J.C. Beall in [1] proposes to read the middle-value of some Weak Kleene logics as of-topic. In such a frame, propositions can be true-and-on-topic, false-and-on-topic or (simply) of-topic. This reading has two obvious consequences. The first is that it is not true, as it has been argued for a long time, that logic is topic-neutral. The second is that it is crucial to know what a topic is to clarify the ontological commitments of a logical theory, and more generally to understand how logic interacts with ontology. Thus, the aim of this paper is to elaborate on Beall's notion of topic and compare it with the three main approaches to topics or subject matter that are currently available in the literature.
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First order logic is notoriously neutral with respect to ontological choices: when a
logical language is used with the purpose of modelling a particular aspect of reality,
the set M of all its models is usually much larger than the set Mi of the intended
ones, which describe only those states of afairs which are compatible with some
underlying ontological commitment.
[Guarino et al. [
Traditionally, Kleene’s three-valued logics divide into two families: strong and weak.1 Weak
Kleene logics, K3 , originate from weak tables (see table 1 below). Arguably, the two most
important K3 are Bochvar [
What does the semantic non-classical value 0.53 mean in B? Some diferent interpretations
are available, such as nonsense, meaninglessness, and undefined .4 Among them, a new proposal
by Beall [
This paper is divided into four sections. In §1 we introduce Bochvar’s logic. In §2 we briefly present the main approaches to topics that are currently available in the literature. In §3 we discuss and further develop Beall’s interpretation of B. Finally, in §4 we elaborate on Beall’s notion of topic and compare it with the three main approaches presented in §2.
2There is increasing interest in K3 . To give some examples, Coniglio and Corbalan [
The language of Bochvar’s logic is the standard propositional language, . Given a nonempty countable set Var = {, , , … } of atomic propositions, is defined by the following Backus-Naur Form:
Φ ∶∶= ∣ ¬ ∣ ∨ ∣ ∧ We use , , , … to denote arbitrary formulas, and Γ, Φ, Ψ, Σ, … for sets of formulas. As usual, ⊃ = ¬ ∨ .
Propositional variables are interpreted by a valuation function ∶ Var ⟼ {1, 0.5, 0} that assigns one out of three values to each atomic proposition in Var. The valuation extends to arbitrary formulas according to the following definition: Definition 1.1 (Valuation). A valuation ∶ Φ ⟼ {1, 0.5, 0} is the unique extension of a mapping ∶ Var ⟼ {1, 0.5, 0} that is induced by the tables from table 1. Definition 1.2. For any ∈ Φ , is a mapping from Φ to the power set of Var, which can be defined inductively as follows: ∎ ∎ ∎ ∎ ( ) = { }, (¬ ) = ( ), ( ∨ ) = ( ) ⋃ ( ), ( ∧ ) = ( ) ⋃ ( ), Then, ( ) is the set of all and only the propositional variables occurring in , for all ∈ Φ . We can also apply to sets of sentences by stipulating that for any set of sentences of , ( ) = ⋃{ ( ) ∣ ∈ }.
Then, the following fact expresses contamination very clearly: Fact 1.1 (Contamination). For all formulas ∈ Φ and any valuation : ( ) = 0.5 if ( ) = 0.5 for some ∈ ( ) The left-to-right direction is shared by all the most common three-valued logics; the right-to-left direction is a peculiar property of K3 .
Bochvar [13] interprets the third value 0.5 as meaningless – or nonsensical.7 This interpretation goes along with the way 0.5 propagates to a compound formula from its components: the sense of a compound sentence depends on that of its components, and if some component makes no sense, the sentence as a whole will make no sense either.8 In his jargon, nonsensical sentences are expressions that are syntactically well-formed and yet fail to convey a proposition, such as the sentences from which the Russell paradox and the Grelling–Nelson paradox are derived.9
The logical consequence relation of B is defined as preservation of truth – i.e., the only designated value is 1. In other words: Definition 1.3. Γ ⊨ if there is no valuation such that ( ) = 1 for all ∈ Γ , and ( ) ≠ 1. Let us now move on and provide a brief overview of the main conceptions of topic that are currently available.
