=Paper=
{{Paper
|id=Vol-3354/short5
|storemode=property
|title=TGTS Based Argumentation
|pdfUrl=https://ceur-ws.org/Vol-3354/short5.pdf
|volume=Vol-3354
|authors=Massimiliano Carrara,Filippo Mancini,Difei Xu,Wei Zhu
|dblpUrl=https://dblp.org/rec/conf/aiia/CarraraMX022
}}
==TGTS Based Argumentation==
https://ceur-ws.org/Vol-3354/short5.pdf
TGTS Based Argumentation
Massimiliano Carrara1,∗,† , Filippo Mancini2,† , Difei Xu3,† and Wei Zhu4,†
1
FISPPA, University of Padua, Piazza Capitaniato 3, Padova 35139, Italy
2
FISPPA, University of Padua, Piazza Capitaniato 3, Padova 35139, Italy
3
School of Philosophy, Renmin University of China, Haidian District, Beijing, 100872, P.R. China
4
FISPPA, University of Padua, Piazza Capitaniato 3, Padova 35139, Italy
Abstract
Suppose two discussants hold opposite views about the level of pollution in Pinarella, and start arguing
about that. Then, one of them says something like “The winner of the next UEFA Champions League final
match will be Hellas Verona”, which is clearly out of topic with respect to the discussion topic. How do
we model such a situation in an argumentation process? Our aim here is to provide a framework capable
of handling such a phenomenon, namely a situation where one of the discussants in an argumentation
process goes out of topic and gives rise to a certain reaction from the other. The ingredients of such a
model are: a game-theoretical-semantics with a verifier and a falsifier, a discussion and a discussion
topic. We develop our framework using a Paraconsistent Weak Kleene logic (PWK), with the off-topic
reading of its non-classical value, and a topic-game-theoretical-semantics.
Keywords
Topic, Weak Kleene Logic, Game-Theoretical Semantics, Topic Game-Theoretical Semantics, Argumenta-
tion theory, Argumentation Process
1. Introduction
Imagine the following situation: there are two discussants holding an opposite view about
the level of pollution in Pinarella. They start arguing about that. Then, one of the two says:
“Pinarella is a sweet prime number”, which is clearly out of topic with respect to the discussion
topic, the level of pollution in Pinarella. Suppose we need a framework capable of handling
such a phenomenon, namely a situation where one of the discussants in an argumentation
process goes out-of-topic. In this short paper we sketch a model to give rise of this kind of
situations in an argumentation process. We first briefly introduce the paraconsistent Weak
Kleene logic (PWK), which belongs to the family of the Weak Kleene logics (WK3), where the
third value, u – traditionally understood as nonsense, meaninglessness or undefined 1 – has been
recently interpreted as off-topic [4]. Then, we present a new semantics for such a logic, the topic
6th Workshop on Advances In Argumentation In Artificial Intelligence (AI³ 2022)
∗
Corresponding author.
†
These authors contributed equally.
$ massimiliano.carrara@unipd.it (M. Carrara); filippo.mancini@unipd.it (F. Mancini); difeixu@163.com (D. Xu);
wei.zhu@unipd.it (W. Zhu)
https://sites.google.com/fisppa.it/massimilianocarrara/ (M. Carrara)
0000-0002-1061-7982 (F. Mancini); 0000-0002-6905-0482 (W. Zhu)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
CEUR Workshop Proceedings (CEUR-WS.org)
Proceedings
http://ceur-ws.org
ISSN 1613-0073
1
See e.g. Bochvar and Bergmann [1], Halldén [2], and Ciuni and Carrara [3].
game-theoretical semantics (TGTS), consisting of the following parts: (i) a set of discussants; (ii)
a set of on-topic sentences with respect to the discussion topic; (iii) a set of discussion rules;
and (iv) a strategy for the discussion capable of allowing us to distinguish on-topic and off-topic
discussions.
