Alternative Axiomatization for Logics of Agency in a G3 Calculus
Sara NegriFoundations of Science
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In a recent paper, Negri and Pavlović (Studia Logica 1–35, 2020) have formulated a decidable sequent calculus for the logic of agency, specifically for a deliberative see-to-it-that modality, or dstit. In that paper the adequacy of the system is demonstrated by showing the derivability of the axiomatization of dstit from Belnap et al. (Facing the future: agents and choices in our indeterminist world. Oxford University Press, Oxford, 2001). And while the influence of the latter book on the study of logics of agency cannot be overstated, we note that this is not the only axiomatization of that modality available. In fact, an earlier (and arguably purer) one was offered in Xu (J Philosophical Logic 27(5):505–552, 1998). In this article we fill this lacuna by proving that this alternative axiomatization is likewise readily derivable in the system of Negri and Pavlović (Studia Logica 1–35, 2020).
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Foundations of Science (2023) 28:205–224
https://doi.org/10.1007/s10699-021-09784-3
Alternative Axiomatization for Logics of Agency in a G3
Calculus
Sara Negri1
· Edi Pavlović2
Accepted: 30 December 2020 / Published online: 6 July 2021
© The Author(s) 2021
Abstract
In a recent paper, Negri and Pavlović (Studia Logica 1–35, 2020) have formulated a decidable
sequent calculus for the logic of agency, specifically for a deliberative see-to-it-that modality,
or dstit. In that paper the adequacy of the system is demonstrated by showing the derivability
of the axiomatization of dstit from Belnap et al. (Facing the future: agents and choices in
our indeterminist world. Oxford University Press, Oxford, 2001). And while the influence of
the latter book on the study of logics of agency cannot be overstated, we note that this is not
the only axiomatization of that modality available. In fact, an earlier (and arguably purer)
one was offered in Xu (J Philosophical Logic 27(5):505–552, 1998). In this article we fill
this lacuna by proving that this alternative axiomatization is likewise readily derivable in the
system of Negri and Pavlović (Studia Logica 1–35, 2020).
Keywords Logic of agency · Stit modalities · Deliberative stit · Sequent calculus · G3
1 Introduction
There has been plenty of interest in the seeing-to-it-that (stit) modalities in recent years.
Notable examples of such modalities include cstit (c for Chellas), an agent-relative box-type
modality as well as more involved dstit (d for deliberative) and astit (a for achievement).
There is a good reason for this interest, as they can be used to unify a great variety of linguistic
expressions (Belnap et al. 2001).
The treatment of stit modalities in terms of proof theory remains relatively scant, notable
exceptions being (Wansing 2006, 2017; Olkhovikov and Wansing 2018, 2019), which all
This paper was written with support of the Academy of Finland research project No. 1308664.
B
Edi Pavlović
[email protected]
Sara Negri
[email protected]
1
Department of Mathematics, University of Genova, Via Dodecaneso 35, 16146 Genoa, Italy
2
Department of Philosophy, History and Art Studies, University of Helsinki, P.O. Box 24, 00014
Helsinki, Finland
123
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S. Negri, E. Pavlović
involve a tableaux approach. Approaches utilizing a sequent calculus have been developed
recently, in particular (using a simplification of a tableux) van Berkel and Lyon (2019); Lyon
and van Berkel (2019) and Negri and Pavlović (2020) (which builds the calculus based on the
semantics from Belnap et al. 2001). The system from the latter will be used in this paper, since
in addition to the usual range of desirable proof-theoretic properties (like the admissibility
of contraction and cut), it also offers a structural proof of multi-agent decidability and,
even though it focuses on dstit, provides a uniform basis for the treatment of multiple stit
modalities.
Even with those upsides, we nonetheless find that paper incomplete in one respect. It
demonstrates the adequacy of the system for the treatment of dstit by showing that the
axioms of dstit from Belnap et al. (2001) are derivable in it. This, however, is not the only
axiomatization of that modality, with another notable formulation found in Xu (1998) (also
investigated in Balbiani et al. 2008). The axiomatization there contains only the dstit operator,
in contrast with the four types of modal operators occurring in Belnap et al. (2001). And while
this system is clearly expressively weaker, it is also quite obviously the logic of dstit proper,
as opposed to being a logic merely containing dstit.
It is the purpose of this article to demonstrate that these alternative axioms are likewise
readily derivable in the system of Negri and Pavlović (2020). As such, in the proceeding we
will lean heavily on the results present therein. In order to avoid repeating large swathes of
that paper we will not present its proofs, but for ease of reference the results relevant for
this paper will be noted in the following section. Throughout the paper we will mark any
propositions, theorems or lemmas from Negri and Pavlović (2020) with an asterisk.
Here we are dealing with three systems—axiomatic ones of Belnap et al. (2001); Xu
(1998) and a sequent calculus from Negri and Pavlović (2020). Given that all three are sound
and complete with respect to the background semantics, one should prima facie expect the
results of this paper to hold.
However, there is a conceptual priority of an existence of proof to metatheoretical results—
a system is complete because certain proofs exist (the other way around would be putting
the cart before the horse), and it is standard practice in proof theory to prefer a direct proof
to a roundabout one. Case in point—in Negri and Pavlović (2020) it would not be, strictly
speaking, necessary to demonstrate completeness via a failed proof search after already
showing that the axioms of Belnap et al. (2001) are derivable. However, the former approach,
in addition to being elegant, provides the correct conceptual connection between the system
and its underlying semantics. The demonstration of axioms is required since it shows, in light
of the fact that axioms are the most common way of presenting a logic, that it adequately
captures a logic of dstit. And insofar as it is not the only logic of dstit around, with the version
from Xu (1998) arguably purer (although at the expense of brevity), the results of the present
paper are required to present the full picture.
After the introduction we lay out the systems we will be comparing, and after that dedicate
each section to one of the consecutive expansions of the axiomatization from Xu (1998), and
give a separate subsection to the more involved proofs. Moreover, since some are outright
preposterously long, their full proof will be deferred to the appendix.
2 Preliminaries
We begin with a note on notation—Belnap et al. (2001), Xu (1998) write “agent α deliberately
sees to it that A” as “[α : A].” For brevity and following Negri and Pavlović (2020)—the
123
Alternative Axiomatization for Logics of Agency in a G3 Calculus
207
decision there was likewise made to shorten the notation—we will write this as “D α A.”
We prefer to mark agents by numbers and thus use letters i, j, . . . instead of Greek letters
for agents, except for the axioms AIAn and APCk , where two different sets of agents are
involved. The full language of dstit that we utilize is defined as
A ::= ⊥ | p | A ◦ A | i A | S A | P A | Di A
where ◦ stands for connectives, i stands for cstit (Chellas 1980), S for settled true and P
for possible, the last two agent-independent (Belnap et al. 2001).
The formulas are labelled by moment/history pairs so that m/h : A states that the formula
A holds at a moment m of history h (m ∈ h indicates history h contains the moment m). In
the paper the reader will also frequently encounter the choice-equivalence relation h ∼im h ′ ,
which can be intuitively understood as saying that at moment m, the agent i cannot choose
between the histories h and h (it is beyond the agent’s control which one comes about). Finally,
the abbreviation Diff (i 1 , . . . , i k ) stands for distinctness of a finite number k of agents, i.e.
inequalities i j = i n where 1 ≤ j < n ≤ k.
2.1 The Alternative Axiomatization
The axiom systems for dstit presented in Xu (1998) are as follows, adjusted for the notation
of this paper. Each subsequent system extends the previous ones.
Logic L0 :
A0
A1
A2
A3
A4
A5
A6
Truth functional tautologies
Di A ⊃ A
Di A ⊃ Di Di A
Di A ⊃ Di ¬Di ¬Di A
(Di A & Di (B & C)) ⊃ Di (A & C)
(Di A & ¬Di B) ⊃ Di (A & ¬Di B)
Di (¬Di A & (B & ¬Di B)) ⊃⊂ Di ¬Di (A & (B & ¬Di B)).
Logic L1 :
A7
A8
A9
A10
A11
A12
i =i
i = j ⊃ (A ⊃ A(i/ j))
Di A ⊃⊂ (Di (A & j = k) ∨ Di (A & j = k))
(Di A & Dk (B & (C & ¬Dk C))) ⊃ Di (A & (C & ¬Di C))
(Di A & ¬Di (A & ¬Dk B)) ⊃ (Dk B ∨ Dk ¬Dk B)
Di ¬Dk B ⊃ i = k
AIAn Da A & (¬Da (A & ¬Db1 B1 ) & . . . & ¬Da (A & ¬Dbn Bn )) & Diff (b1 , . . . , bn ) ⊃
¬Da (A & ¬(Db1 B1 & . . . & Dbn Bn ))
where Diff (b1 , . . . , bn ) indicates pairwise inequality ranging over all pairs of non-identical
agents.
