Lumping and Grounding:A Critical Defence of Premise Semantics for Counterfactuals

Duen-Min Deng

Department of Philosophy,National Taiwan University

dmdeng@ntu.edu.tw

Lumping and Grounding:A Critical Defence of Premise Semantics for Counterfactuals

Duen-Min Deng

Department of Philosophy,National Taiwan University

dmdeng@ntu.edu.tw

.The basic idea of premise semantics is roughly this:A counterfactual conditional asserts that its consequent follows from its antecedent when we add some suitable premises. To make it a viable analysis,Kratzer(1981,1989)introduces the notion of‘premise sets’representing suitable ways of adding premises,such that‘p□→q’is true iff every premise set is a subset of some premise set that logically implies the consequent q.The project,then,is to find appropriate constraints on premise sets that can give us the correct truth-conditions for counterfactuals.

Kratzer(1989)proposes invoking the notion of‘lumping’to specify the required constraints on premise sets.The idea is that propositions are sometimes‘lumped’together,such that when we add one proposition in our counterfactual reasoning we will thereby add all the propositions it lumps.The lumping relation is then defined under the framework of a situation semantics by some part-whole relation on situations.This approach has the advantage of being able to solve many problems that Kratzer’s previous accounts encounter.But it also suffers from enormous difficulties,as Kanazawa et al.(2005)show in their triviality results.Kratzer herselfalsoabandonedthisapproachverysoon(Kratzer1990,2002),reformulatingherpremise semantics in terms of the notion of‘natural’propositions instead of the lumping relations on propositions.

In this paper,I will offer a critical defence of Kratzer’s‘lumping’semantics in her 1989 version.My proposal is that we understand the notion of lumping in terms of the notion of‘grounding’instead of the part-whole relations on situations.More precisely,on my proposal a proposition p lumps a proposition q if and only if p is grounded in q,and accordingly,when we add a proposition into a counterfactual scenario we will thereby add all the propositions that are its grounds.In this way,I argue,we can accommodate all the examples that motivate the lumping semantics,whilst avoiding the triviality results of Kanazawa et al.(2005).

1 Premise Semantics and Its Motivations

Thebasicideaofpremisesemanticsisroughlythis:Acounterfactualconditional asserts that its consequent follows from its antecedent with some suitable premises. For instance,we think that the counterfactual conditional‘If the match had been scratched,it would have lighted’asserts something true,because we think that the match’s lighting follows logically from its being scratched with some suitable premises,including some background facts about the match and the laws of nature thatgovern it.This idea is intuitively plausible.The problem is that we still need to specify what count as‘suitable’premises to be added in our counterfactual reasoning.

Quite naturally,the premises to be added in our counterfactual reasoning should berequiredtobetrueinourworldandcompatiblewiththeantecedent.Butinaddition tothesetwoconstraints,whatelseshouldwerequireoftheaddedpremisestobecount as suitable?An early attempt to answer this question was proposed by Goodman in [3].On this view,besides truth and compatibility with the antecedent,the added premises are required also to be cotenable with the antecedent,in the sense that they should not be those propositions that could be false if the antecedent were true.1In fact,Goodman’s definition states that q is cotenable with p iff it is not the case that q would not be true if p were.([3],p.15)This amounts to saying that q is cotenable with p iff～(p□→～q), which is equivalent to p?→q.However,we have reason to think that something stronger is required for cotenability.For even if q might be true if p were,it could still be the case q might be false if p were.In such cases,we are still not justified in adding q in our counterfactual reasoning.So I here take a slightly different definition from Goodman’s:On my definition,q is cotenable with p iff it is not the case that q might be false if p were,i.e.,～(p?→～q),which is equivalent to p□→q.For suppose the added premises might be false if the antecedent were true,we would no longer regard the counterfactual as asserting something true simply on the basis that its antecedent plus the added premises logically imply its consequent.So,it is plausible to exclude propositions not cotenable with the antecedent from the added premises.

For instance,consider the following counterfactual conditional:

(1)If Boris had gone to the party,he would have met Olga there.

Suppose that Boris did not come to the party but Olga did,and suppose that the party wassmall,suchthatallpeopleinthepartymeteachother.Thefollowingpropositions are thus true:

(2) a.Boris was not in the party.

b.Olga was in the party.

c.All people in the party met each other.

Based on this information,we naturally regard(1)as true because its consequent follows logically from its antecedent together with the true premises(2b)and(2c). However,suppose Olga wanted to avoid Boris.In this case,we would no longer regard(1)as true.Even if Boris had gone to the party,he would not have met Olga there,for she would not have been in the party if Boris had gone there.In Goodman’s term,(2b)is not cotenable with the antecedent of(1),and hence should be excluded from the premises to be added in our evaluation of(1).

Now,apremisesemanticsbasedonGoodman’saccountcanbeformalisedasfollows.2The formalisation is reconstructed based on Kratzer’s semantics and the terminology used in[5]to facilitate our discussions.To evaluate any counterfactual conditional,our conversational context shouldprovide us with a set F of true propositions relevant for our evaluation.Call such a set F a base set.(In the above example,the base set in our context should include at least 2a,2b,and 2c.)To evaluate whether a counterfactual conditional is true(relative to a base set),we add propositions from the base set F to our premises to see whether they, together with the antecedent,should logically imply the consequent.On Goodman’s account,the added propositions are required to be cotenable with the antecedent.We maythusdefineaGoodmanpremiseset,whichissupposedtorepresentanysuchway ofaddingpropositionsmeetingthecotenabilityconstrainttoourpremises,asfollows.

