Robert Pasternak (2018), Thinking Alone and Thinking Together - - PDF document

Robert Pasternak (2018), “Thinking Alone and Thinking Together”

Minghui Yang

• 1. Distributive versus non-distributive readings of attitude ascriptions

w There are two possible ways to interpret a sentence containing plural NP. w Distributive reading: the predicate is applied to each of the individual members of the “group” designated by the NP. Under this reading, each of the individuals designated by the NP must instantiate the relevant property. e.g. Alexis and Brian are running entails Alexis is running. w Non-distributive/collective reading: the predicate is only applied to the “group” understood as a whole. e.g. Alexis and Brian ate a whole pie does not entail Brian ate a whole pie. 1.1 Distributive reading of attitude ascriptions w Attitude ascriptions usually take distributive readings: (1) a. Alexis and Brian think that Cass left.

• b. Alexis and Brian want Cass to leave.
• c. Alexis and Brian wish that Cass had left.
• d. Alexis and Brian regret that Cass didn’t leave.

w Sentence (1a) entails: (1a*) Alexis believes that Cass left, and Brian believes that Cass left. w Similarly, (1b)-(1d) entail that both Alexis and Brian have the desire, wish and regret, respectively. 1.2 Non-distributive reading of attitude ascriptions w Attitude verbs can also take non-distributive readings, as illustrated in the following case: Case: Sam owns a construction company and has six clients, none of whom know of the others’

• existence. She has convinced each client that she would build a house for him. In reality, she is a

con artist and built no houses at all. (2) (In total,) Sam’s six clients think she built six houses for them. Intuition: (2) is true in the context.

w The story has made it clear that none of the clients have the belief that Sam will build six houses. Thus a distributive reading of (2) would render it false. w Moreover, (2) cannot be explained away by appealing to de re, since the following sentence is also false: (3) There are six houses that Sam’s six clients think she built for them. w Thus, sentence (2) must have a non-distributive reading. 1.3 “Conjunction” versus “Disjunction” 1.3.1 Collective beliefs obtained by conjunction w What would be a non-distributive reading of (2)? A natural thought: Each of the six clients believes that Sam will build one house. Now their beliefs could somehow be “added up” conjunctively into one “cumulative” belief that Sam will build six houses. Notes: (i) “Cumulative” beliefs are not mental states proper. (ii) Conjoining “Sam will build a house for client i” (i = 1-6) six times will get you a house-count

• f six only if all instances of “for client i” are non-overlapping (i.e. if house x is for client i, then

it is not for client j where i ≠ j). This assumption is tacit in the story. w One more example that confirms the “conjunction” intuition: Case: Paul just got married, and his cousins Arnie and Beatrice, who have never met, just caught wind of it. Arnie suspects that Paul’s husband is rich, and has no other relevant opinions. Beatrice thinks he’s a New Yorker, and has no other relevant opinions. (4) Paul’s cousins think he married a rich New Yorker. Intuition: Sentence (4) is true. w None of Paul’s cousins individually has the belief that Paul married a rich New Yorker. Thus we are

• nce again forced into a non-distributive reading of (4).

w The most straightforward way to account for the truth of (4) once again appeals to conjunction: the conjunction of Arnie’s belief and Beatrice’s belief is that Paul’s husband is a rich New Yorker. 1.3.2 Collective beliefs obtained by disjunction w There are cases in which the conjunction strategy doesn’t work, for instance: Case: Paul just got married, and his cousins Arnie and Beatrice, who have never met, just caught wind of it. Arnie thinks that Paul married a rich Marylander, while Beatrice thinks he married a poor New Yorker.