Logic Lk : Each subsequent logic Lk is determined by k in APCk .
APCk ¬Da (A & ¬(Db B1 & ¬B2 & ... & ¬Bk )) & ... & ¬Da (A & ¬(Db Bk−1 & ¬Bk )) &
¬Da (A & ¬Db Bk ) & Da A & a = b ⊃ Db B1 ∨ ... ∨ Db Bk
Moreover, the rules used are those of modus ponens and RE:
A ⊃⊂ B
Di A ⊃⊂ Di B
RE
123
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S. Negri, E. Pavlović
2.2 G3DSTIT and G3Ldmk
We now present the sequent calculus to be used in this article. The sequent calculus is a
G3-style calculus (Negri 2005; Negri and von Plato 1998, 2001, 2011), with the treatment
of auxiliary modalities mirroring that of Negri and Sbardolini (2016), defined as in (Fig. 1).
This system will be used for demonstration of the adequacy of axioms up to and including
L1 . For any subsequent system we add the appropriate rule APCk to G3DSTIT to produce
the system G3Ldmk .
2.2.1 Previous Results
The results from Negri and Pavlović (2020) used in this article are:
1. First and foremost, all the axioms of dstit from Belnap et al. (2001) are derivable
(Proposition∗ 3.1),
2. Axiom generalization, m/h : A, Ŵ ⇒ , m/h : A for any formula A (Lemma∗ 4.2),
used here without special reference,
3. Height-preserving substitution (Proposition∗ 4.3),
4. (Lemma∗ 5.1): h ∼im h ′ , m/h : Di A ⇒ m/h ′ : Di A,
5. Impossibility of meta-agency (Theorem∗ 5.2): m/h : Di D j A ⇒ and m/h : Di ¬D j A ⇒,
when Diff (i, j),
6. (Proposition∗ 5.3): Di ¬Di A ⇔ PDi A & ¬Di A
7. Doing is equivalent to refraining from refraining (Proposition∗ 5.4): m/h : Di A ⇔ m/h :
Di ¬Di ¬Di A,
8. Finally, the system with the rule APCk added is decidable (Theorem∗ 6.8), and decidable
without the rule for a system with a single agent (Lemma∗ 6.11).
It is easy to show that the rules used in Xu (1998) are admissible—modus ponens is trivial,
and RE uses a slight modification of the proof of axiom generalization for the case of Di .
3 Logic L0
Since this system is formulated for only a single agent, we know by Lemma∗ 6.11 that all of
its axioms are decidable. This has been particularly useful in the proof search for A6 (given
its length, the proof of this axiom is found in the appendix). In fact, (the sequent calculus
versions of) all of the following are derivable in G3DSTIT:
A0 Truth functional tautologies
A1 Di A ⊃ A.
Proof
h ∼im h, m/h : i A, m/h : A ⇒ m/h : A, m/h : S A
h ∼im h, m/h : i A ⇒ m/h : A, m/h : S A
m/h : i A ⇒ m/h : A, m/h : S A
m/h : Di A ⇒ m/h : A
Li
Re f l∼im
L Di
⊔
⊓
123
Alternative Axiomatization for Logics of Agency in a G3 Calculus
209
Fig. 1 G3DSTIT
123
210
S. Negri, E. Pavlović
A2 Di A ⊃ Di Di A.
Proof
(1)
h ′′ ∼im h, h ′′ ∼im h ′ , h ′ ∼im h, m/h : i A, m/h ′′ : A ⇒ m/h ′′ : A, m/h : S A
Li
E T rans∼im
h ′′ ∼im h, h ′′ ∼im h ′ , h ′ ∼im h, m/h : i A ⇒ m/h ′′ : A, m/h : S A
h ′′ ∼im h ′ , h ′ ∼im h, m/h : i A ⇒ m/h ′′ : A, m/h : S A
Ri
h ′ ∼im h, m/h : i A ⇒ m/h ′ : i A, m/h : S A
(2)
m/h ′′′ : A, m ∈ h ′′′ , m/h ′ : S A, h ′ ∼im h, m/h : i A ⇒ m/h ′′′ : A
m ∈ h ′′′ , m/h ′ : S A, h ′ ∼im h, m/h : i A ⇒ m/h ′′′ : A
m/h ′ : S A, h ′ ∼im h, m/h : i A ⇒ m/h : S A
(1)
h ′ ∼im h, m/h : i A ⇒ m/h ′ : Di A, m/h : S A
m/h : i A ⇒ m/h : i Di A, m/h : S A
LS
RS
R Di
Ri
(3)
m/h ′ : A, h ′ ∼im h ′ , m/h ′ : i A, m ∈ h ′ , m/h : SDi A, m/h : i A ⇒ m/h ′ : A, m/h ′ : S A
h ′ ∼im h ′ , m/h ′ : i A, m ∈ h ′ , m/h : SDi A, m/h : i A ⇒ m/h ′ : A, m/h ′ : S A
m/h ′ : i A, m ∈ h ′ , m/h : SDi A, m/h : i A ⇒ m/h ′ : A, m/h ′ : S A
m/h ′ : Di A, m ∈ h ′ , m/h : SDi A, m/h : i A ⇒ m/h ′ : A
m ∈ h ′ , m/h : SDi A, m/h : i A ⇒ m/h ′ : A
m/h : SDi A, m/h : i A ⇒ m/h : S A
(2)
(3)
m/h : i A ⇒ m/h : Di Di A, m/h : S A
m/h : Di A ⇒ m/h : Di Di A
Li
Re f l∼im
L Di
LS
RS
R Di
L Di
⊔
⊓
A3 Di A ⊃ Di ¬Di ¬Di A.
Proof Proposition∗ 5.4.
⊔
⊓
A4 (Di A & Di (B & C)) ⊃ Di (A & C).
Proof
(1)
h ′ ∼im h, . . . , m/h ′ : B, m/h ′ : C ⇒ m/h ′ : C, . . .
L&
h ′ ∼im h, . . . , m/h ′ : B & C ⇒ m/h ′ : C, . . .
i
L
h ′ ∼im h, . . . , m/h : i (B & C) ⇒ m/h ′ : C, . . .
LDi
h ′ ∼im h, . . . , m/h : Di (B & C) ⇒ m/h ′ : C
R&
h ′ ∼im h, m/h : Di A, m/h : Di (B & C) ⇒ m/h ′ : A & C
′
i
R
m/h : Di A, m/h : Di (B & C) ⇒ m/h : i (A & C)
h ′ ∼im h, m/h ′ : A, . . . ⇒ m/h ′ : A, . . .
Li
h ′ ∼im h, m/h : i A, . . . ⇒ m/h ′ : A, . . .
LDi
h ′ ∼im h, m/h : Di A, . . . ⇒ m/h ′ : A
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Alternative Axiomatization for Logics of Agency in a G3 Calculus
211
(2)
m/h ′ : A, m/h ′ : C, m ∈ h ′ , m/h : S(A & C), m/h : i A, m/h : Di (B & C) ⇒ m/h ′ : A
m/h ′ : A & C, m ∈ h ′ , m/h : S(A & C), m/h : i A, m/h : Di (B & C) ⇒ m/h ′ : A
m ∈ h ′ , m/h : S(A & C), m/h : i A, m/h : Di (B & C) ⇒ m/h ′ : A
m/h : S(A & C), m/h : i A, m/h : Di (B & C) ⇒ m/h : S A
(1)
m/h :
(2)
:
Di (B
LS
RS
LDi
m/h : S(A & C), m/h : Di A, m/h : Di (B & C) ⇒
Di A, m/h
L&
R Di
L&
& C) ⇒ m/h : Di (A & C)
m/h : Di A & Di (B & C) ⇒ m/h : Di (A & C)
⊔
⊓
A5 (Di A & ¬Di B) ⊃ Di (A & ¬Di B).
Proof
(1)
h ′ ∼im h, m/h : i A, m/h ′ : A ⇒ m/h ′ : A, . . .
h ′ ∼im h, m/h : i A ⇒ m/h ′ : A, . . .
h′
∼im
h, m/h :
Di
A⇒
m/h ′
: A, . . .
