With the increasing integration of automated machines in our everyday lives, the development of formal decision-making tools, which take into account moral and legal considerations, is of critical importance [1, 9, 10]. Unfortunately, one of the fundamental hazards of incorporating ethics into decision-making processes, is the apparent incomparability of quantitative and qualitative information—that is, moral problems most often resist quantification .
In contrast, utility functions are useful quantitative tools for the formal analysis of decision-making. Initially formulated in , the influential theory of utilitarianism has promoted utility calculation as a ground for ethical deliberation: in short, those actions generating highest utility, are the morally right actions. For this reason, utilitarianism has proven itself to be a fruitful approach in the field of formal deontic reasoning and multi-agent systems (e.g. [7, 12, 15]).
In particular, in the field of STIT logic—agency logics developed primarily for the formal analysis of multi-agent choice-making—the utilitarian approach has received increased attention (e.g. [7, 15]). Unfortunately, each available utility function comes with its own (dis)advantages, giving rise to several puzzles (some of them addressed in [12, 13]). To avoid such problems, we provide an alternative approach: instead of settling these philosophical issues, we develop a neutral formalism that can be appropriated to different utilitarian value assignments.
The paper’s contributions can be summed up as follows: First, we provide a temporal deontic STIT logic called (Sec. 2). With this logic, we answer a long standing request for temporal embeddings of deontic STIT [3, 12, 15]. Second, although is based upon the atemporal utilitarian STIT logic from , the semantics of will be neutral: instead of committing to utilitarianism, we prove soundness and completeness of with respect to relational frames not employing any utilitarian function (Sec. 3). This approach also extends the results in [2, 11, 14] by showing that can be characterized without using the traditional branching-time (BT+AC) structures (cf. ). Third, we show how neutral frames can be transformed into utilitarian frames, while preserving validity (Sec. 4). Last, we discuss the philosophical ramifications of employing available utility functions in the extended, temporal setting. In particular, we will argue that binary utility assignments can turn out to be problematic.
2 A Neutral Temporal Deontic STIT Logic
In this section, we introduce the language, semantics, and axiomatization of the temporal deontic STIT logic . In particular, we provide neutral relational frames characterizing the logic, which omit mention of specific utility functions. The logic will bring together atemporal deontic STIT logic, presented in , and the temporal STIT logic from .
Definition 1 (The Language )
Let be a finite set of agent labels and let be a countable set of propositional variables. The language is given by the following BNF grammar:
where and .
The logical connectives disjunction , implication , and bi-conditional are defined in the usual way. Let be defined as and define to be . The language consists of single agent STIT operators , which are choice-operators describing that ‘agent sees to it that’, and the grand coalition operator , expressing ‘the grand coalition of agents sees to it that’. Furthermore, it contains a settledness operator
, which holds true of a formula that is settled true at a moment, and thus, holds true regardless of the choices made by any of the agents at that moment. The operatorsand have, respectively, the usual temporal interpretation ‘always going to be’ and ‘always has been’. Last, the operator expresses ‘agent ought to see to it that’. We define and as the duals of and , respectively (i.e. iff , etc.). Furthermore, let iff and iff , expressing ‘ holds somewhere in the future’ and ‘ holds somewhere in the past’, respectively. Finally, deliberative STIT and deliberative ought are obtained accordingly: iff and iff . For a discussion of these operators we refer to [12, 14].
In line with [2, 5, 11, 14], we provide relational frames for instead of introducing the traditionally employed, BT+AC frames (cf. ). Explanations of the individual frame properties of Def. 2 can be found below.
Definition 2 (Relational Frames and Models)
A -frame is defined as a tuple . Let for where . Let be a non-empty set of worlds and:
|For all , are equivalence relations such that:|
|(C2)||For all , if for all , then .|
|(C3)||For all , .|
|is a transitive and serial binary relation and is the converse of , such that:|
|(T4)||For all , if and , then , , or .|
|(T5)||For all , if and , then , , or .|
|(T6)||(relation composition is defined as usual).|
|(T7)||For all , if , then .|
|For all , are binary relations such that:|
|(D9)||For all there exists a such that and for all , if then .|
|(D10)||For all , if and , then .|
|(D11)||For all , if then there exists s.t. , , and for all , if then .|
A -model is a tuple where is a -frame and is a valuation mapping propositional variables to subsets of , that is, .
We label the properties of Def. 2 referring to choice (Ci), those relating to temporal aspects (Ti), and those capturing deontic properties (Di). Observe that, since is an equivalence relation, we obtain equivalence classes . Furthermore, by condition (C1) we know that is an equivalence relation partitioning the equivalence classes of . We call a moment and for each in a moment , we refer to as a choice-cell for agent at moment . In the following, we shall frequently refer to moments and choices in the above sense. Condition (C2) captures the pivotal independence of agents principle for STIT logics, ensuring that at every moment, any combination of different agents’ choices is consistent: i.e., simultaneous choices are independent (see [3, 7C.4]). (C3) ensures that all agents acting together is a necessary condition for the grand coalition of agents acting.111In future work, we aim to study condition (C3) strengthened to equality, as in . In such a setting, completeness is obtained by proving that each -frame can be transformed into a frame (satisfying the same formulae) with strengthened (C3); hence, showing that the logic does not distinguish between the two frame classes.
The conditions on and establish that the frames we consider are irreflexive, temporal orderings of moments. Properties (T4) and (T5) guarantee that histories—i.e., maximally ordered paths of worlds passing through moments—are linear. Condition (T6) ensures the STIT principle of no choice between undivided histories: if two time-lines remain undivided at the next moment, no agent has a choice that realizes one time-line and excludes the other (see [3, 7C.3]). Consequently, this principle also ensures that the ordering of moments is linearly closed with respect to the past and allows for branching with respect to the future: in other words, -frames are treelike.222The main reason why the grand coalition operator is added to our language, is because it will allow us to axiomatize the no choice between undivided histories principle (see A25 of Def. 4). For a discussion of we refer to . Last, (T7) ensures the temporal irreflexivity of moments; i.e., the future excludes the present. For an elaborate discussion of the temporal frame conditions we refer to .
Last, the criteria (D8)-(D11) guarantee an essentially agentive characterization of the obligation operator (cf. the impartial ‘ought to be’ operator in ). Condition (D8) ensures that ideal worlds are confined to moments: i.e., the ideal worlds accessible at a moment neither lie in the future nor in the past. (D9) ensures that, for each agent there is at every moment a choice available that is an ideal choice (cf. the corresponding ‘ought implies can’ axiom ). Furthermore, (D10) expresses that, for each agent, if a world is ideal from the perspective of a particular world at a moment, that world is ideal from the perspective of any world at that moment: i.e., ideal worlds are settled upon moments. Condition (D11) captures the idea that every ideal world extends to a complete ideal choice: i.e., no choice contains both ideal and non-ideal worlds. Last, note that conditions (C2) and (D9) together ensure that every combination of distinct agents’ ideal choices is consistent, i.e., non-empty.
Definition 3 (Semantics for )
Let be a -model and let of . The satisfaction of a formula in at is defined accordingly:
Global truth, validity, and semantic entailment are defined as usual (see ).
Definition 4 (Axiomatization of )
For each we have,
All propositional tautologies.
implies , given
A derivation of in from a set , written , is defined in the usual way (See [6, Def. 4.4]). When , we say is a theorem, and write .
The axioms, , and express the S5 behavior of , (for each ) and , respectively. is the independence of agents axiom. captures that ‘all agents acting together implies the grand coalition of agents acting’. is a bridge axiom linking to and to (cf. (C1) and (D8) of Def. 2). corresponds to the ‘ought implies can’ principle (cf. (D9) of Def. 2). ensures that, when possible, obligatory choices are settled upon moments (cf. (D10) of Def. 2). can be understood as a conditional monotonicity principle for ideal choices (cf. (D11) of Def. 2). Axioms and , together with the necessitation rule , ensure that is a normal modal operator.
With respect to the temporal axioms, capture the KD4 behavior of , whereas, axioms and ensure that is the converse of . and capture connectedness of histories through moments and characterizes no choice between undivided histories. Last, is a variation of Gabbay’s irreflexivity rule (the proofs of Thm. 3.1 and 3.2 give an indication of the rule’s functions).
3 Soundness and Completeness of
In this section, we prove that is sound and complete relative to the class of -frames. In the next section, we show how such frames are transformable into frames employing utility assignments. This allows one to model and reason about utilitarian scenarios in a more fine-grained manner, while obtaining completeness of the logic without commitment to particular utility functions.
Unless stated otherwise, all proofs in this section can be found in App. 0.A.
(soundness of ) , implies .
A set of formulae is a maximally consistent set (MCS) iff (i) , and (ii) for any set , if , then .
(canonical model for ) Let and let be the operator dual to . We define the canonical model to be the tuple such that:
for all , iff for all , if , then (for each );
is a valuation function s.t. , .
(diamond saturated set ) Let be a set of MCSs and let be dual to . We say that is a diamond saturated set iff for all , for each there exists a such that and .
In order to ensure that our canonical model will be irreflexive, we introduce a mechanism that allows us to encode MCSs with information that impedes reflexive points in the model. We call these encoded sets IRR-theories and restrict our canonical model to consist of these sets only. Last, we use the notation to indicate a model whose domain is restricted to the set (see [8, Ch.6]).
Let be a diamond saturated set with , , and let be the canonical model restricted to . Then, iff .
Proven in the usual manner by induction on (see [6, Lem. 4.70]).
Following , we let IRR-theories be those sets of formulae that (i) are maximally consistent, (ii) contain a label , uniquely labeling a moment and (iii) for any world that is reachable through any ‘zig-zagging’ sequence of diamond operators, that is, every zig-zagging formula of the form,
where is dual to with , there exists a corresponding zig-zagging formula (where is a propositional variable) of the form,
labeling reachable worlds. Let us make the above formally precise:
(irr-theory)  Let be the set of all zig-zagging formulae in and let name(p) where is a propositional variable. A set of formulae is called an IRR-theory iff the following hold:
is a MCS and , for some propositional variable ;
if , then , for some propositional variable .
Henceforth, we refer to as the set of all IRR-theories in .
We now present lemmata relevant to the use of IRR-theories in canonical models.
Let be a consistent formula. Then, there exists an IRR-theory such that .
(existence lemma) Let be an IRR-theory and let be dual to . For each there exists an IRR-theory such that .
Subsequently, it must be shown that the canonical model restricted to the set of IRR-theories (i.e., ) is in fact a model (henceforth, we use and interchangeably). First, we provide lemmata ensuring that the model satisfies the desired temporal and deontic properties of Def. 2. The first two follow from  and the latter four results are proven in App. 0.A.
Lemma 4 ()
(property (C2)) Let such that for all . Then, there exists a such that .
Lemma 5 ()
(property (T6)) Let such that and . Then, there exists a such that and .
(property (D9)) Let . Then, there exists a such that and for every , if , then .
(property (D11)) Let such that . Then, there exists a such that , , and for all , if , then .
The canonical model belongs to the class of models.
(completeness) If is a consistent formula, then is satisfiable on a -model.
4 Transformations to Utilitarian Models
In this section, we investigate a truth preserving transformation from models to utilitarian STIT models, embedded in a temporal language. In particular, we are concerned with the semantic characterization of the dominant ought [12, Ch.4]. We start with defining the semantic machinery needed to treat these oughts. In particular, we will introduce a utility function that maps natural numbers (i.e. utilities) to worlds in our domain. In contrast to [12, 15], we do not restrict the assignment of utilities to complete histories where all worlds on a maximal linear path have identical utility. The reason will be addressed at the end of the section, where we discuss a problem related to utility assignments over histories, arising in temporal extensions of STIT.
The pivotal notion involved in the dominant ought is that of a state: Agent cannot influence the choices of all other agents and, for this reason, one can regard the joint interaction of all agents excluding , as a state (of nature) for . To be more precise, we define a state for i at accordingly,
Consequently, all possible combinations of choices available to the agents , are the different states available at that moment to agent .
