1 The Puzzle of Sea Battle
The puzzle of sea battle is from Aristotle. Either there will be a sea battle tomorrow or not. If there is a sea battle tomorrow, it is necessarily so. If there is no sea battle tomorrow, it is necessarily so. So either necessarily there will be a sea battle tomorrow or necessarily there will be no sea battle tomorrow. Here “necessary” is understood as “inevitable”: something is necessary if it is the case no matter what we will do. The conclusion seems fatalistic and unacceptable.
There are two ways out: either arguing that the argument is not sound or arguing that its premises are not all true. The argument is a special case of reasoning by cases: from “ or ”, “if then ” and “if then ”, we get “ or ”. The argument has three premises. The first one may be called the principle of excluded future middle. The second and third may be called the principle of necessity of truth: true propositions are necessary.
The previous solutions to this puzzle presuppose the validity of the argument and adopt the latter strategy. They focus on the following issue: how do we ascribe truth values to the statements such as “there will be a sea battle tomorrow”? These statements are called future contingents in the literature: they are about the future but do not have an absolute sense.
These solutions include Lukasiewicz’s threevalued logic [5], Prior’s Peircean temporal logic [8], Prior’s Ockhamist temporal logic [8], the true futurist theory [6], the supervaluationist theory [11] and the relativist theory [4]. In the first two, the principle of excluded future middle fails. In others, the principle of necessity of truth does not hold. Except Lukasiewicz’s threevalued logic, other solutions use branching time models. In a branching time model, time is represented as a tree. At any state, there is only one history but might be many possible futures. We refer to [7] for detailed comparison between these solutions.
There are two senses of necessity, absolute and relative necessity, determined by how we understand whatever we will do. Decision makes the difference. In reality we make decisions to do something or not to do something. So we will not do whatever we are able to do. We may read whatever we will do relative to the domain consisting of the things that we are able to do. We may also read it relative to the domain consisting of the things that we are able to do but have not decided to avoid. Note that doing a thing in the first domain but not in the second involves changing mind. A proposition is absolutely necessary if it is the case no matter which we choose to do from the big domain. A proposition is relatively necessary if it is the case no matter which we choose to do from the small domain. Accordingly, we have two principles of necessity of truth.
The principle of absolute necessity of truth does not hold for future contingents. Assume that the admiral is able to do two things: and . Doing will cause a sea battle tomorrow but doing will not. He decides to do . In this case, it sounds plausible to say that there will be a sea battle tomorrow. However, it is strange of saying that there will be a sea battle tomorrow no matter what the admiral will do in the absolute sense.
We think that the principle of relative necessity of truth holds for future contingents. Assume that it is not necessary that will be the case. Then someone has a way to act to make not the case in the future without changing his decision. In this situation, it seems hard to say that will be the case. Therefore, if will be the case, it is necessary that will be the case.
The puzzle of sea battle disappears under the absolute sense of necessity because the principle of necessity of truth fails for future contingents. However, it is still puzzling under the relative sense of necessity. The principle of excluded future middle is quite intuitive. So all the three premises of the puzzle seem valid. But the conclusion is still problematic. Suppose that doing will cause a sea battle tomorrow but doing will not. Assume that the admiral has not made the decision to do or . Then it is wrong to say that there will necessarily be a sea battle tomorrow. It is also wrong to say that there will necessarily not be a sea battle tomorrow. So the conclusion of the puzzle is false. How come we get a false conclusion from three valid premises by a sound inference?
The soundness of reasoning by cases presupposes that the two conditionals in it are material implications. However, if the two conditionals are something else, then reasoning by cases might not be a sound argument.
In linguistics, a different view on conditionals is common, that is, the restrictor view. The conditional if then is not a connective connecting two sentences. There is no such a connective in natural languages. Utterance of a sentence is always w.r.t. a specific discourse domain. The ifclause if is a device for restricting the domain. The conditional if then is true w.r.t. a domain if is true w.r.t. the resulted domain. Conditionals collapse to material implications when discourse domains are a singletons. This view can be found in various works including [3], [2] and [13]. It can be tracked back to Ramsey Test in [9].
Reasoning by cases is not generally valid under the restrictor view on conditionals. Let be a discourse domain. Assume that if then and if then are true w.r.t. . Let and be the respect results of restricting with if and if . Then what the two conditionals say is just that is true w.r.t. and true w.r.t. . If neither the truth of nor the truth of is upward monotonic relative to discourse domains, then it is possible that neither nor is true w.r.t. .
The discourse domain of the puzzle of sea battle consists of possible futures. The ifclauses “if there is a/no sea battle tomorrow” restrict the domain. However, relative necessity is not a upward monotonic notion w.r.t. the class of possible futures. Therefore, the argument concerning sea battle is not sound. In what follows we present a logic to formalize this idea.
2 Formal Settings
Let be a countable set of atomic propositions and range over it. Define a language as follows:
where is from , the sublanguage of generated from under and .
The featured formulas of are read as follows:

