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My question is in regard to semantic type theory in connection with syntax. I understand the underlying structure of what type a certain type of phrase is ( i.e., proper names are <e>, verbs can be <et>, <e<et>>, or <e<e<et>>>. This seems to follow a straightforward pattern of having a set rule as to what type can be the input and what is the output. However, what of conjunctions? Looking at it much like a puzzle, conjunctions can take different types based on the XP they are connecting (DP= <e<e<e>>>, VP <et,<et,<et>>>) But what about Quantificational DPs? When I try, I get that the conjunction AND has the type <<et,<t>>,<<et,<t>>,<et,<t>>>>. Can we have such complex types? Surely there is a better type theory allowing for less complex types?

My theory is that conjunctions can only connect identical XPs and thus connect identical types. I'm trying to come up with a sentence which the conjunction cannot conjoin to disprove my theory, yet am struggling to do so. Can someone help me? Or is my theory correct and thus there is no sentence that can disprove it?

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Your idea can be expressed by generalizing over types using variables, and saying that conjunctions are of type <σ,<σ,σ>>: They take two arguments of the same type and return the conjoined phrase which will again have the same type. For σ, you may insert the type of proper names, verbs, QDPS, sentences or whatever else you like.

Note that type ≠ XP; e-t type theory does not systematically distinguish between certain syntactic categories in the same way X-bar theory does: For instance, type-theoretically, according to the above specification, "cat and wash the dishes" should be a well-formed expression where σ = <e,t>: The input NP "cat" and the input VP "wash the dishes" both have the type <e,t>, and can thus be conjoined with and: <<e,t>,<<e,t>,<e,t>>> to a complex phrase of type <e,t>. But X-bar theory correctly predicts that it is not a well-formed expression, because you should not be allowed to conjoin an NP ("cat") with a VP ("wash the dishes") -- a distinction hat e-t type theory doesn't know.
So in some respects, type theory is too coarse-grained to correctly capture syntactic constraints: It can be said to over-generate in the sense that it dubs expressions as type-theoretically well-formed that we would otherwise judge to be ungrammatical.
On the other hand, on the top of my head I can not come up with an example of where conjunction of two XPs with X = X would be illegal.

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There are numerous kinds of coordinate structures that challenge such an approach to coordination in terms of types. The main difficulty concerns the fact that the strings that can be coordinated need not be phrases/constituents, which means it is very difficult to assign any sort of type to them. An unstated premise in the question is, namely, that the conjuncts of coordinate structures are necessarily units of syntax, i.e. phrases/constituents, to which one can assign types. There are many cases where doing this is not possible. What follows is a brief, incomplete inventory of the kinds of coordinate structures that challenge theories of syntax based on semantic types, X-bar theory, or any other approach to syntax that deems the phrase/constituent to be the basic (and only) unit of syntactic analysis.

Syntactically distinct conjuncts: When the coordinated strings occur as (part of) the predicate, they need not match at all in syntactic category. The next examples are taken from the Wikipedia article on coordination (https://en.wikipedia.org/wiki/Coordination_(linguistics)):

(1) Sarah is [a CEO] and [proud of her job]. - NP + AP

(2) Is Jim [conservative] and [a closet Republican]? - A + NP

(3) Bill is [in trouble] and [trying to come up with an excuse]. - PP + VP

(4) Sam works [evenings] and [on weekends]. - Adv + PP

(5) They are leaving [due to the weather] and [because they want to save money]. - PP + Clause

It should be apparent that some sort of super-type would be needed to accommodate these cases.

Right node raising: The phenomenon known as right node raising (RNR) is such that the coordinated strings are clearly not constituents on the surface. The next examples are from the Wikipedia article on RNR (https://en.wikipedia.org/wiki/Right_node_raising):

(6) [Fred prepares] and [Susan eats] the food.

(7) [Larry has promised] but [Jim refuses] to support reform.

(8) [Jim can] but [Jerry cannot] make the meeting.

(9) [When did he] and [why did he] suffer a setback.

(10) [Sometimes she carefully reads] and [at other times she merely skims] the report they produce.

It should be apparent that it would be impossible to acknowledge the conjuncts indicated in these examples as constituents to which one could assign types. At the very least, the analysis needs to be augmented with some notion of ellipsis.

