula does not change its truth value. The construction of partial predicates $\phi$0, then lifted predicates

$\phi$1

and finally a global predicate $\phi$2, yields a theory proving both axiom-soundness and the Global Reflection Principle

$\phi$3

The paper states that $\phi$4 is not conservative over $\phi$5, and that the modified theory actually proves global reflection over the base theory (Łełyk et al., 2017).

In these arithmetic settings, self-application is mediated by arithmetization, coding, and compositional recursion rather than by an unrestricted global biconditional for all formulas containing $\phi$6. The papers nevertheless show that even limited-looking global interaction principles for $\phi$7 can substantially increase proof-theoretic strength (Enayat et al., 2018, Łełyk et al., 2017).

4. Meaningfulness, assertibility, and intuitionistic control

Weaver’s “Truth and meaningfulness” introduces truth together with two auxiliary predicates: $\phi$8 Truth is not presented by an ordinary first-order T-schema, but by two global principles: $\phi$9

$T(\ulcorner \phi\urcorner)$0

The strategy is to insist intuitionistically on tracking meaningfulness, to state Convention T only under the assumption that a sentence is meaningful, and to reason under the assertibility predicate $T(\ulcorner \phi\urcorner)$1, admitting only the capture direction and not a universal release from $T(\ulcorner \phi\urcorner)$2 to truth (Weaver, 11 Jul 2025).

Within that framework, compositionality is described as automatic. For example, from meaningfulness of $T(\ulcorner \phi\urcorner)$3 and $T(\ulcorner \phi\urcorner)$4 one derives under $T(\ulcorner \phi\urcorner)$5 the biconditional

$T(\ulcorner \phi\urcorner)$6

and the same pattern extends to propositional connectives and quantifiers for any set-sized language of known-meaningful sentences (Weaver, 11 Jul 2025). The classical liar is blocked because a putative liar may fail to be definitely meaningful, so one never forms $T(\ulcorner \phi\urcorner)$7 as an unconditional object-language axiom. The constructive liar is blocked because the system has capture $T(\ulcorner \phi\urcorner)$8 but no general release $T(\ulcorner \phi\urcorner)$9, so the assertible liar remains “anomalous” rather than contradictory (Weaver, 11 Jul 2025).

The later ATM system gives a formal propositional realization of the same triad of truth, assertibility, and meaningfulness. Its language contains term symbols $T$00 naming sentences and monadic predicates $T$01. Groundedness is defined by least fixed-point clauses, and the central truth axiom is the meaningfulness-relative T-scheme

$T$02

ATM also includes

$T$03

together with a release rule from $T$04 to $T$05 that can be used only when no undischarged assumptions are in play. The paper’s LP-consistency theorem states that ATM is consistent, and the liar-type sentence $T$06 is treated as “anomalous”: neither definitely meaningful nor definitely meaningless, but not contradictory (Weaver, 9 Oct 2025).

These systems retain global self-application while weakening unconditional access to the T-biconditional. The central technical device is not the abandonment of self-reference, but the insertion of a meaningfulness or groundedness discipline between syntactic self-reference and assertible truth (Weaver, 11 Jul 2025, Weaver, 9 Oct 2025).

5. Internal truth in restricted or nonclassical logics

Some constructions obtain a self-applicative truth predicate by changing the ambient logic. In EFPL, formulas are built from conjunction, disjunction, existential quantification, atomic negation on negatable relations, and the least-fixed-point constructor $T$07. Blass and Gurevich define a ternary predicate $T$08 inside EFPL itself by a simultaneous least-fixed-point definition over clauses for atomic formulas, negated atomic formulas, conjunction, disjunction, existential formulas, atomic formulas using extra head-symbol predicates, and induction assertions. Because each clause refers only positively to the provisional predicate $T$09, the induced operator is monotone and has a least fixed point. Theorem 3.2 states the EFPL-Tarski biconditionals: $T$10 Corollary 3.3 then defines $T$11 as $T$12, with

$T$13

for every closed EFPL formula, including formulas mentioning $T$14 itself (0811.0964).

The paper’s paradox analysis relies on least-fixed-point semantics. For an attempted liar $T$15, the biconditional $T$16 has no fixed-point solution except the least one assigning $T$17 false, so consistency is maintained (0811.0964).