The Goodman Crucial Set and Premise Sets

Let F be a base set provided by the context,and p be a proposition. We define GF,p,called the Goodman crucial set,as the set of all subsets A of F∪{p}satisfying the following conditions:

Intuitively,acounterfactualconditionalistrueifitsconsequentfollowslogicallyfrom the maximal premise sets—i.e.,if its consequent follows when we add as many suitable premises as consistency allows to its antecedent.But such a characterisation would require the existence of maximal premise sets,which for some technical reason we are not guaranteed to have.As a remedy,we may formulate it instead by saying that a counterfactual is true iff every premise set can be extended to a premise set that logically imply the consequent.So we have the following truth-conditions for Goodman’s premise semantics.

Goodman’s premise semantics

Let F be a base set provided by the context.Then a counterfactual conditional p□→q is true(in this context)if and only if for every A in GF,p,there is some B in GF,psuch that A?B and B logically implies q.

However,this premise semantics is utterly unsatisfactory,as Goodman himself notices.For in order to apply it to evaluate whether p□→q is true,we need to determine first which propositions in the base set are cotenable with p(so as to determine the Goodman crucial set).But to determine which propositions in the base set are cotenable with p,we need to evaluate the truth or falsity of some counterfactual conditionals again.For according to our definition,any proposition r is said to be cotenable with p if and only if it is not the case that r might be false if p were true, or equivalently,iff p□→r is true.But to evaluate whether p□→r is true on this semantics,we again need to know which propositions in the base set are cotenable withp.So we are led to an infinite regress or a circle,and can never apply this semantics to evaluate any counterfactual conditional.

Such a‘cotenability’problem is well-known.One common response is to suggest that we abandon premise semantics and try something else.So an alternative semantics was developed by Lewis and Stalnaker,the central idea of which is to invoke a set of entities called possible worlds,together with a system of ordering imposed on it.On this view,a counterfactual conditional p□→q is true in a world w if and only if some world where p and q are both true is closer to w(according to the ordering system)than any world where p is true but q is false.Which world is closer to w will then be determined by their overall similarity to w.

So how does the possible-worlds semantics as developed by Lewis and Stalnaker avoid the‘cotenability’problem that confronts Goodman’s semantics?As we have seen,the root of the problem is that in Goodman’s semantics the truthconditions we provide for counterfactuals directly rely on something about the truth or falsity of counterfactuals,thus rendering it regressive or circular.Now,in Lewis and Stalnaker’s semantics,the truth-conditions are specified in terms of the similarities amongst possible worlds,which do not directly involve the truth or falsity of counterfactuals.This avoids the‘cotenability’problem,but still faces difficulties. For in order for the semantics to yield the correct predictions,the invoked notion of‘similarity’must be a technical one quite unlike our intuitive idea of similarity.But what can that be?Lewis in[10]has attempted a rather complicated story about how to measure the similarities amongst worlds to get the truth-conditions of counterfactualsright,but manyremainscepticalandunsatisfied.Duetothelimitedspacehere,I cannot go through the details of Lewis’s story and its difficulties,which have already been much discussed in the literature.But I would just mention one case against Lewis that will be relevant to our discussions below,which was raised in[5].

Suppose a zebra escaped from the Hamburg zoo due to a forgetful keeper who forgottoclosethedoorofacompoundcontainingzebras,giraffes,andgazelles.Now, consider the following counterfactual conditionals:

(3)If a different animal had escaped instead,it would have been a zebra.

(4)If a different animal had escaped instead,it might have been a gazelle.

Intuitively,(3)is false and(4)is true in our context,but it is difficult to see how Lewis’saccountcanaccommodatethiswithastoryaboutsimilaritiesamongstworlds. For in the actual world it is a zebra that escaped,and so intuitively,a world where the escaping animal was a different zebra should be more similar to the actual world than aworldwheretheescapinganimalwasofadifferentspecies.Ifso,Lewis’ssemantics should predict that(3)is true and(4)is false,which seems wrong.To make the right prediction,we should have a notion of‘similarity’on which the apparent similarities with the actual escaping zebra do not count.But it is very difficult to see what such a notion of‘similarity’can be:‘It must then be a very special sort of similarity’([5],p.626).

The zebra case therefore presents a difficulty for Lewis’s semantics.We need an explanation,absent in Lewis’s account,of why similarity with the actual zebra should not count in evaluating such counterfactuals like(3)and(4).It is somehow striking to note that Goodman’s semantics can nevertheless offer some explanation.We can explainwhyinevaluating(3)and(4),wedonot holdfixedtheactual propertiesofthe escaping animal in our counterfactual scenarios.For in evaluating(3)and(4),such propositions that describe the actual features of the escaping animal are not allowed to be added to our premises:

(5) a.The animal that escaped was striped.

b.The animal that escaped was black and white.

c.The animal that escaped was a zebra.

……

The reason is that these propositions are not contenable with the antecedent of(3)and (4),in the sense that they could fail to be true if the antecedent were true.That is to say,the following seem true:

(6) a.If a different animal had escaped instead,it might not have been striped.

b.If a different animal had escaped instead,it might not have been black andwhite.

c.If a different animal had escaped instead,it might not have been a zebra.

……

This explains why in evaluating(3)and(4),we do not hold fixed the actual features of the escaping animal,as they are not cotenable with our counterfactual supposition.

Such an explanation,of course,is not completely satisfactory due to the cotenability problem.For to say that(5c)is not cotenable with the antecedent,according to our definition,is just to say that(6c)is true,which is equivalent to saying that(3)is false.Here the circularity is quite obvious.However,if we can have an explanation for why such propositions like(5c)should be excluded from the added premises without relying on our intuitions about counterfactuals,it seems that we can still have a defensible approach.Kratzer offered one such account in[5],which invokes the notion of‘lumping’instead of the problematic idea of cotenability.The idea is that propositions are sometimes‘lumped’together,such that when we add one proposition in our counterfactual reasoning we will thereby add all the propositions it lumps. In this way,(5c)can be excluded from the added premises,not because it is not cotenable with the antecedent,but because it lumps some propositions incompatible with the antecedent.The detail of this account will be examined in the next section.

