Stop beating the donkey! Conditionals, Quantifiers and Donkey Sentences

María J. Frápolli

Department of Philosophy University of Granada (Spain)

frapolli@ugr.es

1. Introduction

Classical predicate logic and the philosophy that supports it have shown to be

powerful tools in the task of analysing how language works. It has been definitive in the

foundations of a new way of thinking that has become known as Analytic Philosophy.

Nevertheless, even the best tools have their limitations, and the predicate calculus is no

exception. All-purpose tools, publicity notwithstanding, do not exist. It is not stubbornly

resisting any change that we best honour these products of human thought that have made

of us what we are. Sometimes our respect is better shown by allowing them a peaceful rest.

Not only political constitutions can be changed without treachery, the scope and limits of

logical calculi can also be exposed without any trace of disgracefulness on our part.

That the first order predicate calculus has its shortcomings when we want to

understand the complexities of natural languages is no news. This is the background in

which extended and non-standard logics find their rationale. But apart from adding new

operators or rejecting some of the rules governing them, there are deeper ways of

contesting the logical path that the predicate calculus has converted in the paradigm of

logical reasoning. All the theoretical approaches that are nowadays classified under the

general heading of “Dynamic Semantics” have displayed the difficulties of a static,

sententially-based, predicate calculus to account for the subtleties of our actual use of

language. Our aim in the following pages will be to focus in a particular kind of sentences,

the so-called “Donkey sentences”, to expose a further point in which the complexities of

natural languages will advise finer and more flexible methods of analysis. Again, nothing

new under the Sun. Logicians, linguists and philosophers in the last century has overtly

defended or covertly suggested that such a change would be desirable.

Donkey sentences have proved to be particularly reluctant to logical treatment. Their

trademark is the co-existence of logical operators, i.e., connectives and quantifiers, together

with anaphoric links among singular terms. The main difficulty for a suitable analysis,

capable of being smoothly accepted by the varied group of theories that have risked a

proposal, lies in the issue of the identification of (i) the operators (and operations) involved

in their logical form and (ii) their relative scope.

Although the focus on Donkey sentences typically suggests concern about anaphora,

we will give Donkey sentences a different use, i. e. we will employ them to illustrate a

general proposal about conditionals. Conditionals, as second-order binary operators, should

be understood, we will maintain, as some kind of quantifiers. The general thesis is the

following, that when there is no adverbial operator that modifies their force, conditionals

are general quantifiers ranging over sets of situations. When adverbially modified, the

quantifier might cease to be a general quantifier and turn into one of the non-standard

weaker quantifiers, “most”, “a few”, “seldom”, and in the limit case of negation, it becomes

an existential quantifier. Conditional sentences are thus quantified sentences, and the

adverb “then” that usually lies implicit in the consequent clause works as an anaphoric pro-

noun. Particles that undertake the task of anaphorically referring to an antecedent are

commonly dubbed as “pronouns”, but in this case the particle “then” that, from a syntactic

point of view is an adverb, acts, from a logical point of view, as an anaphoric “pro-adverb”,

linked to its head, the situation variable bound by the general (conditional) quantifier. From

a semantic point of view, quantifiers, conditionals included, can be seeing as circumstance-

shifting operators; they address the participants in the communicative act to a circumstance

that may differ from the circumstance determined by the context of the utterance. In the

case of conditionals, the new circumstance must be one in which the antecedent of the

conditional holds, and it is the locus in which the proposition in the consequent has to be

assessed.

This way of looking at conditional logic, i.e., considering the conditional particle if

as a universal quantifier with the meaning of “in every (some, almost all, etc.) situation”

and interpreting the particle then as an anaphoric pro-adverb pointing to the situation at

issue, allows to take a fresh look at the problem of Donkey sentences. The argument in this

paper will be thus somehow upwards. Instead of presenting a completely worked out view

on conditionals and quantifiers and going then orderly, step by step, to its consequences, we

will focus on a particularly rebellious corner of the semantic debate and illustrate at it the

view we are putting forward. Donkey sentences will be then our experimentum crucis. If

you become persuaded that a treatment of conditionals as quantifiers together with a

dynamic interpretation of them is a possibility worth to be pursued, presenting then the

whole theory will become much easier. The theoretical framework is available even if, we

acknowledge, it is not systematically presented and developed anywhere. We are convinced

that reconstructing it will worth the effort, although this is not the aim of the present paper.

Here we will only present the main lines.

