Abstract
Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature P = {P_i}_{i in I_P} be given. For a set Phi = {phi _i}_{i in I_{Phi }} of L_P-formulas, we introduce a corresponding set Q = {Q_i}_{i in I_{Phi }} of new relation symbols and a set of explicit definitions of the Q_i in terms of the phi _i. This is called a definition system, denoted d_{Phi }. A definition system d_{Phi } determines a translation functiontau _{Phi } : L_Q rightarrow L_P. Any L_P-structure A can be uniquely definitionally expanded to a model A^{+} models d_{Phi }, called A + d_{Phi }. The reduct A + d_{Phi } to the Q-symbols is called the definitional imageD_{Phi }A of A. Likewise, a theory T in L_P may be extended a definitional extension T + d_{Phi }; the restriction of this extension T + d_{Phi } to L_Q is called the definitional imageD_{Phi }T of T. If T_1 and T_2 are in disjoint signatures and T_1 + d_{Phi } equiv T_2 + d_{Theta }, we say that T_1 and T_2 are definitionally equivalent (wrt the definition systems d_{Phi } and d_{Theta }). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories.
Highlights
Sometimes theories are formulated with different sets of primitives and yet are definitionally equivalent
An extension T + in LP,Q of T in LP is called a conservative extension of T with respect to LP -formulas iff, for any LP -formula α, T+ α ⇒ T α
Given the set Φ = {φi}i∈I of LP -formulas, we introduce a disjoint set Q = {Qi}i∈I of new relation symbols, with card Q = card Φ, and with the arity of Qi matching the arity of φi, and let ni be a(φi)
Summary
Sometimes theories are formulated with different sets of primitives and yet are definitionally equivalent. Suppose we introduce a new binary relation symbol, Q, and give an explicit definition of it—call this definition d1—in terms of P as follows: d1 : ∀x∀y(Q(x, y) ↔ (P (x, y) ∧ x = y)). If A1 and A2 are LP -structures, an isomorphism f : A1 → A2 is a bijection from dom(A1) to dom(A2) satisfying the preservation condition that f [(Pi)A1 ] = (Pi)A2 , for each relation symbol Pi in the signature P. An extension T + in LP,Q of T in LP is called a conservative extension of T with respect to LP -formulas iff, for any LP -formula α, T+ α ⇒ T α
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