In this section, we introduce the main approaches to the notion of topic that have been developed
so far. We will basically follow Hawke [17], who distinguishes three main strategies to account
for subject matter: the way-based approach (WBA), the atom-based approach (ABA) and the
subject-predicate approach (SPA).10 All the main theories of topic we can find in the literature
fall into one of these three categories. To be short, we just select and present the basic version
7Bochvar [13] uses meaningless as an umbrella term including paradoxical statements such as the Liar and Russell’s
paradoxes, vague sentences, denotational failure, and ambiguity. Though, there are some issues concerning the
interpretation of 0.5. This is the reason why some alternative interpretations have been recently proposed, e.g.
Beall [
Before presenting them, let us make some assumptions and define a simple notation to facilitate the discussion. We use bold letters for topics, such as s, t, etc. ≤ is the inclusion relation between topics, so that s ≤ t expresses that s is included into (or is a subtopic of) t. Given that, we define a degenerate topic as one that is included in every topic.11 Also, we define the overlap relation ○ between topics as follows: s ○ t if there exists a non-degenerate topic u such that u ≤ s and u ≤ t. Further, it is assumed that every meaningful sentence comes with a least subject matter, represented by ( ). ( ) is the unique topic which is about, such that for every topic is about, ( ) is included into it. Thus, we say that is exactly about ( ).12 But can also be partly or entirely about other topics: is entirely about t if ( ) ≤ t, whereas is partly about t if ( ) ○ t. Lastly, + (combination) is a – for now, unspecified – binary operator between topics that outputs another topic, i.e. their combination. Thus, not only does any given approach consist in defining what a topic is, but also provides a specific interpretation for ≤, ○ and (if needed) +.
Also, for ease of comparison we introduce Hawke’s criteria of adequacy which serve as constraints to evaluate diferent theories of topic. 13 They should be suficiently clear and require no comment. But those who wish to know more about them can refer to Hawke [17, §3]. Thus, here are Hawke’s conditions:
H2) If is entirely about s and is entirely about s, then ∨ is entirely about s. H3) If is entirely about s, then ∧ is partly about s.
H4) The subject matter of ∧ includes that of ∨ .
H5) Expressions of the form ∨ are about something but not necessarily about everything. H6) If and have diferent subject matter and is contingent, then difers in topic to ∧ ( = ) and ∧ ( ∨ ¬ ).
H7) If and have diferent subject matter and is contingent, then difers in topic to ∨ ( ≠ ) and ∨ ( ∧ ¬ ).
H8) A claim of the form ∨ ¬ is about something (for example, a) but not about everything (at least if is about something but not everything).
H9) A claim of the form ∧ ¬ is about something (for example, a) but not about everything (at least if is about something but not everything).
H10) Expressions of the form and are not necessarily about the same topic. Nor is the subject matter of necessarily identical to that of an expression of the form . H11) If is part of , then: if is entirely about a, then is entirely about b. H12) A question Q serves (in some sense) as a subject matter.
Thus, a theory of topic that aspires to be correct must meet (at least) all such criteria. 11The inclusion relation ≤ is usually taken to be reflexive, so that every topic includes itself. 12Throughout this paper, when we talk about the topic of a sentence we mean its least topic. In case we want to refer to one of its topics that is not the least one, we will make it clear. 13Though, they are not exhaustive: there might be other criteria a theory of topic must satisfy to be correct. 2.1. The Subject-Predicate Approach On the SPA, a topic is a set of entities. Therefore, the relations we have previously mentioned correspond to their set-theoretic counterparts: s ≤ t is s ⊆ t, s ○ t is s ∩ t and s + t is s ∪ t. ∅ is a degenerate topic. More specifically, the topic of an arbitrary sentence, , is the set of all and only those objects of which a property or a relation is predicated by . As an example, consider the following sentences: ( ) logic is fun ( ) logic is more fun than topology Thus, according to the SPA ( ) = {logic} and ( ) = {logic, topology}. Also, since mathematics = {logic, algebra, topology, …} then ( ) ≤ mathematics and ( ) ≤ mathematics, that is both and are entirely about mathematics. And since philosophy = {logic, ethic, …} then ( ) ○ philosophy, that is is partly about philosophy.