: will be the case in the nextmoment.

: no matter how the agent will act in the future, is the case now, that is, is necessary.

: given , is the case.
The principle of excluded future middle is expressed as and the principle of necessity of truth as .
is a temporal formula. , and are state formulas. is a temporal formula if is, or else a state formula. Later we will see that temporal formulas are evaluated at states into paths and state formulas evaluated just at states.
The other usual propositional connectives and the falsum are defined in the usual way. , defined as , indicates that the agent has a way to act in the future s.t. is the case now, that is, is possible.
It seems strange to say that the agent has a way to act in the future s.t. is the case now. Actually this is fine, as whether a sentence involving future is true or not now might be dependent on how the agent will act in the future. For example, whether a student will pass an exam is dependent on how he will study.
Let be a nonempty set of states and a binary relation on it. A sequence of states is called a sequence if . As a limit case, is a sequence for any . is a tree if there is a s.t. for any , there is a unique sequence from to . is called the root. It can be seen that the root is unique and is irreflexive. is serial if for any , there is a s.t. .
We say that a tree is serial if is. A serial tree is understood as a time structure encoding an agent’s actions (the transitions) and states in time (the nodes). A branching in the tree is interpreted as a situation in which the agent can choose between different possible actions. The seriality corresponds to the fact that the agent can always perform an action at any given time.
Fix a serial tree . Here are some auxiliary notations. A sequence starting at the root is a history of . For any states and , is a historical state of if there is a sequence s.t. , and . is a future state of if is a historical state of . Note that a state can not be a historical or future state of itself.
An infinite sequence is a path. A path starting at the root is a timeline. A path passes through a state if for some . Let be a path. We use to denote the th element of , the prefix of to the th element, and the suffix of from the th element. For example, if , then , and . For any history and path , if , let denote the timeline .
is a model if is a serial tree with as the root and is a function from to . Figure 1 illustrates a model.
Definition 1 (Semantics).
, the formula being true at the state relative to the timeline in the model , meets the following conditions:
By is well given, we mean that is defined and is a path of it. Note that holds trivially if is not well given. is defined in parallel as follows. Fix a model . We say that is achievable at in if is true at relative to a path from . Let . A structure is called the restriction of to if and .
Definition 2 (Update of models).
Let be a model, a formula and a state. Assume that is achievable at . Let be the history of . Define a set of states as follows: for any , (i) is a future state of and (ii) there is no timeline passing through and s.t. . Define as , the restriction of to . is called the result of updating at with .
is undefined if is not achievable at .
The operator can be viewed as a universal quantifier over possible futures. The truth of a state formula at a state relative to a path is not dependent on the path, but this is not the case for temporal formulas. In some cases, if is a state formula, we write without specifying a path. Sometimes, if is a timeline of containing , we use instead of where .
The meaning of lies in how updates models. Updating with at is to shrink by removing the states in . can be understood as follows. Assume that the agent is at and decides to make true. After the decision is made, some future states are not possible anymore. A state becomes impossible if the agent travels to it, there would be no way to make true at , no matter where he goes afterwards. is the collection of these states. Figure 2 illustrates how a formula updates a model.
Suppose that is achievable at . It can be verified that and is closed under . So is a tree with as the root. It can also be verified that is serial. It follows that is a model. Note that it is possible that is not a timeline of . So might not be well given. Later we will see that is a timeline of iff .
A formula is valid if for any , and , . Let be a set of formulas and a formula. , entails , if for any , and , if , then . We in the sequel use to denote the set of valid formulas.
3 The Puzzle Is Solved in a Way
The update with shrinks models. As a consequence, it restricts possible futures. The following theorem indicates that it restricts possible futures as we wish: it exactly excludes the possible futures which does not satisfy .
Let be the set of natural numbers. Define a function as follows:
intuitively indicates how far can see forward.
Lemma 1.
Let be a natural number and a formula in s.t. . Let be a model and a state.

