Indicative Conditionals Have Relative Truth Conditions* Tamina Stephenson Massachusetts Institute of Technology

1 Introduction In this paper, I will propose an account of indicative conditionals that combines two crucial components. First, I assume that the if-clause of an indicative conditionals acts as a restrictor of a covert epistemic modal (Kratzer 1991a). Second, I combine this with a relativist semantics for epistemic modals proposed in previous work (Stephenson 2005, 2006). I will show that this view provides new solutions to two longstanding puzzles about indicative conditionals. The first is related to contradictory statements in Gibbardian standoffs (where one speaker can appropriately utter “if φ then ψ” and another can utter “if φ then not ψ; the second is related to inference patterns from disjunctions to conditionals. The structure of the paper is as follows: In Section 2, I discuss the two puzzles in more detail. In Section 3, I present a relativist semantics for conditionals using Lasersohn’s (2005) “judge” parameter. In Section 4, I show how this proposal provides solutions to the two puzzles from Section 2, and I conclude in Section 5.

2 The Puzzles In this section, I will discuss two puzzles related to the semantics of conditionals, and show why a simple context-dependent view is not sufficient. 2.1 First Puzzle: Gibbardian Standoffs Consider (1), a variation on examples by Gibbard (1981): (1)

[A murder investigation. Sam has narrowed down the suspects to the cook or the butler, and Mary has narrowed them down to the butler or the maid, both making only correct inferences.] Sam: Mary:

If the butler isn’t the murderer, the cook is. (No! / Nuh-uh!) If the butler isn’t the murderer, the maid is.

The dialogue in (1) presents a puzzle. On the one hand, Sam and Mary seem to be contradicting each other and thus disagreeing. At the same time, though, both seem to be justified in making the assertions that they make.

*

I would like to thank Kai von Fintel, Irene Heim, Sarah Hulsey, and audiences at Yale, the University of Maryland, and of course CLS 43 for useful comments and discussion on parts of this work. Some of this material also appears in my Ph.D. thesis (Stephenson 2007, Ch. 3).

Essentially the same puzzle has been observed for epistemic modals (see e.g. Egan, Hawthorne & Weatherson 2005, Weatherson 2005, MacFarlane 2006). We can see this from examples such as (2). (2)

[Same context as (1).] Sam: Mary:

The cook might be the murderer. (No! / Nuh-uh!) The cook can’t be the murderer.

Once again, Sam and Mary seem to be disagreeing, and at the same time both seem to be justified in making their assertions. Put another way, what makes the dialogues in (1)–(2) puzzling is that each speaker seems to be expressing their own state of knowledge, but they are not making first-person knowledge reports. For example, compare these to the dialogue in (3), where Mary uses the first-person indexical I. (3)

[Context: Same as (1).] Sam: I can narrow down the suspects to the cook or the butler. Mary: # Nuh-uh, I can narrow down the suspects to the butler or the maid.

In this case, it sounds odd for Mary to respond as if she is contradicting Sam. On the face of it, the fact that epistemic modals and conditionals show parallel behavior in (1)–(2) supports the view that indicative conditionals involve covert epistemic modals. However, standard accounts of epistemic modals cannot account for their behavior in dialogues like (2), as I will show below. This means in turn that the standard restrictor view of if-clauses cannot account for dialogues like (1). 2.2 The problem with the contextualist view One might think that epistemic modals are simply context-dependent items, so that, for example, might means roughly “compatible with x’s knowledge,” where x is some salient, contextually determined individual or group (see, e.g., DeRose 1991, von Fintel & Gillies 2007). There are a number of reasons to believe that this kind of view is inadequate, and I will outline one. (For other arguments, see Egan, Hawthorne & Weatherson 2005, MacFarlane 2006.) If we assume that epistemic modals are context-dependent, the most obvious possibility is that they have an element of first-person indexicality, so that, for example, might means roughly “compatible with my (the speaker’s) knowledge.” If this were the case, though, we would expect dialogues of the kind in (2) to be incoherent in the same way as those of the kind in (3) (which they’re not), so this can’t be right. A second possibility is that epistemic modals contain an element of first-person indexicality that is plural rather than singular, so that might means roughly “compatible with our knowledge” (where “us” is a group containing the speaker, addressee, and possibly other individuals). This view becomes problematic when

we consider examples like (4), where an epistemic modal is embedded under an attitude predicate. (4)

Sam thinks that the cook might be the murderer.

In (4), might is automatically linked to Sam’s mental state – that is, the sentence seems to mean “it’s compatible with Sam’s beliefs that the cook is the murderer” or “Sam thinks it’s compatible with his own knowledge that the cook is the murderer.” I will have more to say about this link with the attitude holder below, but for now the crucial observation is that the person (or group) whose knowledge is relevant in (4) is clearly not a group containing the speaker and addressee from the context of utterance of (4).1 A third possibility is that epistemic modals simply refer to the knowledge of some contextually relevant group of people. This fails for the same reason, since in (4) the person whose knowledge is relevant is Sam, and not a group of people considered relevant by the speaker and interlocutors in the context of (4). 2.3 Second Puzzle: Or-to-If Inference I turn now to a second puzzle involving conditionals, which has to do with the relationship between disjunction, material implication, and natural language conditionals. This puzzle plays a crucial role in the so-called direct argument for a non-truth-conditional view of indicative conditionals (see e.g. Edgington 1986, 1995, Bennett 2003). Let’s go back to the murder mystery scenario from (1)–(3) above, and imagine that Sam has narrowed down the suspects to the cook or the butler. In this case, Sam believes (and could appropriately assert) the sentence in (5). (5)

The butler is the murderer or the cook is the murderer.

Intuitively, it follows from this that Sam also believes (and could appropriately assert) (6). (6)

If the butler isn’t the murderer, the cook is.

The standard conclusion to draw from this is that (5) entails (6). Generalizing, this means that “φ or ψ” entails “if not φ then ψ.” By substituting some negations, this means that “not φ or ψ” entails “if φ then ψ,” meaning that conditionals in natural language should be true whenever the corresponding material implication is true. In particular, a natural language sentence of the form “if φ then ψ” should be true whenever the antecedent (φ) is false or the consequent (ψ) is true. It is well known that this is empirically wrong (see e.g. Bennett 2003, Edgington 2006 for recent

1

One might object that this could be a shifting indexical (Schlenker 2003, etc.); however, epistemic modals do not behave like typical shifting indexicals in the relevant respects. In particular, the shifted interpretation in (4) is obligatory rather than optional, and moreover must be linked to the closest attitude holder. For example, in (i) might must be linked to Sue’s knowledge. (i) Sam thinks that Sue thinks that the cook might be the murderer.

summaries of the issues). Just a few examples of the absurd consequences can be seen in (7) (from Bennett 2003, Ch. 2). (7)

a. If I ate an egg for breakfast this morning, you ate a million eggs. b. If there are no planets anywhere, the solar system has at least eight planets.

According to this view, (7a) ought to be true provided the speaker didn’t eat an egg for breakfast, and (7b) ought to be true given the fact that the solar system does have at least eight planets.2

3 Semantic and Pragmatic Proposal In this section I will present a view of conditionals that puts together Kratzer’s view of if-clauses as modal restrictors with a relativist semantics for epistemic modals. 3.1 Preliminaries I will assume that beliefs are expressed using the notion of doxastic alternatives (Lewis 1979, Chierchia 1989) defined in (8). (8)

Doxw,x = { : it is compatible with x’s beliefs in w that x (x’s self) is y in w′}

The set of doxastic alternatives of an individual x at a world w is the set of worldindividual pairs such that it’s compatible with x’s beliefs in w that x (that is, x’s own self) is y in w′. (I am ignoring the role of times.) This concept is needed essentially because people can use their beliefs to place themselves within the population of possible individuals, real and imaginary, as well as within possible worlds – that is, for de se attitudes. For example, if I believe that I am in Chicago, the content of my belief is that my own self is located in Chicago. If in contrast, I believe that Tamina Stephenson is in Chicago, I am placing myself in a world where the individual named Tamina Stephenson is located in Chicago, but I may not believe that I myself am in Chicago, for example if I have lost my memory and don’t realize that I am Tamina Stephenson. I propose that knowledge is also expressed using an analogous notion of epistemic alternatives, defined in (9). (9)