2. Conditionals as General Quantifiers. The Background Picture

The picture behind the new look at Donkey sentences that we would like to defend

here can be summarize, in a nutshell, in the idea that unmodified1 conditionals are instances

of universal quantifiers (if as an instance of ∀) and that quantifiers have, in natural

languages, an intralinguistic, non-representational, inferential, import. Making the first

claim more precise, conditionals are, if they have to support an inferential interpretation,

irreducible2 n-ary devices, n >1. For the second claim we will not argue in this paper

although it would be needed to bring into the mind a couple of the consequences of it. One

of them is that we reject the common objectual interpretation of quantifiers, its restriction

of only allowing nominal variables and, with it, the connection between quantification and

ontology3. Another one is that quantified sentences do not primarily have the role of

representing reality; they rather display inferential connections from the speaker’s point of

view. Its primary role does not consist in expressing propositions, and thus they can be said

to be true or false only in a derivative way. This point, that might appear revolutionary or

even patently false, is no more than an alternative rewording, from an inferentialist

perspective, of the representationalist Tractarian insight that there are not complex facts.

1 By “unmodified” we understand conditionals in which no expressions occur that constraint the force of the universal quantifier. Some adverbs, like “usually”, “often”, “seldom” etc. indicate that the quantifier at issue is weaker than generality. 2 In the sense explained by Keenan (1987). 3 See Williams 1981, 1989 and 1992 for a defence of this view

Some effects of these consequences will be patent in the interpretations of Donkey

sentences in section 3 below. Nevertheless our particular view of logical constants as

intralinguistic devices is independent of the task of identifying the logical structure of

Donkey sentences although it affects our global understanding of them.

That conditionals are some sort of quantifiers is something that has been introduced

to the philosophical arena by van Benthem in his (1984). Conditional logic, as it has been

known the research programme initiated by van Benthem’s paper, is the logic of

conditional statements. The author begins with the following words: “Conditional

statements occupy a central place in reasoning, and hence their proper analysis is a

principal task of logic” (1984:303). Van Benthem’s main claim about conditional

statements and its logic is that “if” works as a generalised quantifier, as a function

connecting sets of antecedent and consequent “occasions”, as he says. Notice that van

Benthem’s thesis is that if works as a generalised quantifier, i.e., as a determiner that

expresses a relation between sets of individuals. What we want to put forward here is that if

is a general (not necessarily unrestricted) quantifier, interpreted as an inferential device of

transference of information. Backgrounds behind van Benthem’s position and ours are quite

dissimilar, although we are taking profit of his basic insights to develop a stronger thesis.

The stronger thesis is that, in natural languages, if and all perform the same task, being the

difference between them one in the range of the “entities” quantified over. Conditionals are

universal quantifiers ranging over sets of situations, circumstances, models or possible

worlds, (here every one is invited to choose their favourite theory). Thus, if is an instance of

all for the case of situations (or your favourite alternative). The accompanying particle

“then” works, on van Benthem’s view that we borrow, as an anaphoric pronoun plugged to

the universal if-quantifier. Van Benthem’s conditional logic is a formal project. From the

point of view of its formal properties, one might want to characterise a minimal conditional

logic, and then to continue adding axioms to cover a wider range of determiners.

Nevertheless, if the aim is the meaning of the natural language expressions, all these

mathematical subtleties are inapplicable (although a big deal of pragmatic ones becomes

inescapable). This claim on natural languages does not diminish in the least the interest of

van Benthem’s proposal for formal semantics. We are placed in a pragmatic perspective

that fits the general framework of what Robert Brandom has dubbed “logical

expressivism”. But a pragmatic account can perfectly well benefit from the light thrown to

languages by formal treatments, with the only proviso of acknowledging that natural

languages are living creatures that do not present the limpid structure presupposed by the

mathematical linguistic. Formal semantics and the pragmatics of natural languages are

different enterprises that can, at least in some cases, successfully cooperate. In order to

assess our proposal about Donkey sentences one does not need to buy logical expressivism.

Taking on the basic insight behind van Benthem’s paper would be enough.

2.1 What kind of expressions quantifiers are?

Let us assume in what follows that conditionals are some kind of quantifiers. The

next step is to answer the question of what kind of device are natural language quantifiers.

Our view here will be dynamic. To express the main idea of a dynamic treatment it is

enough to say that quantifiers display some kind of transition from a situation to another,

understanding “situation” as neutrally as possible. This general idea is compatible with

most, if not all, the semantic accounts of logical constant present in the best-reputed actual

positions. It fits perfectly well the possible world treatment, and also the thesis that

quantifiers are circumstance-shifting operators; it is coherent with the analysis of Discourse

Representation Theory (DRT, hereafter) and also with Game Theoretical Semantics (GTS,

hereafter). Moving on to pragmatics, a quantified sentence makes explicit some kind of

permission in order to add the information in the consequent clause to the information

given in the antecedent in those situations in which we are entitled to assert the latter.

