2 - 2-1Predicate Logic1Examples of AxiomatizationsExample...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2-1Predicate Logic1Examples of AxiomatizationsExample 1: Group Theory. Axioms:(i) Associative:xyz((xy)z=x(yz))(ii) Right Identity:e(x(xe=x))(iii) Right Inverse:xy(xy=e)Q:Iseaconstantor avariable?Now we can prove theorems of group theory!Language of group theoryis given by theSignatureG={e:s,:s2s}: function symbol with arity 2e: function symbol with arity 0, i.e., a constant symbolNote: No predicate symbol.Alternate axiomatisation of Group Theory:Replace(iii) by:(iii)x(xx1=e).New signature is:G={e:s,:s2s,1:ss}(using mix-fix).So canvarysignature for a given structure.1S&A, Ch. 1,2; H&R, Ch. 2JZ CAS701 F092-2Note: for now:single-sortedsignatures,single-sortedstructures.Example 2: Number Theory.Dedekind-Peano Axioms: Universal closures of:suc(x)negationslash= 0suc(x) =suc(y)x=y(0) x((x)(suc(x))) x(x)for any formula(x).(N) =Canexpand(N) by adding:function symbolspredicate symbolsExample 3: Expanding(N) with function symbols.(N) =(N) {+,:s2s}Example 4: Expanding(N)with predicate symbols.(N<) =(N) {<,= :s2}2-3Ex(1) For each of the following structures, say whether it is a group or not.If it is not, give a reason.(a) (Z,+,0)(Zis the set of integers)(b) (,+,0)(is the set of naturals)(c) (Z,,1)(d) (,+,0)(is the set of reals)(e) (,,1)(f) (negationslash=0,,1)(negationslash=0is the set of non-zero reals)(g) (+,,1)(+is the set of positive reals)(2) Formalise in predicate logic the statements:(a) All men are liars(b) Some men are liars(c) No men are liars(d) Some men are not liars(e) Every student studies some subject(f) Some student studies every subject(Use predicate symbols M(x), L(x), Stu(x), Sub(x), S(x, y).)2-4Signatures2Definition. A(1-sorted) signatureis a finite setof strings of twokinds:(i)function symbolsof arityn(= 0,1,2, . . .)f:sns(ii)predicate symbolsof arityn(= 0,1,2, . . .)p:snwheresis thesort symbol.Notes....
View Full Document

This note was uploaded on 10/26/2009 for the course CAS 701 taught by Professor Zucker during the Fall '09 term at McMaster University.

Page1 / 11

2 - 2-1Predicate Logic1Examples of AxiomatizationsExample...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online