Minimalist grammars are a class of formal grammars that aim to provide a more rigorous, usually proof-theoretic, formalization of Chomskyan Minimalist program than is normally provided in the mainstream Minimalist literature. A variety of particular formalizations exist, most of them developed by Edward Stabler, Alain Lecomte, Christian Retoré, or combinations thereof.
Lecomte and Retoré's extensions of the Lambek Calculus Lecomte and Retoré (2001) introduce a formalism that modifies that core of the Lambek Calculus to allow for movement-like processes to be described without resort to the combinatorics of Combinatory categorial grammar. The formalism is presented in proof-theoretic terms. Differing only slightly in notation from Lecomte and Retoré (2001), we can define a minimalist grammar as a 3-tuple G = ( C , F , L ) {\displaystyle G=(C,F,L)} , where C {\displaystyle C} is a set of "categorial" features, F {\displaystyle F} is a set of "functional" features (which come in two flavors, "weak", denoted simply f {\displaystyle f} , and "strong", denoted f ∗ {\displaystyle f*} ), and L {\displaystyle L} is a set of lexical atoms, denoted as pairs w : t {\displaystyle w:t} , where w {\displaystyle w} is some phonological/orthographic content, and t {\displaystyle t} is a syntactic type defined recursively as follows:
all features in C {\displaystyle C} and F {\displaystyle F} are (atomic) types, and if X {\displaystyle X} and Y {\displaystyle Y} are types, so are X / Y {\displaystyle X/Y} , X ∖ Y {\displaystyle X\backslash Y} , and X ∘ Y {\displaystyle X\circ Y} . We can now define 6 inference rules:
⊢ w : X {\displaystyle \vdash w:X} , for all w : X ∈ L {\displaystyle w:X\in L}
w : X ⊢ w : X {\displaystyle w:X\vdash w:X} , for all w : X ∉ L {\displaystyle w:X\notin L}
Γ ⊢ a : X / Y Γ ′ ⊢ b : Y Γ ; Γ ′ ⊢ a b : X [ / E ] {\displaystyle {\frac {\Gamma \vdash a:X/Y\qquad \Gamma '\vdash b:Y}{\Gamma ;\Gamma '\vdash ab:X}}[/E]}
Γ ′ ⊢ b : Y Γ ⊢ a : X ∖ Y Γ ′ ; Γ ⊢ b a : X [ ∖ E ] {\displaystyle {\frac {\Gamma '\vdash b:Y\qquad \Gamma \vdash a:X\backslash Y}{\Gamma ';\Gamma \vdash ba:X}}[\backslash E]}
Γ ; Γ ′ ⊢ α Γ , Γ ′ ⊢ α e n t r o p y {\displaystyle {\frac {\Gamma ;\Gamma '\vdash \alpha }{\Gamma ,\Gamma '\vdash \alpha }}entropy}
Γ ⊢ a : X ∘ Y Δ , b : X , c : Y , Δ ′ ⊢ d : Z Δ , Γ , Δ ′ ⊢ d [ b := a , c := a ] : Z [ ∘ E ] {\displaystyle {\frac {\Gamma \vdash a:X\circ Y\qquad \Delta ,b:X,c:Y,\Delta '\vdash d:Z}{\Delta ,\Gamma ,\Delta '\vdash d[b:=a,c:=a]:Z}}[\circ E]}
… excerpt ends here. Continue reading the full article.
