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Combinations of Stit with Ought and Know

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Abstract

This paper presents a short survey of recent developments in stit theories, with an emphasis on combinations of stit and deontic logic, and those of stit and epistemic logic.

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Notes

  1. We will not survey what is often called the logic of actions, a field with Krister Segerberg as a leading figure. The author is not an expert in that field, and the interested readers are referred to [55, 57, 58] and articles in [63] with references therein.

  2. See [52, 60] and [62].

  3. Standard stit theories require a tree-like frame to satisfy the condition “historical connection” ( H C): \(\forall m\forall m^{\prime }\exists m^{\prime \prime }(m^{\prime \prime }\leqslant m\wedge m^{\prime \prime }\leqslant m^{\prime })\). Here we do not enforce H C because we will consider combinations of stit and epistemic notions. Fortunately, standard stit theories do not deal with operators going back and forth between different “trees”, and thus the absence of H C in our presentation of stit theories won’t make any technical difference.

  4. For discussions on the basic notions involved in tree-like frames, as representations of indeterminism, and the reasons why formulas are evaluated at moment/history pairs, see [60, 62] and [10, 14].

  5. Belnap noted in [7]: “The theory of Instant is not as fundamental as that of \(\leqslant \), and perhaps it is too strong, even pre-relativistically. … and perhaps it should be ultimately be weakened. In the mean time, while it is good to be concerned about over-simplification, the assumption can be clarifying when it comes to thinking about certain aspects of agency.” In this paper, we leave out discussions on various fundamental ideas in stit frames and their possible improvements.

  6. In this paper, we will for convenience use α, β etc. ambiguously both as agents in the semantic structures and as terms for agents in the object language.

  7. For discussions on instants, agents and choices, the reader is referred to [7, 8] and [14].

  8. The astit operator was proposed in [11], and discussed in details in [14]. There are actually two accounts of the astit semantics: one is called astit semantics based on “witness by moments”, which we present here, and the other, “witness by chains”, can be found in [6–8, 12], and [14]. An axiomatization and decidability of astit with a single agent and the refref equivalence (which asserts that doing is equivalent to refraining from refraining from doing) is given in [73]. An axiomatization, with a Gabbay-style rule, of the basic theory of astit with a single agent is given in [74], where the proof contains an error that is corrected in [14]. The Gabbay-style rule in the axiomatization is later shown to be unnecessary, in an unpublished manuscript. Astit operators can be defined in terms of some temporal operators and the cstit operators, as shown in [31] and [71].

  9. The dstit operator was developed independently by von Kutschera and Horty in [65] and [39]. The name “deliberative stit” echoes the notion of deliberative obligation in [61]. Note that seeing to something deliberatively is not the same as seeing to it deliberately. For detailed discussions on dstit and its various applications in deontic logic, see [42] and those cited in Section 2 under the names Bartha, Belnap, Horty and Wansing.

  10. Chellas’ original stit operator is proposed in [30]. What is nowadays widely used and referred to as the cstit operator is a variant of the original one, and is developed later in [31]. This operator is also called “the Q-operator” (see [71]). The bstit operator is an approximation of Brown’s notion called “the canof ability” (of a fixed single agent). Brown’s semantics ([25] and [26]) uses neighborhood models for classical modal logic, and the can operator is interpreted as a monotone modal operator (see [56]). In [27], Brown further develops his theory by treating the can operator as a combination of a possibility operator and an action operator.

  11. Technical developments of cstit and dstit often go hand in hand, though not always. An axiomatization and decidability of the cstit/dstit logic with multiple agents is given in [72], in a language containing cstit/dstit operators and \(\Box \), without temporal operators; and an axiomatization of dstit with multiple agents is given in [75], in a language with dstit to be the only non-truth-functional operator. In a language containing \(\mathsf {F} ,\mathsf {P},\Box \), and two extra operators, [71] gives an axiomatization of cstit with multiple agents, using Gabbay-style rule of irreflexivity (see [33] or [34]). The two extra operators are interpreted respectively as “settled throughout the current instant” and “through every history passing through the current moment, except the current history”. [70] develops a tableaux system for dstit with multiple agents, and [3] simplifies the axiomatization in [72], and proves that the problem of deciding satisfiability of formulas in the language of dstit/cstit theory, with multiple agents and without temporal operators, is NEXPTIME-complete.