This interpretation of conditionals as quantifiers, together with an expressivist,

inferentialist, interpretation of quantifiers does not confuse, pace Quine, use with mention,

nor assume a deductive view on conditionals, as we will show below. Far from these

classical rejoinders, it permits an elegant way out from some of the classical difficulties

with which the treatment of conditionals seems to be fraught.

As an illustration, let us have a look at the famous Quinean examples (1959:15),

If Bizet and Verdi had been compatriots, then Verdi would have been French,

If Bizet and Verdi had been compatriots, then Bizet would have been Italian.

Quine’s point can be summarized saying that subjunctive conditionals are

nonsensical because of their contradictory truth-conditions. His arguments surely make an

interesting observation, but nothing for which honest conditionals should be blamed.

Examples (1) and (2) vividly show that conditionals do not represent the reality around.

Counterfactuals do not do it, for sure, but “material”, indicative, conditionals do not do it

either. The puzzlement that (1) and (2) might produce vanishes as soon one recognises their

logical form: their antecedent clauses are general claims and the conditional operator

forces a circumstance-shifting to proceed to the evaluation of their consequent clauses in

every instance of the generalized antecedent. Their common antecedent clause is a covert

generalisation, which has as instances a whole range of possible state-of-affairs, namely all

the situations in which Verdi and Bizet share their birth country. Among these situations

one can count both that Verdi and Bizet were French, that they were Italian, and many

others. And this fact is hardly more puzzling than the sentence “Victoria reacted in a

strange manner” representing the state of affairs of Victoria running out the room, or of

Victoria climbing to the cliffs to sing a song, of Victoria wearing her three years old

brother’s clothes, etc. Thus, the antecedent clause in Quine’s examples opens a branching

structure in which one has to consider any one of the possibilities that are instances of the

covert generalization. In each one, one has to assess the claim that Verdi and Bizet would

have been French, for instance. If one chooses a possible circumstance in which both have

been born in Cannes, the conditional clause will be true. If the generalization were

instantiated by a situation in which both had been born in Kuala Lumpur, the consequent

clause would be false, and so on. There is nothing baffling here.

In the picture we have in mind, all kind of conditionals, natural language

conditionals and formal conditionals, material and strict, indicative and subjunctive, and

also counterfactuals share the same meaning. The differences among them have to be

accounted for attending differences in the scope and range of the hidden quantifiers.

Some of the difficulties that conditional theorists have encountered to develop a

suitable account of this topic, and specially of subjunctive and counterfactual conditionals,

stem, as we see it, from a pervasive confusion between the task of giving the meaning, (the

communicative role, the pragmatic significance, or any other expression you choose to

express the semantic/pragmatic functioning) of a conditional, and that of stating its

justification conditions. This second tasks would require to describe exactly when we are

allowed to affirm that were things so and so, other things would be so and so. This has

commonly felt to require, for instance, discourse about similarity or “distances” among

possible worlds or types of situation. To our mind, this task cannot be accomplished from a

general point of view. The justification or assertion conditions of conditionals, being them

indicative or not, depend on their content and context. And this dependence is not peculiar

to conditionals, any kind of assertibility conditions are content- and context-dependent.

Meaning is another story. Semantically, conditional markers, as “if…, then…”,

“whenever”, “when” etc., can be seeing as circumstance-shifting operators, and the

pragmatic role they perform in natural languages is displaying a(n) (provisional) inferential

link that, in the simplest case, the speaker is ready to make, keeping in mind the contextual

features of the utterance situation, and also the features of the situations pointed to by the

information in the antecedent clause.

We have mentioned some paragraphs above that the dynamic, inferentialist, view on

conditionals should not be confused with the classical, deductive, account. The deductive

view on conditionals roughly sustains that the information in the consequent of a

conditional validly follows, together with some auxiliary premises, from the information in

the antecedent. This view is too strong and also takes a model theoretical perspective that

lies very far from our pragmatic, dynamic, point of view. It amounts to saying that a

conditional is true when the consequent is true in all models in which the antecedent,

together with the relevant premises, is true. It takes an external, semantic, perspective. In

our view a conditional is neither true nor false, conditionals do not represent situations but

rather have the import of authorisations of informational transfer contextually constrained.

Furthermore, we do not defend that true conditionals represent valid inferences but only

that who asserts a conditionals is inviting her audience, inside the contextual restrictions

relevant in the context of utterance, to try a virtual journey to the situation depicted by the

antecedent, and there to assume the information codified in the consequent.