2 Lumping and Its Formality

Intheprevioussection,wehaveexplainedthebasicideaofpremisesemantics:a counterfactual conditional is true if and only if its consequent follows logically from its antecedent with some suitable premises.Intuitively,the suitable premises to be added in our counterfactual reasoning are required to be true in our world and compatiblewiththeantecedent.Butinadditiontothesetworequirements,thereshouldbe some more constraints.We have seen that Goodman tried to capture the constraints by requiring the added premises to be‘cotenable’:they should in some way remain true in our counterfactual scenarios.But the difficulty is that we seem to have no non-circular specification of cotenability without relying on the very notion of counterfactual conditionals.

Ifcotenabilityfailstocapturetherequiredconstraints,whatelsecando?Kratzer in[5]proposes invoking the idea of‘lumping’to capture the constraints.The basic idea is that propositions are sometimes lumped together,such that when we add one proposition to our premises we will thereby add all the propositions it lumps.As a result,some propositions may be true in our world and compatible with the antecedent, yet are unsuitable for being added in our premises because they lump some other propositions that will contradict the antecedent.

To characterise the notion of lumping formally,Kratzer appeals to a situation semantics.The idea is that propositions can be true or false not only in possible worlds,butalsoinpartsofpossibleworlds,whichshecalls‘situations’.Situationsare partsofpossibleworldswithacertainmereologicalstructure:somesituationsmaybe parts of larger situations,and some situations may contain smaller situations as parts. Situations that are thus related by the part-whole relation are not wholly distinct,and hence the corresponding propositions are in some sense‘lumped’together according to the mereological structure of the situations.So if we identify a proposition,not as a setofpossibleworlds,butasasetofpossiblesituations(i.e.,partsofpossibleworlds) in which it is true,we may define a lumping relationship based on the mereological structure of the possible situations.For instance,suppose Paula painted a still life by painting some apples and some bananas.The situation in which Paula was painting the apples is therefore contained as a part of the larger situation in which Paula was painting the still life.In this sense,we may say the proposition that Paula painted a still life lumps(in our world)the proposition that Paula painted apples,and thus in adding the former to our premises we will in some sense also add the latter as a part. To capture this formally,Kratzer offers the following definition([5],p.611):

Lumping

A proposition p lumps a proposition q in a world w if and only if(i) and(ii)both hold:

(i)p is true in w.

(ii)Whenever a situation s is part of w and p is true in s,then q is true in s as well.

So,inourexample,thepropositionthatPaulapaintedastilllifelumpstheproposition that Paula painted apples in our world,for they are true in our world,and the actual situationofourworldwherePaulawaspaintingastilllifecontainsthesituationwhere Paula was painting apples as a part,hence also makes the latter proposition true.

The idea of lumping also provides a nice explanation for our intuitions in the zebra example.Recall that in the zebra case,we intuitively think that the actual features of the escaping zebra should be irrelevant in considering the counterfactual scenario where a different animal had escaped.So we need an explanation for why in so conceiving,we should not add any of such propositions as:

(5) a.The animal that escaped was striped.

b.The animal that escaped was black and white.

c.The animal that escaped was a zebra.

……

We now have a nice explanation in terms of lumping.Suppose the zebra that actually escaped was named‘John’.Then we can show that each of the propositions listed above lumps the propositions that John escaped,for what makes any of these propositions true in our world is supposed to be the actual situation where John escaped from the zoo.This explains why we should not add these propositions,for they lump the proposition that John escaped,which contradicts the counterfactual supposition.

With the notion of lumping in hand,Kratzer then offers a premise semantics accordingly.The idea,again,is that we define‘premise sets’to represent suitable ways of adding propositions to our premises,and then use the set of premise sets to provide truth-conditions for counterfactual conditionals.3See([5],p.635).Here I make some slight modification to fit our discussions.

The Kratzer Crucial Set and Premise Sets

Let F be a base set provided by the context,and p be a proposition. We define KF,p,called the Kratzer crucial set,as the set of all subsets A of F∪{p}satisfying the following conditions:

(i)A is consistent;

(ii)p∈A;

(iii)A is closed under lumping;i.e.,for any q,r∈F,if q∈A and q lumps r in our world,then r∈A;

(iv)*A is closed under logical consequence;i.e.,for any B?A and any q∈F,if B logically implies q,then q∈A.4The requirement of closure under logical consequence may appear to be questionable,and it does lead to some problems(see[4]and the discussions in the next section).Kratzer’s main motivation for making this requirement is to deal with the King Ludwig case(see[5],p.640f).However,her treatment there is still not completely satisfactory.This eventually led Kratzer to remove this logical closure requirement in her 2012 reprinted version of the paper and tried an alternative treatment of the King Ludwig case(see[7],pp.133,143f).

Any such set A in KF,pis called a Kratzer premise set.

Kratzer’s premise semantics

Let F be a base set provided by the context.Then a counterfactual conditional p□→q is true(in this context)if and only if for every A∈KF,p,there is some B∈KF,psuch that A?B and B logically implies q.

Let us use an example to see how the semantics works.5See([5],p.628).NoticethatIhavemadesomeslightmodificationsofthecasetofitmydiscussions here.Suppose Paula and Otto are the only persons in this room,and they are both painters.Suppose Clara is a sculptor,not a painter.So our base set should contain(at least)the following propositions:

(7) a.Paula is in this room.

b.Otto is in this room.

c.Paula and Otto are the only persons in this room.

d.All persons in this room are painters.

e.Clara is not a painter.