  12. See [13] or Ch. 10 of [14].

  13. In [36], it has been shown for both cstit and dstit that the full theory of group stit is neither decidable nor finitely axiomatizable. The main idea in the proof is to reduce the undecidability and non-finite-axiomatizability of group stit with at least 3 agents to the undecidability and non-finite-axiomatizability of the product logic S5 n with \(n\geqslant 3\) (cf. Theorems 8.6 and 8.2 of [35]). [54] is a recent study of complexity of various fragments of stit logic. Another recent work [79] shows the decidability of a large class of logics based on metric temporal logic in branching structures, in a language containing \(\Box \), X (“next”) and Y (“last”). It employs abstract operators in the object language as well as abstract relations in the semantic structures, and proves the decidability of valid formulas over various classes defined by certain types of sentences in a first-order language that describes frames. The abstract operators can be interpreted as (individual or “weakened” group) cstit operators and epistemic operators, and then in such cases, the decidability result can be applied to deal with combinations of stit and epistemic modalities.

  14. In [21, 22] and [37], Broersen, Herzig and Troquard used a combination of [α]c (\([\mathcal {G}]^{\mathsf {c}}\)) and X in their work to connect stit with Coalition Logic ([51]) and ATL (“alternating-time temporal logic”, [1]), which later became a fixed single operator, now known as xstit. I am not sure exactly when the switch first took place from cstit plus X to xstit, but it seems that [18] and [17] should be among the earliest works in which xstit appeared as a single operator. ([18] was published later than [17], but reading them gives me the impression that [18] might have been written before [17].) For a systematic study of combination of cstit and X, see [24].

  15. Broersen gives the following semantic notions for xstit in [17], where Ags is our Agent, and notions not directly for xstit are omitted by using “. . .”: “A frame is a tuple \(\mathcal {F}=\left \langle H,S,R_{\Box },\{R_{A}:A\subseteq \textit {Ags}\},\ldots \right \rangle \) such that

    • H is a non-empty set of histories. Elements of H are denoted \(h,h^{\prime },\) etc.

    • S is a non-empty set of states. Elements of S are denoted \(s,s^{\prime },\) etc.

    • \(R_{\Box }\) is a ‘historical necessity’ relation over the elements of H×S such that \(\left \langle h,s\right \rangle R_{\Box }\left \langle h^{\prime },s^{\prime }\right \rangle \) if and only if \(s=s^{\prime }\).

    • The R A are ‘effectivity’ relations over the elements of H×S such that:

      • R A g s is a ‘next time’ relation such that if \(\left \langle h,s\right \rangle R_{Ags}\left \langle h^{\prime },s^{\prime }\right \rangle \) then \(h=h^{\prime }\), and R A g s is serial and deterministic (the next state is completely determined by the choice made by the complete set of agents). So, histories ‘contain’ linearly ordered sets of states.

      • \(R_{\Box }\circ R_{Ags}\subseteq R_{\varnothing }\) (the empty set of agents is ineffective)

      • \(R_{A}\subseteq R_{\Box }\circ R_{Ags}\) for any A (an action undertaken by A in the present state ensures the next state is element of a specific subset of all possible next states)

      • \(R_{Ags}\circ R_{\Box }\subseteq R_{A}\) for any A (no actions constitute a choice between histories that are undivided in next states)

      • \(R_{A}\subseteq R_{B}\) for \(B\subset A\) (super-groups are at least as effective)

      • if \(\left \langle h,s\right \rangle (R_{\Box }\circ R_{A})\left \langle h^{\prime },s^{\prime }\right \rangle \) and \(\left \langle h,s\right \rangle (R_{\Box }\circ R_{B})\left \langle h^{\prime \prime },s^{\prime \prime }\right \rangle \) and \(A\cap B=\varnothing \) then there is a \(\left \langle h,s\right \rangle R_{\Box }\left \langle h^{\prime \prime \prime },s\right \rangle \) such that both \(\left \langle h^{\prime \prime \prime },s\right \rangle R_{A}\left \langle h^{\prime },s^{\prime }\right \rangle \) and \(\left \langle h^{\prime \prime \prime },s\right \rangle R_{B}\left \langle h^{\prime \prime },s^{\prime \prime }\right \rangle \) (independence of agency)”.