To state it in a nutshell: conditionals, like quantifiers, are ways of expressing rules

of inference of diverse kind and strength. The relevant point here is what to understand by

“rule of inference” and here might lay the difficulties of the interpretation and the origin of

misunderstandings, because so stated it seems utterly false. But it is not4. A rule of

inference, in the sense understood in this paper, is a semantic railway piece. It is a path to

go from some assumptions, claims, commitments and entitlements, to some others, in a

specified communicative situation. It is essential to keep in mind that these contextual rules

of inference do not need to be logically correct, i. e. rules that “preserve the truth” in every

context. They are contextually correct from the point of view of the speaker who, by using

quantifiers and conditionals, is disclosing the inferential links sustained by the web of her

beliefs, semantics or not. In the context from which they are uttered, and inside its scope,

they express permissions to understand any situation in which the antecedent holds as a

situation in which the consequent also holds, or at least explaining that the speaker believes

that this is so. They can transfer, inside their scope, the entitlements the speaker has to

assert the antecedent to the assertion of the consequent, and merely display the commitment

of the speaker with the consequent once she is committed to the antecedent. A quantifier or

a conditional is a mean of making explicit an inferential practice, and by disclosing it a

speaker can be giving reasons for her own behaviour, or else inviting others to accept her

inferential links. Under which conditions it is appropriate to assert the antecedent, and then

when we are entitled to make the transition to the consequent is something that cannot be a

priori determined. A context that constrains the scope of the rule is crucial to distinguish

universally valid inferences, i.e., that expressing necessary relations among concepts, from

those that are contextually limited. Most of our everyday inferences have contextual limits,

dependent on a big amount of shared background, and so are our natural language

conditionals, that usually express non-monotonic inferences. Being unmodified

conditionals universal quantifiers, the feature of non-monotonicity in conditionals

corresponds to the feature of non-persistence in quantifiers.

4 That logical constants in general should be understood as inference rules is something that has being defended during the last century from diverse theoretical frameworks. Some of the attempts have failed mostly because they interpreted the notion of rule in a sense unjustifiably strict. But there are also successful ways of implementing the idea. We are thinking of the pragmatic perspective taken by Ramsey, or by Ryle, and nowadays it has been pointed at by Robert Brandom in his (1994). To dispel the false belief that a pragmatic perspective is incompatible with a developed, solid, logical structure, it will be enough to have a look at the way in which C. J. F. Williams (1992) discloses the role of logical expressions.

That conditionals are naturally seeing as allowing a transition from the antecedent to

consequent and that they have the import of some kind of license have being felt by most of

the theorists that work on conditionals. The way in which we have presented this view is,

we agree, not the standard one, but the intuition that supports it about the meaning of

conditionals is more widely shared than it is openly acknowledged. In any case, it is not our

aim to contest the general view; on the contrary, we attempt to explicitly draw the

consequences that seem to be implicit in the best treatments of the topic. As an illustration,

let have a quick look at the way in which conditionals are presently characterized.

Van Eijck and Kamp (1997:185), proponents of DRT, typify conditional

information as allowing two situations, the situation described in the antecedent clause and

the situation described in the consequent clause, and a conditional sentence asserts that the

information displayed in the consequent is satisfied in the model that satisfy the antecedent.

A conditional, in van Benthem’s words “invites us to take a mental trip to the land

of the antecedent” (1984:311). The same insight can be express from Recanati´s treatment

of conditionals as circumstance-shifting operators. Conditionals and metarepresentation

oblige us, speakers and audience, to perform some sort of simulation and that simulation

enables us to consider remote and even imaginary situations (2000:93). When a conditional

sentence is uttered, the context of the utterance is the situation that supports the conditional

claim. From the actual situation, we travel to the (type of) situation depicted in the

antecedent, and from them we assess the proposition in the consequent.

Another illustration is the case of GTS. Hintikka and Sandu (1997: 391-393) defend

that the meaning of conditionals does not stop on the classical truth-table characterisation.

The authors explicitly state that there is more in a conditional sentence that what can be

derived from its truth-conditions. What a conditional does is rather “to provide a warrant

for a passage from the truth of S1 [the antecedent] to the truth of S2 [the consequent]”

(p.391).

The treatment of Situation Semantics, and also GTS’s and DRT’s characterisation of

the semantics of conditional sentences, seems to make of them the expression of some kind

of permission to extract some consequences under the appropriate circumstances. And this

is again what one should expect if conditionals were quantifiers and quantifiers were

inferentially understood. The connection between universal quantifier and conditional is

seen everywhere and their closeness is re-stated every time. There is though a reluctance to

draw the obvious conclusion, that conditionals are general quantifiers, for which inertial

respect to tradition has to be blamed, in two different respects, (i) because tradition counts

conditionals as a different logical constant and (ii) because the domain of classical

quantifiers consists of “objects”.