Li
L Di
∗
Lemma
.. 5.1
..
m/h ′ : Di B, h ′ ∼im h, m/h : Di A ⇒ m/h : Di B
h ′ ∼im h, m/h : Di A ⇒ m/h ′ : ¬Di B, m/h : Di B
h ′ ∼im h, m/h : Di A ⇒ m/h ′ : A & ¬Di B, m/h : Di B
m/h : Di A ⇒ m/h : i (A & ¬Di B), m/h : Di B
R¬
R&
Ri
(2)
m ∈ h ′ , m/h : S (A & ¬Di B), m/h ′ : A, m/h ′ : ¬Di B, m/h : i A ⇒ m/h : Di B, m/h ′ : A
m ∈ h ′ , m/h : S (A & ¬Di B), m/h ′ : A & ¬Di B, m/h : i A ⇒ m/h : Di B, m/h ′ : A
m ∈ h ′ , m/h : S (A & ¬Di B), m/h : i A ⇒ m/h : Di B, m/h ′ : A
m/h : S (A & ¬Di B), m/h : i A ⇒ m/h : Di B, m/h : S A
m/h : S (A & ¬Di B), m/h : Di A ⇒ m/h : Di B
(1)
m/h :
Di A
⇒ m/h :
(2)
Di (A
& ¬Di B), m/h : Di B
m/h : Di A, m/h : ¬Di B ⇒ m/h : Di (A & ¬Di B)
m/h : Di A & ¬Di B ⇒ m/h : Di (A & ¬Di B)
L&
LS
RS
L Di
R Di
L¬
L&
⊔
⊓
A6 Di (¬Di A & (B & ¬Di B)) ⊃⊂ Di ¬Di (A & (B & ¬Di B))
Proof See “Appendix A”.
⊔
⊓
This concludes the proof of adequacy of G3DSTIT for the logic L0 . Close inspection of the
rules shows that the rule of independence of agents was not required. This will not be the
case for the next system, where the full G3DSTIT will be used.
123
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S. Negri, E. Pavlović
4 Logic L1
Logic L1 can be divided into three separate segments—those axioms concerning identity,
the one that deals with independence of agents, and the remaining axioms. The first group
needs to be addressed separately due to the specificities of identity in G3, while the second
warrants a separate section due to sheer complexity.
4.1 Identity Axioms
The first two axioms deal with the standard properties of identity, each axiom corresponds
to a rule and both are shown straightforwardly. We will discuss the third of these axioms in
greater detail, however.
A7 i = i
A8 i = j ⊃ (A ⊃ A(i/ j)).
Proof Both of these axioms are proven by Proposition∗ 3.1.
A9
Di A
⊃⊂
(Di (A &
j=
k) ∨ Di (A &
⊔
⊓
j = k)).
Proof We first note that i = j and i = j are relational atoms and don’t occur labelled by
points, since identity is taken as rigid. Since the proof of this axiom requires labelled identity
atoms, we can express rigidity in a form of a rule as in Negri and Orlandelli (2019):
m ∈ h ′ , m/h ′ : j = k, m/h : j = k, Ŵ ⇒
Rig
m ∈ h ′ , m/h : j = k, Ŵ ⇒
and thereby we can derive
m/h ′′ : j = k, m ∈ h ′′ , m/h ′ : j = k, Ŵ ⇒ , m/h ′ : j = k
Rig
m ∈ h ′′ , m/h ′ : j = k, Ŵ ⇒ , m/h ′ : j = k
WD
m/h ′ : j = k, Ŵ ⇒ , m/h ′′ : j = k
We now proceed with the proof. Left to right:
(1)
m/h ′ : j = k, h ′ ∼im h ∼im h ′′ , m/h : i A ⇒ m/h ′′ : j = k, m/h : S A
h ′ ∼im h ∼im h ′′ , m/h : i A ⇒ m/h ′′ : j = k, m/h ′ : j = k, m/h : S A
R¬
(2)
..
..
h ∼im h ′′ , m/h : i A ⇒ m/h ′′ : A, . . .
(1)
R&
..
h ′ ∼im h ∼im h ′′ , m/h : i A ⇒ m/h ′′ : A & j = k, m/h ′ : j = k, m/h : S A
..
Ri
′
i
i
i
′
′
i
i
′
h ∼m h, m/h : A ⇒ m/h : (A & j = k), m/h : j = k, m/h : S A
h ∼m h, m/h : A ⇒ . . . , m/h : A, . . .
R&
h ′ ∼im h, m/h : i A ⇒ m/h : i (A & j = k), m/h ′ : A & j = k, m/h : S A
Ri
i
i
i
m/h : A ⇒ m/h : (A & j = k), m/h : (A & j = k), m/h : S A
(2)
..
..
m/h : S (A & j = k), . . . ⇒ . . . , m/h : S A
m/h : i A ⇒ m/h : Di (A & j = k), m/h : i (A & j = k), m/h : S A
R Di
..
..
m/h : S (A & j = k), . . . ⇒ . . . , m/h : S A
m/h : i A ⇒ m/h : Di (A & j = k), m/h : Di (A & j = k), m/h : S A
m/h : Di A ⇒ m/h : Di (A & j = k), m/h : Di (A & j = k)
m/h : Di A ⇒ m/h : Di (A & j = k) ∨ Di (A & j = k)
123
R∨
L Di
R Di
Alternative Axiomatization for Logics of Agency in a G3 Calculus
213
Right to left:
(1a)
m/h ′ : A, m/h ′ : j = k, h ′ ∼im h, m/h : i (A & j = k) ⇒ m/h ′ : A, m/h : S(A & j = k)
L&
m/h ′ : A & j = k, h ′ ∼im h, m/h : i (A & j = k) ⇒ m/h ′ : A, m/h : S(A & j = k)
Li
h ′ ∼im h, m/h : i (A & j = k) ⇒ m/h ′ : A, m/h : S(A & j = k)
Ri
m/h : i (A & j = k) ⇒ m/h : i A, m/h : S(A & j = k)
(1b)
m/h : A, m/h : j = k, h ∼im h, m ∈ h ′ , m/h : S A, m/h : i (A & j = k) ⇒ m/h ′ : j = k
m/h : A & j = k, h ∼im h, m ∈ h ′ , m/h : S A, m/h : i (A & j = k) ⇒ m/h ′ : j = k
h ∼im h, m ∈ h ′ , m/h : S A, m/h : i (A & j = k) ⇒ m/h ′ : j = k
m ∈ h ′ , m/h : S A, m/h : i (A & j = k) ⇒ m/h ′ : j = k
L&
Li
Re f l
(1c)
m/h ′ : A, m ∈ h ′ , m/h : S A, . . . ⇒ m/h ′ : A
m ∈ h ′ , m/h : S A, m/h : i (A & j = k) ⇒ m/h ′ : A
m∈
h ′ , m/h
: S A, m/h :
i (A &
j = k) ⇒
m/h ′
LS
(1b)
: A& j = k
m/h : S A, m/h : i (A & j = k) ⇒ m/h : S (A & j = k)
R&
RS
(1)
(1a)
(1c)
m/h : i (A & j = k) ⇒ m/h : Di A, m/h : S (A & j = k)
m/h : Di (A & j = k) ⇒ m/h : Di A
R Di
L Di
(2) very similar to (1)
(1)
m/h :
Di (A &
j=
(2)
k) ∨ Di (A &
j = k) ⇒ m/h : Di A
R∨
⊔
⊓
Of course, one should only see this proof as a sketch of why the axiom is intuitively
acceptable. Since neither j = k nor j = k occur in the succedent (on the pain of loss of
admisibility of cut, as discussed in Negri and Pavlović 2020), this is not a proof in G3DSTIT,
even with the rule for rigidity added. What this axiom states, namely that j = k is either
settled true or settled false Xu (1998, p. 515), is instead expressed by using j = k and j = k
as a relational atom.
4.2 Remaining Axioms
A10 (Di A & Dk (B & (C & ¬Dk C))) ⊃ Di (A & (C & ¬Di C)).
We begin this proof with a lemma:
Lemma 4.1 m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : C.
123
214
S. Negri, E. Pavlović
Proof We omit applications of WD for brevity.
. . . , m/h ′′ : C, . . . ⇒ . . . , m/h ′′ : C
L&
h ′′ ∼km h, . . . , m/h ′′ : B & (C & ¬Dk C) ⇒ . . . , m/h ′′ : C
Lk
h ′′ ∼km h, . . . , m/h : k (B & (C & ¬Dk C)) ⇒ . . . , m/h ′′ : C
Rk
. . . , m/h : k (B & (C & ¬Dk C)) ⇒ . . . , m/h : k C
m/h ′ : C, . . . ⇒ m/h ′ : C, . . .
LS
m/h : SC, . . . ⇒ m/h ′ : C, . . .