    It seems that in Broersen’s papers, R 1∘R 2 is consistently used for the relation R such that x R y iff x R 1 z and z R 2 y for some z, rather than the relation \(R^{\prime }\) such that \(xR^{\prime }y\) iff x R 2 z and z R 1 y for some z. We observe that the condition “super-groups are at least as effective” is much weaker than what is required concerning choices for groups in the standard stit theories.

  16. In [17] and Broersen’s later papers, X A is defined as [Agent]x A, under the assumption to the effect that for each m, each selection of one choice for every agent at m determines a unique next moment. A model \(\mathcal {M}\) in [17] is a frame (see the note above) together with a valuation assigning to each propositional letter a subset of H×S, with respect to which Broersen interprets \(\Box \) and the xstit operator as follows, where \(A\subseteq Ags\): \(\mathcal {M},\left \langle h,s\right \rangle \vDash \Box \varphi \) iff \(\left \langle h,s\right \rangle R_{\Box }\left \langle h^{\prime },s^{\prime }\right \rangle \) implies that \(\mathcal {M},\left \langle h^{\prime },s^{\prime }\right \rangle \vDash \varphi \); and \(\mathcal {M},\left \langle h,s\right \rangle \vDash [ A\ \mathsf {xstit} ]\varphi \) iff \(\left \langle h,s\right \rangle R_{A}\left \langle h^{\prime },s^{\prime }\right \rangle \) implies that \(\mathcal {M},\left \langle h^{\prime },s^{\prime }\right \rangle \vDash \varphi \).

  17. If we had the “last” operator Y, [α]c A could then be defined as [α]x Y A. Incidentally, if we restrict to rooted stit frames in which all histories are isomorphic to ω with respect to its usual ordering, then [α]a is “reducible” to [α]x in the presence of \(\square ,\mathsf {X}\) and Y. For example, the \([\alpha ]^{\mathsf {a}}A\leftrightarrow \mathsf {Y}([\alpha ]^{\mathsf {x}} A\wedge \lnot \Box \mathsf {X}A)\vee \mathsf {Y}[\alpha ]^{\mathsf {a}}\mathsf {X}A\) is valid in all such frames, which may serve as a “reduction axiom”. Note that our discussion here on inter-definability of stit operators is with respect to incremental and endless stit frames, and is restricted to individual stit operators — group stit operators are excluded (see the observation at the end of note 15).

    Various axiomatic systems involving the xstit operators have been put forward in a series of articles by Broersen (see our discussion in Section 4.1), and are proved complete with respect to various frame conditions. Note that the xstit operators are not S5 operators, and hence the undecidability and non-finite-axiomatizability of the full theory of group agency ([36]) do not apply to Broersen’s systems. Note also that Broersen’s completeness results are with respect to his semantics for multi-modal logics (see notes 15 and 16), and are obtained by applying the fact that all extra axioms are Salhqvist formulas (see, e.g., [15]) that define the required frame conditions.

  18. A refinement of the account in [4] is developed in [5].

  19. The Meinong-Chisholm approach is an analysis that suggests to identify what an agent ought to do with what it ought to be the case that the agent does. For this approach and Horty’s criticism of it, see [40, 42] and [41].

  20. In [16], Broersen suggests that the relation ≤ could be introduced in some other ways.

  21. [41] uses a combination of ⊙ and \([\mathcal {G}]^{\mathsf {c}}\) as the new operator.

  22. [49] shows an axiomatization and the decidability of the theory of dominance ought with multiple individual agents, in a language without temporal operators. It also shows that a number of semantic accounts for ought operators discussed in [41] are syntactically indistinguishable in the following sense: In the language discussed in [49], where temporal operators are absent, no matter which of those accounts we use to interpret the ought operators in the language, the axiomatization of valid formulas with respect to that account remains the same. This should resolve an issue raised in Section 3.3 of [16]. Note that because the full theory of group stit is not decidable or finitely axiomatizable (see [36]), neither is the full theory of dominance group ought (with group stit operators in the language).

