Intensional Many-Sorted First-Order Logic

Updated 7 July 2026

Intensional Many-Sorted First-Order Logic (IFOL) is a framework that combines first-order quantification, explicit ontological sorts, and an intensional layer to distinguish meanings from extensional denotations.
It enforces a disciplined many-sorted syntax with static and dynamic sorts, ensuring that terms, functions, and predicates adhere to precise type constraints and abstraction rules.
IFOL supports applications in natural language processing, ontology, and AI by maintaining two-step semantics that map syntactic structures to intensional entities and world-relative extensions.

Searching arXiv for the directly relevant IFOL papers and closely related foundations. Intensional Many-Sorted First-Order Logic (IFOL) denotes a family of first-order formalisms that combine three ingredients: first-order quantificational structure, explicit ontological or semantic sorting, and an intensional layer in which formulas are associated with meanings or concepts rather than only extensional denotations. In the literature represented here, the most direct formulation is the many-sorted extension of earlier Intensional FOL proposed in 2024, where “the concepts used in IFOL have associated to them a list of sorted attributes, and the sorts are the intensional concepts as well” (Majkic, 2024). That line of work preserves a two-step semantics from syntax to intensional entities and then to world-relative extensions, while restricting variables, terms, predicates, functions, and admissible assignments by sorts. Closely related work develops the same framework over Belnap’s four-valued bilattice for inconsistent and incomplete knowledge (Majkic, 4 Aug 2025), while other papers contribute essential background on intensional semantics without full many-sorted syntax [(Majkic, 2011); (Majkic, 2011)], sorted translations and proof theory (Oddsson, 18 Mar 2026), modal many-sorted fragments (Leustean et al., 2018, Leuştean et al., 2020), and typed intensional architectures that are many-sorted in spirit though higher-order in official formulation (Walsh, 2015).

1. Terminology and scope

A central preliminary distinction is terminological. In one strand of the literature, IFOL means Intensional First-Order Logic, extended in 2024 to a many-sorted form (Majkic, 2024). In another, superficially similar acronym, “iFOL” denotes intuitionistic first-order logic, not intensional logic, and that usage is explicitly unrelated to a logic of intensions, meanings, or many-sorted semantics (Constable et al., 2011). The latter paper studies a single-sorted first-order language with one domain symbol $D$ , fixed-arity relation symbols, and Brouwer–Heyting–Kolmogorov evidence semantics; it does not develop an intensional or many-sorted object language (Constable et al., 2011).

Within the intensional line, the many-sorted extension is presented as a completion of earlier unsorted IFOL. Its motivation is that “natural language is implicitly many-sorted” and that IFOL is intended for applications involving natural language, ontology, databases, epistemic reasoning, and AGI or robotics (Majkic, 2024). The 2024 paper therefore treats many-sortedness not as a mere metalogical convenience but as a conceptual refinement of the original intensional architecture.

A second scope distinction concerns related but nonidentical systems. Walsh’s treatment of Church’s intensional logic is typed and intensional, with entities of type $a$ and senses of type $a'$ , plus presentation relations $\Delta_a$ and representation functions $\nabla_a$ , but it is higher-order rather than first-order in the narrow sense (Walsh, 2015). Majkić’s earlier papers provide a semantic and algebraic template for intensional FOL with propositions, properties, and relations as denotable entities, but their formal syntax is presented as ordinary unsorted FOL with intensional interpretation layered over it [(Majkic, 2011); (Majkic, 2011)]. For that reason, the 2024 many-sorted IFOL paper is the closest explicit realization of the topic as named (Majkic, 2024).

2. Ontological architecture and sorted vocabulary

The background ontology inherited by many-sorted IFOL is the PRP-style intensional domain

$\D = D_{-1} + D_I$

with

$D_I = D_0 + D_1 + \cdots + D_n + \cdots$

where $D_{-1}$ is the domain of particulars, $D_0$ the domain of propositions, $D_1$ the domain of properties, and $a$ 0 for $a$ 1 the domain of $a$ 2-ary concepts or relations (Majkic, 2024). In the earlier intensional semantics, an open formula $a$ 3 denotes an intensional entity $a$ 4, while a closed sentence $a$ 5 denotes an intensional proposition $a$ 6 (Majkic, 2011).

The many-sorted extension adds the claim that concepts have sorted attributes, and that sorts are themselves intensional concepts (Majkic, 2024). A concept is represented as something of the form

$a$ 7

where the $a$ 8 are sorts attached to the argument places. Predicate symbols are linked to these predicate-concepts by a correspondence of the form

$a$ 9

subject to constraints on the sorts of free variables, functional terms, and abstracted terms (Majkic, 2024).

This sorted vocabulary is coordinated by a sort assignment mapping

$a'$ 0

extended to constants, function symbols, predicate symbols, and virtual predicates (Majkic, 2024). If $a'$ 1, then $a'$ 2. If $a'$ 3 has result sort $a'$ 4, then $a'$ 5 and

$a'$ 6

For predicates corresponding to predicate-concepts $a'$ 7,

$a'$ 8

If $a'$ 9 is an open formula with free variables $\Delta_a$ 0, then

$\Delta_a$ 1

(Majkic, 2024).