RDk
m/h : B, m/h : C, h ∼km h, m/h : k (B & (C & ¬Dk C)) ⇒ m/h ′ : C, . . . , m/h : Dk C
L¬
m/h : B, m/h : C, m/h : ¬Dk C, h ∼km h, m/h : k (B & (C & ¬Dk C)) ⇒ m/h ′ : C, . . .
L&
m/h : B & (C & ¬Dk C), h ∼km h, m/h : k (B & (C & ¬Dk C)) ⇒ m/h ′ : C, . . .
k
L
h ∼km h, m/h : k (B & (C & ¬Dk C)) ⇒ m/h ′ : C, . . .
Re f l∼k
m
m/h : k (B & (C & ¬Dk C)) ⇒ m/h ′ : C, m/h : S(B & (C & ¬Dk C))
k
LD
m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : C
⊔
⊓
Proof
(1a)
Lemma
.. 4.1
..
m ∈ h ′′ , m/h ′ : i C, h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′′ : C
m/h ′ : i C, h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : S C
m/h ′ : Di C, h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒
h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : ¬Di C
RS
L Di
R¬
(1b)
Lemma
.. 4.1
..
(1a)
h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : C & ¬Di C
R&
(1c)
h ′ ∼im h, m/h : i A, m/h ′ : A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : A, m/h : S A
h ′ ∼im h, m/h : i A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : A, m/h : S A
h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : A
Li
L Di
(1b)
h ′ ∼im h, m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h ′ : A & (C & ¬Di C)
m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h : i (A & (C & ¬Di C))
m ∈ h ′ , m/h ′ : A, . . . ⇒ m/h ′ : A
m ∈ h ′ , m/h ′ : A & (C & ¬Di C), . . . , m/h : i A, . . . ⇒ m/h ′ : A
m ∈ h ′ , m/h : S (A & (C & ¬Di C)), m/h : i A, . . . ⇒ m/h ′ : A
m/h : S (A & (C & ¬Di C)), m/h : i A, . . . ⇒ m/h : S A
L&
LS
RS
(1c) m/h : S (A & (C & ¬Di C)), m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒
m/h : Di A, m/h : Dk (B & (C & ¬Dk C)) ⇒ m/h : Di (A & (C & ¬Di C))
m/h : Di A & Dk (B & (C & ¬Dk C)) ⇒ m/h : Di (A & (C & ¬Di C))
R&
Ri
L Di
R Di
L&
⊔
⊓
123
Alternative Axiomatization for Logics of Agency in a G3 Calculus
215
A11 (Di A & ¬Di (A & ¬Dk B)) ⊃ (Dk B ∨ Dk ¬Dk B).
Proof
(1a)
∗
Lemma
.. 5.1
..
m/h ′′ : Dk B, h ′′ ∼km h, m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : S A
h ′′ ∼km h, m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h ′′ : ¬Dk B, m/h : S A
m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : k ¬Dk B, m/h : S A
R¬
Rk
(1b)
∗
Lemma
.. 5.1
..
. . . , m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : S A, m/h ′ : Dk B
L¬
m/h ′ : ¬Dk B, . . . , m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : S A
(1a)
m/h : S ¬Dk B, m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : S A
m/h ′ : Dk B, h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A
h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A, m/h ′ : ¬Dk B
W D, L S
R Dk
R¬
(1c)
h ′ ∼im h, . . . , m/h ′ : A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A, m/h ′ : A
h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A, m/h ′ : A
Li
(1b)
h ′ ∼im h, m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A, m/h ′ : A & ¬Dk B
m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A, m/h : i (A & ¬Dk B)
R&
Ri
m ∈ h ′ , . . . , m/h ′ : A, m/h ′ : ¬Dk B, m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h ′ : A
m ∈ h ′ , . . . , m/h ′ : A & ¬Dk B, m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h ′ : A
m ∈ h ′ , m/h : S(A & ¬Dk B), m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h ′ : A
(1c)
m/h : S(A & ¬Dk B), m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A
m/h : i A ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A, m/h : Di (A & ¬Dk B)
m/h : i A, m/h : ¬Di (A & ¬Dk B) ⇒ m/h : Dk B, m/h : Dk ¬Dk B, m/h : S A
m/h : Di A, m/h : ¬Di (A & ¬Dk B) ⇒ m/h : Dk B, m/h : Dk ¬Dk B
m/h : Di A & ¬Di (A & ¬Dk B) ⇒ m/h : Dk B ∨ Dk ¬Dk B
L&
LS
RS
RDi
L¬
LDi
R∨, L &
⊔
⊓
A12 Di ¬Dk B ⊃ i = k.
Proof By contraposition from the second claim in Theorem∗ 5.2.
⊔
⊓
4.3 Axioms AIAn
While the axiom schema AIAn is defined for any 2 ≤ n, only the simplest case of n = 2 will
suffice for the demonstration, as the proofs for any other n differ in size but not in substance.
We begin the proof with a pair of lemmas:
123
216
S. Negri, E. Pavlović
Lemma 4.2 a = b1 , Diff (b1 , b2 ), h 2 ∼am h, h 1 ∼am h, m/h : Da (A & ¬(Db1 B1 & Db2 B2 )),
m/h 1 : Db1 B1 , m/h 2 : Db2 B2 ⇒
Proof For brevity, we will omit writing that Diff (b1 , b2 ), as well as applications of WD and
Ref∼im .
(1) m/h 0 : S B1 ⇒ m/h 1 : S B1 , . . .
(2) same as the upper left sequent of R &
h 0 ∼am h, h 0 ∼bm1
.
.
.
.
h 1 , . . . , m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . . , m/h 0 : b1 B1 (1)
h 0 ∼am h, h 0 ∼bm1 h 1 , . . . , m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . . , m/h 0 : D b1 B1
R D b1
(2)
R&
h 0 ∼am h, h 0 ∼bm1 h 1 , . . . , m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . . , m/h 0 : D b1 B1 & D b2 B2
L¬
h 0 ∼am h, h 0 ∼bm1 h 1 , . . . , m/h 0 : ¬(D b1 B1 & D b2 B2 ), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
h 0 ∼am h, h 0 ∼bm1 h 1 , . . . , m/h 0 : A & ¬(D b1 B1 & D b2 B2 ), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
L&
La
h 0 ∼am h, h 0 ∼bm1 h 1 , . . . , m/h : a (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
h 0 ∼am h 1 , h 0 ∼bm1 h 1 , . . . , a = b1 , h ∼am h 1 , m/h : a (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
h 0 ∼bm1 h 1 , . . . , a = b1 , h ∼am h 1 , m/h : a (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
h 0 ∼bm1 h 1 , h 0 ∼bm2 h 2 , a = b1 , h ∼am h 1 , m/h : Da (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
a = b1 , h 2 ∼am h ∼am h 1 , m/h : Da (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , . . . ⇒ m/h 1 : S B1 , . . .
a = b1
, h2
∼am
h
∼am
h 1 , m/h
:
Da (A & ¬(Db1 B1
& D b2 B2 )), m/h 1
:
Db1 B1 , m/h 2
:
Db2 B2
⇒
T rans
Repl
L Da
I nd2
L D b1 , L D b2
⊔
⊓
Lemma 4.3 Diff (a, b1 , b2 ), h 2 ∼am h, h 1 ∼am h, m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h 1 :
Db1 B1 , m/h 2 : Db2 B2 ⇒.
Proof For brevity we will omit writing Diff (a, b1 , b2 ), as well as applications of WD and
Ref∼im .
(1) m/h 0 : S B1 ⇒ m/h 1 : S B1 , . . .
(2) same as the upper left sequent of R &
.
.
.
.
h0
∼am
h0
h, h 0
∼am
∼bm1
h, h 0
h1, h0
∼bm1
∼bm2
h1, h0
h 2 , . . . , m/h 1
∼bm2
:
b1 B1 , m/h 2
h 2 , . . . , m/h 1
:
: b2 B2 ⇒ m/h 1 : S B1 , . . . , m/h 0 : b1 B1 (1)
b1 B1 , m/h 2
: b2 B2 ⇒ m/h 1 : S B1 , . . . , m/h 0 : D b1 B1
R D b1
(2)
R&
h 0 ∼am h, h 0 ∼bm1 h 1 , h 0 ∼bm2 h 2 , . . . , m/h 1 : b1 B1 , m/h 2 : b2 B2 ⇒ m/h 1 : S B1 , m/h 2 : S B2 , . . . , m/h 0 : D b1 B1 & D b2 B2
h 0 ∼am h, h 0 ∼bm1 h 1 , h 0 ∼bm2 h 2 , . . . , m/h 0 : ¬(D b1 B1 & D b2 B2 ), m/h 1 : b1 B1 , m/h 2 : b2 B2 ⇒ m/h 1 : S B1 , m/h 2 : S B2 , . . .
h 0 ∼am h, h 0 ∼bm1 h 1 , h 0 ∼bm2 h 2 , . . . , m/h 0 : A & ¬(D b1 B1 & D b2 B2 ), m/h 1 : b1 B1 , m/h 2 : b2 B2 ⇒ m/h 1 : S B1 , . . .
h 0 ∼am h, h 0 ∼bm1 h 1 , h 0 ∼bm2 h 2 , . . . , m/h : a (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , m/h 2 : b2 B2 ⇒ m/h 1 : S B1 , . . .
h 2 ∼am h, h 1 ∼am h, . . . , m/h : a (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , m/h 2 : b2 B2 ⇒ m/h 1 : S B1 , . . .