  23. See p. 185 of [46]. McNamara has other challenges to stit theories in [46] and [47], which will not be discussed here.

  24. For this idea to work, there seems a need to require the given utilitarian stit frame to be such a frame in which the procedure of removing “bad” choices for each agent will not eventually remove all choices for the agent.

  25. A recent work [78] continues the decision-theoretic approach by clarifying the notion of independence in a setting that involves strategies with multiple agents and groups of agents.

  26. See examples discussed in [17].

  27. As far as I know, [67] is the first work to bring together stit and epistemic notions. Wansing uses dstit to define voluntarily acquiring a belief, and voluntarily giving up a belief, and then defines that α believes that A as that α once voluntarily acquired that A, and has not voluntarily given it up. See also [69] for another theory with the same spirit. Because Wansing’s main concerning is in philosophy, not particularly in stit logic or epistemic logic, we will not get into that direction of research.

  28. In [23], Broersen, Herzig and Troquard also combines knowledge operators with their strategic stit operators, and interpret their knowledge operators in terms of epistemic equivalence relations between strategy/state pairs. We will not discuss the work in [23] here because the ideas in the work seem different from those in the series mentioned in the main text.

  29. The name K X is for “knowledge of next states” from [20] , where Broersen deals with the xstit version of knowingly doing. The names of frame conditions listed below are all from [20] , except Broersen uses them only for formulas and almost always in capital letters. Here I use the same names for formulas as well as for their corresponding frame conditions, but only the beginning letters are capitalized in the names for formulas, while all letters are capitalized in the names for frame conditions.

  30. [37] discusses group knowledge and operators like \(\textit {Kstit}_{\mathcal {G}}\) with \(\mathcal {G}\) to be a group of agents, but later articles in the series dropped the topic entirely.

  31. In [18], a frame is a tuple \(\mathcal {F} =\left \langle H,S,R_{\Box },\{R_{A}\mid A\subseteq Ags\},\{\sim _{a}\mid a\in Ags\}\right \rangle \), where A g s is our A g e n t, and \(H,S,R_{\Box }\) and R A are as specified in note 15, and “the ∼ a are epistemic equivalence relations over the elements of H×S such that:

    • \(\sim _{a}\circ R_{a}\subseteq {\sim }_{a}\circ R_{Ags}\) (agents cannot know what choices other agents perform concurrently)

    • \(R_{Ags}\circ {\sim }_{a}\subseteq {\sim }_{a}\circ R_{a}\) (agents recall the effects of the actions they knowingly perform themselves)”. (see p. 49, Def. 3.)

    Note that Broersen’s formulation of independence of agents in [18] appears to contain errors, which are corrected in [17] and other papers.

  32. The following shows that \(\mathsf {K}_{\alpha }[\alpha ]^{\mathsf {x}}A\rightarrow \mathsf {X}\Box \mathsf {K}_{\alpha }A\) follows from K x, E r and \([\alpha ]^{\mathsf {x}}A\rightarrow \mathsf {X}\Box A\) (which corresponds to the condition N C), at the presence of S4 axioms for K α :

    \([\alpha ]^{\mathsf {x}}\mathsf {K}_{\alpha }A\rightarrow \mathsf {X} \Box \mathsf {K}_{\alpha }A\)

    instance of \([\alpha ]^{\mathsf {x}} A\rightarrow \mathsf {X}\Box A\)

    \( \mathsf {K}_{\alpha }[\alpha ]^{\mathsf {x}}\mathsf {K}_{\alpha } A\rightarrow \mathsf {X}\Box \mathsf {K}_{\alpha }A\)

    T-axiom for K α , classical logic

    \( \mathsf {K}_{\alpha }\mathsf {K}_{\alpha }[\alpha ]^{\mathsf {x}} A\rightarrow \mathsf {K}_{\alpha }\mathsf {XK}_{\alpha }A\)

    E r, K-axiom for K α

    \( \mathsf {K}_{\alpha }[\alpha ]^{\mathsf {x}}A\rightarrow \mathsf {K}_{\alpha }[\alpha ]^{\mathsf {x}}\mathsf {K}_{\alpha }A\)