An important peculiarity is the special sort nested sentence, assigned to abstracted terms $\Delta_a$ 2 when they occur as arguments in intensional constructions (Majkic, 2024). This reflects the role of reified sentential content in IFOL and aligns with earlier abstraction-based intensional FOL, where formulas can be converted into terms denoting propositions or higher-arity concepts (Majkic, 2011).

3. Two-step semantics and possible worlds

The semantic core of IFOL is the two-step interpretation

$\Delta_a$ 3

where $\Delta_a$ 4 is a fixed intensional interpretation and $\Delta_a$ 5 ranges over extensionalization functions (Majkic, 2024). In the unsorted background semantics, these functions have the form

$\Delta_a$ 6

with $\Delta_a$ 7, $\Delta_a$ 8, and $\Delta_a$ 9 for $\nabla_a$ 0 (Majkic, 2024). In the earlier semantic presentation, this yields the characteristic factorization of extensional truth by

$\nabla_a$ 1

for a Tarskian interpretation $\nabla_a$ 2 associated with a world (Majkic, 2011).

The many-sorted refinement preserves this architecture but restricts admissible extensionalizations to a subset $\nabla_a$ 3 such that for any $\nabla_a$ 4-ary concept

$\nabla_a$ 5

one has

$\nabla_a$ 6

Thus extensionalizations must respect the declared sort profile of each concept (Majkic, 2024).

Possible worlds remain tied to extensionalization functions. The paper states the Bealer–Montague correspondence

$\nabla_a$ 7

and proves the many-sorted analogue of the standard correspondence

$\nabla_a$ 8

for $\nabla_a$ 9 (Majkic, 2024). This is the key preservation result: many-sorted IFOL refines, rather than replaces, ordinary truth-conditional semantics.

A common misunderstanding is that any modalization of FOL suffices for intensionality. The 2011 semantic paper rejects this explicitly: quantifiers can be recast modally without yielding genuine intensionality if the “intension” of each formula collapses to a constant function over worlds equal to its Tarski extension (Majkic, 2011). IFOL instead insists on a prior intensional layer of concepts, with worlds acting only at the extensionalization stage.

4. Static sorts, dynamic sorts, and formation rules

The 2024 paper distinguishes static sorts from dynamic sorts (Majkic, 2024). Static sorts are assigned by $\D = D_{-1} + D_I$0 to variables, terms, and concepts. Dynamic sorts $\D = D_{-1} + D_I$1 classify actual domain elements $\D = D_{-1} + D_I$2, and the admissible domain for a static sort $\D = D_{-1} + D_I$3 is defined by

$\D = D_{-1} + D_I$4

This allows values whose dynamic sort is a subsort of the declared static sort, and the adequacy condition is

$\D = D_{-1} + D_I$5

when $\D = D_{-1} + D_I$6 is the value of a term (Majkic, 2024).

The syntax of many-sorted IFOL is then constrained accordingly. For each sort $\D = D_{-1} + D_I$7, the set $\D = D_{-1} + D_I$8 of terms of static sort $\D = D_{-1} + D_I$9 is generated by two clauses. First, if $D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 0, then $D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 1. Second, if $D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 2 has sort

$D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 3

and $D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 4 with

$D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 5

then

$D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 6

(Majkic, 2024).

Atomic formulas are similarly sort-constrained. If $D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 7 has sort

$D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 8

and the arguments have admissible static sorts, then

$D_I = D_0 + D_1 + \cdots + D_n + \cdots$ 9

The rest of formula formation is inherited from unsorted IFOL (Majkic, 2024).

Assignments are many-sorted functions $D_{-1}$ 0 satisfying

$D_{-1}$ 1

for each variable $D_{-1}$ 2 of sort $D_{-1}$ 3 (Majkic, 2024). Term evaluation is extended recursively. For variables,

$D_{-1}$ 4

For functional terms $D_{-1}$ 5,

$D_{-1}$ 6

where

$D_{-1}$ 7

Equivalently, in graph form,

$D_{-1}$ 8

for the resulting value $D_{-1}$ 9 (Majkic, 2024).

This strongly typed discipline aligns with broader research on many-sorted syntax. Oddsson’s 2026 paper isolates the standard translation of many-sorted logic into unsorted FOL via sort predicates $D_0$ 0 and relativized quantifiers

$D_0$ 1

showing constructively that derivability is preserved and reflected even with equality and overloaded signatures (Oddsson, 18 Mar 2026). That work is not about intensionality, but it provides a precise proof-theoretic substrate for sorted IFOL encodings.

5. Abstraction, propositions as terms, and intensional predication

A defining feature of this IFOL tradition is the intensional abstraction operator

$D_0$ 2

which turns a formula into a term denoting an intensional entity (Majkic, 2011). If $D_0$ 3 is a sentence, the notation simplifies to $D_0$ 4, a term denoting a proposition. If $D_0$ 5 has free variables, the abstraction denotes a property or relation concept determined by the hidden variables $D_0$ 6 and visible variables $D_0$ 7 (Majkic, 2011).