2. PWK and the Off-topic Interpretation
Paraconsistent Weak Kleene (PWK) belongs to the family of the Weak Kleene logics (WK3).2 The
language of PWK is the standard propositional language, ℒ. Given a nonempty countable set
Var = {𝑝, 𝑞, 𝑟, . . . } of atomic propositions, the language is defined by the following Backus-Naur
Form:
Φℒ ∶∶= 𝑝 ⋃︀ ¬𝜑 ⋃︀ 𝜑 ∨ 𝜓 ⋃︀ 𝜑 ∧ 𝜓 ⋃︀ 𝜑 ⊃ 𝜓
We use 𝜑, 𝜓, 𝛾, 𝛿 . . . to denote arbitrary formulas, 𝑝, 𝑞, 𝑟, . . . for atomic formulas, and
Γ, Φ, Ψ, Σ, . . . for sets of formulas. Propositional variables are interpreted by a valuation
function 𝑉𝑎 ∶ Var z→ {t, u, f } that assigns one out of three values to each 𝑝 ∈ Var. The
valuation extends to arbitrary formulas according to the following definition:
Definition 2.1 (Valuation). A valuation 𝑉 ∶ Φℒ z→ {t, u, f } is the unique extension of a
mapping 𝑉𝑎 ∶ Var z→ {t, u, f } that is induced by the tables from Table 1.
𝜑 ¬𝜑 𝜑∨𝜓 t u f 𝜑∧𝜓 t u f
t f t t u t t t u f
u u u u u u u u u u
f t f t u f f f u f
Table 1
Weak tables for logical connectives in ℒ
Table 1 provides the full weak tables from Kleene et al. [5, §64]. As usual, 𝜑 ⊃ 𝜓 =𝑑𝑒𝑓 ¬𝜑 ∨ 𝜓, and
its table follows accordingly. The way u transmits is called contamination, since for all formulas
𝜑 in ℒ and any valuation 𝑉 , 𝑉 (𝜑) = u iff 𝑉𝑎 (𝑝) = u for some 𝑝 ∈ 𝑣𝑎𝑟(𝜑), where 𝑣𝑎𝑟(𝜑) is the
set of all and only the atomic propositions occurring in 𝜑.
The logical consequence relation of PWK is defined as preservation of non-false values – i.e.,
the designated values are both u and t.
The third value, u has been traditionally understood as nonsense, meaninglessness or undefined.
Recently, [4] gives a new interpretation of it. He 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” [4, p. 140].3 Unfortunately, Beall [4] is silent about what a topic is.
But we can make some assumptions and develop his proposal in order to make it complete and
suitable for our purposes.
2
On these systems see e.g. [5], [1], [6], and [7].
3
Of course, 1, 0 and 0.5 correspond to t, f and u, respectively.
We assume that topics can be represented by sets. 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.4 Given that, we define a degenerate topic as one that is included in every
topic. 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). 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.
Next, we assume the following conditions concerning how topics behave with respect to the
logical connectives:
1. 𝜏 (𝜑 ∧ 𝜓) = 𝜏 (𝜑) ∪ 𝜏 (𝜓).
2. 𝜏 (𝜑 ∨ 𝜓) = 𝜏 (𝜑) ∪ 𝜏 (𝜓).
3. 𝜏 (¬𝜑) = 𝜏 (𝜑).
As shown in Carrara et al. [8, §2], from these assumptions we can also prove that the topic of a
complex sentence boils down to the union of the topics of its atomic components.
Further, not only do sentences have a topic, but also sets of sentences do. More in detail, we
have the following:
Definition 2.2. Given a set 𝑆 of sentences of ℒ, i.e. 𝑆 ⊆ Φℒ , the topic of 𝑆, that is 𝜏 (𝑆), is such
that 𝜏 (𝑆) = ⋃{𝜏 (𝜑) ⋃︀ 𝜑 ∈ 𝑆}.
