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2 - 2-1 Predicate Logic1 Examples of Axiomatizations...

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2-1 Predicate Logic 1 Examples of Axiomatizations Example 1: Group Theory . Axioms: ( i ) Associative: x y z (( x y ) z = x ( y z )) ( ii ) Right Identity: e ( x ( x e = x )) ( iii ) Right Inverse: x y ( x y = e ) Q: Is e a constant or a variable ? Now we can prove theorems of group theory! Language of group theory is given by the Signature Σ 0 G = { e : s, : s 2 s } : function symbol with arity 2 e : function symbol with arity 0, i.e., a constant symbol Note : No predicate symbol. Alternate axiomatisation of Group Theory: Replace ( iii ) by: ( iii ) x ( x x 1 = e ). New signature is: Σ G = { e : s, : s 2 s, 1 : s s } (using mix-fix). So can vary signature for a given structure. 1 S&A, Ch. 1,2; H&R, Ch. 2 JZ CAS701 F09
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2-2 Note: for now: single-sorted signatures, single-sorted structures. Example 2: Number Theory . Dedekind-Peano Axioms : Universal closures of: suc ( x ) negationslash = 0 suc ( x ) = suc ( y ) x = y ϕ (0) ∧ ∀ x ( ϕ ( x ) ϕ ( suc ( x ))) → ∀ ( x ) for any formula ϕ ( x ). Σ ( N 0 ) = Can expand Σ ( N 0 ) by adding: function symbols predicate symbols Example 3: Expanding Σ ( N 0 ) with function symbols . Σ ( N ) = Σ ( N 0 ) ∪ { + , × : s 2 s } Example 4: Expanding Σ ( N ) with predicate symbols . Σ ( N < ) = Σ ( N ) ∪ { <, = : s 2 }
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2-3 Ex (1) For each of the following structures, say whether it is a group or not. If it is not, give a reason. ( a ) ( , + , 0) ( is the set of integers) ( b ) ( , + , 0) ( is the set of naturals) ( c ) ( , × , 1) ( d ) ( , + , 0) ( is the set of reals) ( e ) ( , × , 1) ( f ) ( negationslash =0 , × , 1) ( negationslash =0 is 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 ).)
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2-4 Signatures 2 Definition . A (1-sorted) signature is a finite set Σ of strings of two kinds: ( i ) function symbols of arity n (= 0 , 1 , 2 ,... ) f : s n s ( ii ) predicate symbols of arity n (= 0 , 1 , 2 ,... ) p : s n where s is the sort symbol .
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