Now,consider the following counterfactuals:

(8)If Clara were also in this room,she would(still)not be a painter.

(9)If Clara were also in this room,she might be a painter.

Here the antecedent of the counterfactuals is this:

(10)Clara is in this room.

Intuitively,(8)is true and(9)is false.But presumably,any set that contains(10)and (7d)will have no consistent superset that logically implies that Clara is not a painter. So if we are to make the correct prediction,we should exclude(7d)from being added to our premise sets.Kratzer’s lumping semantics can explain this.For arguably,(7d) is true in all and only those actual situations that contain this room as a part,and(7a), (7b),and(7c)are all true in all these situations.So,arguably,(7d)lumps(7a),(7b), and(7c)in our world.But(7a),(7b),and(7c)form a set that is inconsistent with our antecedent(10),and hence no Kratzer premise set should include(7d),lest it lump propositions inconsistent with our antecedent.

So,what should Kratzer’s lumping semantics predict about(8)and(9)?As we have explained,(7d)lumps(7a),(7b),and(7c)in our world,and hence should not be included in any premise set.For similar reasons,(7c)lumps(7a)and(7b)in ourworld,and should not be included in any premise set either.But no further lumping relationship holds amongst these propositions.So,presumably,all the propositions listed in(7)except(7c)and(7d)can be added to our premise sets,and hence every premise set can be extended to a larger premise set that includes(7e).As a result,(8) is true and(9)is false according to Kratzer’s semantics.

Now,compare this with another case(see[3],p.37).Suppose again that Paula and Otto are the only persons in this room,and suppose that everyone in this room is safe from freezing.Consider a certain Eskimo,say Eason,who is at this moment nearly frozen to death somewhere in the Arctic.Intuitively,we should regard the following counterfactual as true:

(11)If Eason were now in this room,he would be safe from freezing.

The case appears to be structurally similar to the previous one.Our base set includes (at least)the following propositions:

(12) a.Paula is in this room.

b.Otto is in this room.

c.Paula and Otto are the only persons in this room.

d.All persons in this room are safe from freezing.

e.Eason is nearly frozen to death.

Now if the case is really the same as the previous one,we might similarly expect that (12d)should lump(12a),(12b)and(12c),and hence should be unsuitable for being added in any Kratzer premise set on pain of inconsistency.But then,the semantics would predict(11)to be false,which is contrary to our intuition.

To deal with such cases like this,Kratzer suggests making a distinction between two readings of a universally quantified sentence(see[5],pp.620ff).Roughly speaking,on an accidental reading,a universally quantified sentence‘All Fs are G’is true inasituationsonlyifsisbigenoughtocontaintheF-situations.Onanon-accidental reading,by contrast,‘All Fs are G’is true in all or none of the situations of a world. This is meant to capture the idea that a non-accidental generalisation asserts some genuine connection on a generic level,and hence should be true everywhere in the world if true at all.So if we take an accidental reading,(12d)is true in all and only those actual situations that contain this room as a part.Since(12a),(12b)and(12c) are all true in these situations,they are indeed lumped by(12d)on such an accidental reading.However,if we understand(12d)as a non-accidental generalisation(which seems more natural here),it should be true in all situations of the actual world.As a result,(12d)on this reading lumps no other propositions in our list according to our definition of lumping,but is lumped by each of them instead.Consequently,each Kratzer premise set that contains any proposition in the list should also contain(12d). But(12d)together with the antecedent of(11)logically implies that Eason is safe from freezing.As a result,(11)is true according to Kratzer’s semantics,which agreeswith our intuition.

Kratzer’s distinction between accidental and non-accidental readings is intended to capture the idea that in considering a counterfactual scenario we usually hold the laws of nature fixed.Whereas Goodman explains this idea in terms of cotenability (i.e.,that the laws of nature are usually cotenable with our counterfactual suppositions),Kratzer explains it in terms of lumping(i.e.,that the laws are non-accidental generalisationswhichlumpveryfewpropositionsbutarelumpedbyeverytrueproposition).We usually can bring lawlike generalisations into counterfactual scenarios because they usually lumps nothing that will contradict the counterfactual suppositions.We usually do bring lawlike generalisations into counterfactual scenarios because they are lumped by every true proposition,such that in adding any proposition at all we thereby also add the lawlike generalisations it lumps.In this way,the roles that laws of nature are supposed to play in our counterfactual reasoning can be nicely explained by their lumping features.

Kratzer’s‘lumping’semantics therefore presents a promising approach to work out the idea of premise semantics whilst avoiding the problems confronting Goodman’s account.But there may still be some worries.In the next section,I will briefly examine a problem presented in[4],and then raise my own worries about lumping.

3 Worries about the Lumping Semantics

In[4],some‘triviality’resultsarepresentedbyKanazawatoshowthatKratzer’s semantics fails to rule out certain problematic cases that would collapse the truthconditions of counterfactuals to triviality.The problematic cases are roughly as follows.

Consider the tautologous proposition T that is true in all possible worlds but in no situations strictly smaller than a world.Consider also the‘world-proposition’pwthat is true in w and in no other world.Now,suppose w is our actual world.Certainly, both T and pware true propositions in our world.If our base set contains T and pwas relevant propositions,then it can be shown that Kratzer’s truth-conditions would collapse to triviality.For consider a counterfactual q□→r,where q is in fact false in our world w.Then there will be no Kratzer premise set at all that satisfies the constraints Kratzer demands.Recall that a Kratzer premise set A for this counterfactual is supposed to meet the following conditions:

(i)A is consistent;

(ii)q∈A;

(iii)A is closed under lumping;i.e.,for any p1,p2∈F,if p1∈A and p1lumps p2in our world,then p2∈A;

(iv)A is closed under logical consequence;i.e.,for any B?A and any p∈F,if B logically implies p,then p∈A.Now,since T is tautologous,T is logically implied by any set,the empty set included. So by(iv),we should have T∈A.But since T is true in no situations but worlds,T lumps pwin our world w because pwis true in the only situation of w where T is true. But then,by(iii),we should have pw∈A.However,by(ii)we also have q∈A.But there is no world in which both pwand q are true(because pwis true in w and in no other world,and q is false in w).This contradicts(i).As a result,there is no Kratzer premise set that satisfies all the constraints(i)–(iv),and consequently,Krazter’s truthconditions for counterfactuals in terms of Kratzer premise sets collapse to triviality.

