=Paper=
{{Paper
|id=Vol-3637/paper28
|storemode=property
|title=A Meta-Ontological Inquiry on Topics for Non-Topic-Neutral Logics
|pdfUrl=https://ceur-ws.org/Vol-3637/paper28.pdf
|volume=Vol-3637
|authors=Massimiliano Carrara,Filippo Mancini
|dblpUrl=https://dblp.org/rec/conf/jowo/CarraraM23
}}
==A Meta-Ontological Inquiry on Topics for Non-Topic-Neutral Logics==
https://ceur-ws.org/Vol-3637/paper28.pdf
A Meta-Ontological Inquiry on Topics for
Non-Topic-Neutral Logics
Massimiliano Carrara1 , Filippo Mancini2
1
Università degli Studi di Padova, Dipartimento FISPPA, Italia
2
University of Bonn, Institute of Philosophy, Chair of Logic and Foundations, Germany
Abstract
J.C. Beall in [1] proposes to read the middle-value of some Weak Kleene logics as off-topic. In such a
frame, propositions can be true-and-on-topic, false-and-on-topic or (simply) off-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.
Keywords
Off-topic interpretation of the third truth-value, topic, subject matter, Weak Kleene Logics, Bochvar’s
logic
Introduction
Topic-neutrality is taken as a standard, implicit typical trait of logic. Such a feature is connected
to various ideas and views about logic: the Kantian view that logic is constitutive of thought,
Wittgenstein’s view that logical truths are not properly meaningful – for they do not restrict
the possibility space –, the view that logical truths are analytic – which is held, for example, by
neo-empiricists like Carnap –, and so on.
A way to specify topic-neutrality is by arguing that logic is ontologically neutral. Take, for
example, Varzi’s reading of topic-neutrality as ontological neutrality: “[a]s a general theory of
reasoning, and especially as a theory of what is true no matter what is the case (or in every
‘possible world’), logic is supposed to be ontologically neutral. It should be free from any
metaphysical presuppositions. It ought to have nothing substantive to say concerning what
there is, or whether there is anything at all” (Varzi [2]).
Topic-neutrality of logic as ontological neutrality is usually considered a standard also in
formal ontology. Consider, for example, what Guarino, Carrara and Giaretta wrote at the start
of their paper on ontological commitments:
FOUST VII: 7th Workshop on Foundational Ontology, 9th Joint Ontology Workshops (JOWO 2023), co-located with FOIS
2023, 19-20 July, 2023, Sherbrooke, Québec, Canada.
Envelope-Open massimiliano.carrara@unipd.it (M. Carrara); filippo.mancini@unipd.it (F. Mancini)
GLOBE https://sites.google.com/fisppa.it/massimilianocarrara/home (M. Carrara);
https://sites.google.com/view/filippomancini/home-page (F. Mancini)
Orcid 0000-0002-0877-7063 (M. Carrara); 0000-0002-1061-7982 (F. Mancini)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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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 affairs which are compatible with some
underlying ontological commitment.
[Guarino et al. [3], p. 560]
However, nowadays topic-neutrality of logic is widely questioned. Here we focus on some
logics where this problem is particularly evident because topicality can be considered as part of
the very same logical theory: Weak Kleene logics.
Traditionally, Kleene’s three-valued logics divide into two families: strong and weak.1 Weak
Kleene logics, K𝑤3 , originate from weak tables (see table 1 below). Arguably, the two most
important K𝑤3 are Bochvar [5] and Halldén [6]’s logics (B and H, respectively), which differ
in the designated values they take on.2 B assumes that classical truth is the only value to be
preserved by valid inferences. H includes also the non-classical value among the designated
ones. In this paper we will focus on B.
What does the semantic non-classical value 0.53 mean in B? Some different interpretations
are available, such as nonsense, meaninglessness, and undefined.4 Among them, a new proposal
by Beall [1] suggests to read 0.5 as off-topic. Thus, a proposition that obtains this value should
be understood as being off-topic. In general, each proposition could be true-and-on-topic, false-
and-on-topic or (simply) off-topic. Such an interpretation of 0.5 has been criticised by Francez
[12] pointing out that 0.5 as off-topic does not satisfy the pre-theoretic understanding of a
truth-value in a truth-functional logic.5 If Francez’s criticism is sound, a characterization of
topic (or a subject matter) is needed to evaluate pros and cons of Beall’s proposal. But Beall [1]
is silent about what a topic is.6 Our goal here is to help to remedy this deficiency, giving a way
of understanding the fact that in a certain logic a proposition is in or out of a certain topic. We
do this by specifying what topics could be, specifically in B logic.
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.
1
See Kleene et al. [4].