Perry [18]’s theory of topic makes use of the SPA in combination with situation theory. For a situation to be the case is for certain objects to stand in certain relations and certain objects to fail to stand in certain relations. Therefore, may be represented by a partial valuation , assigning 1 (true), 0 (false), or nothing (undetermined) to every atomic claim, in accord with . Then, an arbitrary claim is verified or falsified by as follows: • verifies atomic if ( ) = 1. falsifies atomic if ( ) = 0. • verifies ¬ if falsifies . falsifies ¬ if verifies . • verifies ∧ if verifies and verifies . falsifies ∧ if falsifies or falsifies . • verifies ∨ if verifies or verifies . falsifies ∨ if falsifies and falsifies . Then, Perry proposes that an object is in ( ) if is part of every situation that verifies . As an example, this results in the following intuitive consequences: ( ∧ ) = {, }; ( ) = { } = (¬ ). But Hawke [17, pp. 705-706] shows that some of the conditions H1-12 are not satisfied. We summarize his evaluation in table 2. 2.2. The Atom-Based Approach In general, the ABA of topic resorts to (i) a class of objects (e.g. sets of worlds), , and (ii) a topic function, . A subject matter is any subclass of . Again, ∅ is a degenerate topic, ≤ is ⊆, and ○ is ∩. is a function that assigns a topic to every atomic formula, . The topic of complex formulas is a combination (+) of the topics of their atomic components. yes yes yes
A particular atom-based theory of topic specifies , + and . As an example, a basic atombased theory14 takes to be the class of all sets of possible worlds,15 so that a member of can be interpreted as a piece of information or a truth set.16 Thus, a topic is a set of sets of worlds – i.e. a set of pieces of information. In particular, the topics of atomic sentences are sets of just one set of worlds; instead, the topics of complex sentences are sets of more than one sets of worlds. Then, assigns a topic ( ) = { } to every atomic sentence, , where is the truth set of – i.e. is the set of the worlds at which holds. Finally, + is ∪ between the topics of atomic formulas. Therefore, the subject matter of, say, ∧ ( ∨ ) is { } ∪ { } ∪ { } = { , , }, where , and are the sets of worlds at which , and are respectively true.
As shown in table 3, such a theory meets many but not all the H1-12 criteria. 2.3. The Way-Based Approach According to the WBA, a topic is a comprehensive set of ways things can be, where “a way things can be” is again a truth set or a piece of information – i.e. a set of possible worlds. Therefore, this is essentially the same situation we have in the atom-based approach, but with one crucial diference: for a set of sets of possible worlds to be a subject matter it must be comprehensive. A set of ways W is comprehensive if the union of the members of W is equal to the set of possible worlds, . In other words, a subject matter essentially corresponds to a partition of .
In general, ≤ is not ⊆, but some sort of refinement relation between topics: s ≤ t if s refines t. Such a refinement relation can be precisely defined in diferent ways. For instance, let us consider Lewis [19]’s theory. Here, s refines t means that every member of t is the union of some members of s. In other words, every member of s is a subset of some members of t – i.e. no members of s cuts across any member of t. In particular, given any arbitrary sentence (atomic or complex) Lewis [19] takes ( ) to be the binary partition consisting of the truth set of and its complement.17
As for the previous accounts of topic, we show which constraints are met by Lewis [19]’s theory in table 4. 14See Hawke [17, §4.2.1] 15Let us call the set of possible worlds . Thus, is the power set (or class) of , = P( ). 16A set of worlds can be viewed as a truth set to the extent that the members of such a set are the worlds at which a/some given proposition(s) is/are true. The piece of information represented by this set of worlds is the conjunction of the propositions that are true at those worlds. 17This is not entirely correct, but we can gloss over that. See Hawke [17, fn. 24]. yes
H6 no
H7 no
H8 no
H9 no
H10 yes
H11 yes
H12 yes
As we already mentioned, according to Halldén [
Following Beall’s proposal, What is a topic? becomes a crucial question to be addressed. For
depending on how we answer this question we may have diferent consequences on such a
reading. Unfortunately, Beall [
Beall [
We can arguably formalize Beall’s motivating ideas as follows:19 Definition 3.1. Let be a B-theory and ( ) be its topic. The following conditions regiment how the topics of the compound ∈ Φ and ( ) behave: 1. ( ) = ⋃{ ( ) ∣ ∈ }. 2. a) ( ∧ ) = ( ) ∪ ( ).
b) ( ∨ ) = ( ) ∪ ( ).
c) (¬ ) = ( ).
Corollary 3.1. For any ∈ Φ , ( ) = ⋃{ ( ) ∣ ∈ ( )}.