L¬
L&
La
I nd3
L Da
∼am h, h 1 ∼am h, m/h : Da (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , m/h 2 : b2 B2 ⇒ m/h 1 : S B1 , . . .
L D b2
h 2 ∼am h, h 1 ∼am h, m/h : Da (A & ¬(D b1 B1 & D b2 B2 )), m/h 1 : b1 B1 , m/h 2 : D b2 B2 ⇒ m/h 1 : S B1
b1
L
D
2
b
1
b
b
2
a
1
a
a
b
h ∼m h, h ∼m h, m/h : D (A & ¬(D 1 B1 & D 2 B2 )), m/h : D 1 B1 , m/h : D 2 B2 ⇒
h2
⊔
⊓
We can now show
AIA2
Da A & (¬Da (A & ¬Db1 B1 ) & ¬Da (A & ¬Db2 B2 )) & Diff (b1 , b2 ) ⊃
¬Da (A & ¬(Db1 B1 & Db2 B2 ))
123
Alternative Axiomatization for Logics of Agency in a G3 Calculus
217
Fig. 2 G3Ldmk
Proof We again omit writing Diff (b1 , b2 ) for brevity.
(1) Using Lemma 4.2 for the case where a = b1 and again for the case where a = b2 ,
Lemma 4.3 and an instance of A9, we obtain
m/h : Da A, h 2 ∼am h, h 1 ∼am h, m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h 1 : Db1 B1 , m/h 2 : Db2 B2 ⇒
(2), (3) h ∼am h ′ , m/h : Da A ⇒ m/h ′ : A
(4)
.
.
.
.
m/h ′ : Db1 B1 , . . . , m/h ′ : ¬Db1 B1 ⇒
m/h ′ : Db1 B1 , m/h ′ : Db2 B2 , m ∈ h ′ , . . . , m/h ′ : A & ¬Db1 B1 ⇒
m ∈ h ′ , . . . , m/h ′ : A, . . . ⇒ m/h ′ : A
m ∈ h ′ , . . . , m/h ′ : A & ¬Db1 B1 ⇒ m/h ′ : A
m ∈ h ′ , . . . , m/h : S(A & ¬Db1 B1 ) ⇒ m/h ′ : A
L&
LS
m/h ′ : Db1 B1 , m/h ′ : Db2 B2 , m ∈ h ′ , . . . , m/h : S(A & ¬Db1 B1 ) ⇒
m/h ′ : Db1 B1 & Db2 B2 , m ∈ h ′ , . . . , m/h : S(A & ¬Db1 B1 ) ⇒
m ∈ h ′ , . . . , m/h : S(A & ¬Db1 B1 ) ⇒ m/h ′ : ¬(Db1 B1 & Db2 B2 )
m ∈ h ′ , m/h : a (A & ¬(Db1 B1 & Db2 B2 )), m/h : S(A & ¬Db1 B1 ) ⇒ m/h ′ : A & ¬(Db1 B1 & Db2 B2 )
m/h : a (A & ¬(Db1 B1 & Db2 B2 )), m/h : S(A & ¬Db1 B1 ) ⇒ m/h : S(A & ¬(Db1 B1 & Db2 B2 ))
L&
LS
L&
R¬
R&
RS
LDa
m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : S(A & ¬Db1 B1 ) ⇒
(5) The same as (4).
(1)
.
.
..
h 1 ∼am h ∼am h 2 , m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A, m/h 1 : Db1 B1 , m/h 2 : Db2 B2 ⇒
R¬
(2) h 1 ∼am h ∼am h 2 , m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h 1 : ¬Db1 B1 , m/h 2 : ¬Db2 B2
R&
(3) h 1 ∼am h ∼am h 2 , m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h 1 : A & ¬Db1 B1 , m/h 2 : ¬Db2 B2
R&
h 1 ∼am h ∼am h 2 , m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h 1 : A & ¬Db1 B1 , m/h 2 : A & ¬Db2 B2
a
R
h 1 ∼am h, m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h 1 : A & ¬Db1 B1 , m/h : a (A & ¬Db2 B2 )
a
R
(4)
m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h : a (A & ¬Db1 B1 ), m/h : a (A & ¬Db2 B2 )
m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h : Da (A & ¬Db1 B1 ), m/h : a (A & ¬Db2 B2 )
RDa
m/h : Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da A ⇒ m/h : Da (A & ¬Db1 B1 ), m/h : Da (A & ¬Db2 B2 )
m/h : Da A ⇒ m/h : ¬Da (A & ¬(Db1 B1 & Db2 B2 )), m/h : Da (A & ¬Db1 B1 ), m/h : Da (A & ¬Db2 B2 )
(5)
RDa
R¬
m/h : Da A, m/h : ¬Da (A & ¬Db1 B1 ), m/h : ¬Da (A & ¬Db2 B2 ) ⇒ m/h : ¬Da (A & ¬(Db1 B1 & Db2 B2 ))
L¬
⊔
⊓
As already mentioned, the extension to the case of n agents is long-winded, but straightforward.
5 Logic Lk
These logics are defined by the axiom APCk for an appropriate k. To capture them, we now
utilize logic G3Ldmk (Fig. 2). As with independence of agents, we will demonstrate the
proof of this axiom for the case of k = 2, noting that the extension to k is straightforward.
We begin with a lemma. Note that the Lemma∗ 5.1 refers to the result from Negri and
Pavlović (2020).
Lemma 5.1 m/h 1 : Db B1 , m/h 1 : ¬B2 , m/h 2 : Db B2 ⇒ m/h : Db B1 , m/h : Db B2 .
123
218
S. Negri, E. Pavlović
Proof (1)
∗
Lemma
.. 5.1
..
h 1 ∼bm h, m/h 1 : Db B1 , m/h 1 : ¬B2 , m/h 2 : Db B2 ⇒ m/h : Db B1 , m/h : Db B2
(2) same as (1)
(3)
h 1 ∼bm h 2 , m/h 1 : Db B1 , . . . , m/h 1 : B2 ⇒ m/h : Db B1 , m/h : Db B2 , m/h 2 : S B2 , m/h 1 : B2
L¬
h 1 ∼bm h 2 , m/h 1 : Db B1 , m/h 1 : ¬B2 , . . . , m/h 1 : B2 ⇒ m/h : Db B1 , m/h : Db B2 , m/h 2 : S B2
Lb
h 1 ∼bm h 2 , m/h 1 : Db B1 , m/h 1 : ¬B2 , m/h 2 : b B2 ⇒ m/h : Db B1 , m/h : Db B2 , m/h 2 : S B2
LDb
h 1 ∼bm h 2 , m/h 1 : Db B1 , m/h 1 : ¬B2 , m/h 2 : Db B2 ⇒ m/h : Db B1 , m/h : Db B2
(1)
m/h 1
:
(2)
Db B1 , m/h 1
: ¬B2
, m/h 2
:
(3)
Db B2
⇒ m/h :
Db B1 , m/h
: Db B2
A PC2
⊔
⊓
We can now show
APC2
¬Da (A & ¬(Db B1 & ¬B2 )) & ¬Da (A & ¬Db B2 ) & Da A & a = b ⊃ Db B1 ∨ Db B2
Proof
(1)
h 1 ∼am h, . . . , m/h : a A, m/h 1 : A, . . . ⇒ . . . , m/h 1 : A, . . . , m/h : S A
h 1 ∼am h, . . . , m/h : a A, . . . ⇒ . . . , m/h 1 : A, . . . , m/h : S A
La
L Da
h 1 ∼am h, . . . , m/h : Da A, . . . ⇒ . . . , m/h 1 : A, . . .