2
There is increasing interest in K𝑤3 . To give some examples, Coniglio and Corbalan [7] develop sequent calculi for
K𝑤3 , Paoli and Pra Baldi [8] introduce a cut-free calculus (a hybrid system between a natural deduction calculus and
a sequent calculus) for one of K𝑤3 , PWK, Ciuni [9] explores some connections between H and Graham Priest’s Logic
of Paradox, LP, and Ciuni and Carrara [10] focus on logical consequence in H.
3
0.5 means undefined as in Kleene et al. [4]. This value might also be referred to as the third value, the middle value,
or 0.5.
4
For a survey on new interpretations of 0.5 see Ciuni and Carrara [11].
5
Specifically, Francez [12] claims that any notion that aspires to qualify as an interpretation of a truth-value has to
satisfy certain requirements, and that Beall’s interpretation of 0.5 as off-topic does not do that.
6
“Topic” and “subject matter” are just synonyms, and throughout the paper we will use them interchangeably.
1. B Logic
The language of Bochvar’s logic is the standard propositional language, 𝐿. Given a nonempty
countable set Var = {𝑝, 𝑞, 𝑟, … } of atomic propositions, 𝐿 is defined by the following Backus-Naur
Form:
Φ𝐿 ∶∶= 𝑝 ∣ ¬𝜙 ∣ 𝜙 ∨ 𝜓 ∣ 𝜙 ∧ 𝜓
We use 𝜙, 𝜓 , 𝛾 , 𝛿 … to denote arbitrary formulas, and Γ, Φ, Ψ, Σ, … for sets of formulas. As usual,
𝜙 ⊃ 𝜓 =𝑑𝑒𝑓 ¬𝜙 ∨ 𝜓.
Propositional variables are interpreted by a valuation function 𝑉𝑎 ∶ 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 𝑉 ∶ Φ𝐿 ⟼ {1, 0.5, 0} is the unique extension of a
mapping 𝑉𝑎 ∶ Var ⟼ {1, 0.5, 0} that is induced by the tables from table 1.
Table 1
Weak tables for logical connectives in 𝐿
¬𝜙 𝜙∨𝜓 1 0.5 0 𝜙∧𝜓 1 0.5 0
1 0 1 1 0.5 1 1 1 0.5 0
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0 1 0 1 0.5 0 0 0 0.5 0
Table 1 provides the full weak tables as they can be derived from Bochvar and Bergmann [5].
The way 0.5 transmits is usually called contamination (or infection), since the value propagates
from any 𝜙 ∈ Φ𝐿 to any construction 𝑘(𝜙, 𝜓), independently from the value of 𝜓 (here, 𝑘 is any
complex formula made out of some occurrences of both 𝜙 and 𝜓 and whatever well-formed
combination of ∨, ∧, and ¬). To better capture the way 0.5 works in combination with the other
truth-values, let us introduce the following definition:
Definition 1.2. For any 𝜙 ∈ Φ𝐿 , 𝑣𝑎𝑟 is a mapping from Φ𝐿 to the power set of Var, which can
be defined inductively as follows:
∎ 𝑣𝑎𝑟(𝑝) = {𝑝},
∎ 𝑣𝑎𝑟(¬𝜙) = 𝑣𝑎𝑟(𝜙),
∎ 𝑣𝑎𝑟(𝜙 ∨ 𝜓) = 𝑣𝑎𝑟(𝜙) ⋃ 𝑣𝑎𝑟(𝜓),
∎ 𝑣𝑎𝑟(𝜙 ∧ 𝜓) = 𝑣𝑎𝑟(𝜙) ⋃ 𝑣𝑎𝑟(𝜓),
Then, 𝑣𝑎𝑟(𝜙) is the set of all and only the propositional variables occurring in 𝜙, for all 𝜙 ∈ Φ𝐿 .
We can also apply 𝑣𝑎𝑟 to sets of sentences by stipulating that for any set 𝑋 of sentences of 𝐿,
𝑣𝑎𝑟(𝑋) = ⋃{𝑣𝑎𝑟(𝜙) ∣ 𝜙 ∈ 𝑋}.
Then, the following fact expresses contamination very clearly:
Fact 1.1 (Contamination). For all formulas 𝜙 ∈ Φ𝐿 and any valuation 𝑉:
𝑉(𝜙) = 0.5 iff 𝑉𝑎 (𝑝) = 0.5 for some 𝑝 ∈ 𝑣𝑎𝑟(𝜙)
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 K𝑤3 .
Bochvar [13] interprets the third value 0.5 as meaningless – or nonsensical.7 This interpre-
tation 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 ⊨𝐵 𝜓 iff there is no valuation 𝑉 such that 𝑉(𝜙) = 1 for all 𝜙 ∈ Γ, and 𝑉(𝜓) ≠ 1.