1. Let be an atomic sentence . Then ( ) = ( ). 2. Let = ¬ and ( ) = ⋃{ ( ) ∣ ∈ ( )}. Since ( ) = (¬ ) = ( ), we have (¬ ) = ( ) = ⋃{ ( ) ∣ ∈ ( )}. 3. Let = ∧ , ( ) = ⋃{ ( ) ∣ ∈ ( )}, and ( ) = ⋃{ ( ) ∣ ∈ ( )}. Since ( ) = ( ) ∪ ( ), we can derive ( ) = ( ) ∪ ( ) = ⋃{ ( ) ∣ ∈ ( ( ) ∪ ( ))}.
That is, ( ) = ⋃{ ( ) ∣ ∈ ( )}.
4. For = ∨ , we can prove that ( ) = ⋃{ ( ) ∣ ∈ ( )} in the same way as above. Corollary 3.2. ( ) = ⋃{ ( ) ∣ ∈ ( )}.
This corollary follows from the definition 3.1 and corollary 3.1. It states that what a theory is about boils down to the union of what the atomic components of each claims in are about. Moreover, even if Beall does not mention what an arbitrary set (i.e. not necessarily a theory) is about, we opt for the following very plausible definition: Definition 3.2. Let be a set of sentences. Such a set is about all and only what its elements are about. That is, ( ) = ⋃{ ( ) ∣ ∈ }.
Thus, the following corollary follows from definition 3.2 and corollary 3.2: Corollary 3.3. Let be a set of sentences. Then ( ) = ⋃{ ( ) ∣ ∈ ( )}. 19Note that we use the same notation, (…), both for the topic of a sentence and the topic of a theory – i.e. a set of sentences.
By virtue of corollary 3.3 and definition 3.1, we get: Corollary 3.4. For any , ∈ Φ , ( (, )) = ({, }).
Proof. By induction and definition 3.1, ( (, ({, }) = ( ) ∪ ( ). Hence, ( (, )) = ({, }). )) = ( ) ∪ ( ).20 According to definition 3.2, Corollary 3.5. For any , ∈ Φ , ( (, )) = ( ( )) ∪ ( ( )).
)) =
({, Proof. According to corollary 3.4, ( (, }). By virtue of corollary 3.3, we can derive ( (, Hence, ( (,
( )}. Since ( ( )) ∪ ( ( )) by definition 3.2.
({, ( )) = ⋃{ ( ) ∣ ∈
})}. According to definition 1.2, )) = ⋃{ ( ) ∣ ∈ ( ( ) ∪ ( ))} = ⋃{ ( ) ∣ ∈ ({, }) =
( )∪ ( )} ∪ ⋃{ ( ) ∣ ∈ ( ). ( ) = ( )) and ( ) = ( ( )), we can derive ( (, )) = Corollary 3.6. For any ∈ Φ and B-theory , ( ) = ⋃{ ( ( )) ∣ ∈ }. Proof. From definition 3.2 and substituting ( ) with ( ( )). definition 3.1 claims that ( ) = ⋃{ ( ) ∣ ∈ }, we can derive ( ) = ⋃{ ( ( )) ∣ ∈ } by ( ) = ( ( )), we can derive ( ) = ( ( )). Since Lemma 3.1. For any ∈ Φ and B-theory , if ( ) ⊆ ( ), then ( ) ⊆ ( ). Proof. Since for any set of sentences of , ( ) = ⋃{ ( ) ∣ ∈ }, then ( ) = ( ( )) = ⋃{ ( ) ∣ ∈
( )}. Then ( ), then ∈ ( ). Since ( ) = ⋃{ ( ) ∣ ∈ ( ) = ⋃{
( ) ∣ ∈ ( )}, then ( ) ⊆ ⋃{
( ) ∣ ∈ }, ( )}. If ( ) ⊂ ( ). counterexample.
However, the result does not hold in the opposite direction. That is, if ( ) ⊆ ( ), it might not be the case that ( ) ⊆ ( ). To understand this point, consider the following Example 3.1. For any , ∈ Φ and B-theory , let ( ) = ( ) ⊆ ( ). Therefore, even though ( ) ⊆ ( ), ( ) ⊈ ( ). ( ) ⊈ ( ), ( ) ⊆ ( ), and This counterexample is possible because is not necessarily bijective. As a justification, consider the following line of reasoning. Suppose that if ( ) ⊆ ( ), then case, we get that whatever sentence is in the theory, it is also on-topic; and that whatever ( ). In that ( ) ⊆ sentence is not in the theory, it is also of-topic. But then, Beall’s reading of the truth-value 0 as 20Read (, ) as defined on page 3. false-and-on-topic is not available anymore. In other words, allowing for the bijection results in a conflict between Beall’s conception of topic and his reading of B’s truth-values.