    K x, 4-axiom for K α , classical logic

    \( \mathsf {K}_{\alpha }[\alpha ]^{\mathsf {x}}A\rightarrow \mathsf {X} \Box \mathsf {K}_{\alpha }A\)

    classical logic

  33. Broersen uses U n i f- S t r for our U- s above. The frame condition in [17] (p. 143, Def. 3) corresponding to U- s is the following:

    • “if \(\left \langle h,s\right \rangle R_{\Box }\left \langle h^{\prime },s^{\prime }\right \rangle \) and \(\left \langle h,s\right \rangle \sim _{a}\left \langle h^{\prime \prime },s^{\prime \prime }\right \rangle \) then there is a \(\left \langle h^{\prime \prime \prime },s^{\prime \prime \prime }\right \rangle \) for which \(\left \langle h^{\prime },s^{\prime }\right \rangle R_{\Box }\left \langle h^{\prime \prime \prime },s^{\prime \prime \prime }\right \rangle \) and if \(\left \langle h^{\prime \prime \prime },s^{\prime \prime \prime }\right \rangle R_{a}\left \langle h^{\prime \prime \prime \prime },s^{\prime \prime \prime \prime }\right \rangle \) then \(\left \langle h^{\prime },s^{\prime }\right \rangle R_{a}\left \langle h^{\prime \prime \prime \prime },s^{\prime \prime \prime \prime }\right \rangle \) (uniformity of conformant action)”.

    Note that in the frame condition above, “\(\left \langle h^{\prime },s^{\prime }\right \rangle R_{a}\left \langle h^{\prime \prime \prime \prime },s^{\prime \prime \prime \prime }\right \rangle \)” is replaced in [19] by “\(\left \langle h^{\prime },s^{\prime }\right \rangle (\sim _{a}\circ R_{a} )\left \langle h^{\prime \prime \prime \prime },s^{\prime \prime \prime \prime }\right \rangle \)”.

  34. I am not sure whether \(\Diamond \mathsf {K}_{\alpha }A\rightarrow \mathsf {K}_{\alpha }\Diamond A\) is provable in the system presented in [17] and [19]. Although it can be settled after a somewhat tedious calculation applying Broersen’s semantic conditions, we may look at the matter in the following way without getting into the trouble of semantics. If we add the “last” operator Y to the language and extend Broersen’s system by adding an extra axiom \(A\leftrightarrow \mathsf {XY}A\), then \(\Diamond \mathsf {K}_{\alpha }A\rightarrow \mathsf {K}_{\alpha }\Diamond A\) is provable in the new system. This is because \(\Diamond \mathsf {K}_{\alpha }\mathsf {X}B\rightarrow \mathsf {K}_{\alpha }\Diamond \mathsf {X}B\) is provable in Broersen’s system (by U- s, K x 1, \([\alpha ]^{\mathsf {x} }B\rightarrow \mathsf {X}B\) and the K-axioms for K α and \(\Box \)), and then so is \(\Diamond \mathsf {K}_{\alpha }\mathsf {XY}A\rightarrow \mathsf {K}_{\alpha }\Diamond \mathsf {XY}A\), and hence \(\Diamond \mathsf {K}_{\alpha }A\rightarrow \mathsf {K}_{\alpha }\Diamond A\) (by \(A\leftrightarrow \mathsf {XY}A\) and the K-axioms for K α and \(\Box \)), in the new system.

  35. Let us drop the subscript “ α”, and use J for the dual operator of K. The following shows that \(\Diamond \mathsf {K} A\rightarrow \mathsf {K}\Diamond A\) follows from \(\Box \mathsf {K}A\rightarrow \mathsf {K}\Box A\) (\(\mathsf {J}\Diamond A\rightarrow \Diamond \mathsf {J}A\)) when K is a KB operator and \(\Box \) is a normal operator:

    ∙\( \mathsf {J}\Diamond \mathsf {K}A\rightarrow \Diamond \mathsf {JK}A\)

    instance of \(\mathsf {J}\Diamond A\rightarrow \Diamond \mathsf {J}A\)

    ∙\( \mathsf {KJ}\Diamond \mathsf {K}A\rightarrow \mathsf {K}\Diamond \mathsf {JK}A\)

