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# 5 - 5-1 Semantics of Predicate Logic1-structures Given a...

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5-1 Semantics of Predicate Logic 1 Σ -structures Given a signature Σ = { c : s,...,f : s n s,...,p : s m ,... } a Σ -structure or Σ -interpretation A consists of a non-empty set dom ( A ) = A (e.g., dom ( N ) = ), for each c : s in Σ , an element c A A, for each f : s n s in Σ ( n> 0), an n -ary function: f A : A n A, for each p : s n in Σ ( n> 0), an n -ary relation: p A A n . Put A = ( A ; c A ,...,f A ,...,p A ,... ) . Notes . (1) f A is an interpretation of f in A . f is the name of f A . (2) A is called a Σ -algebra if Σ has no predicate symbols other than =. (3) For any Σ , there are many Σ -structures, for example B = ( B ; c B ,...,f B ,...,p B ,... ) . 1 S&A, Chapter 2 JZ CAS701 F09

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5-2 Examples . (1) For Σ = Σ 0 G or Σ G , a Σ -structure is a group if it satisfies the group axioms (p.2-1). (2) Consider the signature Σ ( N 0 ) = { 0: s, suc : s s } . Then one Σ ( N 0 )–structure is: N 0 = ( ; 0 N , suc N ) ( ) where 0 N and suc N : . Although there are many Σ ( N 0 )-structures, ( ) is the intended structure or intended interpretation .
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5 - 5-1 Semantics of Predicate Logic1-structures Given a...

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