2.2 Variably strict counterfactuals
2.2.2 Two-dimensional similarity
As discussed in Ch. 1, Lewis’s (1973) variably strict approach only predicts what we call the low reading. This follows straightforwardly from its semantics, as defined in (48) and (49): Since we are only concerned with the validity of the material conditional in the most similar antecedent-worlds, if there is a particularly likely way of verifying the antecedent, we will only be concerned with the validity of the material conditional in worlds where the antecedent is satisfied in that particular way. However, the entailments in (7) for (6) – repeated below as (53) and (54) – suggest that we do in fact need to consider at least one world per individual that satisfies the indefinite noun phrase in the high reading.
(53) If John had owned axdonkey, hejwould have beaten itx. (54) a. If John had owned donkeya, John would have beatena.
b. If John had owned donkeyb, John would have beatenb.
c. If John had owned donkeyc, John would have beatenc.
d. If John had owned donkeyd, John would have beatend.
e. etc.
As van Rooij (2006) points out, this suggests an equivalence very similar to the one observed in dynamic semantics for indicative donkey sentences – see (33) –, but in the counterfactual domain:
(55) ∃xPxQx≡ ∀x(Px Qx)
In order to obtain the high reading, van Rooij (2006) combines Lewis’s (1973) variably strict semantics with DPL. Where in the standard static semantics – and consequently, the Lewisian analysis –, the meaning of a sentence could be thought of as a set of worlds (i.e. the worlds that verify the truth conditions), the dynamic analysis treats sentences as functions
from sets of (input) assignments to sets of (output) assignments. Combin-ing these views yields a picture in which sentence meanCombin-ings are functions from world-assignment pairs to world-assignment pairs, where the up-date can both be eliminatory (removing world-assignment pairs that do not verify the sentence) and enriching (where the sentence updates the assignment, e.g. in the case of existential quantification).
Consequently, in considering the semantics of counterfactuals, we need to reconsider similarity: Instead of simply relating worlds, it now has to relate world-assignment pairs. And instead of the one-dimensional ordering in (50), we now obtain a set of orderings that can be represented by a two-dimensional picture, with the original sphere model on the x-, and the set of input assignments on the y-axis. Note that the assignments in this example essentially correspond to the various donkeys the indefinite could refer to, that is, for (53), we can assume that g1 = ga/x,g2 = gb/x, etc. As worlds are ordered by similarity, and as we inherit the original ordering, the x-axis in the picture is ordered by similarity from left to right, essentially a copy of the one-dimensional picture in (50). However, assignments are not ordered in any particular way, so that the order of the elements on the y-axis is arbitrary.
(56) g4
g3
g2
g1
w0 w1 w2, w3, w4 w5, w6 w7, w8
In a system like the one represented by (56), the antecedent of a coun-terfactual is not verified by a world alone, but by a world-assignment pair.
Assuming that p is verified, for example, by {hw2,g1i, hw3,g2i, hw5,g3i, hw5,g2i, hw7,g4i, hw8,g4i}, we can represent the domain for our selection function as follows:
(57)
But now, in contrast to the one-dimensional picture, we are faced with a choice. Our selection function can either remain a classical Lewisian one, returning only the world-assignment pairs from the first sphere (now represented by an entire column instead of a single cell), as in (58). This yields the standard low reading.
Alternatively, we can let the selection function select the world-as-signment pairs from the leftmost cell containing an antecedent-verifying world-assignment pair for each row separately, as in (59). This yields the high reading.
With these graphical representations in mind, let us now approach the formal implementation suggested by van Rooij (2006). The two selection functions represented by (58) and (59) can be based on defining the simi-larity orderings (60) and (61) respectively:
(60) hv,hi≤hloww,gihu,kiiffv<w u
(61) hv,hi≤hhighw,gihu,kiiffh=k∧v<wu
(60) simply lifts the standard Lewisian similarity relation to world-assignment pairs but changes nothing about the conditions: Pairs are compared based on their respective worlds. ≤hloww,gi ranks pairs exactly in the way<w ranks the worlds of those pairs. In (61), however, we add a second condition: h = k, that is, world-assignment pairs can only be ranked with respect to each other if they share an assignment. If they do, they are again ranked based on their worlds, according to <w. This results in a partialization of the standard ordering: We obtain an ordering of pairs for each assignment separately, and the selection function selects the bottom element of all the orderings obtained in this way. This results in quantifying over at least one world-assignment pair for each individual that can be assigned as the referent of an indefinite noun phrase in the antecedent.
In order to generalize these definitions to antecedents with possibly more than one indefinite noun phrase, where each indefinite can obtain either a high or a low reading, we combine (60) and (61) into the lexical entry in (62). In (62), a contextually given setXof variables modulates the behaviour of the similarity relation in the following way: To be compared, world-assignment pairs are required to agree in the values their assignment functions assign to the variables inX.
(62) hv,hi≤hXw,gihu,kiiffh↑X=k↑X∧v<wu
For an emptyX, the requirementh↑X=k↑Xis vacuous, reducing to the low reading. Adding variables toXpartializes the similarity ordering for each such variable, yielding the respective high readings.
Finally, the lexical entries for the counterfactual itself and the selection function, based on the similarity relation defined in (62), are the following3:
(63) ~if p would qhw,gi=1 iff∀hv,hi ∈ fhw,gi(/p/g) :hv,hi ∈/q/g
3 These lexical entries are slightly adapted to make them comparable to the definitions of strict and variably strict approaches in (44) and (48). Specifically, (63) is relativized to an input world-assignment pair; instead of returning the output pair (which would be identical to the input), we consider it to be true iffthere is such an output. For the DPL-style notations, see the following derivation in 2.2.3.
(64) /φ/g ={hu,ki | ∃hv,hi ∈ {hv,hi | v∈W∧h=g}: hu,ki ∈~φ(hv,hi)} (65) fhw,gi(/φ/g)={hv,hi ∈/φ/g :¬∃hu,ki ∈/φ/g: hu,ki≤hXw,gihv,hi}
According to (63), the counterfactual is true relative to a world-assign-ment pairhw,giif and only if all world-assignment pairs returned by the selection function relative tohw,giand the antecedent also verify the con-sequent. Verification is defined as in (64): /φ/g returns the set of pairs obtained by interpretingφrelative to the set of all world-assignment pairs that have g as their assignment, while their worlds can be any member of W. This ensures that we can assess non-actual worlds. The selection function returns all those world-assignment pairs that verify the supp-plied antecedent for which there is no other antecedent-verifying pair that is more similar to hw,giaccording to the X-relative similarity relation in (62)4.