    K-axiom for K

    ∙\( \Diamond \mathsf {K}A\rightarrow \mathsf {K}\Diamond \mathsf {JK}A\)

    B-axiom for K, classical logic

    ∙\( \mathsf {K}\Diamond \mathsf {JK}A\rightarrow \mathsf {K}\Diamond A\)

    B-axiom for K and K-axioms for \(\Box \) and K

    ∙\( \Diamond \mathsf {K}A\rightarrow \mathsf {K}\Diamond A \)

    classical logic

    The following shows that \(\mathsf {K}\Box A\rightarrow \Box \mathsf {K}A\) follows from \(\Diamond \mathsf {K}A\rightarrow \mathsf {K}\Diamond A\) when K is a normal operator and \(\Box \) is a KB operator:

    ∙\( \Diamond \mathsf {K}\Box A\rightarrow \mathsf {K}\Diamond \Box A\)

    instance of \(\Diamond \mathsf {K}A\rightarrow \mathsf {K}\Diamond A\)

    ∙\( \Box \Diamond \mathsf {K}\Box A\rightarrow \Box \mathsf {K}\Diamond \Box A\)

    K-axiom for \(\Box \)

    ∙\( \mathsf {K}\Box A\rightarrow \Box \mathsf {K}\Diamond \Box A\)

    B-axiom for \(\Box \), classical logic

    ∙\( \Box \mathsf {K}\Diamond \Box A\rightarrow \Box \mathsf {K}A \)

    B-axiom for \(\Box \) and K-axioms for K and \(\Box \)

    ∙\( \mathsf {K}\Box A\rightarrow \Box \mathsf {K}A \)

    classical logic

    The following shows that \(\Box \mathsf {K}A\rightarrow \mathsf {K}\Box A\) follows from \(\mathsf {K}\Box A\rightarrow \Box \mathsf {K}A\) (\(\Diamond \mathsf {J}A\rightarrow \mathsf {J}\Diamond A\)) when both K and \(\Box \) are KB operators:

    ∙\( \Diamond \mathsf {J}\Box \mathsf {K}A\rightarrow \mathsf {J}\Diamond \Box \mathsf {K}A\)

    instance of \(\Diamond \mathsf {J}A\rightarrow \mathsf {J}\Diamond A\)

    ∙\( \Box \Diamond \mathsf {J}\Box \mathsf {K}A\rightarrow \Box \mathsf {J} \Diamond \Box \mathsf {K}A\)

    K-axiom for \(\Box \)

    ∙\( \mathsf {J}\Box \mathsf {K}A\rightarrow \Box \mathsf {J}\Diamond \Box \mathsf {K}A\)

    B-axiom for \(\Box \), classical logic

    ∙\( \mathsf {J}\Diamond \Box \mathsf {K}A\rightarrow \mathsf {JK}A \)

    B-axiom for \(\Box \) and K-axiom for K

    ∙\( \Box \mathsf {J}\Diamond \Box \mathsf {K}A\rightarrow \Box A\)

    B-axiom for K, classical logic, K-axiom for \(\Box \)

    ∙\( \mathsf {KJ}\Box \mathsf {K}A\rightarrow \mathsf {K}\Box A\)

    classical logic, K-axiom for K

    ∙\( \Box \mathsf {K}A\rightarrow \mathsf {K}\Box A\)

    B-axiom for K, classical logic

    This justifies our claim in the main text.

    In a private conversation, Yan Zhang observed that U- S is equivalent to a frame condition that corresponds to the validity of K- s in epistemic stit frames. Now is K- s provable in the system in [17]? According to what we said above, the answer is positive iff \(\Diamond \mathsf {K}_{\alpha }A\rightarrow \mathsf {K}_{\alpha }\Diamond A\) is provable in that system. (See note 34.)

  36. In cases like games, one may assume for the sake of simplicity that all agents act or make their choices publically, and that they all keep observing what others do no matter what they themselves may do. It then makes sense, at least to a certain degree, to treat each doing of an agent α not only as a knowingly doing of α, but also as a doing whose consequences include α’s knowledge of what all agents have done.

  37. The static knowledge operator defined in [37] is \(\square \textit {Kstit}_{\alpha }\), but K α and Kstit α are the same operator by interpretation, as we observed earlier.

