Attribute-value logic and the theory of grammar by Mark Johnson

By Mark Johnson

End result of the ease in their implementation, attribute-value dependent theories of grammar have gotten more and more renowned in theoretical linguistics as a substitute to transformational debts and in computational linguistics. This booklet presents a proper research of attribute-value constructions, their use in a idea of grammar and the illustration of grammatical family members in such theories of grammar. It presents a classical therapy of disjunction and negation, and explores the linguistic implications of other representations of grammatical family members. Mark Johnson is assistant professor in cognitive and linguistic sciences at Brown collage. He was once a Fairchild postdoctoral fellow on the Massachusetts Institute of expertise throughout the 1987-88 educational yr.

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Let M be any model and let A be any wff of AVL. If M. |= A v A, then by the definition of V, M 1= A. Thus (A v A) r> A is a tautology for all wffs A. Now consider the equality axiom schemata (El) and (E2). For any model :W = (F,C,5,<)>,x) ar»d any variable x, >(x) e F, and thus M N x » x. Similarly, for any constant symbol c, JcJ = %(c) e F, and thus fW 1= c = c. Consider the instances of axiom schema (E3). For any model M = (F,C,<5,q>,%), % is an injective function, thus for q ^c 2 , [[cj = %(q) * X(c2) = Icj]], and hence M N q 2 c^ Now consider axiom schema (E4).

The validity of axioms that are instances of axiom schemata (PI) through (P6) and the fact that Modus Ponens preserves validity can be proven using the standard techniques of the prepositional calculus (Andrews 1986). I show here that all instances of axiom schema (P3) are valid; the other axioms can be shown valid using similar techniques. Let M be any model and let A be any wff of AVL. If M. |= A v A, then by the definition of V, M 1= A. Thus (A v A) r> A is a tautology for all wffs A. Now consider the equality axiom schemata (El) and (E2).

It is convenient to have metalanguage symbols that represent arbitrary terms or wffs, just as in mathematics a letter such as x is sometimes used to represent an arbitrary real number. Henceforth I will use x, y, z to represent arbitrary variables; a, b, c, to represent arbitrary constants; s, t, u, v, w to represent arbitrary terms, and A, B, C to represent arbitrary wffs. #• This is so I can use '=' to denote identity of terms, wffs or entities. Thus t, = t2 is a wff of A, while ta = t2 is a metalevel statement asserting the (string) identity of the terms denoted by the variables over terms tj and t^.

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