Let us now move on and provide a brief overview of the main conceptions of topic that are
currently available.
2. Conceptions of Topic
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
7
Bochvar [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 [1], Boem and Bonzio [14], and Carrara and Zhu [15].
8
The principle is also endorsed by Goddard [16].
9
Russell sentence: The set of all sets which are not members of themselves is a member of itself.
Grelling–Nelson sentence: ’Heterological’ is heterological.
10
The assumption here is that the notion of topic can actually license a unified theory.
– or theory – of each approach: Perry [18] for the SPA, Hawke [17, §4.2.1] for the ABA, and
Lewis [19] for the WBA.
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 iff there exists a non-degenerate topic u
such that u ≤ s and u ≤ t. Further, it is assumed that every meaningful sentence 𝛼 comes with a
least subject matter, represented by 𝜏(𝛼). 𝜏(𝛼) is the unique topic which 𝛼 is about, such that for
every topic 𝛼 is about, 𝜏(𝛼) is included into it. Thus, we say that 𝛼 is exactly about 𝜏(𝛼).12 But 𝛼
can also be partly or entirely about other topics: 𝛼 is entirely about t iff 𝜏(𝛼) ≤ t, whereas 𝛼 is
partly about t iff 𝜏(𝛼) ○ 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 different theories of topic.13 They should be sufficiently 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:
H1) If 𝜙 is entirely about s, then ¬𝜙 is entirely about s.
H2) If 𝐹 𝑎 is entirely about s and 𝐺𝑏 is entirely about s, then 𝐹 𝑎 ∨ 𝐺𝑏 is entirely about s.
H3) If 𝐹 𝑎 is entirely about s, then 𝐹 𝑎 ∧ 𝐺𝑏 is partly about s.
H4) The subject matter of 𝐹 𝑎 ∧ 𝐺𝑏 includes that of 𝐹 𝑎 ∨ 𝐺𝑏.
H5) Expressions of the form 𝐹 𝑎 ∨ 𝐹 𝑏 are about something but not necessarily about everything.
H6) If 𝐹 𝑎 and 𝐺𝑏 have different subject matter and 𝐹 𝑎 is contingent, then 𝐹 𝑎 differs in topic to
𝐹 𝑎 ∧ (𝑏 = 𝑏) and 𝐹 𝑎 ∧ (𝐺𝑏 ∨ ¬𝐺𝑏).
H7) If 𝐹 𝑎 and 𝐺𝑏 have different subject matter and 𝐹 𝑎 is contingent, then 𝐹 𝑎 differs in topic to
𝐹 𝑎 ∨ (𝑏 ≠ 𝑏) and 𝐹 𝑎 ∨ (𝐺𝑏 ∧ ¬𝐺𝑏).
H8) A claim of the form 𝐹 𝑎 ∨ ¬𝐹 𝑎 is about something (for example, a) but not about everything
(at least if 𝐹 𝑎 is about something but not everything).
H9) A claim of the form 𝐹 𝑎 ∧ ¬𝐹 𝑎 is about something (for example, a) but not about everything
(at least if 𝐹 𝑎 is about something but not everything).
H10) Expressions of the form 𝑎𝑅𝑏 and 𝑏𝑅𝑎 are not necessarily about the same topic. Nor is the
subject matter of 𝐹 𝑎 necessarily identical to that of an expression of the form 𝐺𝑎.
H11) If 𝑎 is part of 𝑏, then: if 𝜙 is entirely about a, then 𝜙 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.
11
The inclusion relation ≤ is usually taken to be reflexive, so that every topic includes itself.
12
Throughout 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.
13
Though, they are not exhaustive: there might be other criteria a theory of topic must satisfy to be correct.
Table 2
Evaluation of Perry [18]’s theory of topic based on Hawke’s twelve criteria of adequacy.
Theory H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12
Perry [18] no yes yes yes no no yes yes no no yes yes
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, 𝜙, is the set of all and
only those objects of which a property or a relation is predicated by 𝜙. As an example, consider
the following sentences:
(𝑝) logic is fun
(𝑞) logic is more fun than topology
Thus, according to the SPA 𝜏(𝑝) = {logic} and 𝜏(𝑞) = {logic, topology}. Also, since mathemat-
ics = {logic, algebra, topology, …} then 𝜏(𝑝) ≤ mathematics and 𝜏(𝑞) ≤ mathematics, that is
both 𝑝 and 𝑞 are entirely about mathematics. And since philosophy = {logic, ethic, …} then
𝜏(𝑞) ○ philosophy, that is 𝑞 is partly about philosophy.