(5) Paul’s cousins think he married a rich New Yorker. (6) Paul’s cousins think he either married a rich Marylander or a poor New Yorker. Intuition: Sentence (5) is false, sentence (6) is true. w It seems that we are adding things up by disjunction this time. w Quibble: maybe we are not forced into read (6) collectively in the first place. Indeed, if we adopt a distributive reading of (6) instead, we would also predict that (6) is true. Note: Here we need the additional assumption that beliefs of Arnie and Beatrice are suitably closed under entailment. The assumption is significant from an epistemological perspective, but tacit and ubiquitous in semantics. w To rule out the distributive reading, we consider yet another case: Case: Paul has three cousins, Arnie, Beatrice, and Kate. Arnie and Beatrice’s beliefs are as in the

• riginal wedding scenario: rich and New Yorker, respectively, and otherwise agnostic. Kate, like

Beatrice, is not opinionated about Paul’s husband’s wealth, but she thinks he’s from Iowa, not New York. (7) Paul’s cousins think he married a rich man from either New York or Iowa. Intuition: (7) is true. w Our intuition in (7) is not compatible with the suggestion that the attitude ascription should be read distributively: Arnie is not committed to Paul’s husband’s being from New York or Iowa, while Beatrice and Kate are not committed to Paul’s husband’s being rich. Thus the distributive reading would render (7) false. And yet (7) seems true. w So we are committed to a collective reading of (7). 1.3.3 Separating conjunctive and disjunctive cases w Question: what separates conjunctive cases from disjunctive cases? Tentative answer: when the individual beliefs of group members are compatible, we add them up by conjunction; otherwise we add them up by disjunction. w To flesh out a little more details of this: The conjunction case seems relatively uncontroversial. When we have a bundle of individual beliefs that are consistent, we simply conjoin them together. The case in which the bundle of beliefs are inconsistent, however, deserves more scrutiny. Suppose we have a set of individual beliefs {P, Q, R} where P is consistent with both Q and R but Q is inconsistent with R. {P, Q, R} is an inconsistent set. But we don’t want the cumulative belief to be (P or Q or R), the straightforward disjunction of all the relevant beliefs--it seems too weak! What we really want, as illustrated by (7), is rather:

(P and (Q or R)) w What is the way to do this? Pasternak has discussed two accounts. Due to the limited time, however, I shall talk about only one of them, i.e. the “premise negotiation” analysis.

• 2. Background: semantics of “believe” (and “think”)

w The lexical entry of “believe”: (8) [[believe]]! = λp.λe. ∀w ∈ Dox(e)[p(w)] (9) [[Arnie believes that Paul married a rich man]]! = 1 iff ∃e [Exp(e) = a ∧ ∀w ∈ Dox(e)[rich-man(w)]] w Transcribe (9) into plain English: in any possible world compatible with Arnie’s doxastic state (which is designated by the event-variable e), Paul married a rich man. w Exp (e) is the function that returns the subject of the doxastic state e. w Ordinary doxastic states have only one subject. But we may introduce “cumulative” doxastic states (notification: 𝑓" ⊔ 𝑓#) which has multiple subjects. Accordingly we extrapolate the definition of Exp (e) to allow plural subjects like x ⊔ y. (10) [[Paul’s cousins believe that p]]! = 1 iff ∃e[Exp(e) = a⊔b ∧ ∀w ∈ Dox(e)[p(w)]] w We may just define x ⊔ y as the mereological sum of x and y. But what is 𝑓" ⊔ 𝑓#? It must be generated from individual doxastic states 𝑓" and 𝑓#. For now we introduce an operation J as a placeholder for this operation: (11) Dox(e) = J({Dox(e’)) | e’ ∈ Atm(e)}) where Atm(e) is defined as the set of individual doxastic states that compose e.

• 3. Premise negotiation

3.1 Motivating ideas w The basic idea of premise negotiation is to “make the most of it” even in a pessimistic situation. Suppose you have a list of obligations to fulfill. When the list is consistent (i.e. it is possible to fulfill them all), the way to “make the most of it” is to fulfill all of them. But what if the list is inconsistent? In this case it is not possible to fulfill all of the listed obligations. Yet it still makes sense to ask them to fulfill their obligations as much as possible. What does this mean? Let’s consider a concrete example: Say, that the person is asked to pick out a card. The relevant obligations are (i) picking out a diamond (D), (ii) picking out a queen (Q) and (iii) picking out a king (K).

