# Appendix for: Cut-free Calculi and Relational Semantics for Temporal STIT logics

This paper is an appendix to the paper "Cut-free Calculi and Relational Semantics for Temporal STIT logics" by Berkel and Lyon, 2019. It provides the completeness proof for the basic STIT logic Ldm (relative to irreflexive, temporal Kripke STIT frames) as well as gives the derivation of the independence of agents axiom for the logic Xstit.

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## Appendix 0.A Completeness of Ldm

We give the definitions and lemmas sufficient to prove the completeness of relative to frames [4, 2]. We make use of the canonical model of (obtained by standard means [3, 1]) to construct a model. A truth-lemma is then given relative to this model, from which, completeness follows as a corollary.

###### Definition 1 (Ldm-Cs, Ldm-Mcs)

A set is a consistent set (-CS) iff . We call a set a maximally consistent set (-MCS) iff is a -CS and for any set such that , .

###### Lemma 1 (Lindenbaum’s Lemma [3])

Every -CS can be extended to a -MCS.

###### Definition 2 (Present and Future Pre-Canonical Tstit Model)

The present pre-canonical model is the tuple defined below left, and the future pre-canonical model is the tuple defined below right:

• is the set of all -MCSs;

• iff for all , ;

• iff for all , ;

• .

• ;

• ;

• ;

• .

###### Definition 3 (Canonical Temporal Kripke STIT Model)

We define the canonical temporal Kripke STIT model to be the tuple such that:

• 111Note that we choose to write each world as to simplify notation. Moreover, we write to mean that the formula is in the -MCS associated with .;

• iff (i) and , or (ii) and ;

• iff (i) and , or (ii) and ;

• ;

• ;

• ;

• .

###### Lemma 2

For all , if for , then .

###### Proof

Follows by definition of the canonical model.

###### Lemma 3

For all with , iff .

###### Proof

This follows from the fact that iff iff for each iff iff for any .

###### Lemma 4 ([3])

(i) For all , iff for all , if , then . (ii) For all , iff for all , if , then .

###### Lemma 5 (Existence Lemma [3])

(i) For any world , if , then there exists a world such that and . (ii) For any world , if , then there exists a world such that and .

###### Lemma 6

The Canonical Model is a temporal Kripke STIT model.

###### Proof

We prove that has all the properties of a model:

• By lemma 1, the consistent set can be extended to a -MCS, and therefore is non-empty. Since is non-empty as well, is a non-empty set of worlds.

• We argue that is an equivalence relation between worlds of , and omit the arguments for and , which are similar. Suppose that . We have two cases to consider: (i) , and (ii) . (i) Standard canonical model arguments apply and is an equivalence relation between all worlds of the form (See [3] for details). (ii) If we fix a , then will be an equivalence relation for all worlds of the form since the intersection of equivalence relations produces another equivalence relation. Last, since is an equivalence relation for each fixed , and because each is disjoint from each for , we know that the union all such equivalence relations will be an equivalence relation.

• Let be in and assume that . We split the proof into two cases: (i) , or (ii) . (i) Assume that . Since is a -MCS, it contains the axiom , and so, as well. Since (because ), we know that by the definition of the relation; therefore, , which implies that by definition. (ii) The assumption that implies that by definition, which implies that .

• Let and assume that for all . We split the proof into two cases: (i) , or (ii) . (i) We want to show that there exists a world such that . Let . Suppose that is inconsistent to derive a contradiction. Then, there are ,…, such that . For each , we define . Observe that for each , because and . Since by assumption for all , this means that for any we pick (with ), for each by lemma 4; hence, . By the axiom, this implies that . By lemma 5, there must exist a world such that and . But then, since by reflexivity, , and , it follows that , which is a contradiction since is a -MCS. Therefore, must be consistent and by lemma 1, it may be extended to a -MCS . Since for each , , we have that for each . Hence, , and so, . (ii) Suppose that , so that iff = . By assumption then, for all and each . Hence, for all . If we therefore pick any , it follows that , meaning that the intersection is non-empty.

• Follows by definition.

• is a transitive and serial by definition, and is the converse of by definition as well.

• For all , suppose that and . Then, and , and since is linearly ordered, we have that , , or , implying that , , or .

• Similar to previous case.

• Suppose that with . By definition of , is the only element in associated with , and so, (By lemma 2 no other with can relate to in .). Since , iff iff . By lemma 3, . This implies that, and since by definition, we have that .

• Follows from the definition of the relation.

• Last, it is easy to see that the valuation function is indeed a valuation function.

###### Lemma 7 (Truth-Lemma)

For any formula , iff .

###### Proof

Shown by induction on the complexity of (See [3]).

## Appendix 0.B G3Xstit Derivation of IOAx Axiom

We make use of the system of rules , to derive the IOA axiom in .

 R□w1w2,R□w1w3,R□w1w4,RAw4w5,RAw2w5,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w5:ϕ,w5:¯¯¯ϕ R□w1w2,R□w1w3,R□w1w4,RAw4w5,RAw2w5,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w5:ϕ (IOA−U1) R□w1w2,R□w1w3,R□w1w4,RAw4w5,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w5:ϕ R□w1w2,R□w1w3,R□w1w4,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w4:[A]xϕ D1

 R□w1w2,R□w1w3,R□w1w4,RBw4w6,RBw3w6,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,....  w6:ψ,w6:¯¯¯¯ψ R□w1w2,R□w1w3,R□w1w4,RBw4w6,RBw3w6,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w6:ψ (IOA−U2) R□w1w2,R□w1w3,R□w1w4,RBw4w6,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w6:ψ R□w1w2,R□w1w3,R□w1w4,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,...  w4:[B]xψ D2

 D1D2R□w1w2,R□w1w3,R□w1w4,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,w1:◊([A]xϕ∧[B]xψ),w4:[A]xϕ∧[B]xψ R□w1w2,R□w1w3,R□w1w4,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,w1:◊([A]xϕ∧[B]xψ) (IOA−E) R□w1w2,R□w1w3,w2:⟨A⟩x¯¯¯ϕ,w3:⟨B⟩x¯¯¯¯ψ,w1:◊([A]xϕ∧[B]xψ) w1:□⟨A⟩x¯¯¯ϕ,w1:□⟨B⟩x¯¯¯¯ψ,w1:◊([A]xϕ∧[B]xψ) w1:□⟨A⟩x¯¯¯ϕ∨□⟨B⟩x¯¯¯¯ψ∨◊([A]xϕ∧[B]xψ)

## References

• [1] Balbiani, P., Herzig, A., Troquard, N.: Alternative axiomatics and complexity of deliberative STIT theories. Journal of Philosophical Logic, 37(4), pp.387–406. Springer (2008)
• [2]

Berkel, K., Lyon, T.: Cut-free Calculi and Relational Semantics for Temporal STIT logics. In Logics in Artificial Intelligence - 16th European Conference, JELIA 2019, Rende, Italy, May 7-10, 2019, Proceedings, To appear.

• [3] Blackburn, P., de Rijke, M., Venema, Y.: Modal logic. Cambridge University Press, Cambridge (2001)
• [4] Lorini, E.: Temporal STIT logic and its application to normative reasoning. Journal of Applied Non-Classical Logics 23 (4), pp. 372–399 (2013)