Epistw,x = { : it is compatible with x’s knowledge in w that x (x’s self) is y in w′}

An individual x’s epistemic alternatives in a world w (again ignoring the role of time) is the set of world-individual pairs such that it’s compatible with 2

There have, of course, been attempts to explain the behavior of natural language conditionals using Gricean maxims, but I will not discuss these here and take it as established that the material implication view of natural language conditionals is wrong.

what x knows in w that x’s own self is y in w′. The key difference between doxastic and epistemic alternatives, for our purposes, is the factivity of knowledge: a person’s knowledge cannot rule out the actual individual that they are in the world in which they are actually located, meaning that an individual x’s epistemic alternatives in a world w (that is, the set Epistw,x) must always include the pair itself. (Similarly, a person’s epistemic alternatives will include any pairs that are ruled out by their beliefs without the kind of justification needed to constitute knowledge; however, this will be less important here.) 3.2 Semantics of epistemic modals and conditionals My view of conditionals owes its main idea to a proposal by Lasersohn (2005) for expressions such as tasty and fun. Lasersohn relativizes the truth of sentences to an individual “judge” parameter. In other words, indices on this view are worldindividual pairs rather than worlds, and propositions are sets of word-individual pairs (type ) rather than sets of worlds (type ). For Lasersohn, the “judge” is the person whose experience or judgment is relevant. For example, if the single contextually relevant cake tastes good to x but not to y in world w, then the sentence The cake is tasty is true at the pair but false at the pair . In recent work (Stephenson 2005, 2006), I have extended Lasersohn’s system to epistemic modals. My proposal there is that epistemic modals such as might and must express the epistemic state of the judge, so that, for example, epistemic might means roughly “compatible with what the judge knows.” Essentially, then, sentences of the form “might φ” and “must φ” (interpreted epistemically) have the meanings given in (10). (10) a. [[might φ]]w,j = 1 iff ∃∈Epistw,j: [[φ]]w′,x = 1 b. [[must φ]]w,j = 1 iff ∀∈Epistw,j: [[φ]]w′,x = 1 For example, (10b) says that a sentence of the form “must φ” is true at a worldindividual pair iff in all of j’s epistemic alternatives, φ is true. Note that we can talk about a proposition being true “at” an epistemic alternative (a world-individual pair) because we are treating propositions as sets of world-individual pairs. Now, I assume (following Kratzer 1991a) that the if-clause in an indicative conditional such as (6) (If the butler isn’t the murderer, the cook is) acts as a restrictor of a silent epistemic necessity modal. In other words, an indicative conditional of the form “if φ then ψ” has the structure in (11), where [MUST] is a silent necessity modal and REPIST contributes the epistemic “flavor” of the modality.3

3

I have not broken down the contribution of epistemic might and must into the parts contributed by the modal and by the epistemic restrictor, but this would be done along the lines of Kratzer (1991b). For more details in the system I am proposing, see Stephenson (2007).

(11) ψ [MUST] [REPIST]

if φ

Thus the meaning of an indicative conditional of the form “if φ then ψ” is given in (12). That is, “if φ then ψ” is true at a world-individual pair iff in all of j’s epistemic alternatives in w where φ is true, ψ is also true. (12) [[if φ then ψ]]w,j = 1 iff ∀∈Epistw,j: [[φ]]w′,x = 1 ⊃ [[ψ]]w′,x = 1 As I mentioned above, I assume that the meaning of the attitude predicate believe (or think) involves doxastic alternatives. A lexical entry for believe is given in (13), which is equivalent to the rule in (14). (13) [[believe]]w,j = [[think]]w,j = [λp . [λze . ∀∈Doxw,z, p(w′)(y) = 1] ] (14) [[z believes / thinks (that) φ]]w,j = 1 iff ∀∈Doxw,z, [[φ]]w′,y = 1 According to either (13) or (14), a sentence of the form “z thinks that φ” is true iff φ is true at all of z’s doxastic alternatives – where, again, we can talk about a sentence being true at a doxastic alternative since propositions are treated as sets of world-individual pairs. Notice that φ is evaluated at z’s doxastic alternatives, so in a certain sense z is the “judge” of the embedded clause. In effect, then, “z thinks that φ” is equivalent to “z thinks that φ is true as judged by z.” 3.3 Pragmatics Having changed the view of such a basic thing as propositions (treating them as sets of world-individual pairs rather than sets of worlds), something more needs to be said about how these are used in conversation. In particular, we need to make sense of the notions of assertion and the context set in this framework. At this point I will sketch a view of conversation that works together with the proposed semantics for conditionals.4 My starting point is the Stalnakerian theory of conversation (see e.g. Stalnaker 1978, 2002). On this view, the context set of a conversation (at a given time) is the set of worlds compatible with what is commonly believed by all the participants in the conversation – that is, what all the participants believe, what they all believe they all believe, and so on ad infinitum.5 Relatedly, a proposition is said to be in the common ground if it’s true in every world in the context set. An assertion of p is a proposal to reduce the context set to worlds where p is true.