Perry [18]’s theory of topic makes use of the SPA in combination with situation theory. For a
situation 𝑆 to be the case is for certain objects to stand in certain relations and certain objects
to fail to stand in certain relations. Therefore, 𝑆 may be represented by a partial valuation 𝜌𝑆 ,
assigning 1 (true), 0 (false), or nothing (undetermined) to every atomic claim, in accord with 𝑆.
Then, an arbitrary claim is verified or falsified by 𝑆 as follows:
• 𝑆 verifies atomic 𝑝 iff 𝜌𝑆 (𝑝) = 1. 𝑆 falsifies atomic 𝑝 iff 𝜌𝑆 (𝑝) = 0.
• 𝑆 verifies ¬𝜙 iff 𝑆 falsifies 𝜙. 𝑆 falsifies ¬𝜙 iff 𝑆 verifies 𝜙.
• 𝑆 verifies 𝜙 ∧ 𝜓 iff 𝑆 verifies 𝜙 and verifies 𝜓. 𝑆 falsifies 𝜙 ∧ 𝜓 iff 𝑆 falsifies 𝜙 or falsifies 𝜓.
• 𝑆 verifies 𝜙 ∨ 𝜓 iff 𝑆 verifies 𝜙 or verifies 𝜓. 𝑆 falsifies 𝜙 ∨ 𝜓 iff 𝑆 falsifies 𝜙 and falsifies 𝜓.
Then, Perry proposes that an object 𝑥 is in 𝜏(𝛼) iff 𝑥 is part of every situation that verifies
𝛼. As an example, this results in the following intuitive consequences: 𝜏(𝐹 𝑎 ∧ 𝐺𝑏) = {𝑎, 𝑏};
𝜏(𝐹 𝑎) = {𝑎} = 𝜏(¬𝐹 𝑎). 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), 𝑈, and (ii) a topic
function, 𝜏. A subject matter is any subclass of 𝑈. Again, ∅ is a degenerate topic, ≤ is ⊆, and ○ is
∩. 𝜏 is a function that assigns a topic to every atomic formula, 𝑝. The topic of complex formulas
is a combination (+) of the topics of their atomic components.
Table 3
Evaluation of Hawke [17, §4.2.1]’s theory of topic based on Hawke’s twelve criteria of adequacy.
Theory H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12
Hawke [17] yes yes yes yes yes yes yes no no yes yes yes
A particular atom-based theory of topic specifies 𝑈, + and 𝜏. As an example, a basic atom-
based theory14 takes 𝑈 to be the class of all sets of possible worlds,15 so that a member of 𝑈 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, 𝜏 assigns a topic 𝜏(𝑝) = {𝑃} to every atomic sentence, 𝑝, where 𝑃 is the truth set
of 𝑝 – i.e. 𝑃 is the set of the worlds at which 𝑝 holds. Finally, + is ∪ between the topics of atomic
formulas. Therefore, the subject matter of, say, 𝑝 ∧ (𝑞 ∨ 𝑟) is {𝑃} ∪ {𝑄} ∪ {𝑅} = {𝑃, 𝑄, 𝑅}, where
𝑃, 𝑄 and 𝑅 are the sets of worlds at which 𝑝, 𝑞 and 𝑟 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
difference: 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, 𝑊. In other words, a subject matter essentially corresponds to a partition of 𝑊.
In general, ≤ is not ⊆, but some sort of refinement relation between topics: s ≤ t iff s refines
t. Such a refinement relation can be precisely defined in different 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 𝜙
(atomic or complex) Lewis [19] takes 𝜏(𝜙) to be the binary partition consisting of the truth set
of 𝜙 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.
14
See Hawke [17, §4.2.1]
15
Let us call the set of possible worlds 𝑊. Thus, 𝑈 is the power set (or class) of 𝑊, 𝑈 = P(𝑊).
16
A 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.
17
This is not entirely correct, but we can gloss over that. See Hawke [17, fn. 24].
Table 4
Evaluation of Lewis [19]’s theory of topic based on Hawke’s twelve criteria of adequacy.
Theory H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12
Lewis [19] yes yes no no yes no no no no yes yes yes
3. Beall’s Interpretation of B
As we already mentioned, according to Halldén [6] and Bochvar and Bergmann [5], the third
value in K𝑤3 is to be interpreted as meaningless or nonsense. However, such interpretations
seem to suffer from some problems. For example, it is not at all obvious that we can make the
conjunction or the disjunction of a meaningless sentence with one with a traditional truth-value.