D, Q and K can never be fulfilled simultaneously. Yet one may fulfill two of them. One may fulfill D and Q by picking out a diamond queen, or one may fulfill D and K by picking out a diamond

• king. It would seem that this is the best they can do. That is, one may fulfill (D and (Q or K)).

Observation: the behavior of these obligations is very similar to the behavior of individual beliefs

• f multiple subjects in the previously noted cases of (5), (6) and (7).

3.2 Implementing the idea w The most common way of capturing the idea in a possible-world semantics framework originates from Lewis (1981) and Kratzer (1981). w We introduce the following order over possible worlds with respect to a set of propositions Q: (12) w1 ≾$ w2 iff {p ∈ Q | p(w1)} ⊇ {p ∈ Q | p(w2)} w Matching the order with the degree of idealness: w1 is at least as ideal as w2 iff w1 ≾ w2. Rationale: if w1 ≾ w2, then every proposition that is true in w2 is true in w1. So whatever makes w2 a somewhat-ideal world will also make w1 ideal. w We apply this to the toy example of picking cards. Worlds are arranged in the following way: {D, Q} ≾$ {D} {D, Q} ≾$ {Q} {D, K} ≾$ {D} {D, K} ≾$ {K} And given that K∩Q is the null set, no world is more ideal than either {D, Q} (i.e., a world in which

• ne picks out a diamond queen) or {D, K} (i.e., a world in which one picks out a diamond queen).

Now we may see that the set of possible worlds in which the obligations are maximally is (D∩Q)∪(D ∩K), which is exactly D∩(Q∪K). w Just a little more technicality: we define BEST (≾$) as the set of possible worlds that are maximally ideal under the relation ≾$: (13) BEST(≾) ≡ {w | ¬∃w’ [w’ ≾ w & w’ ≠ w]} And of course this also signifies a proposition …

• 4. Applying all these to beliefs

w Now we just identify the content of the cumulative belief of the group with the “best” proposition generated from individual doxastic states and the relation ≾: (14) Dox(e) = BEST(≾{Dox(e’) | e’∈Atm(e)})

(where BEST(≾) ≡ {w | ¬∃w’[w’ ≾ w & w’ ≠ w]}) w Let’s go through how this works. Starting from the case of Sam’s six clients. Use e1-6 and k1-6 as respective shorthands for e1 ⊔ …⊔ e6 and k1 ⊔ … ⊔ k6. The sentence: (2) (In total,) Sam’s six clients think she built six houses for them. Is interpreted as: (15) ∃e[Exp(e) = k1–6 ∧ ∀w ∈ Dox(e)[six-houses(w)]] The doxastic state e1-6 satisfies the condition that Exp (e) = k1–6. The only remaining question is what Dox(e) is. In the setting of the story, each of the clients believe that Sam will build a house, but none of them know the existence of other clients. Thus the individual doxastic states e1 … e6 are compatible. {Dox(e) | e ∈ {Dox(ei) | 1 ≤ i ≤ 6}} = BEST(≾{Dox(e’) | e’∈ {Dox(ei) | 1 ≤ i ≤ 6}}) = ∩{Dox(ei) | 1 ≤ i ≤ 6} (If Q is a consistent set, BEST(≾(Q)) =∩Q) w Next, we consider the case of: (4) Paul’s cousins think he married a rich New Yorker. Which is interpreted as: (16) ∃e[Exp(e) = a⊔b ∧ ∀w ∈ Dox(e)[rich-NYer(w)]] In the story, Arnie believes that Paul’s husband is wealthy, and Beatrice believes that he is a New

• Yorker. We thus have Dox(e) to be ∩{ Dox(e-arnie), Dox(e-beatrice)} for the same reason.

