Alternative Axiomatization for Logics of Agency in a G3 Calculus

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).

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 resultsa 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.

G3DSTIT and G3Ldm k
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, 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 L 1 . For any subsequent system we add the appropriate rule APC k to G3DSTIT to produce the system G3Ldm k .

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) Doing is equivalent to refraining from refraining (Proposition * 5.4): m/h : . Finally, the system with the rule APC k 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 D i .

Logic L 0
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: This concludes the proof of adequacy of G3DSTIT for the logic L 0 . 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.

Logic L 1
Logic L 1 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.

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.
Proof Both of these axioms are proven by Proposition * 3.1.
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): and thereby we can derive We now proceed with the proof. Left to right: (2) very similar to (1) 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.

Remaining Axioms
We begin this proof with a lemma: Proof We omit applications of WD for brevity.
Proof By contraposition from the second claim in Theorem * 5.2.

Axioms AIA n
While the axiom schema AIA n 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: Proof For brevity, we will omit writing that Diff (b 1 , b 2 ), as well as applications of WD and Proof For brevity we will omit writing Diff (a, b 1 , b 2 ), as well as applications of WD and We can now show AIA 2 (1) Using Lemma 4.2 for the case where a = b 1 and again for the case where a = b 2 , Lemma 4.3 and an instance of A9, we obtain (5) The same as (4). (3) (1) . . . .
As already mentioned, the extension to the case of n agents is long-winded, but straightforward.

Logic L k
These logics are defined by the axiom APC k for an appropriate k. To capture them, we now utilize logic G3Ldm k (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).

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.

Conflict of Interest The authors declare that they have no conflict of interest.
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Appendix A: Axiom A6
In this appendix we give a full proof for the axiom A6. Left to right: