1 Introduction
On October 13th, 2011, a twoyearold girl was run over by a van in the city of Foshan in Guangdong Province, China. 18 people passed by and ignored her before a stranger moved the girl to the side of the road and notified her mom [周松 et al.2011, Bristow2011]. The girl, nicknamed “Little Yueyue” by Chinese media, later died in hospital [BBC2017]. This tragic incident is not unique. On April 21, 2017, a woman was crossing a street in the city of Zhumadian in the Chinese Province of Henan. She was struck by a taxi and laid on the road until an SUV run over her, neither drivers nor pedestrian stopped to help her. She later died from injuries [熊浩然2017, Johnson2017]. Although many Chinese were outraged by these events and blamed the passersby for not helping, others did not. Trying to understand the people who does not blame passersby, the New York Times quotes a post on Weibo, a Chinese social network: “If I helped her to get up and sent her to the hospital, doctors would ask you to pay the medical bill. Her relatives would come and beat you up indiscriminately.” [Johnson2017]. This explanation refers to an even earlier event. On November 20, 2006, in the city of Nanjing, Jiangsu Province, an old woman fell down on a bus stop. A young man helped her to get up, escorted her to a hospital, and stayed there until she was examined by a doctor. He was later sued by the woman and her relatives. The court eventually ordered him to pay 40 percent of the medical costs. The verdict said that “according to common sense” it is highly likely that the man is responsible for the woman’s fall because otherwise he “would have left soon after sending the woman to the hospital instead of staying there for the surgical check” [Shinan2011]. New York Times cites Dali L. Yang, a political scientist at the University of Chicago, saying “In the aftermath of the Nanjing case, many Chinese worry about the victims turning around to blame the helpers, and thus feel unable to offer direct help.” [Johnson2017].
The cited above post on Weibo essentially says that the people should not be blamed for not doing something that would require them to sacrifice a lot. A similar position is argued by Shelly Kagan in his book The Limits of Morality, where he suggests that sacrifice could be one of possible ways to measure the degree of moral blameworthiness: the higher sacrifice is required to prevent something the less a person (or a group of people) should be blamed for not doing it [Kagan1991]. Nelkin argues that the sacrifice can be used as a degree of blameworthiness and praiseworthiness [nelkin16nous]. [Halpern and KleimanWeiner2018] suggested to measure sacrifice by its cost
. They also noted that there are other ways to define degree of blameworthiness; for example, through probability with which the harm could have been prevented.
The dilemma of balancing costs and moral responsibilities is being faced not only by passersby in China. Insurance companies need to choose between refusing to pay for very expensive drug and potentially saving life of a patient. Car engineers choose between better safety equipment and its higher cost. Governments have to balance public support of lifesaving medical research with its cost.
In this paper we propose a sound and complete logical system that describes modality meaning “coalition is blameable for with degree ”, where the degree is defined as the cost of the sacrifice the coalition would have made to prevent . Our work heavily builds on [Alechina et al.2011] logic for resource bounded coalitions and [Naumov and Tao2019] logic of blameworthiness in strategic games. In turn, both of these papers rooted in widely studied [van der Hoek and Wooldridge2005, Borgo2007, Sauro et al.2006, Ågotnes et al.2010, Ågotnes et al.2009, Belardinelli2014, Goranko et al.2013, Goranko and Enqvist2018] Marc Pauly’s logic of coalition power [Pauly2002]. Pauly gave a complete axiomatization of modality that stands for “coalition has a strategy to achieve ”. [Alechina et al.2011] did the same for modality meaning “coalition has a strategy to achieve using resources ”. [Naumov and Tao2019] proposed a complete axiomatization of modality , “coalition is blamable for outcome ”. They used a common approach of defining blamable as “statement is true but coalition had a strategy to prevent it”. This approach is known as the principle of alternative possibilities [Frankfurt1969, Widerker2017] or counterfactual [Cushman2015] definition of blameworthiness. It is also a part of HalpernPearl formal definition of causality as a relation between sets of variables [Halpern2016]. Counterfactuals could also be used to define regret and several other emotions [Lorini and Schwarzentruber2011]. Although the principle of alternative possibilities is the most common way to define blameworthiness, other approaches has been explored too. [Xu1998] introduced a complete axiomatization of a modal logical system for reasoning about responsibility defined as taking actions that guarantee a certain outcome. [Broersen et al.2009] extended Xu’s work from individual responsibility to group responsibility.
In this paper we propose a logic of blameworthiness with sacrifice that describes universal property of modality , meaning “statement is true but coalition had a strategy to prevent it at cost no more than ”. Our main technical results are the soundness and the completeness theorems for this logical system.
2 Syntax and Semantics
In this paper we assume a fixed set of agents and a fixed set of propositional variables . By a coalition we mean an arbitrary subset of set .
Definition 1
is the minimal set of formulae such that

for each variable ,

for all formulae ,

, for each coalition , each real number , and each formula .
In other words, language is defined by grammar:
Informally, modality means “coalition is blameable for statement with degree ”. The other modality in our system is , which stands for “statement is universally true in the given game”. By we mean formula . We assume that conjunction and disjunction are defined in the standard way.
By we mean the set of all functions from set to .
Definition 2
A game is a tuple , where

is a nonempty set of “actions”,

is a set of “outcomes”,

is a nonnegative real number for each , called the cost of action ,

is a zerocost action: ,

a set of “plays” is an arbitrary set of pairs such that is an outcome and is a function from set to set , called “complete action profile”,

is a function that maps into subsets of .
For example, in the case of Little Yueyue from the introduction, each of the passersby had two available actions: to help or to ignore. Thus, . For the purpose of this example, we assume that any passerby volunteering to help, would have to pay ¥1000 towards Little Yueyue’s medical bill: . We also assume that ignoring is a zerocost option: . There are two possible outcomes: either Yueyue stays alive or dies. Thus, . The set of plays describes all possible combinations of actions and outcomes in the game. In our case, these combinations are listed as separate lines of the table in Figure 1. Although there have been 19 passersby mentioned in Little Yueyue tragic story (18 of who decided to ignore and one who decided to help), to keep our example simple, we assume that there were only three agents: , , and . If any of the first two of them decided to help, Little Yueyue would be alive. We assume that the third agent arrived too late to save her life, see Figure 1.
ignore  ignore  ignore  dead 
ignore  ignore  help  dead 
ignore  help  ignore  alive 
ignore  help  help  alive 
help  ignore  ignore  alive 
help  ignore  help  alive 
help  help  ignore  alive 
help  help  help  alive 
Note that maps propositional variables not into set of outcomes, but into sets of plays. This is because we allow statements represented by atomic propositions to refer not only to outcome, but to actions as well. An example of such statement in our case is “if agent helps, then Little Yueyue stays alive”.
One can imagine a fixed “tax” added to costs of all actions in the game. Such uniform tax constitutes fixed overhead costs and should not be used to measure the sacrifice. To avoid this situation in Definition 4 we assume existence of zerocost action . This assumption is significant because without it the Monotonicity axiom of our system would not be valid. A similar assumption is made in the logic for resource bounded coalitions [Alechina et al.2011].
Definition 3
For any action profile of a coalition by we mean the total cost of the action profile to the coalition: .
For any functions and , we write , if for each .
Definition 4
For any formula and any play of a game , the satisfiability relation is defined recursively as follows:

if , where ,

if ,

if or ,

if for each play ,

if and there is such that and for each play , if , then .
Note that item 5 of Definition 4 takes into account potential costs under action profile to coalition and ignores the actual costs under the action profile . We refer to this way of defining the sacrifice as the absolute sacrifice. Alternatively, by the relative sacrifice we mean the difference between the costs under these two profiles or how much more it would cost the coalition to prevent undesired outcome comparing to the current costs.
The use of the absolute sacrifice as a degree of blameworthiness makes sense in many situations. For example, let us assume that in the Little Yueyue example agent was heading to a medical supply store to buy a new ¥1000 wheelchair for his ill child. The agent is now facing a moral choice between (a) zerocost option of doing nothing, (b) spending ¥1000 on a new wheelchair, and (c) spending ¥1000 on a medical bill. If the agent were to choose to help instead of buying the wheelchair, his relative sacrifice would be zero. In this case, the absolute sacrifice of ¥1000 is probably a better measure of the degree of blameworthiness. At the same time, relative sacrifice makes sense in situation like blame for the result of cornercutting in safety, when a small additional expense could prevent a tragic incident. [Halpern and KleimanWeiner2018] use relative sacrifice as their measure of degree of blameworthiness.
3 Axioms
In addition to the propositional tautologies in language , our logical system contains the following axioms:

Truth: and ,

Distributivity: ,

Negative Introspection: ,

None to Blame: ,

Monotonicity: , where and ,

Joint Responsibility: if , then
, 
Blame for Cause:
, 
Fairness: .
These axioms are the same as the axioms of the logic of blameworthiness [Naumov and Tao2019] except for the sacrifice superscript being added. The Truth, the Distributivity, and the Negative Introspection axioms for modality capture the fact that this is an S5modality. The Truth axiom for modality states that a coalition can only be blamed for something which is true. The None to Blame axiom says that the empty coalition can not be blamed for anything.
The Monotonicity axiom states that if a smaller coalition can be blamed for not preventing an outcome at cost at most , then any larger coalition can also be blamed for not preventing the outcome at cost at most , where . This axiom is valid because each agent in set could use the zerocost action. One may question our underlying assumption that a larger coalition should be blamed for wrongdoings of its part. This assumption is consistent, for example, with how the entire millennial generation is blamed in the media for decline in sales of beer, paper napkins, and motorcycles [Scipioni2018].
The Joint Responsibility axiom shows how blames of two disjoint coalitions can be combined into a blame of their union. It resembles the Cooperation axiom for resourcebounded coalitions [Alechina et al.2011]: if , then .
To understand the Blame for Cause axiom note that formula means that implies for each play of the game. In this case we say that is a cause of . The axiom says that if a coalition is responsible for a statement, then it is also responsible for its cause as long as the cause is true.
The Fairness axiom states that if a coalition is blamed for , then it should be blamed for each time when is true.
We write if formula is provable from the axioms of our system using the Modus Ponens and the Necessitation inference rules:
The next lemma generalizes the Joint Responsibility axiom from two to multiple coalitions. Its proof is identical to the proof of the corresponding result in [Naumov and Tao2019, Lemma 5] with the superscript added.
Lemma 1
For any integer ,
where sets are pairwise disjoint.
The following two lemmas capture wellknow properties of S5 modality. Their proofs, for example, could be found in [Naumov and Tao2018].
Lemma 2
If , then .
Lemma 3 (Positive Introspection)
.
We conclude this section with an example of a formal proof in our logical system. This example will be used later in the proof of the completeness.
Lemma 4
For any integer and any disjoint sets if , then
Proof. By Lemma 1,
Hence, by the Monotonicity axiom,
Then, by the Modus Ponens inference rule,
Hence, by the Truth axiom and the Modus Ponens rule,
At the same time,
is an instance of the Blame for Cause axiom. Thus, by the Modus Ponens inference rule applied twice,
Then, by the Modus Ponens inference rule,
Hence, by the deduction lemma,
Thus, by Lemma 2,
Then, by the definition of modality , the Negative Introspection axiom, and the Modus Ponens inference rule,
Therefore, by Lemma 3 and the Modus Ponens inference rule, the statement of the lemma is true.
4 Soundness
In this section we prove soundness of our logical system. The soundness of S5 axioms (the Truth, the Distributivity, and the Negative Introspection) for modality is straightforward. Below we prove the soundness of the remaining axioms for any arbitrary play of an arbitrary game .
Lemma 5
, for all .
Proof. Suppose that . Thus, by Definition 4, and there is such that and for each play , if , then . Consider play and note that statement is vacuously true. Thus, , which is a contradiction.
Lemma 6
For all sets and all , if , , and , then .
Proof. By Definition 4, assumption implies that (i) and (ii) there is such that and for each play , if , then . Define action profile as follows:
Because is a zerocost action, by Definition 3 and the assumption . Consider any play such that . By Definition 4 and because , it suffices to show that . Indeed, . Therefore, by the choice of action profile .
Lemma 7
For all and all , if , , , and , then .
Proof. By Definition 4 and the definition of modality , assumption implies that there is a play such that . Thus, by Definition 4, there is an action profile such that and
(1) 
Similarly, assumption implies that there is an action profile such that and
(2) 
Consider action profile such that
(3) 
Action profile is welldefined because by the assumption of the lemma. Note that by Definition 3 and inequalities and .
Then, by Definition 4 and the assumption , it suffices to show that for each play , if , then . Indeed, equality implies . Thus, by equation (3). Hence, by statement (1). Similarly, by statement (2). Then, , by Definition 4.
Lemma 8
If , , and , then .
Proof. By Definition 4, assumption implies that there is an action profile such that and for each play , if , then .
Thus, by Definition 4 and assumption , for each play , if , then . Therefore, assumption implies that again by Definition 4.
Lemma 9
If , then .
5 Completeness
We start the proof of the completeness by defining the canonical game for each maximal consistent set of formulae .
Definition 5
Set consists of a zerocost action and all triples such that is a formula, is a nonempty coalition, and is a nonnegative real number.
Informally, we consider actions as “votes” of agents. Zerocost action could be interpreted as abstaining from voting. Action by an agent means that agent is voting as a part of coalition to prevent at the total cost to the whole coalition. If agent votes , then statement is not necessarily false in the outcome. The vote aggregation mechanism is given in Definition 8. Definition 5 is substantially different from a similar definition in [Naumov and Tao2019], where each action consists of just a single formula .
Definition 6
For each action , let if and if .
