I am currently working on Paul Pietroski’s Events and Semantic Architecture, and after reading the first chapter, I am a bit puzzled by a few things he says as to in what the underlying structure of a semantic theory should consist. As I am still working through the beginning sections of the book, I would just like to get clear on a few things he says here, as well as some potential pitfalls that I hope to see him address as I get further into the work. The main concern is related to his criticisms and responses to how one should go about assigning semantic content to expressions. The traditional picture as developed by Montague in the 70s assigns semantic content to an expression through a set of compositional axioms which provide expressions particular interpretation functions, and these functions take as input either entities or functions from entities to truth values and map them either to truth values or further functions from entities to truth values. Pietroski would like us to reject this framework, and there are two particular reasons for this rejection that I would like to examine in some detail. First, he argues for an event semantics that replaces function application with concatenation through conjunction and existential closure over events. Now, this is a fairly prevalent alternate framework currently, and his departure would not be so spectacular were it not for his further commitment to the replacement of the set theoretic foundations of the functionist programme with a kind of property evaluation model. In what follows I first flag some remarks by Pietroski which provide the most significant departures from the functionist programme, then I note that the actual implementation of his programme retains some very similar moving parts to that of the model that he is attacking.
To begin, note that Pietroski defines the `Semantic Value’ of an expression as a set of semantic properties which serve to explain the way in which competent speakers evaluate said expression: “to say that expression Σ has the Value(s) it has is just to say that Σ has certain semantic properties , and is thus evaluated in a certain way by competent speakers.”1 So, rather than a particular expression having as its semantic content an intension or extension, it has a set of properties which cause a particular mode of evaluation. Now, given this view, he goes on to specify some notation such that the semantic value of an expression φ as Val(x,φ), which relates the particular θ-role of an expression to an event in the most basic cases. This move by Pietroski is interesting in that it draws a tight connection between the psychological aspects of meaning with that of an expressions semantic content. So the semantic properties in question are actually relational properties between the thematic role of an expression and an event, and this relational property supervenes on the (potential?) evaluation of speakers of the language.
At this point we have lost one nice thing about the functionist view: compositionality. Given that the semantic content of certain expressions could just be functions from functions to functions, the function application account allows for basic compositionality without the need for any additional machinery. Pietroski, on the other hand, has to posit two additional principles in order to get the individual expressions that comprise a particular sentence in a language to compose. These are (1) concatenation is conjunction and (2) existential closure. I bracket existential closure for the moment, as the details involved here come later in the book, and instead limit my comments to how certain criticisms Pietroski levels against the function application view interact with his view of semantic content coupled with (1).
The first step in problematizing this account is to note Pietroski’s view on vagueness and its relation to natural language semantics. The argument runs by first citing Benaceraff and then noting that vagueness exists in natural language to the preliminary conclusion that there is no fact of the matter as to whether there are precise extensions of vague predicates:
If `{x: x is bald}’ specifies a set, there is a set that it specifies. So given this set and some others that `{x: x is bald}’ might specify, for all we know, there is a fact of the matter as to which set it does specify. But given some individuals who are (intuitively) neither clearly bald nor clearly not bald, many sets are equally good–and equally bad–candidates for being the alleged set of bald things. There seems to be no fact of the matter as to which of these is the alleged set. So perhaps we should conclude that `{x: x is bald}’ does not specify any set. [...] In my view, there is no fact of the matter about which of the candidates is specified by `{x: x is bald}’.2
Now, putting aside the many ways in which the literature has gone about fixing up this problem, if we just grant Pietroski this claim, and his further conclusion that “the apparent fact of vagueness creates a difficulty for even stating Functionist axioms that actually assign values to predicates of natural language.”3, then a problem surfaces for the positive account Pietroski is sketching simply due to the apparent dearth of formal methods now available.
The problem in question relates to his account of concatenation as conjunction. The general strategy is Davidsonian, and it is one that I am amenable to. The problem, however, is that without the mathematical tools of analysis that Pietroski seems to be denying the linguist access too, the particular methods involved in composing lexical strings by conjunction becomes a somewhat magical process. Differently put, conjunction is a form of function application. Pietroski seems to assent to this:
Conjunctivists do not deny that concatenation corresponds to a function, since predicate-conjunction can obviously be so described: ||^|| = λo.v iff ∃F∃G[o=<F,G> & v=λo'.t iff F(o')=t & G(o')=t]. One can also encode Functionism in terms of assigning a semantic value to concatenation: ||Π^α||=||^||(<||Π||,||α||>); where ||^||=λo.v iff ∃F∃x[o=<F,x> & F(x)=v], and `F’ ranges over functions from individuals to truth-values (and `o‘ ranges over ordered pairs of functions and elements to the relevant domains).4
Now, given his preceding rejection of the possibility of even stating functionist axioms for a natural language semantics, he owes an explanation of how conjunction works, if this particular strategy fails. But the only account he gives is precisely that which he later denies. So it looks like Pietroski owes us an explanation of concatenation as conjunction, where conjunction is not a boolean operation as it is standardly conceived yet still provides equivalent results.
Update 2/7/2010
Through correspondence with Pietroski, I have cleared up the primary concern which I raised in this post. It appears that when Pietroski was writing this book he put some of the terminology in such a way (due to external pressures) as to cause me to have some misleading assumptions. That is to say, Pietroski’s arguments are more coherently taken to be suggesting a move away from model theoretic semantics on the whole. The problem I was having is that, assuming we are doing MTS, I don’t see how you can specify the meaning of `&’ in your recursive definition of the model without function application. But Pietroski would rather think of the job of natural language semantics as illuminating a conceptual model (in the Chomskian vein), where formal tools are used for the purposes of illumination and clarity. Given this shift, he would like to say that we can just take conjunction as basic, and then use plural quantification to directly refer to objects in the world, circumventing the mathematical representation via sets.
Here T2 > T1 in terms of informational content. The relationship between theories need not–indeed, is often not–be so direct. T2 does not have to nest in T1, for example:
In this case T1>E and T2>E, but there is no direct relationship between T1 and T2. T2, however, is denser than T1 in the sense that it provides an equivalent explanation in a smaller logical space (less axioms, for example). A third kind of relationship could occer where T1 and T2 are equal in size, but the explanatory range itself shifts, where T1>E1 and T2>E1, but T2>E2 and while E1:{p_1,…,p_n}, E2:{p_1,…,p_m} where m>n
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