Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K. On Algebraization of Classical First Order Logic
Nabila M. Bennour, Kahtan H. Alzubaidy
Abstract - Algebraization of first order logic and its 3) SSS )()()( deduction are introduced according to Halmos approach. Application to functional polyadic algebra is done. 4) )( ididS 5) JSJS )()()()( if JIJI Index Term: Polyadic algebra, Polyadic ideal, Polyadic filter, 6) 1 JSSJ ))(()()()( if is one to one on Functional polyadic algebra. 1 J .The cardinal number I is called the degree of the
I. INTRODUCTION algebra. If I = n we get the so called n -adic algebra. If There are mainly three approaches to the algebraization of I I first order logic. One approach is to develop cylindric I then 2 and idI . Therefore we get algebras[1]. The second approach is by polyadic algebra[2]. only the identity quantifier )( id . Thus 0 -adic the third one is by category theory [3]. In 1956 P. R. Halmos introduced polyadic algebra to express algebra is just the Boolean algebra B . If I 1 then first order logic algebraically. I Polyadic algebra is an extension of Boolean algebra with idI . Therefore there are only two quantifiers )( operators corresponding to the usual existential and universal and I )( . Thus 1-adic algebra is the monadic algebra. quantifiers over several variables together with endomorphi- sms to represent first order logic algebraically. Certain polyadic filters and ultra filters have been used to Polyadic Ideals And Filters
express deduction in polyadic algebra. These ideas are A subset U of a Boolean algebra B is called a Boolean applied to the specific case functional polyadic algebra. ideal if
II. POLYADIC ALGEBRA i) 0U General definition [3] ii) Uba for any , Uba Suppose that B is a complete Boolean algebra. An iii) If UbthenabandUa existential quantifier on B is a mapping : BB such that A subset F of a Boolean algebra B is called a Boolean filter if i) 0)0( i) 1 F ii) aa )( for any Ba . ii) Fba for any , Fba iii) baba )()())(( for any , Bba . iii) If FbthenabandFa . B ),( is called monadic algebra. A subset U of a polyadic algebra B is called a polyadic
ideal of B if Let I : aisIII transformation. Denote the i) U is a Boolean ideal set of all endomorphisms on B by BEnd )( and the set of ii) If UaJthenUaandIJ all quantifiers on B by BQant )( . iii) If I UaSthenIandUa A polyadic algebra is SIB ),,,( where A subset F of a polyadic algebra B is called a polyadic : I BEndIS )( and I BQant )(2: such that for filter of B if I I any KJ 2, and , I we have i) F is a Boolean filter 1) id ii) If FaJthenFaandIJ 2) KJKJ iii) If I FaSthenIandFa .
Manuscript received February 06 2014; revised March 24 2014 Nabila M. Bennour is a lecturer at Mathematics department, faculty of Proposition 1) [3] science, Benghazi university; e-mail: [email protected] A subset U of a polyadic algebra B is an ideal of it if and Kahtan H. Alzubaidy is a lecturer at Mathematics department, faculty of science, Benghazi university; e-mail: [email protected] only if U is an ideal of the Boolean algebra B and
ISBN: 978-988-19253-5-0 WCE 2014 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K.
UaI for every Ua . F is a filter of B if and Therefore 21 Jbb . If only if is a filter of the Boolean algebra and F B 21 .... xxxba n , then Ja . Then J is FaI whenever Fa . a Boolean ideal containing . Therefore JU . Proof If Jb , then ... xxxb where x Let U be an ideal of a polyadic algebra , by the definition 211 n i i.e. Ux . Then Ub . Thus JU as a U is an ideal of a Boolean algebra and UaI . i
Let U be an ideal of a Boolean algebra B and Boolean ideal. Let Ja then 21 ... xxxa n
UaI for every Ua . 21 ... n 1 2 ... xIxIxIxxxIaI n If IJ , then aIaJ , so that Therefore UaI . UaJ . Now we verify that UaS , if
I Ua and I . We have aIa and ii) A similar argument leads to (ii).