3. Donkey Sentences: A Case of Study

We cannot think of any range of sentences that had proved more resistant to logical

and semantic analysis than Donkey sentences. Our diagnosis is that, apart from the

difficulties related to anaphora, there has been a poor identification of the operators

involved. In order to understand the point, one has just to assume that conditionals

represent some sort of quantifiers ranging on situations. The rest of the background picture,

logical expressivism, a pragmatist stance and a dynamic view on meaning, are all added

features of our picture, although independent of analysis of Donkey sentences that we put

forward. Thus, one might buy the analysis and inserting it in a different general setting.

Donkey sentences were introduced in the philosophical debate by Peter Geach in the

sixties (Geach 1962: 155-156), a debate that Geach brings in from the Middle Age, and

since then they have been a field in which every theory of quantifiers and pronouns in

natural languages has had to test its merits.

Examples of Donkey sentences are:

(1) If Pedro owns a donkey, he beats it,

(2) When a farmer owns a donkey, he beats it,

(3) Always that a farmer owns a donkey, he beats it now and then.

Examples (1) and (2) are the classical cases, and example (3) is a more complex

version due to D. Lewis (1975).

One of the pervasive difficulties listed in the bibliography has been how to explain

why, for a question of scope, (1) cannot be translated as

(1.a) ∃x [(Donkey (x) & Owns (Pedro, x) ) → Beats (Pedro, x)]

This rendering is inappropriate because it would give priority to the existential

quantifier over the conditional, converting (1) in an existential claim, which it is not. The

other option (1.b),

(1.b) ∃x (Donkey (x) & Owns (Pedro, x) ) → Beats (Pedro, x),

i.e., making the conditional the dominant operator, would leave the variable in the

consequent clause unbound and thus would convert it in an open formula, which it is not

either.

To overcome this difficulty, it has been a common place to translate (1) using a

general quantifier instead, as in (1.c),

(1.c) ∀x [(Donkey (x) & Owns (Pedro, x) ) → Beats (Pedro, x)].

Regarding (1.c), a point of discomfort has been how (and why) the expression “a

donkey” is now translated using a general quantifier, whereas in narrower contexts, like for

instance (1.d),

(1.d) Pedro owns a donkey,

it would have been understood as involving an existential quantifier binding the

variable that accompany the monadic first order predicate, as in (1.e),

(1.e) ∃x [Donkey (x) & Owns (Pedro, x)],

which is the standard translation of (1.d).

To illustrate the debate that lies behind the difficulties listed above, it will be enough

to recall the treatment of donkey sentences in three major contemporary theories: Dynamic

predicate logic (DPL, hereafter), Game theoretical semantics (GTS) and Discourse

representation theory (DRT).

DPL and GTS deal with (1) in a similar way that consists in the distinction of two

notions of scope (GTS) or of binding (DPL), that in this case amounts to the same, although

there are notorious differences between the two views. One of them is the role that each one

concedes to the Principle of Compositionality (PC). Thus, while DPL sticks to PC (and

prides itself on it), GTS rejects the principle as a general desideratum of any semantic

theory. PC is hardly reconcilable with the branching quantifiers characteristic of GTS. DPL

and GTS also share some of their shortcomings, being both unable to explain the

mechanisms behind anaphoric links when the head is either in a general or a negative

sentence.

One of the picks of GTS is the identification of the two separable tasks that the

traditional notion of scope performs. GTS denounces that the notion of scope that we use in

our standard calculi embodies, in fact, two different notions that answer for different

relations among the ingredients of propositions. One of them accounts for the relations of

logical priority among logical operators, the other accounts for the binding scope of

quantifiers. In complex cases like those of Donkey sentences, the two scopes do not go

together and this divergence explains the difficulties that Donkey sentences pose to be

translated into a calculus that, like standard first order calculus, does not make room for the

distinction. With roughly the same range of problems in mind, DPL introduces two kinds of

binding for variables. An occurrence of a variable in a formula can be free in the standard

sense, because from a syntactical point of view it lies outside the scope of quantifiers in the

formula, while anaphorically referring to some of these quantifiers that acts as the

anaphoric heads. These variables would be, using Sandu’s terminology (Sandu 1997),

syntactically free but semantically bound to their head. Bifurcation, in scope or binding

respectively, is meant to differentiate the logical priority of operators, that in Donkey

sentences like (1)-(3) belongs to the conditional over any quantifier, from the scope of the

explicit quantifiers that reach all variables in the formula. DPL, on its turn, would render

(1) as (1.b), and GTS would use its bidimensional language to display the independence of

both tasks.