For any timelines and of passing through , if they share the same elements after , then iff .

For any timeline of passing through , is wellgiven iff .
Proof.
We show the results (a) and (b) in parallel by inductions on and . Note that if , then is defined and a path of by Definition 2. So the rightleft direction of (b) holds. This direction will not be considered in the sequel.
Case . The subcase is trivial and the subcase impossible.
Subcase .
(a) Clearly iff .
(b) Assume . Then is undefined and not well given.
Subcase .
(a) By the inductive hypothesis, iff . Then iff .
(b) Assume . Then . Let be an arbitrary timeline of through . By the inductive hypothesis, . Then . Then is undefined and not well given.
Subcase .
(a) By the inductive hypothesis, iff , and iff . Then iff .
(b) Assume . Then or . Assume the former. Let be an arbitrary timeline of through . By the inductive hypothesis, . Then . Then is undefined and not well given. In a similar way, we can get that if , then is not well given either.
Subcase .
(a) Assume . Then if is well given, . By the inductive hypothesis, is well given iff , iff , iff is well given, and iff . Then if is well given, . Then . Similarly, we can show that if , then .
(b) Assume . Then is well given but . Let be a timeline of through . In a similar way as (a), we know . Then is undefined and not well given.
Case . The subcases and are impossible.
Subcase .
(a) Similar arguments with the case .
(b) Assume . Then . Let be an arbitrary timeline of passing through and sharing the same elements after with . By the inductive hypothesis, . Then . Let be the th element of after . By Definition 2, is not in if is defined. Then is not well given.
Subcase .
(a) Similar arguments with the case .
(b) Assume . Then or . Assume the former. Let be an arbitrary timeline of passing through and sharing the same elements after with . By the inductive hypothesis, . Then . Let be the th element of after . Then is not in if is defined. Then is not well given. In a similar way, we can get that if , then is not well given either.
Subcase .
(a) Let . By the inductive hypothesis, iff iff iff .
(b) Let . Assume . Then . Let be an arbitrary timeline of passing through and sharing the same elements after with . Then shares the same elements after with . By inductive hypothesis, . Then . Let be the th element of after . Then is not in if is defined. Then is not well given.
Subcase .
(a) Similar arguments with the case .
(b) Assume . Then is well given but . Let be a timeline of passing through and sharing the same elements after with . In a similar way as (a), we get . Let be the th element of after . Then is not in if is defined. Then is not well given. ∎
Theorem 1.
Let be a model, a state and a formula of achievable at . For any timeline of passing through , is a timeline of iff .
Proof.
As is achievable at , is defined. Let be a timeline of passing through . Assume . By the definition of , all the elements of are in . So is a timeline of . Assume that is a timeline of . Let and . Let . As is a future state of and is in , there is a timeline passing through s.t. . share the same elements after with . By Lemma 1, . ∎
We now show that the principle of necessity of truth, , is valid. Let denote the sublanguage of generated from under .
Lemma 2.
For any in , there is a in equivalent to .
From Lemma 1 we can get a simple fact: for any and in , is equivalent to . Based on this fact, we can show this lemma in an inductive way.
If a formula contains no and , then its truth value at a state relative to a timeline is determined by the timeline itself. As can be reduced to , what follows is true.
Lemma 3.
Let be well given. For any in , iff .
Theorem 2.
is valid.
Proof.
Let denote that there is a sea battle. The puzzle of sea battle can be formalized as the inference . It is easy to see that is not valid. It is also easy to get that the principle of excluded future middle, , holds. Therefore, the puzzle of sea battle is not a sound argument.
It can be verified that . For example, . So the notorious problem with material implication is not a problem here. collapses to the material implication if is a state formula. The reason is as follows. Assume that is a state formula. Fix a model and a state . Then is true or false at . Suppose that is true at . Then the update with at does not change . Then the truth conditions of and at relative to any timeline are the same. Suppose that is false at . Then the update with at fails. Then both and are trivially true at relative to any timeline.
4 Completeness by Reduction
The idea of showing the completeness of is to reduce the dynamic operator in . This idea is from dynamic epistemic logic [12]. To reduce , a strategy is to massage into deeper and deeper until it meets atomic propositions. A difficulty arises when meets the operator . To handle this, some pretreatment of is needed.
Let be the set of formulas of Propositional Calculus.
Lemma 4.
For any in , there are in and in s.t. is equivalent to .
The operator can freely go into and out of conjunctions and disjunctions: and are valid. By making a reflection we can see that this lemma holds.
Lemma 5.
is valid.
Proof.
Firstly, we show that is defined iff is defined. Assume that is defined. Then for some through . By Lemma 1, is defined and is a path of . By Lemma 3, . Then is defined. Now assume that is defined. Then for some through . By Theorem 2, for any through . Then . By Lemma 3, . Then is defined.
Secondly, we show . It suffices to show that and have the same timelines through . By Lemma 1 and Lemma 3, what follows is the case: is a timeline of through is a timeline of through and is a timeline of through .
By the previous results, what follows is the case: . ∎
From this result and Lemma 2, it follows that is equivalent to .
Lemma 6.
Let be in and . Then the generated submodels of and at are identical.
Lemma 7.





