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By J. Bhasker

Moment version describes extra positive aspects, has elevated try out bench modeling part, extra examples explaining constructs and has routines to each bankruptcy.

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Since 2 every element of σL is of the form τ and every element of ρL is of the form h(τ ), (h is a bijection from (M od(L) =L ) onto {a∗ | a ∈ ExprL }), it follows 2 onto ρL . that h is a bijection from σL (ii): Suppose that L has conjunction. It is clear that P T h(L) = {a∗ | a ∈ ExprL }. , an ) holds i≤n a∗i = c∗ : Suppose that T ∈ c∗ . Then c ∈ T . T = T hL (A) for some model A. Since A is a model of c, it is a model of ai , for every i ≤ n. Whence, ai ∈ T = T hL (A), for all i ≤ n. Whence, T ∈ i≤n a∗i .

If L has conjunction, then the topological spaces 2 ) and (P T h(L), ρL ) are homeomorphic. ((IntL =L ), σL Proof. (i): It is easy to see that h is well-defined, injective and surjective. h is also a bijection from (M od(L) =L ) onto {a∗ | a ∈ ExprL }: For a ∈ ExprL , we have h(M odL (a) =L ) = {h([A]L ) | [A]L ∈ (M od(a) = {T hL (A) | [A]L ∈ (M odL (a) = {T hL (A) | A ∈ M odL (a)} = {T ∈ P T h(L) | T ∈ a∗L } = a ∗L . =L )} =L )} A Topological Approach to Universal Logic 43 It is clear that h is surjective.

Abstract logic without negation There are many examples of abstract logics L = (S, F, |=, V ) with a natural occurrence relation which are not closed under negation, see Figure 4. Several concepts of abstract model theory have definitions which are equivalent if we have negation but otherwise different. It is not immediately obvious which of these definitions are the most natural ones when we do not have negation. For the Interpolation Theorem it seems that the Separation Theorem is the right formulation in the absence of negation.

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