Then, Beall [1] proposes to “[...] read the value [1] not simply as true but rather as true and
on-topic, and similarly [0] as false and on-topic. Finally, read the third value [0.5] as off-topic” [1,
p. 140]. Thus, an arbitrary sentence 𝜙 is either true and on-topic, false and on-topic, or off-topic.
And note that since both on-topic and off-topic sentences are arguably meaningful, the problem
concerning the conjunction/disjunction of meaningless sentences vanishes.
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 different consequences on such a
reading. Unfortunately, Beall [1] is silent about that. But he gives some constraints we can use
to explore how topics behave and how they relate to the K𝑤3 truth-values.
Beall [1]’s new interpretation starts from setting a terminology concerning a theory, 𝑇. 𝑇 is a
set of sentences closed under a consequence relation, 𝐶𝑛. That is, 𝑇 = 𝐶𝑛(𝑋) = {𝜙 ∣ 𝑋 ⊢ 𝜙}
where 𝑋 is a given set of sentences and ⊢ is the consequence relation of the logic we are
working with. As for B, theories (or B-theories) are sets of sentences closed under B’s logical
consequence. Then, Beall puts forward the following motivating ideas for his proposal:
1. A theory is about all and only what its elements – that is, the claims in the
theory – are about.
2. Conjunctions, disjunctions and negations are about exactly whatever their
respective subsentences are about:
a) Conjunction 𝜙 ∧ 𝜓 is about exactly whatever 𝜙 and 𝜓 are about.
b) Disjunction 𝜙 ∨ 𝜓 is about exactly whatever 𝜙 and 𝜓 are about.
c) Negation ¬𝜙 is about exactly whatever 𝜙 is about.
[1, p. 139]18
18
To be precise, Beall states also a third condition concerning a theory being about every topic. We do not report it
here since it is not relevant for our discussion.
We can arguably formalize Beall’s motivating ideas as follows:19
Definition 3.1. Let 𝑇 be a B-theory and 𝜏(𝑇) be its topic. The following conditions regiment
how the topics of the compound 𝜙 ∈ Φ𝐿 and 𝜏(𝑇) behave:
1. 𝜏(𝑇) = ⋃{𝜏(𝜙) ∣ 𝜙 ∈ 𝑇}.
2. a) 𝜏(𝜙 ∧ 𝜓) = 𝜏(𝜙) ∪ 𝜏(𝜓).
b) 𝜏(𝜙 ∨ 𝜓) = 𝜏(𝜙) ∪ 𝜏(𝜓).
c) 𝜏(¬𝜙) = 𝜏(𝜙).
From definition 3.1 we can get some further results.
Corollary 3.1. For any 𝜙 ∈ Φ𝐿 , 𝜏(𝜙) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝜙)}.
Proof. We can prove it by induction.
1. Let 𝜙 be an atomic sentence 𝑝. Then 𝜏(𝜙) = 𝜏(𝑝).
2. Let 𝜙 = ¬𝜓 and 𝜏(𝜓) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝜓)}. Since 𝑣𝑎𝑟(𝜙) = 𝑣𝑎𝑟(¬𝜓) = 𝑣𝑎𝑟(𝜓), we have
𝜏(¬𝜓) = 𝜏(𝜓) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝜙)}.
3. Let 𝜙 = 𝛾 ∧ 𝛿, 𝜏(𝛾) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝛾)}, and 𝜏(𝛿) = ⋃{𝜏(𝑞) ∣ 𝑞 ∈ 𝑣𝑎𝑟(𝛿)}. Since
𝑣𝑎𝑟(𝜙) = 𝑣𝑎𝑟(𝛾) ∪ 𝑣𝑎𝑟(𝛿), we can derive 𝜏(𝜙) = 𝜏(𝛾) ∪ 𝜏(𝛿) = ⋃{𝜏(𝑟) ∣ 𝑟 ∈ (𝑣𝑎𝑟(𝛾) ∪ 𝑣𝑎𝑟(𝛿))}.
That is, 𝜏(𝜙) = ⋃{𝜏(𝑟) ∣ 𝑟 ∈ 𝑣𝑎𝑟(𝜙)}.
4. For 𝜙 = 𝛾 ∨ 𝛿, we can prove that 𝜏(𝜙) = ⋃{𝜏(𝑟) ∣ 𝑟 ∈ 𝑣𝑎𝑟(𝜙)} in the same way as above.
Corollary 3.2. 𝜏(𝑇) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝑇)}.
This corollary follows from the definition 3.1 and corollary 3.1. It states that what a theory 𝑇 is
about boils down to the union of what the atomic components of each claims in 𝑇 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 𝑋 be a set of sentences. Such a set is about all and only what its elements
are about. That is, 𝜏(𝑋) = ⋃{𝜏(𝜙) ∣ 𝜙 ∈ 𝑋}.