4

This is a very slight variant on my proposal in Stephenson (2006); cf. also Egan (2007) for a related but somewhat different view. 5 The relevant attitude is often taken to be something like acceptance for the purpose of the conversation rather than actual belief, but that distinction will not be important here.

To adapt this view of conversation to a semantic system with a judge parameter, all notions involving sets of worlds need to be changed to involve sets of world-individual pairs. For example, the context set should be a set of worldindividual pairs, and an assertion should be a proposal to restrict the context set to world-individual pairs where the proposition asserted is true. There are different ways to do this, though, and two particular assumptions I make will be crucial. First, I propose that the context set is a set of world-time pairs where, crucially, J is always the plurality of the group of participants in the conversation. An assertion, again, is a proposal to restrict the context set to world-individual pairs in which the proposition asserted is true – but because all of the world-individual pairs in the context set involve the group of participants in the conversation, an assertion that is accepted by the hearers will have the effect of making it common ground that the asserted proposition is true as judged by the entire group of participants. The second crucial assumption has to do with the norm of assertion. I assume that in order for a speaker to be justified in making an assertion, they don’t need to know (or believe) that the sentence is true as judged by the entire group of participants in the conversation. Rather, they only need to know that the sentence is true as judged by their own self. More precisely, the asserted proposition must be true in all of the speaker’s own epistemic alternatives. This is repeated in (15). (15) Norm of justified assertion: For speaker A to be justified in asserting a sentence S in w it must be the case that ∀∈Epistw,A: [[S]]w′,x = 1 On an intuitive level, what I am proposing with this pragmatic view is that people in conversation together are on a joint venture to place themselves, as a group, not only in the space of possible worlds (as assumed on the standard Stalnakerian view) but also in the space of possible groups of people in these possible worlds. For instance, a group of participants in a conversation might establish that they are a group of people who exist in a world where it is currently raining, or they might establish that they are a group of people whose knowledge is compatible with the butler being the murderer. Assertion is an efficient way for an individual to help the group place itself, since they can simply make a proposal along the lines of “let’s establish that we are a group of people who…” and see if other participants object.

4 Solving the Puzzles Now I will show how the semantic and pragmatic views I proposed in Section 3 help to solve the two puzzles related to conditionals that I introduced in Section 2. 4.1 Solution to the First Puzzle (Gibbardian Standoffs) Recall that Gibbardian standoffs occur in examples like (16).

(16) [Context: The murder investigation from (1)] Sam: Mary:

If the butler isn’t the murderer, the cook is. (No! / Nuh-uh!) If the butler isn’t the murderer, the maid is.