To sum up: by Beall’s ideas and the way B works we get that (1) the topic of a sentence is completely determined by (is the union of the topics of) its propositional variables, and (2) the topic of a theory is completely determined by (is the union of the topics of) the propositional variables of its sentences. Moreover, we get also the following important result: Theorem 3.1. For any ∈ Φ and B-theory , if ( ) ⊆ ( ), then ( ) ⊆ ( ). Proof. We can prove it by induction.
1. If is an atomic sentence, this theorem holds for by virture of Lemma 3.1. 2. If = ¬ and this theorem holds for ¬ . We can derive that this theorem holds for , because (¬ ) = ( ). 3. If = ∨ , and this theorem holds for and . We can derive this theorem holds for , because ( ∨ ) = ( ) ∪ ( ).
4. We can prove this holds for = ∧ in the same way.
By virtue of theorem 3.1, we can clarify Beall’s on-topic/of-topic interpretation in the following way.
Corollary 3.7. Let be a B-theory and ( ) be its topic. For any ∈ Φ , 1. is on-topic if ( ) ⊆ ( ). But note that this does not guarantee that ∈ . However, if ∈ , by definition 3.1, is definitively on-topic.
2. is of-topic if ( ) ⊈ ( ). This sufices to say that ∉ .
Finally, we can note that such an interpretation fits Beall’s conditions as well as B semantics. To see this, let’s conjoin two propositional variables, and , to get ∧ . Suppose that both are on-topic, i.e. ( ) ⊆ ( ) and ( ) ⊆ ( ). According to 2(a), ( ∧ ) = ( ) ∪ ( ). Thus, ( ∧ ) ⊆ ( ), that is ∧ is on-topic, which is in line with B semantics. Now, suppose that at least one of the conjuncts is of-topic, say . Thus, ( ) ⊈ ( ). Therefore, ( ∧ ) ⊈ ( ), which is also in line with B semantics. Alternatively, we might also be tempted to consider the following diferent interpretation: for to be on-topic means that ( ) ∩ ( ) ≠ ∅, whereas to be of-topic means that ( ) ∩ ( ) = ∅. For instance, this is exactly what Hawke [17, p. 700] suggests: “[t]o say that a claim is somewhat on-topic is to say that its subject matter overlaps with the discourse topic”.21 However, such an interpretation is not compatible with Beall’s constraints. For condition 2(a) clashes with B semantics. To see this, suppose that ( ) ∩ ( ) ≠ ∅ but ( ) ∩ ( ) = ∅. Thus, since ( ∧ ) = ( ) ∪ ( ), it follows that ( ∧ ) ∩ ( ) ≠ ∅ – i.e. ∧ is on-topic. This contradicts B semantics – namely, contamination. Moreover, our observations 21Here, the discourse topic is what we call the topic of reference. match [1, fn. 5]: “[a]n alternative account might explore ‘partially of-topic’, but I do not see this as delivering a natural interpretation of K3 ”. Here, Beall is suggesting to distinguish two notions: of-topic and partially of-topic . The latter might be legitimately taken to correspond to the alternative reading in terms of overlap between topics that we rejected – as indeed he does.
Now that Beall’s alternative interpretation of B and his preliminary remarks about how topics should behave have been clarified, we can make a deeper analysis of his intended notion of topic and a comparison between that and the conceptions of topic described in §2.
Here, we evaluate Beall’s conception of topic22 based on the criteria H1-12, and compare it with the other theories we have introduced in §2. Thus, in what follows we go through H1-12 and see whether Beall’s constraints meet them. (H1) Negation preserves subject matter, and corresponds to Beall’s condition 2(c). Therefore,
H1 is met. (H2) and cannot be expressed in B, since is a propositional language. However, we can consider a generalization of H2: if is entirely about s and is entirely about s, then ∨ is entirely about s. This says that disjunction preserves shared subject matters. Now, because of 2(b) and union idempotence,23 such a generalization is met. Therefore, since H2 is just an instance of that, H2 is also met. (H3) The same strategy applies here. We can take a generalization of H3: if is entirely about s, then ∧ is partly about s. This states that conjunction results in a topic expansion, so that the topics of the conjuncts overlap the topic of the conjunction. Now, because of 2(a) and the basic set-theoretical theorem ∩ ( ∪ ) ≠ ∅, provided = ∅, this is true for Beall’s conception. Therefore, H3 is met. (H4) H4 is met. Consider the following version of this constraint: the topic of ∧ includes that of ∨ . Since the inclusion relation (≤) between topics is interpreted as the subset relation, this says that ( ∧ ) ⊆ ( ∨ ). Because of 2(a) and 2(b), ( ∧ ) = ( ∨ ).