As GTS, DRT also uses bidimensional structures to represent the informational

content of a discourse, although instead of trees, DRT uses complexes of (possibly)

embedded boxes. An interpretation of a sentence in DRT is a set of referential markers and

conditions. This set is called “a Discourse Representation Structure” (DRS). Referential

markers are singular terms and conditions are formulae that the individuals referred to by

the referential markers have to satisfy. A DRS for (1) would be something like the

following:

(DRS.1) {{x}, {y}, Pedro (x), Donkey (y), Owns (x, y), Beats (x, y)},

where “x” and “y” are referential markers, with the conditions of being Pedro and a

donkey, respectably, and the rest are relational conditions on them. Although DRT would

interpret (1.d) as (1.e), it would treat (1) as if it had a structure more like (1.c), with a

universal quantifier. These treatment (see, for instance, van Eijck & Kamp, 1997, p.) needs

to assume that the logical role, and then the operator behind, of indefinite descriptions, as

“a donkey”, vary depending on context, having sometimes the import of a universal

quantifier, sometimes that of an existential quantifier.

Keeping track of what has been said so far, Donkey sentences, while perfectly

grammatical and admissible sentences, cause translation difficulties that reflect failures in

the identification the logical ingredients of the propositions expressed and the relations

among these ingredients. In order to get out from this uncomfortable situation, the semantic

theories mentioned above have introduced modifications of the standard notions of scope or

of binding, or else making of indefinite descriptions context-dependent expressions in an

unusual way.

Before checking our own proposal, it is important to recall that we are not arguing

that these semantic theories, or other contemporary proposals, do not have resources for a

correct treatment of Donkey sentences. Adjusting parts of their theories to the needs of

analysis, all of them accomplish an acceptable picture. But the required adjustments seem

to be ad hoc most of the times, generally unjustified apart from their interest as a way to

solve these particular difficulties of analysis. Our proposal, on the other hand, offers a

general account of conditionals that show its fruitfulness also in the analysis of Donkey

sentences. Furthermore, and this is something that we would like to stress, treating

conditionals as quantifiers is not completely alien to standard dynamic semantics. Kamp

and Reyle for instance, defenders of DRT, say that

There exists an intimate connection between conditionals and the words every, all and each. We get a glimpse of this connection when we compare, say

(2.41) If a farmer owns a Mercedes he thrives and (2.42) Every farmer who owns a Mercedes thrives. Although these sentences may not have exactly the same meaning, their

meanings are certainly very close. In particular, each provides the information on the basis of which we can, whenever we encounter an individual whom we recognise to be a farmer and in possession of a Mercedes, infer that that individual thrives. (1993: 166)

This text suggests our view of conditionals as quantifiers, and also an inferential

view of both. Neither of the two ideas is further implemented in DRT but when, out of the

technicalities of a particular proposal, one is forced to explain what a conditional does, the

general insight goes along the lines we have suggested. In this sense, assuming the view we

are putting forward here would not require to reject the big amount of theoretical and

instrumental benefits of formal semantics.

Having a view on conditionals as general quantifiers, the structure of Donkey

sentences can be displayed in a systematic, straightforward way, which also explains the

difficulties encountered so far, and also the alternative ways of solution, like those of GTS,

DPL, and DRT.

All these proposals agree in that, in (1)-(3), the conditional has the logical priority

over the quantifiers. And they are right. GTS signals it making the conditional independent

of the quantifier. DPL establishes the priority of the conditional maintaining the variables

in the consequent syntactically free (although semantically bound by the existential

quantifiers) and DRT eliminates the existential quantifiers altogether making indefinite

descriptions logically context-dependent. DRT is (accidentally) closer to our view in that it

acknowledges the occurrence of a general quantifier. But, contrary to DRT, we defend that

the universal quantifier does not account for the indefinite noun phrase “a donkey” that,

even in this context, has an existential quantifier at its logical form, but the conditional

particle if, which is a universal quantifier on its own, ranging over sets of situations. This

universal quantifier has indeed the logical priority in (1). Thus, those who have felt that

there were a general quantifier with wider scope were right: the if-quantifier. But also those

who have felt that there must be an existential quantifier with narrower scope are right,

although related to a different logical ingredient, this time the indefinite description. The

quantifiers in (1) are thus (a) a universal quantifier translating the conditional particle if,

meaning “in all situation s”, and having the logical priority, and (b) an existential quantifier

related to the indefinite description “a donkey”. There also are three anaphoric links, one

related to the pronoun “it” which is a variable bound by the existential quantifier that

occurs in “a donkey”, another related to the pronoun “he” which head is the proper name

