That laws play a special role for the truth of would have conditionals is not at all a new idea. Particularly clear on this point are the proponents of cotenability theories for the meaning of conditionals. The cotenability approach dominated the thinking on the semantics of would have conditionals before the similarity approach came about. The founding work of this paradigm is Goodman (1965). He proposes that a would have conditional is true if the consequent can be derived from general laws and the antecedent plus a set of relevant conditions. Hence, the truth depend, in addition to the propositions expressed by the antecedent and the consequent, on two factors: (i) the set G of general laws (ii) a relevant set S of statements true of the evaluation world. According to Goodman, the problem of the meaning of would have conditionals is to specify these ingredients for a concrete example, in particular, to do so in a non-circular way.
In the seventies there was a strong opposition between the defenders of the similarity approach and the proponent of cotenability theories (see Fine 1975). This may seem surprising, because there is an obvious way to relate the two approaches. We can easily define a similarity relation based on cotenability the-
ory by proposing that the worlds most similar to the evaluation world are the worlds where all general laws hold plus the relevant conditions of the evaluation world. Thus, we can interpret cotenability theories as making the similarity re- lation precise. There is one branch of theories for the semantics of would have conditionals that follows this idea of combining the cotenability approach with the similarity theory. This is premise semantics, introduced by Veltman (1976) and Kratzer (1979, 1981a).9 The basic idea behind premise semantics is this. We
define a function, called by Veltman (1985) the premise function, that, given a set of possible worlds W , maps a member w of this set to a set Pw of propositions
in W . Veltman (1985) interprets Pw as “your stock of beliefs in w”; for Kratzer
(1981) it is “everything which is the case in w”. When a would have conditional is evaluated in w the interpreter tries to verify the consequent on those worlds where the antecedent and as many members of Pw as possible are true.10
The premise semantics rule for would have conditionals
Let W be a set of possible worlds and P be a premise function that maps a possible world w ∈ W on a set of propositions Pw in W . We
say that a set of propositions S make a sentence ψ true, if every world w contained in all propositions in S makes ψ true. We say that a set of propositions S admits the sentence ψ if there is some world w contained in all elements of S that makes ψ true. A would have conditional is true in world w iff every maximal subset of Pw that
admits the antecedent makes the consequent true.
It is easy to see that for every premise function you can find some order such that the premise semantics and the similarity approach evaluate exactly the same would have conditionals as true (see Veltman 1985 for discussion). The problem of specifying similarity now becomes a problem of specifying the premise function. Giving the initial idea of Kratzer that Pw is everything that is the case
in w, one could suggest defining this function as the set of true propositions in w. However, this does not work. As Veltman shows, in this case the truth conditions of would have conditionals reduce to something very similar to strict implication (see Veltman, 1985, proposition II.65). Because the truth conditions of a strict conditional approach are not very satisfying, we have to dismiss this option. We can repair the approach based on Kratzer’s suggestion in two ways. First, we may restrict the facts about the actual world that are in Pw. Second, we may localize
the problem in the way Pw is proposed to contribute to the meaning of would
9The name premise semantics has been introduced in Lewis (1981b).
10For more details see Veltman (1985). For simplicity I have chosen here a formulation of
premise semantics that assumes that the premise function satisfies the limit assumption: each ψ-admitting subset of every set Pw is a subset of some maximal ψ-admitting subset of Pw.
Without this assumption, the formulation of the last sentence in the definition would have to be: A would have conditional is true in world w iff every subset of Pwthat admits the antecedent
can be extended to a set P′
have conditionals. Kratzer (1989) proposes that we have to take both options at once. Kratzer (1989) introduces some general restrictions on the set of facts of the actual world that may be relevant for the truth conditions of would have conditionals. But she also proposes some additional constraints on the subsets of Pw∪ A on which the truth of the consequent is checked, besides consistency. We
will not go into the details of her analysis.11 However, it is interesting to observe
that she emphasizes that non-accidental generalizations, i.e. laws, are always in the set of propositions from which the consequent has to be derivable. Thus, she proposes that general laws are facts that cannot be given up by the similarity relation.
In reaction to the result mentioned above (Veltman 1985, proposition II.65) Veltman (1985) proposes that there have to be some asymmetries between the propositions selected by the premise function, i.e the facts that count for simi- larity. Some may count more than others, some may not count at all. Hence, he suggests that the simple distinction in facts that count for similarity (those in Pw) and facts that do not (those not in Pw) that underlies the rule of premise
semantics has to be given up. Now, we have to distinguish different classes of premises and describe their respective impact on similarity. Similar to Kratzer (1981a, 1989), one of these classes Veltman considers to be the class of laws we consider to be valid in a certain context. Elements of this class cannot be given up at any cost by similarity: “The role which laws – and other propositions we treat as such – play is important, since they determine which possible worlds can enter into the relation of comparative similarity and which cannot. Only those worlds in which the same laws hold as in the actual can.” (Veltman, 1985: 121). But Veltman (1985) realizes that more information about the actual world goes into the evaluation of would have conditionals besides what counts as law. Velt- man (1985) tries to describe this additional information as those characteristics of the evaluation world that the interpreter is acquainted with. This would stand in direct tradition with the Ramsey receipe for the interpretation of conditionals. But he cites an example of Tichy (1976) that shows that this is not the correct way to. The following is a slight variation of the original example from Tichy.
Consider a man - call him Jones – who is possessed of the following disposition as regards wearing his hat. If the man on the news pre- dicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case, ... .
(66) If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat.
11Notice, that there exist some strong objections against her proposal (see Kanazawa et al.
In this context Jones wears his hat is a fact of the actual world that we are aware of. There is no reason why when making minimal amendments to what we are aware of concerning the world described in this context in order to make the antecedent of (66) true, this fact should be given up. But then the would have conditional (66) comes out as true, while intuitively it is false.
Hence, for some reason the fact Jones wears his hat seems to be excluded from the facts that count for similarity. Thus, the criterion of awareness does not work. In his book from 1985 Veltman closes with admitting that he does not know how to distinguish between facts that do count and those that do not.
Let me add a final remark on this example. One may again be tempted to propose that temporal properties of the involved facts are relevant for similarity in this case. One may propose, for instance, that facts about the future of the evaluation time of the antecedent in general do not count for similarity. Then that Mr. Jones is wearing his hat in the evaluation world would have no impact anymore and one would correctly predict that (66) is false in the given context. Example (67) shows that this will not do. The would have conditional (67) is intuitively true. But that means that the outcome of the chance event, that lies in the future of my betting, has to count for similarity. This example clearly shows that the future of the evaluation time of the antecedent matters for similarity.
A coin is going to be thrown and you have bet $5 on heads. Fortu- nately, heads comes up and you win. You say
(67) If I had bet on tails, I would have lost.