  38. Let J α be the dual operator of K α . For \(\Box \mathsf {K}_{\alpha }\) to be an S5 operator, \(A\rightarrow \Box \mathsf {K}_{\alpha }\Diamond \mathsf {J}_{\alpha }A\) would be provable, and so would be \(\mathsf {J}_{\alpha }\Diamond A\rightarrow \Diamond \mathsf {J}_{\alpha }A\), and then \(\Box \mathsf {K}_{\alpha } A\rightarrow \mathsf {K}_{\alpha }\Box A\), by the KB-axioms for K α and \(\Box \), and hence so would be K- s (see note 35). On the other hand, having K- s as a theorem would suffice for \(\Box \mathsf {K}_{\alpha }\) to be an S5 operator, for \(\Box \mathsf {K}_{\alpha }A\rightarrow A\) follows from the T-axioms for K α and \(\Box \), and \(A\rightarrow \Box \mathsf {K}_{\alpha }\Diamond \mathsf {J}_{\alpha }A\) and \(\Box \mathsf {K}_{\alpha }A\rightarrow \Box \mathsf {K}_{\alpha }\Box \mathsf {K}_{\alpha }A\) follow from K- s and the K4B-axioms for K α and \(\Box \).

  39. Broersen expressed his own concerns about K- s in [20]. With respect to our discussions in earlier notes, even if \(\Diamond \mathsf {K}_{\alpha }A\rightarrow \mathsf {K}_{\alpha }\Diamond A\) is provable in the system in [18] (see note 34), in which case so is K- s (see note 35), we are still left with a question why a static knowledge operator should depend on a condition like U- S, not to mention that U- S is not exactly what uniform strategy requires.

  40. Interested readers are referred to [28, 29, 76] and [77]. The simple ideas below of actions are the same as “primitive actions” in [76] and [77], except here we use sets of choices at moments while [76] and [77] use sets of immediate transitions.

  41. Such a condition may not seem necessary for a general notion of action type. For example, actions performed by female agents form a type in the intuitive sense. In a game between two female players, nevertheless, this type does not distinguish any action from any other actions of the players. Our current subject matter requires that actions of an agent at a moment be distinguished by different types, which is why we have the condition above.

  42. Each action type in [50], is tagged to a unique agent (p. 319), and at any time only one agent can perform any action (p. 230). The treatment of histories in [50] also allows any finite initial segment of a history to be concatenated to an action, and then to be “extended” to other histories (p. 320). All these assumptions seem convenient and harmless for many applications, but do not seem necessary for a general theory. Here for convenience in our formulation of type-actions, we make use of their first assumption, without which an account of action types is still possible. For example, we may let \(\mathcal {C}\) be the set of all choices for individual agents at moments, and let Act be a subset of the powerset of \(\mathcal {C}\) such that \(\bigcup \textit {Act}=\mathcal {C}\), and for each π∈ Act, each α and each m, \(\left \vert \pi \cap \textit {Choice}_{\alpha }^{m}\right \vert \leqslant 1\).

  43. The object language in [50] contains operators \([\pi ],[\pi ^{\prime }]\) etc., for type-actions \(\pi ,\pi ^{\prime }\) etc., whose interpretations make them ability operators, similar to the bstit operators [α, π]b and \([\beta ,\pi ^{\prime }]^{\mathsf {b}}\) below if π and \(\pi ^{\prime }\) are tagged to α and β respectively. Note that the interpretations presuppose a condition mentioned in note 42, which our current semantic structure does not satisfy.

  44. [37] (Section 5.2) has suggested an intuitively reading of Kstit α A, which is close to our suggestion: “ Kstit a ϕ is an agency operator with an implicit epistemic feature. It means that a knows that if he performs his chosen action, he ensures that ϕ.” Here “what a knows” should be understood to be settled in order for the suggestion to make sense, because if it is understood to be relative to histories in which a performs his chosen action, the antecedent “he performs his chosen action” would be oddly redundant.

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Xu, M. Combinations of Stit with Ought and Know . J Philos Logic 44, 851–877 (2015). https://doi.org/10.1007/s10992-015-9365-7

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