Sikter’s Turing-Verifiable Logic (TVL) achieves a different kind of internal truth. Its modified language $T$18 allows conjunction, disjunction, existential quantification, and bounded universal quantification, but has no unbounded $T$19, no negation $T$20, no implication, and no biconditional. The domain contains pure objects and relation names, and truth of relational atoms is tied to halting of Turing-machine programs. The truth predicate is defined by

$T$21

where $T$22 asserts that $T$23 codes a complete halting computation of the program coded by $T$24. For every closed $T$25-formula $T$26, the paper states the schema

$T$27

The liar is blocked because $T$28 lacks full negation and unbounded universal quantification, so one cannot form $T$29 (Sikter, 2022).

These two approaches make different trade-offs. EFPL keeps self-application by embedding truth in a monotone least-fixed-point logic; TVL keeps it by restricting the syntax so that the classical diagonal mechanism is unavailable. Both are explicit counterexamples to the idea that every formal language with self-reference must inherit Tarski-style undefinability in its original form [(0811.0964); (Sikter, 2022)].

6. Paradox management, scope of “globality,” and disputed formulations

The papers use “global” in different senses. In Heikkilä’s MTT, globality means that every sentence of $T$30, including those mentioning $T$31, falls under the truth predicate (Heikkilä, 2013). In the 2015 and 2017 fixed-point languages, $T$32 applies to all numerals naming sentences of the constructed language, and the biconditional $T$33 is stated for every sentence $T$34 of that language (Heikkilä, 2015, Heikkilä, 2017). In EFPL and TVL, the global domain is all closed formulas of the restricted logic [(0811.0964); (Sikter, 2022)]. In Weaver’s account, the paper states that one recovers the full Tarski biconditional “for any set-sized language of known-meaningful sentences,” so the operative domain is mediated by meaningfulness rather than by unconditional sentencehood (Weaver, 11 Jul 2025).

The methods of paradox avoidance are likewise heterogeneous. Čulina uses a three-valued primary semantics with an undetermined value $T$35, then a final two-valued classical semantics (Čulina, 2021). Heikkilä’s constructions use recursive operators $T$36 and $T$37, consistency preservation, and transfinite stabilization at a least fixed point [(Heikkilä, 2013); (Heikkilä, 2015); (Heikkilä, 2017)]. Weaver and ATM use meaningfulness and assertibility to block unrestricted release and to classify liar-like sentences as anomalous or ungrounded rather than contradictory (Weaver, 11 Jul 2025, Weaver, 9 Oct 2025). EFPL uses monotone least-fixed-point semantics (0811.0964). TVL blocks diagonal contradiction by refusing precisely the logical resources that classical liar constructions require (Sikter, 2022).

Several recurrent misconceptions are explicitly rejected by these works. One is that a self-applicative truth predicate must always be stratified into a metalanguage hierarchy; MTT states that there is “no need to build a separate ‘metalanguage’,” and EFPL defines satisfaction within the very logic whose formulas it evaluates [(Heikkilä, 2013); (0811.0964)]. Another is that Tarski’s undefinability theorem bars every self-contained truth theory; TVL responds by observing that Tarski’s argument is formulated for classical first-order arithmetic with full $T$38 and unrestricted $T$39, and then deliberately removes those features (Sikter, 2022).

A further point of dispute appears in the presentation of Weaver’s 2025 paper. The abstract of “Truth and meaningfulness” states that “The correct, non-paradoxical form of Frege’s Basic Law V is given,” whereas the detailed exposition supplied here states that the paper “does not mention Frege” and that “there is no Frege-style Basic Law V to quote,” describing the account instead as “entirely in the style of Tarski + intuitionistic assertibility, not Fregean abstraction” (Weaver, 11 Jul 2025). This leaves the status of Basic Law V within that presentation textually unsettled.

Taken together, these results suggest that the decisive question is not whether truth can be global and self-applicative at all, but which semantic, syntactic, or proof-theoretic constraints make such a predicate stable. The surveyed literature answers that question in mutually incompatible ways: by partiality and fixed points, by transfinite recursion, by compositional arithmetic with strong reflection consequences, by meaningfulness and restricted release, or by redesigning the logic so that truth can be defined internally without reintroducing the classical liar.