To this triviality result,it can be replied that the proposition pwin question is highly dubious,and hence no conversational context would include it as a relevant proposition in our base set.But Kanazawa et al.have a second triviality case that invokes no such proposition.Again,consider the counterfactual q□→r,where q is in fact false in our world w.Accordingly,both～q and q∨～q are true in our world w.Now,if our base set contains these two propositions(i.e.,～q and q∨～q) as relevant propositions,then it can be shown again that Kratzer’s truth-conditions would collapse to triviality.For arguably,the disjunctive proposition q∨～q lumps～q in our world w(because what makes the disjunction true in the actual world is the situation where～q is true).So for any putative Kratzer premise set A,by the fourth condition(iv),the tautology q∨～q should be in A;but then we should have～q∈A by the condition(iii)about closure under lumping.As a result,A should contain both q and～q,which violates the first condition about consistency.Consequently, thereisnoKratzerpremisesetthatsatisfiesalltheconditions(i)–(iv),whichcollapses Kratzer’s truth-conditions again to triviality.

How damaging are these triviality results?In a certain sense,they seem to be quite harmless,for we can always have our base set not to include such tautologous propositions like T or q∨～q.The triviality results critically rely on there being these propositions to lump something that will contradict the antecedent,6In fact[4]has a third triviality case that invokes no tautologous propositions.However,the case is even much less damaging than the first two,for it requires the base set to include still more propositions that may look quite unnatural.and hence by having our base set to include no such propositions,we can be immune from the problem.But on the other hand,do we really have any positive reason to banish them from our base set?In her original paper,Kratzer only provides some very rough idea about what constraints we should impose on the base set:the propositions in the base set should be true in our world,humanly graspable,and persistent7A proposition is persistent just in case whenever it is true in a situation,it is also true in all larger situations that contain it as a part.For more discussions,see[5].([5],p. 634)(where these three constraints are not meant to be exhaustive).But this does not clearly rule out such propositions like T or q∨～q.So it may seem that Kratzer’s semantics is still not completely satisfactory.

However,even in her original formulation,Kratzer already has some tools thatcansatisfactorilydealwiththeproblematiccasesraisedbyKanazawaetal.Theideais torecognisethatinasituationsemantics,tautologiescanbeambiguouswithrespectto theirlumpingcapacities.Forinstance,letT(asbefore)bethetautologousproposition that is true in all possible worlds but in no situations strictly smaller than a world; and let Tsbe the tautologous proposition that is true in all possible worlds and in all situations thereof.Then T and Tshave quite different lumping capacities:in any world w,T lumps every true proposition,whereas Tslumps only those true nonaccidental propositions,8Recall that a proposition is non-accidental if and only if it is true in all or none of the situations of a world.butislumpedbyeverytrueproposition.Logicaltautologies like Tsare very poor lumpers,and can be added to our premise sets without any problem(just as any other lawlike generalisations).But T is quite different;we have every reason to forbid adding it to our premise sets lest it lump all true propositions. Similarly,q∨～q,if interpreted as a non-accidental proposition that is true in all situations,can be added to our premise set without causing any problem;but if it is interpreted on an accidental reading as Kanazawa et al.take it to be,we have good reason to forbid adding it to our premise sets.So,this is how Kratzer replies to the triviality cases in a footnote to the revised version of the paper.

…Since logical tautologies are non-accidental generalisation par excellence,Kanazawa et al.made a mistake by putting[T],rather than[Ts], in some of the critical premise sets they consider.[T]is an extremely strong lumper that will wreak havoc in any premise set for counterfactuals.That’s the kind of phenomenon that lumping semantics is all about [9,p.134 fn17].

In a certain sense,Kratzer’s reply is just to dismiss those‘extremely strong lumpers’(e.g.,T or q∨～q interpreted accidentally)as highly dubious propositions to be banished from our consideration,for they will only‘wreak havoc in any premise setforcounterfactuals’.Whilstthiscouldbeasensiblewayofimprovingthelumping semantics,it also points to a very different approach to premise semantics in general. For if problematic propositions are to be ruled out anyway,we may also banish them in a much more direct manner than invoking the‘closure under lumping’requirement to constrain our premise sets.This leads Kratzer to think that she can have a better proposal,and she does develop such a new approach very soon in[8]and[7],as she reports later:

I abandoned the approach of[5]as early as 1990 because I had found a better way of characterising the constraints for admissible premise sets. ([6],p.153)

The main idea of[8]and[7]was to construct premise sets for counterfactuals in such a way that whenever they contain propositions p andq such that p lumps q,p also logically implies q.The requirement that premise sets be closed under lumping became superfluous,then.From the present perspective,the work reported in[5]was a useful intermediate step that eventually helped me find a way of getting rid of closure under lumping…The unpleasant results of[4]are all brought about by applying the closure conditions of[5]to premise sets containing propositions like singletons,tautologies,negations,and disjunctions.The new approach excluded the dangerous kinds of premises on general grounds, and at the same time eliminated the need for closure under lumping,too. ([6],p.154)

On this new approach,premise sets are to be constructed from‘natural’propositions only.(Roughly,a proposition is natural if and only if it can be projected from a single actual situation s,in the sense that it should be true only in those situations that are maximally similar to some actual situation that contains s as a part.)In this way, we can directly rule out such problematic propositions like T,or accidental disjunctions like q∨～q,or accidental generalisations like(7d),etc.,by showing that they are not‘natural’propositions thus projectable from a single actual situation.As a result, we can drop the‘closure under lumping’condition altogether from our formulation of the semantics.In this sense,this new approach can provide an elegant formulation of premise semantics.