Non-constituent conjuncts (NCC): The cases above could perhaps be rectified in terms of some sort of super-type and/or some approach that acknowledges ellipsis. There are other cases, however, where these additions are less plausible. The following coordinate structures are such that the intonation contour is normal (unlike in cases of RNR):

(11) Sam bought [wine yesterday in the local store] and [beer today in the big supermarket].

(12) Sam bought wine [yesterday in the local store] and [today in the big supermarket].

(13) Sam bought [wine yesterday] and [beer today] in the local store.

Most theories of syntax do not view the conjuncts marked in these examples as constituents. It would therefore be very difficult to assign any semantic type to them. The following further examples of NCC are taken from Wikipedia (https://en.wikipedia.org/wiki/Coordination_(linguistics)):

(14) [Did he] or [did he not] do that?

(15) She [has] or [has not] understood the task.

(16) Susan [asked you] but [forced me] to read the book on syntax.

(17) Jill [has been promising] and [is now actually trying] to solve the problem.

(18) [The old] and [the new] submarines submerged side-by-side.

(19) [Before the first] and [after the second] presentation, there will be coffee.

(20) Fred sent [Uncle Willy chocolates] and [Aunt Samantha earrings].

(21) We expect [Connor to laugh] and [Jill to cry].

Some of these examples might be construed as instances of RNR and could hence be grouped to the examples of RNR further above. In any case, they all help illustrate the challenge posed by NCC to any theory of coordination that assumes that the conjuncts of coordinate structures necessarily qualify as phrases/constituents.

In sum, the approaches to coordination suggested in the question (in terms of types) and in Lemontree’s answer (in terms of X-bar theory) are hopelessly outmatched by the data. Note that the challenging cases can be expanded if examples of gapping (https://en.wikipedia.org/wiki/Gapping) and stripping (https://en.wikipedia.org/wiki/Stripping_(linguistics)) are included. My personal stance is that to be plausible, the theory of coordination should eject the premise that coordination operates on constituents and allow it to operate on strings instead, whereby many of the relevant strings are non-constituents.

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  • Strings i.e. any string is too broad, surely. Why not borrow the structure you proposed for supplementary constructions and diagram them as parallel sentences. That way you capture the fact that deleting either of the coordinates plus the conjunction must always leave you with a well-formed sentence (at least, that's what strikes me looking at your example sentences - I don't think (18) is really a counterexample but perhaps there are others). – rchivers Mar 27 at 7:18
  • @rchivers That doesn't work in many cases, e.g. [Frank saw a girl] and [Bill greeted a girl] (two distinct girls) vs. [Frank saw] and [Bill greeted] a girl (probably just one girl). What you propose is known as the "large conjunct approach". There are many problems with the large conjunct approach. You are certainly right, though, that there restrictions on the strings that can be coordinated. – Tim Osborne Mar 27 at 8:02
  • I'm not seeing the problem there - it's just that the parallel tracks have to rejoin the main line after a girl in the first example (because it appears twice) whereas they rejoin before a girl in the second example, which makes that version felicitous when you are thinking of one and the same girl. I don't know about the large conjunct approach but it seems to me that the kind of analysis I had in mind actually explains your example quite neatly. – rchivers Mar 27 at 9:24
  • @rchivers The two have distinct meanings. If the account is to be plausible, the two ways of expressing the content should have the same one meaning. Here's another example illustrating the same problem: ?[Frank hummed the same tune] and [Sam sang the same tune] vs. [Frank hummed] and [Sam sang] the same tune. For the coordination of complete clauses to work at all, we have to assume that the same tune denotes a contextually relevant song mentioned in previous discourse. That reading is not natural for the shorter version. – Tim Osborne Mar 27 at 10:16
  • The two have distinct meanings but a plausible account of conjunctions has to make them the same? I don't think that can be what you mean but I'm not sure how else to read your comment. Same thing for the last sentence - you seem to be saying that it's not natural to read the same tune, in Frank hummed and Sam sang the same tune, as referring to a contextually relevant song mentioned in previous discourse, but surely that's not what you mean. – rchivers Mar 28 at 18:12

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