In this paper, we extend the mereology of slots with an order relation, of particular importance for characterizing the structure and identity of informational entities such as texts. We propose a theory for linear informational entities. We show the theory consistent using Alloy, a model finder. We discuss under which condition two informational entities that share mereological and order structure should be considerd as identical or not. The ontology of informational entities is currently receiving attention [1, 2, 3]. It has been shown in such work that classical mereology is not adequate for accounting for their parthood structure, and that one rather needs to resort to slot mereology [4], initially developed for structural universals [5]. In this paper, we intend to supplement this work by proposing an extension involving order relations, of particular importance for two kinds of informational entities, texts (which will be the main focus here) and procedures. Structural universals are less prominent in applied ontology, which usually considers only domains of particulars. However, the ontology of structural universals also crucially requires additional relations on top of mereology to characterize their identity. In classical extensional mereology [6], adequate in particular for material entities, two entities are identical if they share exactly the same proper parts-this is extensionality. As argued in [ 7], the identity of structural universals and informational entities cannot be grasped by mereology alone and extensionality doesn't apply. For instance two diferent texts can have exactly the same words but arranged diferently: 'Lea loves Leo' is not the same sentence as 'Leo loves Lea' while they are composed of exactly the same three words. Additional structural relations are needed to characterize the identity of structural universals and informational entities; in the case of texts, we need word order. In this paper, we propose to extend the mereology of slots proposed in [7] with order or sequence relations, building on Van Benthem's classical theory of time based on periods, parthood and temporal precedence [8]. We first briefly present the mereology of slots, and then discuss and motivate choices for new primitives and axioms to extend this mereology of slots with order.
Slot mereology was introduced by Bennett in 2013 [
Standard mereology cannot represent such facts. Slot mereology addresses this by introducing new entities—called “slots” of a whole—to account for cases where the same part occurs multiple times within that whole. Each slot is filled by a unique “filler” and identifies one of the possibly multiple specific contexts in which this filler appears as a part in the whole. A same part, then, can fill diferent slots of a given whole. The domain is thus partitioned into fillers, which could be, e.g., structural universals or informational entities, and slots, that characterize the mereological structure of such fillers.
Bennett’s slot mereology was recently thoroughly investigated from a theoretical point of view,
leading to a revised and expanded theory named “mereology of slots” [
For lack of space, we will not reproduce the whole theory here, which has 3 primitive relations: , and a ternary primitive of “contextualization” (omitted here), 18 axioms, 18 definitions and more than 60 theorems. We’ll use only some of its vocabulary, namely: • • • • • • • • • + • (, ) reads “filler fills slot ” (, ) reads “slot is a parthood slot of filler ” or “filler owns slot ”
() reads “ is a slot”. This is defined as ∃ (, ) (, ) reads “filler is part of filler ”. This is defined as ∃( (, ) ∧ (, )) (, ) reads “slots and share the same owner”.
(, ) reads “slot is part of slot ” (, ) reads “slot is a proper part of slot ”
(, ) reads “slot overlaps slot ” = + reads “slot is the sum of slots and ” () reads “ is the fusion of the slots having property ”
We’ll also use relevant axioms and theorems, in particular that parthood between slots is transitive, antisymmetric and (conditionally) reflexive (T18, T19, T20), 1 that is an equivalence relation and that sums and fusions (conditionally) exist. Other important axioms and theorems state that each slot is filled by a unique filler (BA7) and that each slot is owned by a unique filler (A1). In addition, each filler has a unique improper slot, filled by itself (A2, A3). The transitivity of parthood between fillers (T13) is guaranteed by a mechanism of duplication of slots between levels, based on the ternary primitive of contextualization. What is important for the remainder of the paper is that only slots belonging to a same owner can be mereologically related: the mereology of slots is local. Each filler generates a separate extensional mereology of its slots, whose universe is the improper slot of the filler (T62).