Then, since both theories and arguments can be represented by sets of sentences, we can
legitimately speak about their topics. Moreover, as shown by Carrara et al. [8, Corollary 2.2],
what a set of sentences 𝑆 is about boils down to the union of what the atomic components of
each claims in 𝑆 are about: that is, 𝜏 (𝑆) = ⋃{𝜏 (𝑝) ⋃︀ 𝑝 ∈ 𝑣𝑎𝑟(𝑆)}, where 𝑣𝑎𝑟(𝑆) is the set of
all and only the atomic variables occurring in the sentences that belong to 𝑆.
Finally, let us set a reference (or discourse) topic, 𝜏𝑅 , that is the topic that one or more agents
discuss/argue about. Then, a sentence 𝜑, or an argument 𝐴, or a theory 𝑇 5 are off-topic with
respect to 𝜏𝑅 iff 𝜏 (𝜑), 𝜏 (𝐴), 𝜏 (𝑇 ) ⊈ 𝜏𝑅 – i.e. iff 𝜑, 𝐴 and 𝑇 are not entirely about 𝜏𝑅 . Given
such a regimentation of the notion of topic and Beall’s off-topic interpretation of u, our aim
now is to use them to get an argumentation framework based on PWK.
3. TGTS
In this section we present a new PWK semantics, the topic game-theoretical semantics (TGTS),
which is based on Hintikka’s game-theoretical semantics (GTS). TGTS consists of the following
parts: (i) a set of discussants, 𝐼 = {verifier, falsifier}; (ii) a set Φ𝑅 of on-topic sentences with
respect to the discussion topic (𝜏𝑅 ), that essentially depends on the set of the on-topic atomic
4
The inclusion relation, ⊆, is usually taken to be reflexive, so that every topic includes itself.
5
Here, arguments and theories are taken to be sets of sentences.
propositions, Var𝑅 ; (iii) a set of discussion rules, {𝑅0 , 𝑅1 }; and (iv) a non-losing strategy
condition.
Let us now discuss these parts in more details. At the beginning of the discussion, the verifier
and the falsifier hold opposite (classical) opinions – i.e., t and f – about a given proposition,
say 𝜑. We denote such a discussion with 𝐷(𝜑). Thus, assume there is a reference/discussion
topic, 𝜏𝑅 , which generates a partition on Φℒ that separates all the on-topic sentences from the
off-topic ones. As before, any sentence 𝜓 ∈ Φℒ is on-topic with respect to the discussion topic
iff 𝜏 (𝜓) ⊆ 𝜏𝑅 . If this is not the case, then 𝜓 is off-topic. We call Φ𝑅 the set of all and only the
on-topic sentences with respect to 𝜏𝑅 . Then, Φℒ ∖ Φ𝑅 is the set of all and only the off-topic
sentences. Moreover, also the set of atomic propositions, Var, divides into the set of on-topic
atomic propositions, Var𝑅 , and the set of the off-topic ones, Var ∖ Var𝑅 .
The following rules constrain how the discussion is made:
Definition 3.1 (Discussion Rules). For any 𝜑 ∈ Φℒ , the discussion 𝐷(𝜑) is divided into two
sub-discussions, 𝐷0 (𝜑) and 𝐷1 (𝜑), which will each take place in turn.
The rules for 𝐷0 (𝜑) are as follows:
• (𝑅0 .𝐴𝑡) If 𝑝 ∉ Var𝑅 , then the two discussants reach a draw and close the discussion.
Otherwise, the two discussants move on to discussion 𝐷1 (𝑝).
• (𝑅0 . ⊛ (𝜑, 𝜓)) If ⊛(𝜑, 𝜓) ∉ Φ𝑅 , then the two discussants reach a draw and close the
discussion. Otherwise, the two discussants move on to discussion 𝐷1 (⊛(𝜑, 𝜓)) (here, ⊛ is
any well-formed formula which combines 𝜑, 𝜓, ¬, ∨, ∧, ⊃).
