5 - 5-1Semantics of Predicate Logic1-structuresGiven a...

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Unformatted text preview: 5-1Semantics of Predicate Logic1-structuresGiven a signature={c:s, . . . , f:sns, . . . , p:sm, . . .}a-structureor-interpretationAconsists ofa non-empty setdom(A) =A(e.g.,dom(N) =),for eachc:sin, an elementcAA,for eachf:snsin(n >0), ann-ary function:fA:AnA,for eachp:snin(n >0), ann-ary relation:pAAn.PutA= (A;cA, . . . , fA, . . . , pA, . . .).Notes.(1)fAis aninterpretationoffinA.fis thenameoffA.(2)Ais called a-algebraifhas no predicate symbols other than =.(3) For any, there aremany-structures, for exampleB= (B;cB, . . . , fB, . . . , pB, . . .).1S&A, Chapter 2JZ CAS701 F095-2Examples.(1) For=GorG, a-structure is agroupif it satisfies the groupaxioms (p.2-1).(2) Consider the signature(N) ={0:s,suc:ss}.Then one(N)structure is:N= (; 0N,sucN)()where 0NandsucN:.Although there are many(N)-structures, () is theintendedstructureorintended interpretation.(3) For(N1) =(N) {+,:s2s}the intended(N1)-structure is:N1= (; 0N,sucN,+N,N)....
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5 - 5-1Semantics of Predicate Logic1-structuresGiven a...

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