“Pedro”, and still another related to the (implicit) then particle which also is a variable

bound by the universal quantifier signalled by the if particle. Predicate calculus is only able

to represent static, linear, structures. For this reason, the subtleties of Donkey sentences

cannot be displayed in it. Let us try a semi-formalized translation of (1), (1.f), in which we

signal the quantifiers involved,

(1.f) Every situation s such as ∃x (Donkey (x) & Owns (Pedro, x)) in s, is a situation

in which Beats (Pedro, x) in s.

Apart from the quantifier ranging on situations, the basic semantic idea coincides

with the analysis that we have encountered in the quoted text by Kamp and Reyle.

Form a syntactic point of view, “in s” is an adverbial phrase, and from a logical

point of view it is an operator that shifts the circumstance. Thus, what we have in (1.f) is

not a formula of the predicate calculus and, in general, it is not a formula at all if for

“formula” we understand a structure capable of being statically interpreted into a model and

able to acquire a semantic value as a truth bearer. Quantified sentences express potential

inferential moves, and do not primarily represent states of affairs. Their import can be

projected into a model and, if we insist, they can be understood as representational devices,

i. e., as devices that express propositions, but this procedure would neglect their dynamic,

transitional, intralinguistic peculiarities.

The rationale behind DPL, GTR and GTS is precisely to offer a way of dealing with

the dynamic aspects of our language. Another proposal is Situation Semantics. For the sake

of perspicuity, let us represent the movements involved in a conditional sentence using a

sophisticated version of Situation Semantics, Recanati’s δ-structures. A stated conditional

is always stated from a particular situation that defines the general context. What the

conditional expresses is that the speaker assumes that (all, most, a few, none) situations of

the kind depicted in the antecedent clause, are also situations in which what is said in the

consequent holds. Something like that

(1.g) [s’] |=@ << s |=∀w σ >>

where there is a situation, s´, the exercised situation from which the speaker utters

her conditional sentence, and from it, she points to a cluster of situations characterised by

the features in s, in which we are allowed (or so the speakers thinks) to affirm σ. Applied to

example 1,

(1.h) [s’] |=@ << [s ≡ Donkey (x) & Owns (Pedro, x)] |=∀s Beats (Pedro, x) >>

taking into account that the information after the turnstile is added to the model that

satisfies the sentences before this sign. At the end, what one should have is the merging of

several models and the resolution of the anaphoric operations.

Donkey sentences are much more complex than that, and it is no wonder that they

have deserved the enormous amount of ink and paper already devoted to them. We have

already listed the if-quantifier, the existential quantifier, and the three anaphoric links

marked respectively by “then”, “he” and “it”. The anaphoric links cannot be completely

characterised as repeated bound variables, but require their own operator that, as in DRT, it

is sometimes substituted by identity. This move will add three more operators to the logical

form of (1)5. The total sum will be five higher-order6 functions, which explain the

difficulties enclosed in this kind of examples.

With this analysis in mind, let us go back to examples (2) and (3) and see how it

behaves. Example (2) above,

(2) When a farmer owns a donkey, he beats it,

adds a new existential quantifier to the logical form of (1) corresponding to the

indefinite description “a farmer”. This quantifier also falls under the scope of the universal

conditional operator.

Example (3) poses an interesting challenge. Lewis uses it to discuss the range of the

quantifiers involved, and says that in most interpretations, (3) will express a contradictory

proposition due to the divergent meanings of “always” and “now and then”. Independently

of the weight we give to Lewis’ conclusion, our proposed account of conditionals as

quantifiers will explain why (3) is not contradictory, although it is somehow paradoxical in

the sense that the adverb “always” does not add anything to the content. (3),

(3) Always that a farmer owns a donkey, he beats it now and then,

is only a stylistic variant of (4),

(4) When a farmer owns a donkey, he beats it now and then.