(2) very similar to (1)
(3)
m ∈ h ′ , m/h ′ : A, m/h ′ : ¬Db B2 , m/h : S(A & ¬Db B2 ), m/h : a A, . . . ⇒ . . . , m/h ′ : A
m ∈ h ′ , m/h ′ : A & ¬Db B2 , m/h : S(A & ¬Db B2 ), m/h : a A, . . . ⇒ . . . , m/h ′ : A
m ∈ h ′ , m/h : S(A & ¬Db B2 ), m/h : a A, . . . ⇒ . . . , m/h ′ : A
m/h : S(A & ¬Db B2 ), m/h : a A, . . . ⇒ . . . , m/h : S A
m/h : S(A & ¬Db B2 ), m/h : Da A, . . . ⇒ . . .
L&
LS
RS
LDa
(4) very similar to (3)
∗
Lemma
. 5.1
.
.
.
m/h 1 : Db B1 , m/h 1 : ¬B2 , m/h 2 : Db B2 , . . . , m/h : Da A, m/h : a = b ⇒ m/h : Db B1 , m/h : Db B2
m/h 1 : Db B1 & ¬B2 , m/h 2 : Db B2 , . . . , m/h : Da A, m/h : a = b ⇒ m/h : Db B1 , m/h : Db B2
(1) . . . , m/h :
Da A, m/h
: a = b ⇒ m/h :
D b B1 , m/h
:
D b B2 , m/h 1
:
¬(Db B1
& ¬B2
), m/h 2
:
¬Db B2
(2) . . . , m/h : Da A, m/h : a = b ⇒ m/h : Db B1 , m/h : Db B2 , m/h 1 : A & ¬(Db B1 & ¬B2 ), m/h 2 : ¬Db B2
h1
∼am
h, h 2
m/h :
∼am h, m/h
Da A, m/h :
:
Da A, m/h
: a = b ⇒ m/h :
D b B1 , m/h
:
D b B2 , m/h 1
:
A & ¬(Db B1
L&
R¬
R&
& ¬B2 ), m/h 2 : A & ¬Db B2
a = b ⇒ m/h : Db B1 , m/h : Db B2 , m/h : a (A & ¬(Db B1 & ¬B2 )), m/h : a (A & ¬Db B2 )
R&
Ra
(3)
m/h : Da A, m/h : a = b ⇒ m/h : Db B1 , m/h : Db B2 , m/h : a (A & ¬(Db B1 & ¬B2 )), m/h : Da (A & ¬Db B2 )
m/h :
Da A, m/h
: a = b ⇒ m/h :
D b B1 , m/h
:
D b B2 , m/h
:
Da (A & ¬(D b B1
& ¬B2 )), m/h :
R Da
Da (A & ¬D b B2 )
m/h : ¬Da (A & ¬(Db B1 & ¬B2 )), m/h : ¬Da (A & ¬Db B2 ), m/h : Da A, m/h : a = b ⇒ m/h : Db B1 , m/h : Db B2
(4)
R Da
L¬
⊔
⊓
123
Alternative Axiomatization for Logics of Agency in a G3 Calculus
219
6 Concluding Remarks
The axiomatization of dstit offered in Xu (1998) can be seen as the logic of deliberative
stit proper, as it contains only that modality. However, we have shown in this paper that
the sequent calculus approach to it from Negri and Pavlović (2020), based on the BT + AC
semantics and shown there to capture the axiomatization in Belnap et al. (2001), can also
successfully encompass it.
As the height of the derivation in any G3 system depends on the weight of its endsequent,
the longer axioms invariably lead to extended derivations, which has been very evident in
this paper. So, compared against that common semantic backdrop, we have at the same time
illustrated the benefit of utilizing multiple modalities, as the increase in the complexity of
the language is offset by the gain in the brevity of proofs.
But first and foremost we have in this paper demonstrated that the system presented in
Negri and Pavlović (2020) successfully and readily captures either approach to dstit.
Funding Open access funding provided by University of Helsinki including Helsinki University Central
Hospital.
Compliance with Ethical Standards
Conflict of Interest The authors declare that they have no conflict of interest.
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appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
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Appendix A: Axiom A6
In this appendix we give a full proof for the axiom A6.
Left to right:
(1)
h ′′ ∼im h ′ , m/h ′′ : A, m/h ′′ : B & ¬Di B, h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′′ : A, m/h ′ : Di B
L&
h ′′ ∼im h ′ , m/h ′′ : A & (B & ¬Di B), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′′ : A, m/h ′ : Di B
Li
′′
i
′
′
i
i
′
i
′
i
i
′′
′
i
h ∼m h , m/h : (A & (B & ¬D B)), h ∼m h, m/h : B, . . . ⇒ m/h : S(¬D A & (B & ¬D B)), m/h : A, m/h : D B
Ri
m/h ′ : i (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : i A, m/h ′ : Di B
i
LD
m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : i A, m/h ′ : Di B
(2a) very similar to (1)
(2b)
m/h ′′′ : A, m ∈ h ′′′ , m/h ′′ : i A, m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′′ : A
m ∈ h ′′′ , m/h ′′ : i A, m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′′ : A
RS
m/h ′′ : i A, m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′ : S A
LDi
′′
i
′′
′
′
′
i
i
′
i
′
m/h : D A, m ∈ h , m/h : S B, m/h : S A, m/h : D (A & (B & ¬D B)), h ∼m h, m/h : B, . . . ⇒
R¬
m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′ : ¬Di A
LS
123
220
S. Negri, E. Pavlović
(2c) very similar to (2b)
(2d)
m/h ′′ : B, m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′ : B
LS
m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′ : B
(2c)
R&
m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′ : B & ¬Di B
(2)
(2b)
(2d)
m ∈ h ′′ , m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B)
m/h ′ : S B, m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S(¬Di A & (B & ¬Di B))
(2a)
RS
RDi
m/h ′ : S A, m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : Di B
(3)
(1)
(2)
m/h ′ : Di (A & (B & ¬Di B)), h ′ ∼im h, m/h ′ : B, . . . ⇒ m/h : S (¬Di A & (B & ¬Di B)), m/h ′ : Di A, m/h ′ : Di B
R Di
h ′ ∼im h, m/h ′ : ¬Di A, m/h ′ : B, m/h ′ : ¬Di B, . . . ⇒ m/h ′ : ¬Di (A & (B & ¬Di B)), m/h : S (¬Di A & (B & ¬Di B))
h ′ ∼im h, m/h ′ : ¬Di A & (B & ¬Di B), . . . ⇒ m/h ′ : ¬Di (A & (B & ¬Di B)), m/h : S (¬Di A & (B & ¬Di B))
h ′ ∼im h, m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h ′ : ¬Di (A & (B & ¬Di B)), m/h : S (¬Di A & (B & ¬Di B))
m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : i ¬Di (A & (B & ¬Di B)), m/h : S (¬Di A & (B & ¬Di B))
L¬, R¬
L&
Li
Ri
(4a)
m ∈ h ′ , m/h ′′′′ : S(A & (B & ¬Di B)), m/h ′ : A, m/h ′ : (B & ¬Di B), . . . ⇒ m/h ′ : A, . . .
m ∈ h ′ , m/h ′′′′ : S(A & (B & ¬Di B)), m/h ′ : A & (B & ¬Di B), . . . ⇒ m/h ′ : A, . . .
m ∈ h ′ , m/h ′′′′ : S(A & (B & ¬Di B)), . . . ⇒ m/h ′ : A, . . .
m/h ′′′′ : S(A & (B & ¬Di B)), . . . ⇒ m/h ′ : A, . . .