Thus, the following corollary follows from definition 3.2 and corollary 3.2:
Corollary 3.3. Let 𝑋 be a set of sentences. Then 𝜏(𝑋) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝑋)}.
19
Note that we use the same notation, 𝜏(…), 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, 𝜓 ∈ Φ𝐿 , 𝜏(𝑘(𝜙, 𝜓)) = 𝜏({𝜙, 𝜓}).
Proof. By induction and definition 3.1, 𝜏(𝑘(𝜙, 𝜓)) = 𝜏(𝜙) ∪ 𝜏(𝜓).20 According to definition 3.2,
𝜏({𝜙, 𝜓}) = 𝜏(𝜙) ∪ 𝜏(𝜓). Hence, 𝜏(𝑘(𝜙, 𝜓)) = 𝜏({𝜙, 𝜓}).
Corollary 3.5. For any 𝜙 , 𝜓 ∈ Φ𝐿 , 𝜏(𝑘(𝜙, 𝜓)) = 𝜏(𝑣𝑎𝑟(𝜙)) ∪ 𝜏(𝑣𝑎𝑟(𝜓)).
Proof. According to corollary 3.4, 𝜏(𝑘(𝜙, 𝜓)) = 𝜏({𝜙, 𝜓}). By virtue of corollary 3.3, we can derive
𝜏(𝑘(𝜙, 𝜓)) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟({𝜙, 𝜓})}. According to definition 1.2, 𝑣𝑎𝑟({𝜙, 𝜓}) = 𝑣𝑎𝑟(𝜙)∪𝑣𝑎𝑟(𝜓).
Hence, 𝜏(𝑘(𝜙, 𝜓)) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ (𝑣𝑎𝑟(𝜙) ∪ 𝑣𝑎𝑟(𝜓))} = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝜙)} ∪ ⋃{𝜏(𝑞) ∣ 𝑞 ∈
𝑣𝑎𝑟(𝜓)}. Since 𝑣𝑎𝑟(𝜙) = 𝑣𝑎𝑟(𝑣𝑎𝑟(𝜙)) and 𝑣𝑎𝑟(𝜓) = 𝑣𝑎𝑟(𝑣𝑎𝑟(𝜓)), we can derive 𝜏(𝑘(𝜙, 𝜓)) =
𝜏(𝑣𝑎𝑟(𝜙)) ∪ 𝜏(𝑣𝑎𝑟(𝜓)) by definition 3.2.
Corollary 3.6. For any 𝜙 ∈ Φ𝐿 and B-theory 𝑇, 𝜏(𝑇) = ⋃{𝜏(𝑣𝑎𝑟(𝜙)) ∣ 𝜙 ∈ 𝑇}.
Proof. From definition 3.2 and 𝑣𝑎𝑟(𝜙) = 𝑣𝑎𝑟(𝑣𝑎𝑟(𝜙)), we can derive 𝜏(𝜙) = 𝜏(𝑣𝑎𝑟(𝜙)). Since
definition 3.1 claims that 𝜏(𝑇) = ⋃{𝜏(𝜙) ∣ 𝜙 ∈ 𝑇}, we can derive 𝜏(𝑇) = ⋃{𝜏(𝑣𝑎𝑟(𝜙)) ∣ 𝜙 ∈ 𝑇} by
substituting 𝜏(𝜙) with 𝜏(𝑣𝑎𝑟(𝜙)).
Lemma 3.1. For any 𝑝 ∈ Φ𝐿 and B-theory 𝑇, if 𝑣𝑎𝑟(𝑝) ⊆ 𝑣𝑎𝑟(𝑇), then 𝜏(𝑝) ⊆ 𝜏(𝑇).
Proof. Since for any set 𝑋 of sentences of 𝐿, 𝑣𝑎𝑟(𝑋) = ⋃{𝑣𝑎𝑟(𝜙) ∣ 𝜙 ∈ 𝑋}, then 𝑣𝑎𝑟(𝑇) =
⋃{𝑣𝑎𝑟(𝜙) ∣ 𝜙 ∈ 𝑇}, 𝑣𝑎𝑟(𝑣𝑎𝑟(𝜙)) = ⋃{𝑣𝑎𝑟(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝜙)}. Then 𝑣𝑎𝑟(𝑇) = ⋃{𝑣𝑎𝑟(𝑝) ∣ 𝑝 ∈
𝑣𝑎𝑟(𝑇)}. If 𝑣𝑎𝑟(𝑝) ⊂ 𝑣𝑎𝑟(𝑇), then 𝑝 ∈ 𝑣𝑎𝑟(𝑇). Since 𝜏(𝑇) = ⋃{𝜏(𝑝) ∣ 𝑝 ∈ 𝑣𝑎𝑟(𝑇)}, then 𝜏(𝑝) ⊆
𝜏(𝑇).