w For disjunctive cases, we first inspect the behavior of: (6) Paul’s cousins think he either married a rich Marylander or a poor New Yorker. Which is interpreted as: (17) ∃e[Exp(e) = a⊔b ∧ ∀w ∈ Dox(e)[rich-MDer(w)∨poor-NYer(w)]] In this case, Dox(e-arnie)∩Dox(e-beatrice) is the null set, since Arnie thinks that Paul’s husband is a rich Marylander, but Beatrice thinks he is a poor New Yorker. Given this, the maximally ideal possible worlds for Arnie and Beatrice are non-overlapping sets Dox(e-arnie) ∪ Dox(e-beatrice). This will predict successfully that (6) is true. w We now proceed to the case of: (7) Paul’s cousins think he married a rich man from either New York or Iowa. The truth-condition:

(18) ∃e[Exp(e) = a⊔b⊔k∧∀w ∈ Dox(e)[rich(w)∧(NYer(w)∨Iowan(w))]] Note that the Dox(e-arnie) contains worlds in which Paul’s husband is rich, but agnostic about his hometown, Dox(e-beatrice) and Dox(e-kate) are agnostic about his origin but contains New Yorker-worlds and Iowan-worlds respectively, we now have: Which predicts that the “best” doxastic state in this case is ((Dox(e-arnie)∩Dox(e-beatrice)) ∪ ((Dox(e-arnie)∩Dox(e-kate)) And this is exactly what we want: a doxastic state that contains all and only worlds in which “Paul’s husband is a rich New Yorker or Iowan” is true.

• 5. A worry: irrelevant disagreement

w The motivating intuition: if we intersect the whole doxastic state of a subject, all the beliefs that the subject have would matter to the truth condition. But many of those beliefs are irrelevant to what we are interested in. They may mess things up. w A concrete example of this: Case: As before, Arnie believes that Paul married a rich man, and Beatrice believes that he married a New Yorker, with no other relevant beliefs. In addition, Arnie mistakenly believes that Mozart was born in 1755, while Beatrice correctly believes him to have been born in 1756. (19) Paul’s cousins think he married a rich New Yorker. Intuition: Sentence (19) is true. w The proposed semantics would fail to predict this! w The reason is that, due to the disagreement between Arnie and Beatrice about Mozart, Dox(e-arnie)∩ Dox(e-beatrice) is just the null set … w There is a somewhat stronger counterexample that makes use of disagreements between individual doxastic states: Case: Sam owns a construction company and has six clients, none of whom know of the others’

• existence. She has convinced each client that she would build a house for him and no one else. In

reality, she is a con artist and built no houses at all.

(20) Each client believes that Sam built a house for him and him alone. (21) Sam’s clients thought she had built six houses for them. Intuition: Sentence (20) and (21) are both true. w Once again the previously proposed semantics fails to predict this, since the doxastic states e1-6 have no non-empty intersection. w This case seems trickier than the previous one, since the irrelevance of the disagreement is less

• significant. If the natural way to block the previous counterexample is to rule out irrelevant

disagreements by irrelevance, it might seem unclear whether that strategy works in the current circumstance … w We address the two counterexamples in turn.

• 6. Fixing the irrelevant disagreement problem, step 1: situations

w The motivating idea: the troublemaking disagreements in the Mozart case are irrelevant to the subject matter that we are interested in, so they ought not to be considered in the first place. w We now need to find a way to suppress these disagreements. To do this, we introduce the formalism of situations into the semantics of “believe”: (22) [[believe]]! = λp λe. about(e) = s! ∧ ∀w ∈ Dox(e)[p(w)] s! is the situation s determined by context c. w The definition of doxastic state Dox(e) must accordingly be revised. We introduce doxastic states about specific situations (notification: 𝑓"

% where x is the subject and s is a situation), such that Dox(𝑓" %)

becomes the set of worlds compatible with what the experiencer of e (i.e. x) believes specifically about the situation s. w The ongoing move serves to “expand” the set of possible worlds Dox(e) (and thus making it less specific) from what it was. w To see the details of this, let s&'( be the situation of the wedding of Paul. We spell out the truth condition of (19): (23) ∃e[Exp(e) = a⊔b ∧ about(e) = s&'( ∧ ∀w ∈ Dox(e)[rich-NYer(w)]] w To satisfy (23), we need to find a doxastic state that satisfies the conditions in the bracket. Now, 𝑓)