The many-sorted extension inherits this mechanism directly and assigns abstracted terms the special sort nested sentence when they occur in argument positions (Majkic, 2024). This enables the representation of attitude reports, nested propositions, and natural-language-like structures. The paper’s examples include a nested formulation involving “know,” “tell,” and a geometric sphere formula, where variables such as $D_0$ 8 are naturally constrained to the sort Reals in the many-sorted setting (Majkic, 2024). Another example formalizes “Mario Rossi works to resolve the EN-problem for which the people do not believe there exists somebody who resolved it,” using abstraction to embed quantified sentential content inside attitude predicates (Majkic, 2024).

The semantic treatment of abstraction remains the earlier one. The assignment extension satisfies

$D_0$ 9

and the extension of an abstracted term is given by projection: $D_1$ 0 This makes abstraction the bridge between formula-level structure and concept-level denotation (Majkic, 2011).

A plausible implication is that many-sortedness significantly improves the discipline of abstraction. In the unsorted version, abstraction yields a concept with arity determined by free-variable positions; in the sorted version, the same operation yields a concept whose argument places are already ontologically typed. That consequence is strongly suggested by the 2024 refinement, although the paper frames it through sort restrictions rather than a separate theorem (Majkic, 2024).

The same abstraction-based architecture is reused in later robotics-oriented work, where predicates such as $D_1$ 1 take abstracted terms as arguments and support autoepistemic reasoning, though that presentation is not itself the canonical statement of many-sorted IFOL (Majkic, 2022). The 2025 bilattice variant retains “the same syntax but different semantics,” confirming that abstraction is part of the stable core of the framework (Majkic, 4 Aug 2025).

6. Variants, neighboring frameworks, and open issues

The most direct semantic variant is the 2025 extension of many-sorted IFOL over Belnap’s bilattice, intended for Strong AI and robotics (Majkic, 4 Aug 2025). It preserves the same syntax while replacing two-valued semantics by a four-valued one based on

$D_1$ 2

with both truth-ordering and knowledge-ordering (Majkic, 4 Aug 2025). In that version, a sentence may be true, false, unknown, or inconsistent; extensions explicitly omit $D_1$ 3-cases and recover them through completion. This variant is presented as addressing paradoxes, contradictory information, and incomplete knowledge while retaining the same many-sorted intensional architecture (Majkic, 4 Aug 2025).

Not all relevant work stays within first-order many-sorted syntax. Walsh’s study of Church’s intensional logic introduces the typed contrast between extensional entities of type $D_1$ 4 and intensional entities of type $D_1$ 5, together with presentation relations $D_1$ 6 and optional representation functions $D_1$ 7 (Walsh, 2015). That framework is higher-order, but it offers a closely related design lesson for IFOL: a many-sorted first-order analogue can be obtained by reifying extensional and intensional entities into separate sorts linked by first-order relations, while predicativity is needed to avoid Russell–Myhill-style paradoxes (Walsh, 2015).

Many-sorted modal and hybrid systems contribute additional substrate. A many-sorted polyadic modal logic with typed operators $D_1$ 8, Kripke semantics, canonical completeness, and a Jónsson–Tarski theorem supplies a propositional modal core that can be read as a precursor to sorted intensional reasoning (Leustean et al., 2018). Many-sorted hybrid modal languages add nominals, state variables, satisfaction operators $D_1$ 9, and a standard translation into many-sorted first-order logic, giving a different route from world-indexed intensionality to sorted FOL correspondence (Leuştean et al., 2020).

On the proof-theoretic side, many-sortedness itself is now technically well understood. The 2026 constructive treatment of the standard translation from many-sorted to unsorted predicate logic shows that sorted quantifiers, equality, and even overloaded signatures can be handled by explicit derivation transformations (Oddsson, 18 Mar 2026). This does not solve the intensional problem, but it clarifies the exact many-sorted machinery that an intensional extension requires.

Several limitations and controversies remain clear in the literature represented here. The 2024 many-sorted IFOL paper adds little new proof theory; its contribution is primarily ontological and semantic, and the IS-A relation $a$ 00 is not internalized as an ordinary predicate with axioms but treated as part of the enriched conceptual structure (Majkic, 2024). The use of higher-arity concepts as unary sorts through derived sort-extensions is unconventional and partly ontology-driven rather than standard model-theoretic typing (Majkic, 2024). The special sort nested sentence is not itself an element of $a$ 01, creating a hybrid boundary between internal ontology and external typing (Majkic, 2024). More broadly, the literature shows that “intensionality” is not uniform: in some works it means a PRP-based two-step semantics (Majkic, 2011), in others a typed sense/reference architecture (Walsh, 2015), in others stable-model nonmonotonicity for functions (Bartholomew et al., 2023), and in still others merely modal locality. A persistent misconception is to treat these as interchangeable.

Taken together, these works present IFOL as a structured research direction rather than a single universally standardized calculus. Its clearest encyclopedia-level characterization is: a first-order framework in which formulas denote intensional entities, sorts restrict admissible terms and assignments, worlds or extensionalization functions determine extensions, and abstraction allows propositions, properties, and relations to become first-class objects of predication (Majkic, 2024).