L&
LS
WD
(4b)
m/h ∗ : A, h ∗ ∼im h ′′′′ , m/h ′′′′ : i A, m ∈ h ′′′′ , m/h ′ : S B, . . . , m/h ′ : B ⇒ . . . , m/h ∗ : A, m/h ′′′′ : S A
h ∗ ∼im h ′′′′ , m/h ′′′′ : i A, m ∈ h ′′′′ , m/h ′ : S B, . . . , m/h ′ : B ⇒ . . . , m/h ∗ : A, m/h ′′′′ : S A
h ∗ ∼im h ′′′′ , m/h ′′′′ : Di A, m ∈ h ′′′′ , m/h ′ : S B, . . . , m/h ′ : B ⇒ . . . , m/h ∗ : A
LDi
Li
similar to (4c)
h ∗ ∼im h ′′′′ , m/h ′′′′ : Di A, m ∈ h ′′′′ , m/h ′ : S B, . . . , m/h ′ : B ⇒ . . . , m/h ∗ : A & (B & ¬Di B)
Ri
m/h ′′′′ : Di A, m ∈ h ′′′′ , m/h ′ : S B, . . . , m/h ′ : B ⇒ . . . , m/h ′′′′ : i (A & (B & ¬Di B))
R&
(4a)
RDi
m/h ′′′′ : Di A, m ∈ h ′′′′ , m/h ′ : S B, h ′′ ∼im h ′ ∼im h, . . . , m/h ′ : B ⇒ m/h ′ : A, m/h ′′ : A, m/h ′′′′ : Di (A & (B & ¬Di B))
L¬
′′′′
i
′′′′
′
′′
i
′
i
′′′′
i
i
′
′
′′
m/h : D A, m ∈ h , m/h : S B, h ∼m h ∼m h, m/h : ¬D (A & (B & ¬D B)), . . . , m/h : B ⇒ m/h : A, m/h : A
LS
m/h ′′′′ : Di A, m ∈ h ′′′′ , m/h ′ : S B, h ′′ ∼im h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), . . . , m/h ′ : B ⇒ m/h ′ : A, m/h ′′ : A
m ∈ h ′′′′ , m/h ′ : S B, h ′′ ∼im h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h ′′′′ : ¬Di A, m/h ′ : A, m/h ′′ : A
R¬
(4c)
m ∈ h ∗ , m/h ′′′′ : i B, m/h ′ : S B, m/h ∗ : B, . . . ⇒ . . . , m/h ∗ : B
m ∈ h ∗ , m/h ′′′′ : i B, m/h ′ : S B, . . . ⇒ . . . , m/h ∗ : B
m/h ′′′′ : i B, m/h ′ : S B, . . . ⇒ . . . , m/h ′′′′ : S B
m ∈ h ′′′′ , m/h ′ : S B, m/h ′′′′ : B, . . . ⇒ m/h ′′′′ : B, . . .
m ∈ h ′′′′ , m/h ′ : S B, . . . ⇒ m/h ′′′′ : B, . . .
m/h ′′′′ : Di B, m/h ′ : S B, . . . ⇒ . . .
LS
m/h ′ : S B, . . . ⇒ m/h ′′′′ : ¬Di B, . . .
LS
RS
LDi
R¬
(4b) m ∈ h ′′′′ , m/h ′ : S B, h ′′ ∼im h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h ′′′′ : B & ¬Di B, m/h ′ : A, m/h ′′ : A
R&
m ∈ h ′′′′ , m/h ′ : S B, h ′′ ∼im h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h ′′′′ : ¬Di A & (B & ¬Di B), m/h ′ : A, m/h ′′ : A
m/h ′ : S B, h ′′ ∼im h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : A, m/h ′′ : A
123
R&
RS
Alternative Axiomatization for Logics of Agency in a G3 Calculus
221
(4d)
h ′′′ ∼im h ′′ ∼im h ′ ∼im h, . . . , m/h : i (¬Di A & (B & ¬Di B)), m/h ′′′ : ¬Di A, m/h ′′′ : B, m/h ′′′ : ¬Di B, . . . ⇒ . . . , m/h ′′′ : B
h ′′′ ∼im h ′′ ∼im h ′ ∼im h, . . . , m/h : i (¬Di A & (B & ¬Di B)), m/h ′′′ : ¬Di A & (B & ¬Di B), . . . ⇒ . . . , m/h ′′′ : B
h ′′′ ∼im h ′′ ∼im h ′ ∼im h, . . . , m/h : i (¬Di A & (B & ¬Di B)), . . . ⇒ . . . , m/h ′′′ : B
L &, L &
Li
Ri
h ′′ ∼im h ′ ∼im h, m/h : S ¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h : S (¬Di A & (B & ¬Di B)), m/h ′ : A, m/h ′′ : A, m/h ′ : i B
(4c)
h ′′ ∼im h ′ ∼im h, m/h : S ¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h : S (¬Di A & (B & ¬Di B)), m/h ′ : A, m/h ′′ : A, m/h ′ : Di B
h′
∼im
h, m/h :
S ¬Di (A & (B
& ¬Di B)), m/h
:
i (¬Di
A & (B
& ¬Di B)), m/h ′
: B ⇒ m/h :
S (¬Di
A & (B
& ¬Di B)), m/h ′
:
A, m/h ′
:
i
A, m/h ′
:
R Di
Ri
Di B
(4e)
. . . , m/h ′ : S A, m/h ′ : A ⇒ . . . , m/h ′ : A, . . .
LS
. . . , m/h ′ : S A ⇒ . . . , m/h ′ : A, . . .
(4d)
RDi
h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : B ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : A, m/h ′ : Di A, m/h ′ : Di B
L¬
h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : ¬Di A, m/h ′ : B, m/h ′ : ¬Di B ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : A
L&
h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : ¬Di A & (B & ¬Di B) ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : A
(4)
m/h ′ : B & ¬Di B ⇒ m/h ′ : B & ¬Di B
. . . , m/h ′ : ¬Di A & (B & ¬Di B) ⇒ . . . , m/h ′ : B & ¬Di B
(4e)
L&
h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)), m/h ′ : ¬Di A & (B & ¬Di B) ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : A & (B & ¬Di B)
h ′ ∼im h, m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h ′ : A & (B & ¬Di B)
m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h : i (A & (B & ¬Di B))
R&
Li
Ri
(5a)
h ∼im h ′′ , . . . , m/h : S(A & (B & ¬Di B)), m/h ′′ : A, m/h ′′ : (B & ¬Di B), . . . ⇒ m/h ′′ : A
L&
h ∼im h ′′ , . . . , m/h : S(A & (B & ¬Di B)), m/h ′′ : A & (B & ¬Di B), . . . ⇒ m/h ′′ : A
LS
h ∼im h ′′ , . . . , m/h : S(A & (B & ¬Di B)), . . . ⇒ m/h ′′ : A
i
R
. . . , m/h : S(A & (B & ¬Di B)), . . . ⇒ m/h : i A
(5b)
m/h ′ : A, m/h ′ : B & ¬Di B, m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . ⇒ m/h ′ : B & ¬Di B
m/h ′ : A & (B & ¬Di B), m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . ⇒ m/h ′ : B & ¬Di B
m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . ⇒ m/h ′ : B & ¬Di B
L&
LS
(5)
. . . , m/h ′ : i A, m/h : S A, m/h ′′ : A ⇒ . . . , m/h ′′ : A
. . . , m/h ′ : i A, m/h : S A ⇒ . . . , m/h ′ : S A
RS, LS
LDi
(5a)
. . . , m/h ′ : Di A, m/h : S A ⇒ . . .
RDi
i
′
i
′
h ∼m h, m/h : D A, m ∈ h , m/h : S(A & (B & ¬Di B)), . . . ⇒ m/h : Di A
L¬
h ∼im h, m/h ′ : Di A, m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . , m/h : ¬Di A, m/h : (B & ¬Di B)) ⇒
L&
h ∼im h, m/h ′ : Di A, m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . , m/h : ¬Di A & (B & ¬Di B)) ⇒
i , Re f l
L
m/h ′ : Di A, m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . , m/h : i (¬Di A & (B & ¬Di B)) ⇒
m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . , m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h ′ : ¬Di A
R¬
(5b)
m ∈ h ′ , m/h : S(A & (B & ¬Di B)), . . . , m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h ′ : ¬Di A & (B & ¬Di B)
m/h : S(A & (B & ¬Di B)), . . . , m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : S(¬Di A & (B & ¬Di B))
R&
RS
(6)
(4)
(5)
. . . , m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : S(¬Di A & (B & ¬Di B)), m/h : Di (A & (B & ¬Di B))
m/h : ¬Di (A & (B & ¬Di B)), . . . , m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : S(¬Di A & (B & ¬Di B))
m/h : S¬Di (A & (B & ¬Di B)), m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : S(¬Di A & (B & ¬Di B))
RDi
L¬
LS
123
222
S. Negri, E. Pavlović
(3)
(6)
m/h : i (¬Di A & (B & ¬Di B)) ⇒ m/h : Di ¬Di (A & (B & ¬Di B)), m/h : S(¬Di A & (B & ¬Di B))
m/h : Di (¬Di A & (B & ¬Di B)) ⇒ m/h : Di ¬Di (A & (B & ¬Di B))
RDi
LDi
Right to left:
(1a)
m/h ′′′ : A, . . . , h ′′′ ∼im h ′′ , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′′ : A, . . .
m/h ′′ : i A, . . . , h ′′′ ∼im h ′′ , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′′ : A, . . .
m/h ′′ : Di A, . . . , h ′′′ ∼im h ′′ , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′′ : A, . . .
. . . , h ′′′
∼im
h ′′ , m
∈
h ′ , m/h ′
:
A, m/h ′
: B, . . . ⇒
m/h ′′
:
¬Di
A, m/h ′′′
: A, . . .