*!"#⊔ 𝑓) *!"# fits the need:

Exp (𝑓)

*!"#⊔ 𝑓) *!"#) = a⊔b for obvious reasons;

About (𝑓)

*!"#⊔ 𝑓) *!"#) = s&'(;

Given that s&'( is orthogonal with s+,-)./ (the situation concerning Mozart’s birth), both

Dox(𝑓)

*!"#) and Dox(𝑓) *!"#) are agnostic about when Mozart was born. The disagreement has been

• suppressed. Dox (𝑓)

*!"#⊔ 𝑓) *!"#) = Dox(𝑓) *!"#) ∩ Dox(𝑓0 *!"#) which is non-empty. And for all

possible worlds in Dox(𝑓)

*!"#) ∩ Dox(𝑓0 *!"#) Paul’s husband is a rich New Yorker.

We have rightly predicted that (19) is true in the given context!

• 7. Fixing the irrelevant disagreement problem, step 2: contextual relativity

w We have not yet figured out how to accommodate sentences (20) and (21), in particular (21). For if we just have a coarse-grained situation s1)2 about Sam’s construction promises, we won’t be able to predict the truth of (21) since there is no obvious way to filter out the disagreement. w Pasternak’s solution: we further modify the situation element in our semantics so that we may handle the disagreements between multiple subjects in a more subtle way. w First, we introduce two groups of situations about each client, 𝑡3 and 𝑡3

4:

For each client 𝑙3, 𝑡3 is the situation containing all and only those portions of the contract that state that Sam will build a house for 𝑙3. 𝑡3

4 on the other hand is the situation containing all and only those portions of the contract that state

that Sam will build a house for 𝑙3 and no one else. For all values of i, 𝑡3 is a proper part of 𝑡3

4.

Note: given that 𝑡3 and 𝑡3

4 are defined relative to each client, there won’t be a common situation

that all of Sam’s clients are engaging with. w Next, we revise the semantics of “believe” by introducing sets of situations: (24) [[believe]]! = λpλe. about(e) ∈ S! ∧ ∀w ∈ Dox(e)[p(w)], where S! is the set of situations specified by context c. w The semantics will rightly predict that (20) and (21) are both true in the modified Sam case. The truth condition of (20) is: (25) ∀x : client(x)[∃e[Exp(e) = x∧about(e) ∈ S! ∧ ∀w ∈ Dox(e)[only-house(x,w)]]] Which is satisfied. w The contextually specified set of situations is S! = {𝑡3

4| 𝑗 = 1, … , 6}, Given this, (25) will be

satisfied by 𝑓3

%$

%

for each i. The truth condition of (21) is: (26) ∃e[Exp(e) = k1–6 ∧about(e) ∈ S! ∧ ∀w ∈ Dox(e)[six-houses(w)]] Which is also satisfied.

w The contextually specified set of situations is S! = {𝑡3| 𝑗 = 1, … , 6}, Given this, (26) will be satisfied by 𝑓5

%& ⊔ … 𝑓6 %'.

• 8. Bonus: partitions!

w We now consider an alternative way to handle the irrelevant disagreement case. w The underlying intuition is the same: we filter out irrelevant disagreements by their irrelevance. The disagreement between Arnie and Beatrice in case (19) is about Mozart’s birth, not about Paul’s husband. The disagreement between the clients of Sam in case (20)/(21) is about whether Sam will build houses for someone else, not whether Sam will build houses for the clients themselves. We will get the desired disjunction if we can somehow limit ourselves to talk only about Paul’s husband, or about whether Sam will build houses for the clients themselves. w I find it more natural to think about this in terms of the notion of subject matter. My direct reference here is Yablo 2014, but also see Lewis 1988, among other earlier works. Roughly, the idea is that instead