The two things to explain about this kind of example are, first, that Sam and Mary seem to be contradicting each other and, second, that both Sam and Mary seem to be justified in making the assertions they each make. On the semantic and pragmatic view I have proposed, both of these observations can be accounted for. To see why Sam and Mary are contradicting each other, we only need to look at the propositions expressed by the two assertions. Sam’s assertion expresses the proposition in (17), and Mary’s assertion expresses the proposition in (18). (17) Proposition expressed by Sam’s assertion in (16): {: ∀: ∈Epistw,j & the butler is not the murderer in w′ ⊃ the cook is the murderer in w′} (18) Proposition expressed by Mary’s assertion in (16): {: ∀: ∈Epistw,j & the butler is not the murderer in w′ ⊃ the maid is the murderer in w′} In (16), the proposition asserted by Sam is the set of world-individual pairs such that in all of j’s epistemic alternatives where the butler is not the murderer, the cook is. On the other hand, the proposition asserted by Mary is the set of world-individual pairs such that in all of j’s epistemic alternatives where the butler is not the murderer, the maid is. These are disjoint sets provided that there can only be one murderer, and that the context set does not include any pairs such that the butler is the murderer in all of j’s epistemic alternatives – that is, it is common ground that it has not been established that the butler is the murderer. I take this second condition to be a presupposition of the conditional, as a special case of a general prohibition on vacuous satisfaction of universal quantification. Since the sets are disjoint (under these assumptions), it makes sense that they sound contradictory. At the same time, both Sam and Mary seem to be justified in making their contradictory assertions. This can be explained by the norm of assertion given in (15). First, let’s assume that Sam and Mary’s reasoning from the evidence available to them was justified enough to constitute knowledge in each case. This means that in all of Sam’s epistemic alternatives, either the butler or the cook is the murderer, and in all of Mary’s epistemic alternatives, either the butler or the maid is the murderer. This is repeated formally in (19) (w* is used to indicate the actual world where the example takes place). (19) a. ∀∈Epistw*,Sam: the butler or the cook is the murderer in w′ b. ∀∈Epistw*,Mary: the butler or the maid is the murderer in w′

Assuming that the relation of epistemic alternativeness is transitive (i.e., that if A knows that p, then A knows that A knows that p), this means that the statements in (20) also hold. (20) a. ∀∈Epistw*,Sam: ∀∈Epistw′,x: the butler or the cook is the murderer in w″ b. ∀∈Epistw*,Mary: ∀∈Epistw′,x: the butler or the maid is the murderer in w″ According to the norm of assertion in (15), these are exactly the conditions that need to hold for Sam and Mary to be justified in asserting the propositions from (17)–(18). The pragmatic system I have developed allows for a situation where two speakers can be perfectly justified in making contradictory assertions. 4.2 Solution to the Second Puzzle (Or-to-If Inference) The observation from Section 2.3 was that if a person A believes that not φ or ψ, then A must also believe that if φ then ψ. Given sentences φ and ψ, I’ll refer to a sentence of the form “not φ or ψ” as “the disjunction” and to the corresponding sentence of the form “if φ then ψ” as “the conditional.” Now, since believing the disjunction requires also believing the conditional, the standard conclusion is that the disjunction entails the conditional. We know that this standard conclusion is false, though, since natural language conditionals are not equivalent to material implication. At first this looks like a paradox. However, the move from this observation to the standard conclusion involves an implicit assumption, namely, that if believing p is sufficient for believing q, then p entails q. I will refer to this as preservation of entailments under belief. Although this principle seems obvious, and is valid on standard views of logic and semantics, it turns out to be invalid in the system I have proposed here. The reason for this, as we will see, has to do with the way belief predicates interact with the judge parameter. The task at hand is to show that, on my view, the disjunction (“if not φ then ψ”) does not entail the conditional (“if φ then ψ”), but that nevertheless belief of the disjunction entails belief of the conditional. This will show that entailments are not preserved under belief in my system, and resolve the apparent paradox. Essentially what we will see is that on my view the disjunction is strictly weaker than the conditional, but they become equivalent under belief. First let me show that the disjunction “if not φ then ψ” does not entail the conditional “if φ then ψ.” The propositions expressed by the two statements, in terms of φ and ψ, are given in (21)–(22). (21) not φ or ψ: {: [[φ]]w,j = 0 or [[ψ]]w,j = 1} (22) if φ then ψ: {: ∀∈Epistw,j: [[φ]]w′,x = 1 ⊃ [[ψ]]w′,x = 1} = {: ∀∈Epistw,j: [[φ]]w′,x = 0 or [[ψ]]w′,x = 1}

