Variable-polyadic relation

A variable-polyadic relation can apply one logical relation symbol to finite argument sequences of different lengths under a declared interpretation.

Version

v1 · 2026-10-31 · History

Domain-specific #

14595

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Knowledge Representation, Formal Semantics → Mathematics

Core Idea

A variable-polyadic relation lets one logical relation symbol be applied to finite sequences of different lengths. An interpretation specifies which sequences satisfy it; an allowed length alone does not make a tuple true. In the KIF draft, a relation is a set of finite lists that may have unequal lengths. Particular relations can impose additional minimum-length or type conditions.^[1][2]

How would you explain it like I'm…

Any-Number Friend Word

Some words can be used with different numbers of names. You can say 'Amy and Ben are friends' or 'Amy, Ben and Cara are friends.' A Variable-polyadic relation is like that: one relation word that works with two names, three names, or more. Saying it doesn't make it true, though.

Relations With Flexible Slots

In logic, a relation is like a statement-pattern such as 'is taller than' that you fill in with things. Usually each relation has a fixed number of blanks: 'is taller than' always takes two. A Variable-polyadic relation can take different numbers of things, like 'are in a line' working for three people or five people. What it means decides which lists of things make it true. There can still be rules, like needing at least two things, and a sentence can follow the rules and still be false.

Variable-Arity Relation Symbol

A Variable-polyadic relation is a logical relation symbol that can be applied to finite lists of arguments of different lengths, instead of having one fixed number of places. In standard logic, a relation symbol has a set arity: a two-place relation always takes exactly two arguments. Here, the same symbol can appear with two, three or more arguments in different statements, and its interpretation decides which lists satisfy it. A vocabulary can still impose limits, like a minimum length or type restrictions. And applying it correctly doesn't make the statement true; being well-formed and being true are separate.

 

A Variable-polyadic relation is a logical relation symbol that may be applied to finite argument sequences of varying lengths, with its interpretation determining which sequences satisfy it. This contrasts with the standard practice of assigning each relation symbol a single fixed arity, so that a two-place symbol always takes exactly two arguments. Under the variable-polyadic approach, the same symbol can occur in assertions with different numbers of explicit arguments. The vocabulary may still constrain applications, for example by imposing a minimum length or type restrictions on argument positions. Well-formedness and truth remain distinct: a properly formed application with any permitted number of arguments need not be satisfied by the interpretation.

Examples

Numeric ordering. The KIF draft gives one less-than relation containing both <2,3> and <2,3,4>; the longer list means 2<3 and 3<4. It is a formal illustration in a draft specification, not an observed deployment.^[1]

Ontology partition. Brown, Pease and Urban describe SUMO’s Partition relation as variable arity with at least two class arguments and report a six-argument assertion in the January 2023 ontology file. Their translation carries arity and type information into higher-order set theory. The admissible form is distinct from the truth of any particular class sequence.^[2]

Boundaries

Fixed finite arity assigns a place count to a particular representation; this account allows one logical relation symbol to take different counts. A programming variadic function is a callable interface rather than this truth relation. Neither variable arity nor sequence notation implies an infinite-length tuple, that all permitted tuples satisfy a relation, or that a linguistic valency analysis is identical to this formalism.^[1][3]

Neighborhood in Abstraction Space

Variable-polyadic relation sits in a moderately populated region (45^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Models & Logical Foundations (37 abstractions)

Nearest neighbors

• Valuation (logic) — 0.87
• Well-Formed Formula — 0.87
• Categorial Grammar — 0.86
• Meaning Postulate — 0.86
• Propositional formula — 0.86

Computed from structural-signature embeddings · 2026-10-31

References

[1] Knowledge Interchange Format, draft proposed American National Standard NCITS.T2/98-004 (1998), §5.1. https://www.jfsowa.com/ikl/KIFdpans.htm registry ↩a ↩b ↩c

[2] Chad E. Brown, Adam Pease and Josef Urban, “Translating SUMO-K to Higher-Order Set Theory,” Frontiers of Combining Systems, Lecture Notes in Computer Science 14279 (2023), §2.2. https://doi.org/10.1007/978-3-031-43369-6_14 registry ↩a ↩b

[3] Christopher Menzel, “Knowledge Representation, the World Wide Web, and the Evolution of Logic,” §3.1, author-paper copy. https://jfsowa.com/ikl/Menzel11.pdf registry ↩