• f thinking about parts of possible worlds, we take subject matters to be different ways of grouping

possible worlds1: A subject matter S induces a partition on the logical space, such that worlds are cell-mates of the partition if and only if they are indiscernible (note that this is an equivalence relation) where that subject matter is concerned. Since partitions are naturally correlated with questions, we may alternatively define subject matter as such: a subject matter S induces a partition on the logical space such that each cell of the partition corresponds to one specific answer to a contextually-specified question about the subject matter S. Two partitions S1 and S2 are orthogonal iff each cell of P1 overlaps with each cell of P2. Intuitively, this means that every possible answer of the two relevant questions are compatible. w To apply this idea to the case (19) for a test: The partition Mozart’s birth is one in which each cell corresponds to a specific answer to the question when Mozart was burn. The partition origin of Paul’s husband is one in which each cell corresponds to a specific answer to the question where Paul’s husband is from. Likewise, the partition wealth of Paul’s husband has its cells corresponding to whether Paul’s husband is rich.

1 See Yablo 2014: chap 2 for details about this. Note that the two accounts (i.e. parts of possible worlds versus ways of

grouping possible worlds) would be equivalent if we group possible worlds by some contextually-specified crossworld indiscernibility relation between their parts, but concerns might arise. I won’t pursue the issue any further.

Mozart’s birth is orthogonal with origin of Paul’s husband. For the same reason it is also

• rthogonal to wealth of Paul’s husband. Moreover, origin is orthogonal with wealth, since

wherever you are from you might be rich or poor (let’s assume that there is no logically necessary connection between geometry and economy). w Now we design a function that returns doxastic alternatives of a doxastic state with respect to a specified subject matter Dox1(e): (27) Dox1(e) = {w| w ∈ Cell1 (Dox(e))} Cell1(w) is the cell of partition S where w is located in. Cell1(W) where W is a set of possible worlds is accordingly the cell(s) of partition S in which you may find W-members. Dox(e) is defined “traditionally” as the set of doxastic alternatives compatible with the subject’s (total) doxastic state e. w Intuitively, the idea is that the total doxastic state e of a subject is the self-ascribed “location” of the subject in the logical space. It needs not come down to one specific possible world, of course, but it may come down to a narrow range of possible worlds. In particular, if the subject has some specific belief about S, then she will be located in one specific

• cell. Otherwise she would be located in multiple cells where her doxastic alternatives reside, which

means her beliefs are compatible with multiple ways which S turns out to be. w Now we introduce the semantics of “believe”: (28) [[believe]]! = λp.λe. ∀w ∈ Dox (e)[p(w)] And just take doxastic states of plural subjects to be: (29) Dox1 (e) = BEST(≾{Dox1 (e’) | e’∈Atm(e)}). w We may verify that the foregoing semantics filters out the irrelevant disagreement about Mozart between Arnie and Beatrice! Take S1 to be origin, and S2 to be wealth. Arnie’s doxastic state lies in the cell rich of partition S2, while Beatrice’s doxastic state lies in the cell New Yorker of partition S1. S1 and S2 are

• rthogonal.

Now, consider Dox15 (𝑓) ⊔ 𝑓0), it is transcribed int BEST(≾{Dox15 (𝑓)), Dox15 (𝑓0)). Dox15 (𝑓0) is just the cell New Yorker of S1, while Dox15 (𝑓)) is “all over the place” (since Arnie is agnostic about origin), including the cell New Yorker. We thus rightly predict that the cumulative belief of Arnie and Beatrice about origin is that the husband is a New Yorker. Similarly we can predict that the cumulative belief of Arnie and Beatrice about wealth is that the husband is rich. The final step is to recover the belief that Paul’s husband is a rich New Yorker from the two aspects

• specified. The natural way to do this is to find the intersection of the two cells rich and New

Yorker …

w We may apply the same strategy to the case of Sam’s six clients each believing that Sam will serve him alone. w There seem to me some way to simplify the account. I won’t pursue them here.