Informally, means that the cost of each joint action is divided evenly between all members of the coalition. Note that size of coalition is nonzero by Definition 5.
Definition 7
The set of outcomes is the set of all maximal consistent sets of formulae such that for each formula if , then .
Definition 8
The set consists of all pairs such that for any formula , if for each agent , then .
In other words, for each formula , if each member of coalition votes as a part of to prevent at cost , then is guaranteed to be false in the outcome.
Definition 9
.
As usual, the key part of the proof of the completeness is the induction, or “truth”, lemma. In our case this is Lemma 13. The next three lemmas are auxiliary lemmas used in the proof of Lemma 13.
Lemma 10
For any play , any profile , and any formula , if , then there is a play such that and .
Proof. Consider the following set of formulae:
Claim 1
Set is consistent.
Proof of Claim. Suppose the opposite. Thus, there are formulae
(4)  
(5) 
such that
(6)  
(7)  
(8) 
Without loss of generality, we can assume that formulae are distinct. Thus, assumption (7) implies that sets are pairwise disjoint. Hence, by Definition 6 and formula (7),
Thus, by to the assumption of the lemma,
(9) 
At the same tine, assumption (8) by the laws of propositional reasoning implies that
Thus, by Lemma 2, Hence, by assumption (4). Thus, by Lemma 4, using assumption (5), statement (9), and the fact that sets are pairwise disjoint,
Hence, because set is maximal. Then, by Definition 7. Thus, because set is consistent, which contradicts assumption of the lemma. Therefore, set is consistent.
Let be any maximal consistent extension of set . Thus, by the choice of sets and . Also, by Definition 7 and the choice of sets and .
Let the complete action profile be defined as follows:
(10) 
Then, .
Claim 2
.
Proof of Claim. Consider any formula such that for each . By Definition 8, it suffices to show that .
Case I: . Thus, by the definition of set . Therefore, by the choice of set .
Case II: . Consider any . Thus, by equation (10). At the same time, because . Therefore, , which is a contradiction. This concludes the proof of the lemma.
Lemma 11
For each outcome , there is a complete action profile such that .
Proof. Consider a complete action profile where for all . To show , consider any such formula that for all . Due to Definition 8, it enough to prove that .
Case I: . Hence, by the None to Blame axiom. Then, by the Necessitation inference rule. Thus, by the consistency of the set . Therefore, due to the definition of the modality , which contradicts to the assumption .
Case II: . Hence, set contains at least one agent . Then, by the definition of profile . Thus, , which is a contradiction.
Lemma 12
For each play and each formula , there is a play such that .
Proof. Let be the set . Next, we prove the consistency of set . Assume the opposite. Hence, there are formulae where Thus, due to Lemma 2. Then, because . Thus, by Lemma 3. Hence, it follows from assumption and Definition 7 that . Thus, by the consistency of set , which contradicts the assumption of the lemma. Therefore, set is consistent.
Consider any maximal consistent extension of set . Observe that due to the definition of set . Finally, by Lemma 11, there is a profile where .
Lemma 13
iff for any play and any formula .
Proof. The lemma will be shown by induction on structural complexity of formula . If is a propositional variable, then the statement of the lemma follows from Definition 4 and Definition 9. The cases when is a negation or an implication, as usual, can be proven from the maximality and the consistency of set .
Let formula have the form .
Suppose . Then, by the maximality of set . Thus, there is a play such that , by Lemma 12. Hence, because set is consistent. Then, by the induction hypothesis, . Therefore, by Definition 4.
Suppose . Then, because set is consistent. Thus, by Definition 7. Hence, due to the maximality of set . Then, by the Negative Introspection axiom, . Thus, by the maximality of set . Hence, for each outcome by Definition 7. Hence, by the induction hypothesis, for all plays . Then, by Definition 4.
Let formula have the form .
Suppose that .
Case I: . Thus, by the induction hypothesis. Therefore, by Definition 4.
Case II: . First we prove that . Suppose . Thus, by the Modus Ponens inference rule. Hence, by the maximality of set , we have , which contradicts the assumption .
Since is a maximal set, statement implies that . Hence, by Lemma 10, for any action profile , if , then there is a play where and . Thus, by the induction hypothesis, for each profile , if , then there is a play such that and . Therefore, by Definition 4.
Suppose that . Hence, by the Truth axiom. Then, by the maximality of the set . Thus, by the induction hypothesis.
Define to be an action profile such that for each agent .
Claim 3
.
Statement implies that because set is consistent. Thus, by Definition 7 and because . Hence, due to the maximality of the set . Thus, by the definition of modality . Also, for each . Hence, by Definition 8 and the assumption . Then, by the consistency of set . Therefore, by the induction hypothesis.
Finally, we are prepared to state and prove the strong completeness of our logical system.
Theorem 1
If , then there is a game and a play of the game where for all and .
Proof. Assume that . Hence, set is consistent. Choose to be any maximal consistent extension of set and to be the canonical game defined above. Then, by Definition 7 and the Truth axiom.
6 Conclusion
In this paper we combine the ideas from the logics of resource bounded coalitions [Alechina et al.2011] and blameworthiness [Naumov and Tao2019] into a logical system that captures the properties of a degree of blameworthiness. Following [Halpern and KleimanWeiner2018], the degree of blameworthiness is defined as the cost of sacrifice. The main technical result is the completeness theorem for our system.
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