Let’s illustrate this theory with a simple example: the word ‘ada’ has 2 atomic parts (assuming
letters are considered as atoms), the letter 1 ‘a’ being part of twice. Thus has 3 slots, 1 to 3, filled
by these 2 letters: 1 and 3 by the letter 1 ‘a’ and 2 by the letter 2 ‘d’. All sum combinations of these
atomic slots give rise to three additional slots filled by a combination of two letters. In addition, has
the improper slot 0, filled by itself. This can be (partially) described by the following formulas:
(0, )
(1, )
(2, )
1All numbered axioms, definitions, lemmas and theorems not explicitly introduced in this paper refer to those of [
Notice that the slot 13, sum of 1 and 3, is filled by something having as part 1 twice, and this thing is therefore part of . However, it can be argued that the string ‘aa’ isn’t a part of the word ‘ada’ and that the filler of 13 is not a string. In fact, words and strings are (convex) sequences of letters, a feature impossible to grasp with mereology alone. This is one motivation to add order to the theory.
No order relation can be introduced between fillers since they may repeatedly appear in a word and moreover appear in many words with diferent orderings: in the above example of the word ‘ada’, the letter ‘a’ would be both before and after the letter ‘d’, which is inconsistent with the asymmetry of strict order. But order between slots can be introduced, as they stand for specific contexts in which the letters appear (and are not shared between words). We will address the interplay between this order relation and the mereological relations and operators on slots.
Order considerations are especially prominent in the ontology of time. In the ’80s and ’90s of last century, a number of first-order theories of time have been proposed. Some of them consider intervals (aka periods) as basic elements of the domain, and therefore also handle notions of sub-interval, that is, mereological relations.
One such proposal is the theory of Allen and Hayes [
There are important diferences between axiomatizing the temporal order between temporal intervals and the sequence between slots of an informational entity in the context of the mereology of slots.
First, the explicit or implicit assumption of Van Benthem and Hajnicz, as well as Allen & Hayes, is that
intervals or periods are “one-piece”, i.e., convex, following the mathematical definition of intervals in ,
or ℛ (any point situated in between two points of the interval belongs to the interval). On the other
hand, as seen in the example above with slot 13, the sum operator between slots in the mereology of
slots makes it impossible to adopt this assumption. Such slots can be compared to non-convex intervals
[
Second, time is generally assumed to be unbounded, and this is reflected in axioms in Van Benthem’s, Hajnicz’s and Allen & Hayes’s works. For informational entities, for instance words as sequences of letters and texts as sequences of words, it is clearly impossible to assume that all of them are infinite, so such axioms have to be dropped. Assuming that, on the contrary, there always is a beginning and an end is a step ahead that we shall leave open. Similarly, time is often seen as dense, as with intervals in or ℛ, but this is questionable here, especially with words and texts. Atomicity, discussed by Van Benthem, was not discussed in the mereology of slots. Again, this is an option that will be proposed.
Finally, linearity is adopted by Van Benthem, whose aim is to axiomatize the structures of intervals
in or ℛ. The same assumption is made by Allen & Hayes, but this is discussed by Hajnicz. Linearity
is not always adopted in theories of time and action, as time is sometimes seen as branching (to the
right, to the left, or both—as illustrated e.g. by the work of Broersen [
The order relation < is intended to hold when a slot fully precedes another one, including adjacency (i.e., “meets” in Allen’s algebra). For instance, in the case above of slots of the word (‘ada’), 1 < 2, 2 < 3 and 1 < 3, but also 12 < 3 and 1 < 23.
Just as mereological relations, the order < holds only on slots belonging to the same owner: OA 1. < → (() ∧ () ∧ (, )) OA 2. ( < ∧ < ) → < OA 3. ¬ <
The order < standardly is transitive and irreflexive (and thus asymmetric):
Since the order < covers adjacency, it makes no sense to assume a form of density, there is not necessarily a third slot between two slots preceding one another. But we do assume that the order has no “holes”, there always is (at least) an adjacent slot to the right when there is a slot to the right (and similarly to the left). To do so, we first introduce the definition of Allen’s meets relation, and adopt Allen & Hayes’s axiom guaranteeing the unicity of meeting points:2 OD 1. (, ) ≡ < ∧ ∀ ¬( < ∧ < ) OA 4. ( (, ) ∧ (, ) ∧ (, )) → (, )
Note that, trivially, two slots that meet have the same owner. Then, the equivalent of Van Benthem’s “neighbour” conditions are adopted: OA 5. < → ∃ (, ) OA 6. < → ∃ (, )
As discussed above, we leave it open to add further axioms constraining the order relation, depending on the domain at hand. For informational entities, having an unbound order does not seem to make sense; on the contrary, one could assume the existence of initial and ending slots. We first define initial and ending slots. Note that among the slots of a same filler, there can be several such slots (in our example above, 1 and 12 are both initial slots).
