Heim and Kratzer's "Semantics in Generative Grammar", bases its semantics on some version of transformational syntax. However, it is remarkably inexplicit about formalising the syntax it presupposes beyond the assumption that it applies to the familiar X-bar, binary trees found in the tradition in discussion.

I am looking for a pithy formalisation of the sort of X-bar theory and tree-theoretic concepts undergirding Minimalism, and a formalisation of the key concepts involved in derivation of trees (movement, etc) so that I can provide a new semantic theory for structures of this kind. I have come across the work of Stabler and others, but have not located the precise parts where (esp. formalisation of X bar theory) some of these matters are stated.

Could anyone give me page references of specific papers that may be useful?

  • X-bar theory is now rejected by many who do theoretic syntax, and many such as myself reject strict binarity of branching outright. Thus the comment I have for someone with your goals concerns the value of any formalization that is based on the strictly binary branching structures. What makes you think that is a good way to do syntax and/or semantics? Jul 3 '15 at 5:42

Context free grammar (cfg) satisfies your requirements, in a certain sense, when interpreted in a certain way.

(1) Among transformational theories, it is Minimal, since it has no transformations.

(2) It distinguishes X-bar types from X types in a systematic way, by (at least in my interpretation) not having X types as primitives. To be consistent with traditional grammar, it is usual in cfg to write, e.g., P-bar -> P N-bar; P -> under, ..., but this is a bad move. It should be P-bar -> under N-bar, ... If we still want to refer to traditional parts of speech, we define: (a) a head is the pronunciation part of a context free phrase structure rule (cfpsr), and (b) a P is head of a P-bar. Similarly for V (head of V-bar) and N (head of N-bar).

(3) It is binary-branching, at least in the most straightforward way of doing derivations and displaying derivation trees. There is a rule of replacement, allowing a new cfpsr to be derived from two old ones:

replacement: From A -> xBz and B -> y, derive A -> xyz, where A, B are grammatical types (i.e. nonterminal symbols), and x, y, and z are strings of grammatical types and phonemes (i.e. terminal symbols).

A derivation tree is a branching diagram built up by joining subtrees each of whose mother nodes is a cfpsr derived by the replacement rule and whose two daughter nodes are the two cfpsrs used to do the replacement.

(4) It's pithy. Although I used -bar names above, since there are no non-bar primitive syntactic types to distinguish them from, it's not actually necessary to use the bars.

(5) There are no primitive transformations (other than the replacement rule), but derived transformations can be introduced as relations on the cfpsrs, following the method used in GPSG.


I will take your request at face value, not questioning whether you actually want minimalist syntax for your (semantic) goals or not, and will provide a literature recommendation:

Consider taking a look at Andrew Radford's work, either his books: Minimalist Syntax: Exploring the Structure of English (2004), or English Syntax: An Introduction (2004), or for an online source: Minimalist Syntax Revisited.

For a thorough introduction to the minimalist syntax program, you can do a lot worse than with Radford, in my experience.

Note that the above are not strictly presentations of X-bar theory, as you requested, but they represent the further development of this branch (bare phrase structure). That said, your "core requirement" seemed to be to only admit binary branching trees, which holds as a principle in the above.

  • Those books are insufficiently rigorous for my purposes.
    – user65526
    Jul 13 '15 at 11:16
  • If you're looking for mathematical rigor, your search is doomed. There isn't any. There are no axioms and no proofs, therefore no theorems. Minimalism has no undergirdle.
    – jlawler
    Jul 13 '15 at 17:00
  • That's not quite true. There's the work of Stabler and Graf.
    – user65526
    Jul 14 '15 at 7:46

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