However, the result does not hold in the opposite direction. That is, if 𝜏(𝑝) ⊆ 𝜏(𝑇), it might
not be the case that 𝑣𝑎𝑟(𝑝) ⊆ 𝑣𝑎𝑟(𝑇). To understand this point, consider the following
counterexample.
Example 3.1. For any 𝑟, 𝑞 ∈ Φ𝐿 and B-theory 𝑇, let 𝑣𝑎𝑟(𝑟) ⊈ 𝑣𝑎𝑟(𝑇), 𝑣𝑎𝑟(𝑞) ⊆ 𝑣𝑎𝑟(𝑇), and
𝜏(𝑟) = 𝜏(𝑞) ⊆ 𝜏(𝑇). Therefore, even though 𝜏(𝑟) ⊆ 𝜏(𝑇), 𝑣𝑎𝑟(𝑟) ⊈ 𝑣𝑎𝑟(𝑇).
This counterexample is possible because 𝜏 is not necessarily bijective. As a justification, consider
the following line of reasoning. Suppose that if 𝜏(𝑟) ⊆ 𝜏(𝑇), then 𝑣𝑎𝑟(𝑟) ⊆ 𝑣𝑎𝑟(𝑇). In that
case, we get that whatever sentence is in the theory, it is also on-topic; and that whatever
sentence is not in the theory, it is also off-topic. But then, Beall’s reading of the truth-value 0 as
20
Read 𝑘(𝜙, 𝜓) 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 ∈ Φ𝐿 and B-theory 𝑇, if 𝑣𝑎𝑟(𝜙) ⊆ 𝑣𝑎𝑟(𝑇), then 𝜏(𝜙) ⊆ 𝜏(𝑇).
Proof. We can prove it by induction.
1. If 𝜙 is an atomic sentence, this theorem holds for 𝜙 by virture of Lemma 3.1.
2. If 𝜙 = ¬𝜓 and this theorem holds for ¬𝜓. We can derive that this theorem holds for 𝜙,
because 𝑣𝑎𝑟(¬𝜓) = 𝑣𝑎𝑟(𝜓).
3. If 𝜙 = 𝛾 ∨ 𝛿, and this theorem holds for 𝛾 and 𝛿. We can derive this theorem holds for 𝜙,
because 𝑣𝑎𝑟(𝛾 ∨ 𝛿) = 𝑣𝑎𝑟(𝛾) ∪ 𝑣𝑎𝑟(𝛿).
4. We can prove this holds for 𝜙 = 𝛾 ∧ 𝛿 in the same way.
By virtue of theorem 3.1, we can clarify Beall’s on-topic/off-topic interpretation in the following
way.
Corollary 3.7. Let 𝑇 be a B-theory and 𝜏(𝑇) be its topic. For any 𝜙 ∈ Φ𝐿 ,
1. 𝜙 is on-topic iff 𝜏(𝜙) ⊆ 𝜏(𝑇). But note that this does not guarantee that 𝜙 ∈ 𝑇. However, if
𝜙 ∈ 𝑇, by definition 3.1, 𝜙 is definitively on-topic.
2. 𝜙 is off-topic iff 𝜏(𝜙) ⊈ 𝜏(𝑇). This suffices to say that 𝜙 ∉ 𝑇.
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, 𝑝 and 𝑞, to get 𝑝 ∧ 𝑞. Suppose that both
are on-topic, i.e. 𝜏(𝑝) ⊆ 𝜏(𝑇) and 𝜏(𝑞) ⊆ 𝜏(𝑇). According to 2(a), 𝜏(𝑝 ∧ 𝑞) = 𝜏(𝑝) ∪ 𝜏(𝑞). Thus,
𝜏(𝑝 ∧ 𝑞) ⊆ 𝜏(𝑇), that is 𝑝 ∧ 𝑞 is on-topic, which is in line with B semantics. Now, suppose that
at least one of the conjuncts is off-topic, say 𝑞. Thus, 𝜏(𝑞) ⊈ 𝜏(𝑇). Therefore, 𝜏(𝑝 ∧ 𝑞) ⊈ 𝜏(𝑇),
which is also in line with B semantics. Alternatively, we might also be tempted to consider the
following different interpretation: for 𝑝 to be on-topic means that 𝜏(𝑝) ∩ 𝜏(𝑇) ≠ ∅, whereas to
be off-topic means that 𝜏(𝑝) ∩ 𝜏(𝑇) = ∅. 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 𝜏(𝑝)∩𝜏(𝑇) ≠ ∅
but 𝜏(𝑞) ∩ 𝜏(𝑇) = ∅. Thus, since 𝜏(𝑝 ∧ 𝑞) = 𝜏(𝑝) ∪ 𝜏(𝑞), it follows that 𝜏(𝑝 ∧ 𝑞) ∩ 𝜏(𝑇) ≠ ∅ – i.e. 𝑝 ∧ 𝑞
is on-topic. This contradicts B semantics – namely, contamination. Moreover, our observations
21
Here, the discourse topic is what we call the topic of reference.