Therefore, ( ∧ ) ⊆ ( ∨ ),24 and then H4.
(H5) Again, let us reason in the propositional context of B. Recall that for a sentence to be
about everything means that ( ) = ∅. If the sentence is a disjunction, ∨ , by 2(b) we
have ( ∨ ) = ( ) ∪ ( ). Thus, for ( ∨ ) = ∅ we need ( ) = ( ) = ∅. But this is
not necessary, since it is very plausible to claim that there is at least one sentence of the
form such that ( ) ≠ ∅. For example, consider the sentence “Maths is fun”. Arguably,
this is about mathematics, but is not about literature. Thus, we can conclude that
mathematics ≠ ∅. Therefore, H5 is met.
22To be clear, we reiterate that Beall [
H10. (H11) Beall’s proposal is also silent about H11. For the language of B does not contain any individual constant. (H12) H12 depends on what a topic is taken to be (a set of objects, a set of worlds, a partition on , etc). But Beall does not say anything about what a topic is. Therefore, this issue remains open.
We report the results of our analysis in table 5. Here, we also show a quick comparison between Beall’s conception of topic and Hawke [17, §4.2.1] basic version of the ABA. Thus, as should be clear, this is the closest theory to Beall’s proposal among those we have considered. Essentially, this suggests that Beall’s conception of topic points in the direction of an atom-based approach.
In this paper we have formalized Beall’s ideas about topic and drawn some facts from them, to see what conception of topic is the best one for his of/on-topic reading of the truth-values of B. The result is that, in Beall’s perspective, for a claim to be on-topic means that ( ) ⊆ ( ), where is the B-theory at stake; whereas, for to be of-topic means that ( ) ⊈ ( ). This is in line with B semantics. Further, even if Beall is silent about what exactly a topic is, the basic atom-based conception of topic does seem to be the closest notion of topic to his conception.
Some final questions, to be answered in future works. How does the topicality of a logic interact with the ontological commitments of the theory described? And what contribution does the topic of a logical frame make in terms of the characterisation of the meta-ontological categories of a specific theory? For instance, consider a tool such as OntoClean, i.e. a methodology that by means of certain meta-properties can validate ontologies.25 What is the role given by the topicality of the logical frame in the validation of ontologies? 25See Staab and Studer [20, pp. 201–221] for an overview on OntoClean.
Massimiliano Carrara’s research is partially funded by the CARIPARO Foundation Excellence Project (CARR_ECCE20_01): Polarization of irrational collective beliefs in post-truth societies. How anti-scientific opinions resist expert advice, with an analysis of the antivaccination campaign (PolPost).
Filippo Mancini’s research, including this work, is supported by the DFG (Deutsche Forschungsgemeinschaft), research unit FOR 2495. [13] D. Bochvar, On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus, Matamaticheskii Sbornik 4 (1938) 287–308. [14] F. Boem, S. Bonzio, A logic for a critical attitude?, Logic and Logical Philosophy 31 (2022) 261–288. URL: https://apcz.umk.pl/LLP/article/view/36948. doi:1 0 . 1 2 7 7 5 / L L P . 2 0 2 2 . 0 1 0 . [15] M. Carrara, W. Zhu, Computational Errors and Suspension in a PWK Epistemic Agent,
Journal of Logic and Computation 31 (2021) 1740–1757. [16] L. Goddard, Towards a logic of significance, Notre Dame Journal of Formal Logic 9 (1968) 233–264. [17] P. Hawke, Theories of aboutness, Australasian Journal of Philosophy 96 (2018) 697–723. [18] J. Perry, Possible worlds and subject matter, in: P. Alto (Ed.), The Problem of the Essential
Indexical and Other Essays, CSLI Publications, CA, 1989, pp. 145–160. [19] D. Lewis, Relevant implication, Theoria 54 (1988) 161–174. [20] S. Staab, R. Studer, Handbook on ontologies, Springer Science & Business Media, 2010.