The rules for 𝐷1 (𝜑) are as follows:
• (𝑅1 .𝐴𝑡) If 𝑝 is true, the verifier wins 𝐷(𝑝) and the falsifier loses. If 𝑝 is false, the falsifier
wins 𝐷(𝑝) and the verifier loses it.
• (𝑅1 .¬) 𝐷1 (¬𝜑) is like 𝐷1 (𝜑), except that the roles of the two players (as defined by these
rules) are interchanged.
• (𝑅1 .∨) 𝐷1 (𝜑 ∨ 𝜓) begins with the choice by the verifier of 𝛿 (𝛿 is either 𝜑 or 𝜓). The rest
of the discussion is as in 𝐷1 (𝛿).
• (𝑅1 .∧) 𝐷1 (𝜑 ∧ 𝜓) begins with the choice by the falsifier of 𝛿 (𝛿 is either 𝜑 or 𝜓). The rest
of the discussion is as in 𝐷1 (𝛿).
• (𝑅1 . ⊃) 𝐷1 (𝜑 ⊃ 𝜓) is the same as 𝐷1 (¬𝜑 ∨ 𝜓).
Based on the notion of a winning strategy in GTS, a non-losing strategy in TGTS for PWK is
defined as follows:
Definition 3.2 (Non-losing Strategy). The initial verifier (falsifier) has a non-losing strategy
in 𝐷(𝜑) if either the discussants reach a draw in 𝐷0 (𝜑), or the initial verifier (falsifier) has a
winning strategy in 𝐷1 (𝜑).
From this definition, two facts follow immediately:
Fact 3.1. Both of the initial verifier and falsifier have a non-losing strategy in 𝐷(𝜑) if and only if
the discussants reach a draw in 𝐷(𝜑).
Fact 3.2. Only one of the two initial discussants has a non-losing strategy in 𝐷(𝜑) if and only if
one discussant has a winning strategy in 𝐷1 (𝜑).
4. Suggestions and Concluding Remarks
What is the relation between argumentation and TGTS? According to McBurney and Parsons
[9], McBurney et al. [10], “game-theoretical semantics have also been used to study the properties
of formal argumentation systems and dialogue protocols, such as their computational complexity,
or the extent of truth-convergence under an inquiry dialogue protocol, and to identify acceptable
sets of arguments in argument frameworks." [9, p. 272]. As [11] suggests, there are a number of
mainstream argumentation semantics developed by means of structured discussion. Consider,
for example, the dialogical argumentation: it emphasizes the exchange of arguments and
counterarguments between agents, which includes consideration of protocols and strategies
for the agents to follow. [12] proposes the dialogue-based (or dialetical) approach to logic and
argumentation theory, namely dialogue logic. The same proposal has been summarized in [13],
where the proof theoretical approach of Lorenzen and Lorenz [14] and the model theoretical
approach of Hintikka, GTS, are included.6 If we follow Hintikka’s idea “to consider all reasoning
and argumentation as a question-answer sequence, intersperse by logical (deductive inferences)”
[17, pp. 307–308], we can consider a topic based discussion on a sentence as a sequence for
answering a "yes or no or off-topic" question about a sentence. Here we propose that TGTS
can be regarded as a type of argumentation semantics that is able to deal with the off-topic
phenomenon. As we have introduced in the previous sections, the ingredients of TGTS are a
discussion topic, two discussants, some specific discussion rules, and a non-losing strategy. Not
only it can deal with the off-topic phenomenon, it is also able to account for the existence of a
non-losing strategy in such a type of argumentation. We believe this will provide an innovative
understanding of a particular class of argumentation processes, and this might set a new trend
in formal argumentation.
5. Acknowledgments
We would like to express our sincere gratitude to Roberto Confalonieri and Daniele Porello for
organizing the workshop. We would like to thank the reviewers for their valuable comments
on the article’s previous version. Our article is partially funded by the CARIPARO 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).
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