Now,it is beyond the scope of the present paper to examine this new theory in more detail.But notice that this theory will demand a metaphysics of situations to help us pick up the right kind of‘natural’propositions to construct premise sets:We need situations in our ontology(in the sense that our ontology has to include some entities called‘possible worlds’and some entities called‘situations’that are parts of possible worlds);and we need situations to be properly structured by a part-whole relation and also by a resemblance relation.This is a metaphysical price we need to pay if we want to rule out problematic premises in this way.But it also makes us wonder whether this is a price worth paying.

So here is my general worry for Kratzer’s premise semantics,which is a concern about its commitment to possible worlds and situations.For in a certain sense, premise semantics can be very attractive to philosophers mainly because it offers a viable alternative to the standard‘possible-worlds’semantics.It offers us,so to speak,a semantics for counterfactuals without possible worlds.To evaluate counterfactual conditionals on this semantics,all we need to do is consider what logically follows from the antecedent if we add suitable premises,rather than compare and arrange some putative entities called‘possible worlds’according to some murky notion of‘similarity’for worlds.Surely we still need to have some way to constrain the premises to be added,and the idea of having a‘lumping’structure amongst propositions remains to be a very natural proposal.But precisely at this point,Kratzer goesback to the resource of possible worlds,invoking situations(i.e.,parts of worlds)to help her formulate the required constraints.She tries to define lumping in terms of the mereological structures of situations by which to constrain the premise sets accordingly,and she even tries to get away with lumping and formulate the constraints directly in terms of features of situations.So in the end we still need to compare situations according to their similarities and mereological structures in order for this semantics to work.This may not be a problem if one already has good reason to accept an ontology of possible worlds and situations.But it does undermine a motivation for premise semantics that some may find attractive:i.e.,to have a viable alternative semantics for counterfactuals without possible worlds.

This is not to say that a commitment to possible worlds and their associated metaphysics is so bad that we should try our best to avoid it.Some people may find Lewis’s semantics attractive precisely because its associated Humean metaphysics looks promising to them.Comparing possible worlds or parts of possible worlds according to their intrinsic similarities with regard to local arrangement of qualities may sound a very sensible way of doing semantics.One philosopher’s metaphysical burden can turn out to be another philosopher’s metaphysical benefit,and vice versa. But still,if we can have a semantics for counterfactuals without possible worlds,we can offer an alternative at least for those who are unhappy with possible worlds.

In the next section,I will offer such an alternative semantics without appealing to possible worlds.The idea is to develop premise semantics along the‘lumping’approach that Kratzer originally proposes in[5],and to understand the lumping of propositions directly as a kind of grounding structure amongst propositions rather than in terms of the part-whole relationship of situations.I will argue that it can accommodate all the examples that motivate Kratzer’s semantics without committing us to an ontology of worlds or situations.It therefore offers a viable alternative,or so I claim.

4 Premise Semantics Based on Grounding

In this section,a premise semantics for counterfactuals based on grounding will be provided as a viable alternative.Here is an informal picture.A counterfactual conditionalistrueifandonlyifitsconsequentfollowslogicallyfromitsantecedentwhen weaddsomesuitablepremises.AsinKratzer’slumpingsemantics,wehavethesame constraints for suitable premises:(i)they should be true propositions;(ii)they should be compatible with the antecedent;and(iii)they should not lump any propositions that will contradict the antecedent.But instead of invoking an ontology of situations, we analyse lumping in terms of a grounding relation between propositions.Some propositions are grounded in other propositions.They are thus non-fundamental,in the sense that their beingtrueis something that canbeexplained furtherbysomethingelse:they are true in virtue of something else.So,if p is such a proposition that is grounded in other propositions,then adding p in our counterfactual reasoning should require us to add those propositions that are its grounds,such that p can remain true in the counterfactual scenario in virtue of these propositions.In this way,we can define lumping in terms of grounding:for any true propositions p and q,p lumps q if and only if p is grounded in q.

Here I assume that grounding is a factive relation between propositions and is irreflexiveandasymmetric.Morespecifically,Itakeittobeabinaryrelationbetween propositionsinthesensethatitsbasiclogicalformis‘xisgroundediny’or‘y grounds x’,where‘x’and‘y’range over propositions or pluralities of propositions.(When p is grounded in a plurality of propositions one of which is q,we say that q is a partial ground of p,and that p is grounded partially in q.When p is grounded in q but not partially,wesaythatpisgrounded fullyinq.)Itakegroundingtobeafactiverelation in the sense that only true propositions can be grounded and can be the grounds.

Philosophers are still debating about how to characterise the notion of grounding properly.9For example,see[2]and also other papers in[1].Some are more inclined to take grounding primarily as a relation between facts rather than between propositions.Some are inclined to allow grounding to be nonfactive.Some prefer taking grounding as a primitive relation,whilst some are trying to analyse it in other terms.All these are controversial issues,and I have no intentiontosettlethemhere.Forthepurposeofthispaper,however,Icanonlyassume that there is a sensible notion of grounding applying to true propositions.And if you are inclined to take grounding rather as a relation between facts or other fact-like entities,you can still define a derivative notion of‘grounding’that applies to true propositions.