In both cases, “always” and “when” signal the presence of a universal rule, and the

users of (3) and (4) are displaying their beliefs about all situations of a certain kind. They 5 A promising way of treating anaphoric links is using a second order quantifier-like identity operator, as the one used by Williams (1989). An alternative, and equivalent, way is using Geach’s reflexivity operator, (Geach 1962). A treatment of anaphora using a strategy close to Geach´s operator can be found in Böttner (1992), and he draws the idea back to Suppes (1974) and Keenan (1987). DRT also offers an appropriate account via the first order relator of identity, although when intensional operators are involved, the second-order identity operator shows its power over the traditional relation.

are all those situations in which there is (at least) a farmer and (at least) a donkey, and in

those situations the farmer beats the donkey now and then. Whereas “always” or “when”

encode sets of situations, “now and then” talk about instants of time. A new quantifier,

ranging on instants, is thus required. No contradiction then, although the contradiction

obviously rises if, as happens in every standard analysis, “always” and “now and then” are

interpreted as sharing their range.

As it has been already signalled, the semiformal paraphrases by which we have

explained the structure of (1)-(3) cannot be turned out into formulas belonging to the

predicate calculus. Contrary to the standard path in formal semantics, we are not interested

in showing that our account fits the template of first order logic. We use formal devices

insofar as they help to clarify our analysis, by the aim is rather to understand natural

languages, and one of our points of departure is that not all declarative sentences represent

states-of-affairs. To this extent, models are not our target.

In trouble-free examples, for uncomplicated semantic questions, the formal options

that one chooses are, for most purposes, quite irrelevant, even if the philosophical

backgrounds that support them are not subtle enough to undertake the task of analysing

natural languages. But Donkey sentences are not interpreters-friendly. They resist any

approach that does not incorporate the appropriate tools and perspectives. This proof-

resistant nature is one of the sources of the success of Geach’s examples, and a reliable

guide for sharpening our analytical equipment. Understanding conditionals as general

quantifiers seems to pass the test. This is not its only merit, but we believe that it is at least

a reason to attentively consider this proposal.

One might suppose that DPL, or DRS, both possess mechanisms for representing the

logical import of structures such as (1.f). We are pessimistic about it. Of course, it is always

possible to construct languages with this purpose, but the experience in this field has shown

that all these trials end up with structures that automatically allow the representation into a

model or set of models. Taking into account the inertial burden of tradition, it is advisable

to be patient and not to try to assimilate an analysis as the one we are putting forward to

any of the available projects. We are in the middle of a semantic paradigm shifting that is

6 “Higher order” because we assume that anaphoric links need a second order operator to display all their power. See the previous footnote.

leaving the representationalist treatment of logic and semantics and welcoming a non-

representationalist, dynamic, understanding of them. The proliferation of diverse dynamic

semantics, as the ones we have mentioned here, is a sign of this shifting. But all of them,

DPL, DRT, GTS and the like, are still trapped in the conceptual framework and methods of

the milieu in which they and their proponents were brought up.

To conclude, let us recall that interpreting conditionals as quantifiers is close to the

spirit, if not to the letter, of some of the most promising proposals in formal semantics, and

that actually making the move and unifying the treatment would provide analysis with a

sharper tool. It would not require, on the other hand, a painful rejection of the present

paradigm because, in fact, conditionals and quantifiers are already treated in practice in a

very similar way.

There is, nevertheless, a feature of our view that does not allow to be accommodated

in a simple way even by the most sophisticated formal semantics in the market. It is the

idea that quantified sentences and, in general, sentences in which a logical expression

occurs do not represent neither single states of affairs nor, as in dynamic semantics,

successive states of affairs in which information becomes updated. Quantified sentences

display mental connexions between concepts and propositions from the point of view of the

person who utters them, and thus they give support to contextually bound rules of

inference. At this point we should keep in mind Ramsey’s words: “Many sentences express

cognitive attitudes without being propositions; and the difference between saying yes or no

to them is not the difference between saying yes or no to a proposition” (1931: 237-238).

Conditional sentences are of this kind, they open restricted paths to transport information

from some situations to others. They do not describe the world around but in their barest

form display the speaker’s inferential network.

Nevertheless, as we have tried to make clear, you do not need to buy the whole pack

in order to accept our point about Donkey sentences. The part that is needed is small and

mainly harmless.

Acknowledgments

Part of this paper was written during my stay at the Department of Philosophy, University of Helsinki, Finland, financially supported by the Spanish Ministerio de Educación, Cultura y Deporte, Servicio de Acciones de Promoción y Movilidad. There I had the pleasure of discussing substantial parts of it with Gabriel Sandu, to whom I am deeply grateful. A previous version of this paper was presented at the VI Congreso de la Sociedad Española de Filosofía Analítica (SEFA) that took place in Murcia (Spain), 16-18 December 2004. I thank the audience for an interesting debate. Manuel de Pinedo and Neftalí Villanueva, researchers at the Department of Philosophy, University of Granada, helped me in many ways with profound and insightful comments on previous drafts of this paper.

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