1. First, give an analysis of definite descriptions in first-order predicate logic (i.e., just using E;
and/or A) that accounts for their semantic properties, and show how this analysis explains the
truth conditions and acceptability of sentences like (4)-(6). Be clear and explicit, constructing
additional examples if necessary.
Uniqueness and existence can be captured in first order logic by analyzing the in terms of both existential and universal quantification. Roughly, a sentence of the form ``the P is Q'' is true just in case there is something that is both a P and a Q, and if other thing is a P, then it must be the same as the first thing. This is formalized in the logical representation of ``the P is Q'' in (7). (Note that (7) is equivalent to Ex[(P(x) & Q(x)) & ~Ey[P(y) & ~(y = x)]], which a few people used in their homeworks.)
Under this analysis, the logical representation of a sentence like (4) would be something like (8), where I(e) = Evanston, I(K) = {[x,y] | x is the king of y}, I(g) = the grey aliens, and I(M) = {[x,y] | x met with y for three hours}.
(8) is true if and only if there is a king of Evanston, therefore in any model in which no such individual exists, (8) works out to be false. Under this analysis, then, the ``anomaly'' of (4) is really just due to the fact that it is a false claim.
Similarly, the anomaly of (6) is due to the fact that the logical representation of this sentence, shown in (9), is such that it is true if and only if there is a single student in C72. (In this logical representation, I(S) = {x | x is a student}, I(T) = {[x,y] | x is taking y}, I(c) = C72, I(D) = {[x,y] | x is discussing semantics with y}, and I(g) = two aliens (this last one is for simplicity---really, two aliens should be analyzed as a generalized quantifier).)
Whenever there is more than one student in C72, (9) works out to be false.
The bottom line, then, is that a semantic analysis of definite descriptions in terms of first order predicate logic can come up with representations that accurately characterize the truth conditions of these sentences, but the explanation of facts like (4)-(6) leaves a bit to be desired. There is a strong intuition that these sentences are weird not because they're false, but because some independent requirement associated with the definite NPs isn't being satisfied.
[A historical aside: this analysis is essentially the one given by Bertrand Russell in his famous and influential 1905 paper ``On denoting''. Unlike many semantic theories, this one has very much stood the test of time!]
2. Propose an alternative analysis that builds on the analysis of
indefinite NPs that we developed in class, in which indefinites are not
quantificational, but are rather expressions that introduce restrictions
on free variables whose use is governed by pragmatic principles.
Such an alternative analysis might run like this. In terms of their basic meaning, definite NPs are just like indefinite NPs --- they denote restrictions on free variables. Where they differ is in their pragmatic requirements: while indefinites introduce entities into the discourse (discourse referents) by requiring the object associated with the variable whose value they restrict (determined by the assignment function in the model) to be novel, definite NPs refer to entities already in the discourse. This idea can be implemented by hypothesizing that the definite determiner carries with it a presupposition of familiarity, as well as a uniqueness presupposition. These assumptions together account for the properties listed above: if an object is familiar (i.e., ``discourse old''), then it must exist; if uniqueness is a presupposition, then this explains why it is a requirement even in the scope of negation (as in (5); recall that negation is one of the environments in which presuppositions, but not entailments, come through).
On this view, the truth conditions of an example like (4) are extremely simple, as shown in (10) (using the same symbols as above):
The value of the variable "x" must be determined by the assignment function, but it must meet the discourse requirements of familiarity and uniqueness imposed by the definite determiner.
3. Compare the two analyses. Which provides a more satisfactory semantic analysis of definite descriptions, and why?
The quantificational analysis is fundamentally unsatisfactory, because it explains facts like (4)-(6) in terms of whether the sentences work out to be true or not in the model. That is, the analysis predicts (4)-(6) to be false when either uniqueness or existence isn't satisfied, yet there is a strong sense that something more interesting is going on here. Is it really true to say that (4) is false? Maybe, in some strict sense, but there is a very strong intuition that this is not the reason why (4) is odd. Instead, (4) is odd because some kind of ``condition'' is being violated: if you use a phrase like the king of Evanston, then by god there ought to be a king of Evanston! (Note that indefinites differ in this respect: it is perfectly natural (and even true!) to say ``there isn't a king of Evanston''.)
The alternative analysis arguably provides a better account of the anomaly of (4)-(6), if we take the view that presuppositions areconditions that have to be met in order for an utterance to be felicitous. On this view, a sentence like ``Chris quit smoking'' is felicitous only if it is the case that Chris smoked at some point in the past. This inference isn't part of the truth conditions of the sentence; rather, it's a special felicity condition introduced by one of the lexical items, in this case, quit. What facts like (4)-(6) show is that a definite NP---specifically, the definite determiner the---presupposes that there be some entity that satisfies the description it introduces, and moreover that this entity is the only entity that satisfies the description. When these presuppositions aren't met, as in (4)-(6), we end up with ``presupposition failure'', which results in the sense of anomaly or oddness that we detect.
A third alternative analysis of definite descriptions that ought to be considered is our original analysis, which treated definite descriptions like names, i.e., as expressions that directly denote objects in the model/world. In this appraoch, the explanation for the anomaly of (4)-(5), which fail to satisfy uniqueness, would be something like the following: if there is no object for the definite to refer to, then it can't be used felicitously. It would be like trying to use the name ``Bingus'' to refer to an entity that doesn't exist. (We can ignore imaginary entities---they certainly do exist, just not in the ``real world''!) The explanation of the uniqueness requirement is even simpler: singular referring expressions refer to atomic objects. As a result, the student in C72 must refer to some particular, atomic entity. Using this NP in a context in which there is actually more than one student in C72 would be infelicitous because it would be impossible to tell who is being referred to---it would be like saying ``the aliens spoke to Joseph'' in a room full of people named Joseph.
Although this hypothesis seems attractive at first glance, it just can't be right. Simply put: definite descriptions and names are different! Although they share a number of semantic properties, they also differ in some fundamental ways. Here are two.
First, it actually is possible to felicitously say something like ``the aliens spoke to Joseph'' in a room full of Josephs if the speaker is somehow indicating which Joseph he is referring to (e.g., by pointing to him). Definites aren't like this. The sentence ``the aliens spoke to the student in C72'', uttered in C72 when there is more than one student around, is anomalous no matter how hard you try to point out a particular student. In fact, the only way to make this sentence felicitous is to get rid of the definite article and replace it with e.g. a demonstrative: ``the aliens spoke to this/that student in C72''. Why this difference if definite descriptions are just like names?
Second, we can't escape the fact that even if definite descriptions act like names in terms of their semantic behavior, syntactically, they look a lot more like other (quantificational) NPs: they are constructed out of determiners and other lexical items (nouns, adjectives, etc.), just like quantificational NPs. Moreover, these ``other lexical items'' seem to be performing the same sort of function with definite descriptions that they perform with quantifiers: they restrict the possible value of the thing they refer to, like quantificational NPs (the coach of the Indiana Pacers can only refer to someone who is a `coach').
The analysis of definite descriptions as restrictions on free variables associated with certain presuppositions combines the best of both approaches: it acknowledges the contribution of the lexical material (sometimes called the ``descriptive content'') to the interpretation of the NP, and it accounts for the uniqueness and existence constraints in a way that is somehow ``outside of'' truth conditions proper, and rather part of the conditions that determine whether a particular utterance is felicitous or not.