So,what are the propositions that are supposed to be grounded in other propositions,and what are their grounds?Since I take grounding to be a sort of explanatory relationship,I think it depends on what kind of explanation we have in mind.So, depending on the context,there can be a causal sense of grounding,a nomic sense of grounding,an epistemic sense of grounding,a metaphysical sense of grounding,and so on.But there are also paradigmatic cases of grounding:(i)an(accidental)disjunction is fully grounded in its true disjuncts(i.e.,p∨q is fully grounded in p,given that p is true);(ii)a conjunction is partially grounded in its conjuncts(i.e.,p∧q is partially grounded in p,given that p∧q is true);(iii)an(accidental)existential generalisation is fully grounded in its true instances(i.e.,(?x)φx is fully grounded in φa,given that φa is true);(iv)an(accidental)universal generalisation is partially grounded in its instances(i.e.,(?x)φx is partially grounded in φa,given that(?x)φx is true).

With the notion of grounding in hand,we can now offer an official definition for lumping:

Lumping(a grounding-based account)

A proposition p lumps a proposition q(relative to a context)if and only if

(i)p and q are true;and

(ii)p is fully or partially grounded in q(according to the grounding relation as is understood in the context).

ThisdefinitioncapturessomeofKratzer’sintuitions.Kratzerthinksthataproposition exemplifying a whole situation lumps those propositions exemplifying its parts.We canexplainthisintermsofgrounding.Forarguably,thewholeisgroundedinitsparts. Soapropositionexemplifyingthewholeisgroundedinthepropositionsexemplifying itsparts.Forinstance,supposingthatPaulapaintedastilllifebypaintingsomeapples and some bananas,then the proposition that Paula painted a still life is grounded partially in the proposition that Paula painted apples.Our definition therefore yields the desired result that the former lumps the latter.Hence,our definition agrees with Kratzer’s intuitions about lumping.

Our definition also captures some of Goodman’s intuitions.Goodman thinks that if a proposition fails to be cotenable with the antecedent(in the sense that it could fail to be true if the antecedent were true),then it is unsuitable for being added in our counterfactual reasoning.We can explain this in terms of grounding as well.For arguably,failure of cotenability amounts to some sort of counterfactual dependence, which is a good indication of the obtaining of some grounding relation.So for instance,suppose that Boris did not come to the party but Olga did,but Olga wanted to avoid meeting Boris in such a way that she would not have been in the party if Boris had been there.In this case,the proposition that Olga was in the party fails to be cotenable with the counterfactual supposition that Boris came to the party.But the failure of cotenability also shows that Olga’s presence depends counterfactually upon Boris’s absence,which indicates that the proposition that Olga was in the party should be in some sense grounded in the proposition that Boris was absent from the party.According to our definition,then,the former lumps the latter,whilst the latter is incompatible with the counterfactual supposition that Boris came to the party.This explains why such a proposition is unsuitable for being added in our counterfactual reasoning.

We can now offer our official truth-conditions for counterfactual conditionals.

The Crucial Set and Premise Sets(a grounding-based version)

Let F be a base set provided by the context,and p be a proposition. WedefineHF,p,calledthecrucialset,asthesetofallsubsetsAofF∪{p} satisfying the following conditions:

(i)A is consistent

(ii)p∈A

(iii)A is closed under lumping;i.e.,for any q,r∈F,if q∈A and q is fully or partially grounded in r(according to the grounding relation as is understood in the context),then r∈A.

Any such set A in HF,pis called a premise set.

Premise semantics(a grounding-based version)

Let F be a base set provided by the context.Then a counterfactual conditional p□→q is true(in this context)if and only if for every A∈HF,p,there is some B∈HF,psuch that A?B and B logically implies q.

We now revisit the examples that motivates Kratzer’s semantics as discussed in the previous sections to see how our grounding-based semantics works.Let us examine the zebra case first.Recall that we are to evaluate the following counterfactual conditionals:

(3)If a different animal had escaped instead,it would have been a zebra.

(4)If a different animal had escaped instead,it might have been a gazelle.

We wonder why the following propositions should not be included in our premise sets:

(5) a.The animal that escaped was striped.

b.The animal that escaped was black and white.

c.The animal that escaped was a zebra.

……

Our grounding-based account provides such an explanation.Suppose the zebra that actually escaped was named‘John’.Then it is reasonable to say that(5a)is true because John was the animal that escaped and John was striped.So we may say that (5a)is grounded(partially)in the proposition that John was the animal that escaped, and consequently no premise set should include(5a)on pain of inconsistency.For similar reasons,none of(5b),(5c),and so on,can be included in our premise sets,as each of them is also grounded(partially)in the proposition that John was the animal that escaped.As a result,our semantics can therefore correctly predict that(3)is false and(4)is true.

Now,letusturntocasesinvolvingaccidentalandnon-accidentalgeneralisations. In the first case,our base set contains(at least)these relevant propositions:

(7) a.Paula is in this room.

b.Otto is in this room.

c.Paula and Otto are the only persons in this room.

d.All persons in this room are painters.

e.Clara is not a painter.

We are evaluating the following counterfactuals:

(8)If Clara were also in this room,she would(still)not be a painter.

(9)If Clara were also in this room,she might be a painter.

Here the antecedent is this proposition:

(10)Clara is in this room.

We wonder why(7d)should not be included in our premise sets to make us predict (8)false and(9)true.Our grounding-based account provides an explanation.For arguably,all persons in this room are painters because Paula is in this room and is a painter,and Otto is in this room and is a painter,and Paula and Otto are the only persons in this room.So it is reasonable to say that(7d)is grounded(partially)in (7a),(7b),and(7c).But(7a),(7b),and(7c)form a set that is inconsistent with our antecedent(10).So our grounding-based account can explain why(7d)should not be included in any premise set.