Li
L Di
R¬
like in (1c) below
m/h ′ : S B, . . . ⇒ m/h ′′ : B & ¬Di B, . . .
R&
m/h ′ : S B, . . . , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A, . . .
(1b)
m/h ′′′′ : A, m/h ′′′′ : B, m/h ′′′′ : ¬Di B, h ′′′′ ∼im h ′ , . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . . , m/h ′′′′ : B
L&
m/h ′′′′ : A & (B & ¬Di B), h ′′′′ ∼im h ′ , . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . . , m/h ′′′′ : B
Li
h ′′′′ ∼im h ′ , . . . , m/h ′ : i (A & (B & ¬Di B)), . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . . , m/h ′′′′ : B
Ri
. . . , m/h ′ : i (A & (B & ¬Di B)), . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . . , m/h ′ : i B
(1c)
.
.
.
.
m/h ′ : S B, . . . ⇒ . . . , m/h ′′′ : ¬Di B
.
.
.
.
m/h ′ : S B, . . . ⇒ . . . , m/h ′′′ : B
(1a) m/h ′ : S B, . . . , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : B & ¬Di B, . . .
(1b) m/h ′ : S B, . . . , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . .
R&
R&
R Di
∼im h ∼im h ′′ , m ∈ h ′ , m/h ′ : A, m/h ′ : B, . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . . , m/h ′ : Di B
L¬
h ′′′ ∼im h ∼im h ′′ , m ∈ h ′ , m/h ′ : A, m/h ′ : B, m/h ′ : ¬Di B, . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . .
L&
′′′
i
i
′′
′
′
i
′′
i
i
′′′
i
h ∼m h ∼m h , m ∈ h , m/h : A & (B & ¬D B), . . . ⇒ m/h : ¬D A & (B & ¬D B), m/h : A & (B & ¬D B), . . .
Re f l, Li
h ′′′ ∼im h ∼im h ′′ , m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)), . . . ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B), . . .
L Di
h ′′′ ∼im h ∼im h ′′ , m ∈ h ′ , m/h ′ : Di (A & (B & ¬Di B)) ⇒ m/h ′′ : ¬Di A & (B & ¬Di B), m/h ′′′ : A & (B & ¬Di B)
i , Ri
R
m ∈ h ′ , m/h ′ : Di (A & (B & ¬Di B)) ⇒ m/h : i (¬Di A & (B & ¬Di B)), m/h : i (A & (B & ¬Di B))
h ′′′
(1d)
.
..
.
. . . , m/h ′′ : ¬Di A, m/h ′′ : B & ¬Di B, m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h ′′ : B & ¬Di B, . . . , m/h ′ : Di A
. . . , m/h ′′ : ¬Di A & (B & ¬Di B), m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h ′′ : B & ¬Di B, . . . , m/h ′ : Di A
. . . , m/h : S(¬Di A & (B & ¬Di B)), m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h ′′ : B & ¬Di B, . . . , m/h ′ : Di A
(1e)
123
L&
LS
Alternative Axiomatization for Logics of Agency in a G3 Calculus
223
h ′′′ ∼im h ′ , . . . , m/h ′′′ : A & (B & ¬Di B) ⇒ . . . , m/h ′′′ : A
Li
h ′′′ ∼im h ′ , . . . , m/h ′ : i (A & (B & ¬Di B)) ⇒ . . . , m/h ′′′ : A
m/h ′′ : A, .. ⇒ m/h ′′ : A, . . .
LS
Ri
. . . , m/h ′ : i (A & (B & ¬Di B)) ⇒ . . . , m/h ′ : i A
m/h ′ : S A, .. ⇒ m/h ′′ : A, . . .
RDi
′′
i
′
i
′
′
i
i
′′
′
i
(1d)
h ∼m h, m/h : B & ¬D B, . . . , m ∈ h , m/h : (A & (B & ¬D B)) ⇒ m/h : A, . . . , m/h : D A
h ′′ ∼im h, m/h ′ : B & ¬Di B, . . . , m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h ′′ : A & (B & ¬Di B), . . . , m/h ′ : Di A
m/h ′ : B & ¬Di B, . . . , m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : i (A & (B & ¬Di B)), . . . , m/h ′ : Di A
m/h ′ : ¬Di A, m/h ′ : B & ¬Di B, . . . , m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : i (A & (B & ¬Di B)), . . .
m/h ′ : ¬Di A & (B & ¬Di B), . . . , m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : i (A & (B & ¬Di B)), . . .
m/h : S(¬Di A & (B & ¬Di B)), m ∈ h ′ , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : i (A & (B & ¬Di B)), . . .
m/h : S(¬Di A & (B & ¬Di B)), m ∈ h ′ , m/h ′ : Di (A & (B & ¬Di B)) ⇒ m/h : i (A & (B & ¬Di B))
R&
Ri
L¬
L&
LS
LDi
(1)
(1c)
(1e)
m ∈ h ′ , m/h ′ : Di (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B)), m/h : i (A & (B & ¬Di B))
RDi
(2)
.
..
.
m/h ′′ : A & (B & ¬Di B), m ∈ h ′′ , . . . , m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B)), m/h ′′ : A & (B & ¬Di B)
m ∈ h ′′ , m/h : S(A & (B & ¬Di B)), m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B)), m/h ′′ : A & (B & ¬Di B)
m/h : S(A & (B & ¬Di B)), m/h ′ : i (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B)), m/h ′ : S(A & (B & ¬Di B))
m/h : S(A & (B & ¬Di B)), m/h ′ : Di (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B))
(1)
(2)
m ∈ h ′ , m/h ′ : Di (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B)), m/h : Di (A & (B & ¬Di B))
m ∈ h ′ , m/h ′ : Di (A & (B & ¬Di B)), m/h : ¬Di (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B))
m/h : PDi (A & (B & ¬Di B)), m/h : ¬Di (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B))
m/h : Di ¬Di (A & (B & ¬Di B)) ⇒ m/h : Di (¬Di A & (B & ¬Di B))
LS
RS
LDi
RDi
L¬
LP
Pr op ∗ 5.3
References
Balbiani, P., Herzig, A., & Troquard, N. (2008). Alternative axiomatics and complexity of deliberative stit
theories. Journal of Philosophical Logic, 37(4), 387–406.
Belnap, N., Perloff, M., & Xu, M. (2001). Facing the future: Agents and choices in our indeterminist world.
Oxford: Oxford University Press.
Chellas, B. (1980). Modal logic: An introduction. Cambridge: Cambridge University Press.
Lyon, T., & van Berkel, K. (2019). Automating agential reasoning: Proof-calculi and syntactic decidability for
stit logics. In International conference on principles and practice of multi-agent systems (pp 202–218).
Berlin: Springer.
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34(5–6), 507.
Negri, S., & Orlandelli, E. (2019). Proof theory for quantified monotone modal logics. Logic Journal of the
IGPL, 27(4), 478–506.
Negri, S., & Pavlović, E. (2020). Proof-theoretic analysis of the logics of agency: The deliberative STIT. Studia
Logica 1–35. https://doi.org/10.1007/s11225-020-09914-1.
Negri, S., & Sbardolini, G. (2016). Proof analysis for Lewis counterfactuals. The Review of Symbolic Logic,
9(1), 44–75.
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Negri, S., & von Plato, J. (1998). Cut elimination in the presence of axioms. Bulletin of Symbolic Logic, 4(4),
418–435.
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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional affiliations.
Sara Negri is Professor of Mathematical Logic at the University of Genova. She has been Professor of Theoretical Philosophy at the University of Helsinki since 2015, as well as a Docent of Logic since 1998. After a
PhD in Mathematics in 1996 at the University of Padova and research visits at the University of Amsterdam
and Chalmers, she has been a Research Associate at the Imperial College in London, a Humboldt Fellow in
Munich, and a visiting scientist at the Mittag-Leffler Institute in Stockholm and at the Hausdorff Institute for
Mathematics in Bonn and a visiting professor at the Scuola Normale Superiore of Pisa and the University of
Verona. Her research interests range from mathematical logic and philosophy of mathematics to proof theory
and its applications to modal and non-classical logics.
Edi Pavlović was born in Rijeka, Croatia and studied for his bachelor’s degree at the University of Rijeka.
He continued his graduate studies at the Central European University, then in Budapest, where he earned
his master’s degree in 2012 and a Ph.D. in 2017, both in philosophy. After graduating he joined University
of Helsinki in 2018 as a postdoctoral researcher on the project Modalities and conditionals: Systematic and
historical studies with Prof. Negri the principal investigator. His research interests lie in philosophical logic,
as well as proof theory and its applications to modal and non-classical logics in general, and particularly to
logics of agency and free logics
123
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