Expansion Let i and o be two first order systems 1. If L(FuCL(2), then we say that F2 is an expansion of Fi 2. If L(FCc(F2), then we say that F2 is a proper expansion of F1 3. 2 is an expansion of i iff every constant of F1 is a constant ya Logic in Computer Science p 6/21
Expansion Let F1 and F2 be two first order systems. 1. If L(F1) ⊆ L(F2), then we say that F2 is an expansion of F1. 2. If L(F1) ⊂ L(F2), then we say that F2 is a proper expansion of F1. 3. F2 is an expansion of F1 iff every constant of F1 is a constant F2. Logic in Computer Science – p.6/21
Extension Let i and o be two first order systems T1≌C(万1)andI2SC(F2) 1. If L(FiCC(F2)and Th(F1∪n1)sTh(2∪I2, then we say that F2ur2 is an extension of flury 2. If for every A L(Fi), Iihr A iff T2 HF, A then we say that F2Ur2 is a conservative extension of fFiUr1 Logic in Computer Science p 7/21
Extension Let F1 and F2 be two first order systems. Γ1 ⊆ L(F1) and Γ2 ⊆ L(F2). 1. If L(F1) ⊆ L(F2) and Th(F1 S Γ1) ⊆ Th(F2 S Γ2), then we say that F2 S Γ2 is an extension of F1 S Γ1. 2. If for every A ∈ L(F1), Γ1 `F1 A iff Γ2 `F2 A, then we say that F2 S Γ2 is a conservative extension of F1 S Γ1. Logic in Computer Science – p.7/21