OD 2. () ≡ () ∧ ∀((, ) → ¬ < ) OD 3. () ≡ () ∧ ∀((, ) → ¬ < )
Then, the following definition can be used to characterize the boundedness of the order among the slots of a given filler, and used to impose an axiom claiming that every filler is bounded, if required. OD 4. () ≡ ∃, ((, ) ∧ (, ) ∧ () ∧ ())
We now need to link mereology and order. As proposed by Van Benthem, we adopt axioms of monotonicity. Any part of a slot preceding a slot also precedes , and vice versa: OA 7. < → ∀( (, ) → < ) OA 8. < → ∀( (, ) → < )
As discussed above, slots can be non-convex, but of course there are convex slots. A slot is convex if for any two parts of it that are ordered, every slot in between is also a part of it: OD 5. () ≡ () ∧ ∀(( (, ) ∧ (, )) → ∀(( < ∧ < ) → (, )))
Actually, such a definition makes sense only if we assume a form of linearity of the order (the sum of two slots on diferent branches would be trivially convex, which does not capture the intuition of convexity). We do adopt linearity here, as discussed above. Any two slots of a same owner always are “comparable” in this mixture of order and mereology: they cannot be on separate branches of the order, as may happen in a branching time. A simple way of adopting linearity in the presence of non-convex slots is to exploit the underlying order of implicit meeting “points” of adjacent slots:3 OA 9. ( (, ) ∧ (, ) ∧ (, )) → ( < ∨ < )
That is, if meets , meets and and have the same owner, then either is before or is before . For example, in our former example, 12 meets 3, 1 meets 23 and as it happens, 1 is before 3.
This is not enough to exclude branching at the boundaries of the whole order (if present). We need to impose that any two initial slots are related by parthood, and the same constraint for any two ending slots: OA 10. ((, ) ∧ () ∧ ()) → ( (, ) ∨ (, )) OA 11. ((, ) ∧ () ∧ ()) → ( (, ) ∨ (, ))
In our example, 1 and 12 are initial slots, and indeed, 1 is a part of 12.
Finally, to guarantee linearity, we need to add the constraint that two convex slots being situated after the same slots and before the same slots are identical.4 This avoids diamond structures in which is before both and , which are both before , with no order relation between and . OA 12. ((, ) ∧ () ∧ () ∧ ∀( < ↔ < ) ∧ ∀( < ↔ < )) → =
As discussed above, it makes sense in certain cases to add the optional axiom of atomicity in the
mereology of slots, something not considered in [
OD 6. () ≡ () ∧ ∀( (, ) → = ) ATOM () → ∃ ( (, ) ∧ ())
The mereology of slots (without sum and fusion axioms which generate too many entities to remain
tractable) conjoined with the definitions OD1, OD2, OD3, OD5 and the axioms OA1-OA12 was validated
with Alloy, a model finder. The model presented on Figure 1 was found (where < is coded ‘INF’). Alloy
was also used as a heuristic tool to check whether some unwanted models were found and add axioms
to exclude those when needed (axioms OA10 and OA11 were added in that respect).
3This is akin to Allen & Hayes’s linearity axiom M2 in [
Some of Van Benthem’s axioms on period parthood were already discussed in Sec.2: in the mereology of slots, the transitivity, reflexivity and anti-symmetry of are theorems (with domain restriction to slots for reflexivity). We’ll now show that Van Benthem’s axioms FREE, CONJ, DISJ and DIR on period parthood are theorems in the mereology of slots (with suitable restrictions to slots of a same owner where necessary).