where is in

Proof.
5. Assume . Then and . Assume . As is in , . Then . Then . Assume . As is in , . Then . Then . By Lemma 6, . Then . Then . Then .
Assume . Then and . Then . Then . Then . Assume . Then and . Then . Then . Then . By Lemma 6, . Then . Then . Then .
6. Assume . Then but . By Lemma 1, is in . Then there is a starting at s.t. . Then . Then . Then . Then .
Assume . Then is well given but . By Lemma 1, . Then there is a starting at in s.t. . Then . Then . Then . ∎
Let denote the language generated from under and .
Theorem 3.
The language can be reduced to .
Proof.
Let be a formula in . We pick a subformula of which is in the form of where contains no . Note that if has no such a subformula, then contains no and is already in . Then we do the following things:
We repeat until contains no . ∎
The completeness of the logic for the language is already shown in the literature. Then we can get the completeness of the logic .
5 Conclusive Remarks
The argument in the puzzle of sea battle is a special case of reasoning by cases. Its premises include the conditionals “if there is a/no sea battle tomorrow, it is necessarily so”. It has a fatalistic conclusion: we can not interfere with whether there will be a sea battle tomorrow. The principle expressed by the conditionals are plausible if we read necessity in the relative sense. The conditionals are not material implication. Instead, the ifclauses in them are devices for restricting possible futures. Necessity is not a upward monotonic notion. So the argument is not valid. It is fine that an invalid argument has a fatalistic conclusion. We present a formal way to make this precise.
Reading conditionals as material implication makes reasoning by cases valid. This causes others puzzles. Among them is the Puzzle of Miners presented by [10]. [1] proposes a deontic logic based on an extension of . By applying the approach in this work to the deontic logic, we might get a solution to the Puzzle of Miners. This is our future work.
Acknowledgment
Thanks go to Maria Aloni, Johan van Benthem, Alessandra Marra, Floris Roelofsen, Frank Veltman, and the audience of seminars or workshops at Delft University of Technology, Beijing Normal University and Tsinghua University.
References
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