I will refer to the set in (21) as S(21) and to the set in (22) as S(22). I need to show that in general S(21) is not a subset of S(22). To do this, I will construct a counterexample in the form of a multi-world model. Consider the model in (23), which contains four worlds, w1–w4, and one individual A. Assume that φ and ψ are nonjudge-dependent (that is, the propositions they express depend only on the world for their truth value). Intuitively, this model could represent a situation where A believes that φ and ψ have to either both be true or both be false, and where A is right about this in worlds w1 and w4 but wrong (or not sufficiently justified) in worlds w2 and w3. Note that the four worlds cover the four possibilities for truth and falsity of φ and ψ: both φ and ψ are true in w1, only φ is true in w2, and so on. (23) w1: φ is true, ψ is true; A’s epistemic alternatives are {, } w2: φ is true, ψ is false; A’s epistemic alternatives are {, , , } w3: φ is false, ψ is true; A’s epistemic alternatives are {, , , } w4: φ is false, ψ is false; A’s epistemic alternatives are {, } Restricting ourselves to this particular model, the propositions expressed by “not φ or ψ” and “if φ then ψ” are the sets of world-individual pairs given in (24) and (25), respectively. (24) “not φ or ψ”: {, , } (25) “if φ then ψ”: { , } Clearly, the set in (24) is not a subset of the set in (25) because of the presence of the pair . Therefore S(21) is not in general a subset of S(22), and the disjunction does not entail the conditional.6 Now let me show that, in contrast, belief of the disjunction does entail belief of the conditional. The propositions expressed by the two sentences “A believes that not φ or ψ” and “A believes that if φ or ψ” are given in (26) and (27), respectively. (26) A believes that not φ or ψ: {: ∀∈Doxw,A: [[φ]]w′,x = 0 or [[ψ]]w′,x = 1}

6

Note that the reverse entailment (from the conditional to the disjunction) does hold. Proof: Suppose that ∈S(22). This means that in all of j’s epistemic alternatives in w, either φ is false or ψ is true. Since knowledge is factive, itself must be among j’s epistemic alternatives in w, and so it must be the case that either φ is false or ψ is true at the pair . Thus is also in the set S(21); thus S(22) must be a subset of S(21), so the conditional entails the disjunction.

(27) A believes that if φ then ψ: {: ∀∈Doxw,A : ∀∈Epistw″,y: [[φ]]w′,x = 0 or [[ψ]]w′,x = 1} I assume that the relationship between belief and knowledge is such that to believe something is to believe that you know it. This means that the epistemic alternatives of A’s doxastic alternatives are the same as A’s doxastic alternatives. This equivalence is stated more formally in (28).7 (28) For any world w and individual A: {: ∃ [∈Doxw,A and ∈Epistw′,x] } = Doxw,A Given this assumption, (27) becomes equivalent to (29). (29) {: ∀∈Doxw,A: [[φ]]w′,x = 0 or [[ψ]]w′,x = 1} This is the same as (26), and so “A believes that not φ or ψ” is equivalent to “A believes that if φ then ψ.” In particular, then “A believes that not φ or ψ” entails “A believes that if φ then ψ.” As I noted above, what’s going on here is that the conditional in fact entails the disjunction, and not vice versa, but the two become equivalent when embedded under believe. This is because the meaning of believe involves doxastic alternatives, which gives the effect of shifting the judge of the embedded clause to the attitude holder. This shows, then, that even if believing p is sufficient for believing q, it does not follow in my system that p entails q. This means that we can capture the or-toif inference without making the disjunction as strong as the conditional. To be fair, we might also cut traditional logicians some slack for analyzing conditionals as material implication, if believing one is equivalent to believing the other.

5 Conclusion In summary, in this paper I have given new solutions to two longstanding puzzles involving conditionals by combining a relativist view of epistemic modals (using Lasersohn’s “judge” parameter) with Kratzer’s restrictor analysis of if-clauses. Regardless of whether the particular view I have proposed turns out to be the right one, there are two lessons that we can and should take from this discussion. The first lesson is the importance of maintaining a distinction between truth conditions and assertability conditions. Gibbard examples, for instance, may be telling us less about conditionals than about assertion – namely, that the norms involved in assertion are crucially weak in a particular sense compared to the effect that a successful assertion has on the context set. The second lesson is that, depending on the semantic system we adopt, it may or may not be valid to conclude that p entails q from the fact that believing p is sufficient for believing q. In other words,

7

For a discussion of the exact assumptions needed to derive (28), see Stephenson (2007: 122–23).

we always need to keep in mind that intuitive patterns of inference don’t necessarily translate to entailments in the sense of subset relations between sets of indices.

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