aISaS , so it means to show that every A filter F of a polyadic algebra is called ultrafilter if F is S acts as identity transformation from I I since there is maximal with respect to the property that 0 F . nothing out of I .Then Ultrafilters satisfy the following important properties [4]. aIaIidaIS , therefore
UaS . Proposition 5) A similar argument proves the second part. Let F be a filter of a polyadic algebra B . Then
i) is an ultrafilter of iff for any exactly one Proposition 2) [3] F B Fa There is a one to one correspondence between ideals and of aa ', belongs to F . filters : if U is an ideal , then the set FU of all a ii) F is ultrafilter of B iff 0 F and Fba iff with Ua is a filter . Analogously, if F is a filter , then Fa or Fb for any , Fba . : FaaFU is an ideal. iii) If FBa , then there is an ultrafilter L such that F L and La . Proposition 3) The set of all polyadic ideals and the set of all polyadic Let B . The ultrafilter containing F )( is denoted by filters are closed under the arbitrary intersections. . UF )( Let B be a polyadic algebra and B . LetU )( denote A mapping : BB between two Boolean algebras is the least polyadic ideal containing and F )( denote the 21 called a Boolean homomorphism if least polyadic filter containing .We say that U )( and i) baba F )( are generated by . ii) aa Obviously, baba , 00 , Proposition 4) Let B be a polyadic algebra and B . Then 11 . i) A mapping : BB between two polyadic algebras is : 21 ... n 21 ,...,, xxxsomeforxxxbBbU n 0 21 ii) called polyadic homomorphism if : 21 ... n 21 ,...,, xxxsomeforxxxbBbF n 1 i) is a Boolean homomorphism Proof i) Let ii) I : 21 ... n 21 ,...,, xxxsomeforxxxbBbJ n 0 iii) for any I
0 J . Let , 21 Jbb . Then 211 ... xxxb n Obviously, . and 2 1 yyb 2 ... y m for some , yx . ii 2121 .... n 21 ... yyyxxxbb m .
ISBN: 978-988-19253-5-0 WCE 2014 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K.
III. DEDUCTION aqapaqp )()())(( , Notation: for B and Bb , ⊢ b reads b is aqapaqp )()())(( , apap )()( , deduced from in B . Now , we define ⊢ b in a a 0)(0 , a 1)(1 for any Aa . We have polyadic algebra B iff Fb .
By definitions of filter and deduction ⊢ we have : Proposition 8) [2]
A Proposition 6) B 1,0,,,, is a functional Boolean algebra. i) ⊢1 and ⊬ 0 ii) If ⊢ x , ⊢ y then ⊢ x y . Proposition 9) [5] A A A B , is a monadic algebra, where : BB is General properties of deduction are given by the following: given by :)(sup))(( Aaapap Theorem 7)
i) If b then ⊢ b Proposition 10) [2] ii) If ⊢ b and then ⊢ b A SIB ,,, is a polyadic algebra, where iii) If ⊢ a for any a and ⊢ b , then ⊢ b : I BEndIS A )( and I BQant A )(2: . iv) If ⊢ b , then ⊢ b for any substitution
v) If ⊢ b , then ⊢ b for some finite k 0 0 . Now, let A : k functionaisBAppB ; Proof. k ,...2,1,0 . Define i) b Fb ⊢ b I 1 k aapaapS k )()1( ),...,(),...,()( for any I ii) ⊢ b Fb 21 ... xxxb n for k and 1 k ),...,( Aaa . we have some xi
Fb ⊢ b Proposition 11) iii) Let ⊢ b Fb Ak SIB ,,, is a polyadic algebra 21 ... xxxb n for some xi Proof
⊢ x1 , ⊢ x2 ,…, ⊢ xn i) 1 k 1 k 1 aapaapaap k ),...,(),...,(sup),...,()( ⊢ ... xxx by proposition (6) 21 n )( id
21 ... xxx n F Fb ii) 1 aapMJ k ),...(sup)( ⊢ b MJ
iv) ⊢ b Fb ... xxxb 1 k 1 aapaap k ),...,(sup),...(sup 21 n J K for some xi 1 k 1 aapMaapJ k ),...,()(),...,()(
21 ... n 1 2 ...xxxxxxb n MJMJ )()()( for some i Fx iii) 1 k id kid )()1( 1 aapaapaapidS k ),...,(),...(),...,()( Fb ⊢ b )( ididS v) By proposition 4 (ii) and proposition (6) iv) 1 k )1( k )( ))1(( aapaapaapS k ))(( ),...,(),...,(),...()(
IV. FUNCTIONAL POLYADIC ALGEBRA k )()1( 1 aapSSaapS k ),...,()()(),...,()( Let BB 1,0,,,, be a complete Boolean algebra, SSS )()()( v) A 1 k 1 ,...,(sup)(),...,()()( aapSaapJS k : functionaisBAppB where A is an J A = algebra of type F. For , Bqp define qp , qp , aap k )()1( ),...(sup J p and 1,0 pointwise as follows: ,...,(sup aap k )()1( J
if JIJI
ISBN: 978-988-19253-5-0 WCE 2014 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K.