Subsequently, we define a preference order over choices (and subsets thereof). Let , then weak preference is defined accordingly,
That is, for an agent a choice is weakly preferred over another, when all values of the possible outcomes of the former are at least as high as those of the latter (where is the number assigned to , etc). Strict preference is defined as,
Next, a dominance order over choices is defined as,
We say an agent’s choice weakly dominates another, if the values of the outcomes of the former are weakly preferred to those of the latter choice, given any possible state available to that agent. For a discussion of dominance orderings see [12, Ch. 4]. Again, in the usual way we obtain strict dominance,
On the basis of the above, we now formally introduce temporal utilitarian STIT frames and models, defined over relational Kripke frames.
Definition 9 (Relational Frames and Models)
Let for . A relational Temporal Utilitarian STIT frame (-frame) is defined as a tuple where is a non-empty set of worlds and:
For all , , , are equivalence relations for which conditions (C1)-(C3) of Def. 2 hold.
is a transitive and serial binary relation, whereas is the converse of , and the conditions (T4)-(T7) of Def. 2 hold.
is a utility function assigning each world in to a natural.
A -model is a tuple where is a -frame and is a valuation function assigning propositional variables to subsets of : i.e., .
Notice that the above frames only differ from frames through replacing the relations and corresponding conditions (D8)-(D11) (for each ) with the utility function . We observe that the assignment of utilities to worlds is agent-independent. Nevertheless, since the choices of an agent depend on which worlds are inside the choice-cells available to the agent, the resulting obligations are in fact agent-dependent. Let us define the new semantics:
Definition 10 (Semantics of models)
Let be a -model, of and . We define satisfaction of a formula as follows:
Clause (1)-(10) are the same as those from Def. 3, with the exception of clause (7), which we replace by the following clause ():
Clause is interpreted accordingly: Agent ought to see to it that iff for every choice available to that does not guarantee there (i) exists a strictly dominating choice that (ii) does guarantee and (iii) every weakly dominating choice over also guarantees . In other words, all choices not guaranteeing are strictly dominated only by choices guaranteeing . (We note that clause () is obtained through an adaption of the definition provided in  to relational frames.) We show that the logic is also sound and complete with respect to the class of -frames.
(soundness) , if , then .
We now prove that the class of -frames characterizes the same set of formulae as the class of frames. We prove both directions separately:
we have implies .
We prove by contraposition assuming . Hence, there is a -model, such that for some . We use to construct a model in , such that:
We show that for some . To define let , , , , , , and let be a function assigning each to a natural number, satisfying the following criteria:
, if , , and , then ;
, if and , then ;
, if , then ;
Let , we call an optimal choice for agent . (It can be easily checked that the function can be constructed.)
We state the following useful lemma (the proof of which is found in App. 0.A).
The following holds for any frame:
(1) ; (2) ;
(3) ;(4) we get ;
(5) , either or .
We observe that conditions (C1)-(C3) and (T4)-(T7) will be satisfied in since all of the relations of , with the exception of , are identical to those in . Moreover, complies with Def. 9 and so is in fact a model. The desired claim will follow if we additionally show that and :
We prove the claim by induction on the complexity of .
Base Case. Let be a propositional variable . By the definition of in it follows directly that iff iff iff .
Inductive Step. The cases for the propositional connectives and the modalities are straightforward. We consider the non-trivial case when is of the form . Let us first prove the left to right direction.
() Assume . We show that . By the semantics for (Def. 9) it suffices to prove that: if , then such that the following three clauses hold: (i) ; (ii) ; and (iii) , implies .
Let be arbitrary and assume that . We prove that there is a for which conditions (i)-(iii) hold. First, we prove the existence of such a : By (C1) and (D9) of Def. 2, we know,
We also know by (D9) that . By we know that , i.e., there exists a . Consequently, we obtain the following statement,
Last, by construction of we know . We show that (i)-(iii) hold:
(i) We show , that is, (a) and (b) :
(a) Recall, , we know s.t. . By definition of , and by (IH) we get . Consequently, by the assumption that , and the fact that , it follows that . Hence, we know that , which implies by Lem.9. Therefore, by this fact along with statement (1) above, we know that,
For all , if and , then and .
Let be arbitrary and assume that and . By the statement above, it follows that and , which in conjunction with criterion 1 on the function implies that . Therefore, the following holds,
For all , if and