match [1, fn. 5]: “[a]n alternative account might explore ‘partially off-topic’, but I do not see
this as delivering a natural interpretation of K𝑤3 ”. Here, Beall is suggesting to distinguish two
notions: off-topic and partially off-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.
4. Beall’s Conception of Topic: Evaluation and Comparison
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) 𝐹 𝑎 and 𝐺𝑏 cannot be expressed in B, since 𝐿 is a propositional language. However, we
can consider a generalization of H2: if 𝜙 is entirely about s and 𝜓 is entirely about s, then
𝜙 ∨ 𝜓 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 𝜙 is entirely about
s, then 𝜙 ∧ 𝜓 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 𝑋 ∩ (𝑋 ∪ 𝑌) ≠ ∅, provided 𝑋 = ∅, 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 𝜙 ∧ 𝜓 includes
that of 𝜙 ∨ 𝜓. Since the inclusion relation (≤) between topics is interpreted as the subset
relation, this says that 𝜏(𝜙 ∧ 𝜓) ⊆ 𝜏(𝜙 ∨ 𝜓). Because of 2(a) and 2(b), 𝜏(𝜙 ∧ 𝜓) = 𝜏(𝜙 ∨ 𝜓).
Therefore, 𝜏(𝜙 ∧ 𝜓) ⊆ 𝜏(𝜙 ∨ 𝜓),24 and then H4.
(H5) Again, let us reason in the propositional context of B. Recall that for a sentence 𝜙 to be
about everything means that 𝜏(𝜙) = ∅. If the sentence is a disjunction, 𝜙 ∨ 𝜓, by 2(b) we
have 𝜏(𝜙 ∨ 𝜓) = 𝜏(𝜙) ∪ 𝜏(𝜓). Thus, for 𝜏(𝜙 ∨ 𝜓) = ∅ we need 𝜏(𝜙) = 𝜏(𝜓) = ∅. But this is
not necessary, since it is very plausible to claim that there is at least one sentence of the
form 𝜙 such that 𝜏(𝜙) ≠ ∅. 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.
22
To be clear, we reiterate that Beall [1] dos not offer a specific conception of topic. For he does not say what a topic
is. It may be represented by a set – we claim –, but Beall is silent about what we should take its members to be (e.g.
sets of worlds, objects, etc). Nevertheless, we can partially develop how Beall’s conception of topic works based on
his 1-3 conditions.
23
That is, 𝑋 ∪ 𝑋 = 𝑋.
24
This would not be true if ≤ were taken to be the strict subset relation, ⊂ .
Table 5
A comparison between Beall’s conception of topic (BT) and Hawke [17, §4.2.1] basic version of the ABA.
Theory H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12
Hawke [17] yes yes yes yes yes yes yes no no yes yes yes
BT yes yes yes yes yes yes yes yes yes ? ? ?
(H6-9) As for H6 (H7), note that the only way for Beall to violate it is to take the topic of
tautologies (contradictions) to be ∅. But this is not possible. As an example, consider
𝜙 ∨ ¬𝜙. According to Beall, 𝜏(𝑝 ∨ ¬𝑝) = 𝜏(𝑝) ≠ ∅. In other words, the subject matter of
tautologies (contradictions) is not degenerate – i.e. is not ∅. And this is also in line with
H8 and H9.
(H10) We cannot express 𝑎𝑅𝑏, 𝑏𝑅𝑎, 𝐹 𝑎 and 𝐺𝑎 in B. Thus, Beall’s interpretation is silent about
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 𝑊, 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.
Conclusion
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 off/on-topic reading of the truth-values of
B. The result is that, in Beall’s perspective, for a claim 𝜙 to be on-topic means that 𝜏(𝜙) ⊆ 𝜏(𝑇),
where 𝑇 is the B-theory at stake; whereas, for 𝜙 to be off-topic means that 𝜏(𝜙) ⊈ 𝜏(𝑇). 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 methodol-
ogy 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?
25
See Staab and Studer [20, pp. 201–221] for an overview on OntoClean.
Acknowledgments
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 Forschungs-
gemeinschaft), research unit FOR 2495.
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