Now,consider the second case,which involves a non-accidental generalisation instead of an accidental generalisation.In this case,we are evaluating this counterfactual conditional:

(11)If Eason were now in this room,he would be safe from freezing.

Our base set contains(at least)these relevant propositions:

(12) a.Paula is in this room.

b.Otto is in this room.

c.Paula and Otto are the only persons in this room.

d.All persons in this room are safe from freezing.

e.Eason is nearly frozen to death.

We wonder why in this case,we should add(12d)to our premise sets to predict(11) true.Our grounding-based account also provides an explanation.For if we are to follow our reasoning in the previous case to banish(12d)from our premise sets,we will have to say that(12d)should be grounded in(12a),(12b),and(12c).But this makes the direction of explanation wrong.For unlike the previous case,it is implausible to say that all persons in this room are safe from freezing because Paula is in this room and safe from freezing,and Otto is in this room and safe from freezing,and Paula and Otto are the only persons in this room.On the contrary,we should rather say that Paulaissafefromfreezingbecauseallpersonsinthisroomaresafefromfreezingand Paula is in this room.Presumably,(12d)is intended to be a non-accidental or lawlike generalisation by reference to which we explain some other things,rather than an accidental generalisation which is to be explained by its confirming instances.So it is plausible to assume that(12d)is not grounded in any other propositions listed in (12).Our grounding-based account therefore explains why(12d)can be added to our premise sets without causing any problem.

5 Conclusion

The basic idea of premise semantics is intuitively plausible:A counterfactual conditional is true if and only if its consequent logically follows from its premises together with some suitable premises.But we still need to specify the constraints for what count as suitable premises to be added in our counterfactual reasoning.Goodman offers a‘cotenability’account to capture the constraints,but fails to characterise the very notion of cotenability independently of our intuitions about counterfactual conditionals.Kratzer offers a‘lumping’account to capture the constraints,but her theorycommitsustoametaphysicsofsituations.Inthispaper,Iproposeagroundingbasedaccounttocapturetheconstraint,whichcanavoidinvokingworldsorsituations on the one hand,and can be independent of our intuitions about counterfactuals on the other hand.My grounding-based version of premise semantics therefore provides a viable alternative semantics for counterfactual conditionals.

[1] F.Correia and B.Schnieder(eds.),2012,Metaphysical Grounding:Understanding the Structure of Reality,Cambridge:Cambridge University Press.

[2] K.Fine,2012,“A guide to ground”,in F.Correia and B.Schnieder(eds.),Metaphysical Grounding:Understanding the Structure of Reality,Cambridge:Cambridge University Press.

[3] N.Goodman,1983,Fact,Fiction,andForecast,Cambridge,Mass:HarvardUniversity Press.

[4] M.Kanazawa,S.Kaufmann and S.Peters,2005,“On the lumping semantics of counterfactuals”,Journal of Semantics,22(2),129–151.

[5] A.Kratzer,1989,“An investigation of the lumps of thought”,Linguistics and Philosophy,12(5),607–653.

[6] A.Kratzer,2005,“Constraining premise sets for counterfactuals”,Journal of Semantics,22(2),153–158.

[7] A.Kratzer,2002,“Facts:Particulars or information units?”,Linguistics and Philosophy,25(5-6),655–670.

[8] A.Kratzer,1990,“How specific is a fact?”,In Proceedings of the 1990 Conference on Theories of Partial Information,Center for Cognitive Science and College of Liberal Arts at the University of Texas at Austin.

[9] A.Kratzer,2012,Modals and Conditionals,Oxford:Oxford University Press.

[10] D.Lewis,1979,“Counterfactualdependenceandtime’sarrow”,No?s,13(4),455–476.

聚合與立基：反事實(shí)條件句之前提語義學(xué)的辯護(hù)

鄧敦民

臺(tái)灣大學(xué) 哲學(xué)系

dmdeng@ntu.edu.tw

2016-12-20

前提語義學(xué)的基本想法是主張反事實(shí)條件句宣稱我們可以從前件加上適當(dāng)?shù)那疤嵬茖?dǎo)出后件。Kratzer（1981,1989）發(fā)展了這樣的想法，引進(jìn)了前提集（premise sets）的概念，來表達(dá)我們加入前提的適當(dāng)方式，主張條件句p□→q為真，若且惟若每個(gè)前提集皆可被擴(kuò)張為邏輯上蘊(yùn)含后件的前提集。而Kratzer的計(jì)劃，便是要替前提集找到恰當(dāng)?shù)南拗?，可以使得反事?shí)條件句的真值條件正確。

Kratzer（1989）使用了聚合（lumping）的概念來刻畫前提集的限制。這里的想法是，有些命題會(huì)聚合在一起，使得我們加入一個(gè)為前提，就會(huì)一起加入其余它所聚合的命題。而命題間的聚合關(guān)系，則是用情境語義學(xué)（situation semantics）中，在情境之間的部分與整體來定義。這樣的處理，可以解決Kratzer早期理論遇到的一些問題，但仍會(huì)遇到如Kanazawa等人（2005）所提出的困難。而Kratzer本人也很快的放棄了這條進(jìn)路，改而采用自然命題的概念來重構(gòu)其前提語義學(xué)。

在本文中，我將辯護(hù)Kratzer（1989）的聚合語義學(xué)。我主張將聚合的概念，用立基（grounding）的概念來理解，以取代Kratzer所使用的情境概念。根據(jù)我的定義，命題p聚合命題q，意謂著p立基在q上，因此當(dāng)我們加入一個(gè)命題為前提時(shí)，就會(huì)一起加入它所立基的命題。我主張的這種做法，可以成功處理聚合語義學(xué)的案例，并避開其困難。