FREE. FREE says that if all parts of overlap , then is part of . This is a standard theorem in
classical extensional mereology (see SCT13 in [
OT 1. ∀( (, ) → (, )) → (, )
Proof: suppose ∀( (, ) → (, )) and ¬ (, ). The slot strong supplementation axiom A10 with ¬ (, ) implies that ∃( (, ) ∧ ¬(, )) which contradicts the premise. CONJ. CONJ is the existence of the “intersection” of two overlapping slots—that is, two overlapping slots and have a largest common slot-part .
OT 2. (, ) → ∃( (, ) ∧ (, ) ∧ ∀(( (, ) ∧ (, )) → (, )))
This is standardly obtained in general extensional mereology through the fusion operator applied
to the property of overlapping both these slots. Axiom AS12 in [
DISJ. DISJ is the existence of the sum of the slots and provided they “underlap” (two slots underlap if there is a slot of which they are both part).
OT 3. ∃( (, ) ∧ (, )) → ∃( (, ) ∧ (, ) ∧ ∀(( (, ) ∧ (, )) → (, )))
Proof: Axiom A11 asserts the existence of the sum, with the premise that and have the same owner. Since implies (T21), which by definition D3 and the unicity of the owner A1 is an equivalence relation, from the premise of OT3 we get the premise of A11. Then we can apply Lemma 49 to obtain, considering as the sum, the first two terms of the conclusion of OT3 (the operands are part of the sum), and Lemma L57 to obtain the last part of the conclusion of OT3 (the sum is part of any slot of which the operands are both part).
DIR. DIR is the assumption that any two slots underlap. Here we need to alter Van Benthem’s formula: two slots of a same owner underlap.
OT 4. (, ) → ∃( (, ) ∧ (, ))
Proof: OT4 is implied by the sum existence axiom A11 and Lemma L49 saying that the operands are part of the sum.
SEP. Finally, Van Benthem’s SEP principle, which says that order and overlap are separated, is a theorem: OT 5. < → ¬(, )
Proof: Suppose < and (, ), that is ∃( (, ) ∧ (, )). With axiom OA8, < and (, ) entail < . Applying now OA7 to < and (, ) we obtain < which contradicts irreflexivity axiom OA3.
We can now come back to the discussion on identity that was raised in the introduction. We’ll consider whether, having added order to the mereology of slots, we can characterize the identity of (linear) informational entities. As seen in the introduction, having the same parts, even in the same numbers, is not enough to characterize identity of words as sequences of letters: ‘ada’ and ‘daa’ have the same parts, but in a diferent order.
First, let’s note that if two entities have the exact same slots, they have the same improper slot and are thus identical (and two entities cannot have diferent improper slots that would have the exact same slot-parts, because of slot extensionality).
The issue then regards comparing two fillers without assuming they share the same slots. If the two ifllers have the same structure, both as far as mereology is concerned (and thus have the same parts) and as far as order is concerned, are they identical or not? Of course, if we conclude they are identical, then, they do have the same slots.
Checking isomorphism between structures cannot be done in FOL unless we assume the domain is ifnite, but this assumption is not first-order expressible. Such an assumption remains here implicit, but as evoked before, it really makes sense when textual informational entities are concerned. We will adopt a similar approach as for the notion of “mirroring” between an informational entity and a template (an informational entity used to express structural constraints on other informational entities, e.g., a form or a document model) [16, 17], axiomatizing a new primitive of equivalence EQ between slots. By extension, two fillers will be considered equivalent if their improper slots (their unique slot filled by themselves) are equivalent. Here, we will merely characterize very broadly this relation of equivalence with a few axioms that should be completed in future work.
The relation holds between slots. It is a symmetric relation.
OA 13. (, ) → (() ∧ ()) OA 14. (, ) → (, )
If two slots are equivalent, then each of their proper parts must be equivalent to a single proper part of the other, and these proper parts have the same fillers: OA 15. ((, ) ∧ (, )) → ∃!( (, ) ∧ (, )) OA 16. ((, ) ∧ (, ) ∧ (, ) ∧ (, ) ∧ (, )) → (, )
If two slots are ordered, their equivalent slots of a same owner are similarly ordered: OA 17. ( < ∧ (, ′) ∧ (, ′) ∧ (′, ′)) → ′ < ′
For example, ‘ada’ and ‘daa’ have similar mereological structures (two atomic slots filled by ‘a’ and one atomic slot filled by ‘d’) but with diferent orders and so are not equivalent. On the other hand, if we have two strings and ′ written ‘ada’ with their respective improper slots and ′, each constituted by the three slots 1 < 2 < 3 and ′1 < ′2 < ′3, such that (1, ′1), (2, ′2) and (3, ′3) and such that the filler ‘a’ fills 1, 3, ′1 and ′3 and the filler ‘d’ fills 2 and ′2, then (, ′) is compatible with OA13-17. Assuming that and ′ are equivalent but diferent would require to introduce two strings of characters ( and ′) which share all characteristics but would be diferent, which might be considered as an unwanted violation of a principle of identity of indiscernible. So for words as sequences of letters and for texts as sequences of words, it seems appropriate to assume that implies identity, and that = ′ and = ′. However, this is not the case for all linear informational entities.
Consider now the case where two diferent entities are composed by diferent slots that have exactly similar slot-structures in terms of slot-parthood, slot-order and fillers. This could be useful to represent a situation where two agents utter the same sentence through diferent speech-acts, e.g. Mary and John both uttering ‘I love pasta’. As it happens, we may have to represent those two utterances as diferent entities even if they share the same slot structure and filler proper parts. First, because they have diferent authors (which, for example, is relevant for the identity of informational content entities according to the IAO ontology—note though that other ontologies might have diferent commitments on this matter). Second, because of indexicals : ‘I’ refers to diferent persons in both sentences (respectively Mary and John). Third, because of pragmatic inferences. If Mary is at a bufet when uttering this sentence, she might want to imply that she would love to get now a serving of pasta, whereas if John is discussing food preferences with a friend, it might not lead to the same implicature5. In case authoring, indexical and pragmatic considerations would contribute to the identity of a utterance, a standard assumption in linguistics (see, e.g., [19]), this might be a reason to accept two utterances or discourses with equivalent mereological and order slot structures and identical proper part fillers, but diferent identities. According to this conception, in Borges’s famous example, Cervantes and Pierre Menard could author two diferent novels, both named Don Quixote and with exactly the same words and the same structure. So, one may hold that there is a unique text, as a sequence of words, but two diferent novels, as authored discourses in specific contexts. 5See [18] for a proposal of accounting for an illocutionary dimension of informational entities through slots—in that case, accounting for their directive dimension through directive slots.
We have proposed an axiomatic system that added order to the formerly developed mereology of slots, concentrating on entities such as words, sentences or texts that are linear.
This work would need to be extended to informational artifacts where branching is possible, such as procedures or flowcharts which can branch to the right (depending on a condition, or because of starting parallel processes) but also to the left (as branches might be merged after developing in parallel). We need to leave this possibility for future work, as it would imply assessing whether or not changes in the mereology of slots are needed. Indeed, one may question the relevance of summing slots belonging to diferent branches. In addition, future considerations could investigate how order on elements of a template can constrain the order of elements of an informational entity compliant with this template [17].
The authors wish to thank Claudio Masolo for insightful discussions on the topic of this paper. The comments of two anonymous reviewers helped improving the paper. All remaining errors are ours.
During the preparation of this work, the authors occasionally used DeepL in order to check the phrasing. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content. [16] A. Barton, C. Tarbouriech, L. Vieu, J.-F. Éthier, A mereological system for informational templates roles, in: T. P. Sales, M. M. Hedblom et al (Ed.), Proceedings of the Joint Ontology Workshops 2022 (JOWO 2022). 6th International Workshop on Foundational Ontologies (FOUST), 2022. [17] A. Barton, L. Vieu, J.-F. Éthier, How information complies with a template: A dual mereological system (submitted). [18] A. Barton, L. Vieu, J.-F. Ethier, Expressions, utterances and directive slots, in: 5th Workshop on Foundational Ontology (FOUST 2021)@ JOWO 2021: The Joint Ontology Workshops, volume 2969, CEUR-WS. org, 2021. [19] N. Asher, A. Lascarides, Logics of conversation, Cambridge University Press, 2003.