1 aapS k ),...,(sup)( J = )(sup iapJSiapS )( J 1 aapJS k ),...,()()( JSJS JSJS )()()()( if JIJI vi) )( iapiapJiapSJ )(sup)( J 1 vi) 1 k aapJaapSJ k )()1( ),...,()(),...,()()( iap )(sup if is 1-1 on J )( 1 J )( aap ),...,(sup sup )( iapSiapS )(sup k )()1( 1 1 J J )( J )( 1 if is 11 on 1 J )( iapJS )()( 1 aap k )()1( ),...,(sup JSSJ )( 1 J )(
1 aapS k ),...,()(sup Natural inference rules of polyadic logic are now transferred 1 J )( into certain algebraic rules in B A governing the algebraic 1 aapJS ),...,())(()( A 1 k deduction in F where B . These are given as 1 JSSJ ))(()()()( follows:
Theorem 13) Finally, let I ,...2,1 be a countable set, Let B A .Then AI I : functionaisBAppB , i) ⊢1 and ⊬ 0 . Aa I ; : AIa is a function. For I I , a is ii) ⊢ qp iff ⊢ p and ⊢ q . iii) ⊢ p iff ⊬ p'. defined by )()( Iiiaia . We have iv) ⊢ p q iff ⊢ p or ⊢ q . v) ⊢ p iff ⊢ pJ . Proposition 12) vi) F⊢ ap )( iff F⊢ apJ )( . AI 0 B is a polyadic algebra Proof. i) By proposition 6(i) Proof ii) Follows from definition of filter and proposition 1 i) apapap )()(sup)()( iii) This is so because if both ⊢ p and ⊢ p' we get )( id ap )( and ap '))(( in F. Thus we get 0 ,which is a contradiction. ii) apapapapMJ )(sup)(sup)(sup)()( MJ J M iv)Let ⊢ qp apMapJ )()()()( )()( Faqap MJMJ ')()( Faqap iii) iapiidapiapidS )())(()()( ')(')( Faqap ididS ' Fap or ')( Faq iv) )( )( iapiapiapS )( )( Fap or )( Faq . )( iapSSiapS )( Thus ⊢ p or ⊢ q . SSS Let ⊢ p or ⊢ q v) )()( iapSiapJS iap )(sup)(sup )( Fap or )( Faq J J iap )(sup apaqap )()()( J )()( Faqap . Therefore ⊢ p q . if JIJI v)This is by pp and FF .
ISBN: 978-988-19253-5-0 WCE 2014 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K.
vi)This is so by the definition :)(sup)( AaapapJ . J If we alter the definition of deduction to ⊢ p if UFp we get the following dichotomy by proposition 5
Theorem 14) Either ⊢ p or ⊢ p'.
REFERENCES [1] L. Henkin , J. D. Monk, and A. Tarski, "Cylindric Algebras Part I" . North-Holland publishing company Amsterdam. London (1971). [2] P.R . Halmos, "Algebraic Logic II .Homogenous Locally Finite Polyadic Boolean Algebras Of Infinite Degree". Fundamenta Mathematicae 43, 255-325 (1965). [3] B. Plotkin, "Universal Algebra, Algebraic Logic, and Databases", Kluwer academic publishers (1994). [4] S. Burris, & H.P. Shakappanavor, "A Course in Universal Algebra", Springer Verlag, New York (1981). [5] K.H. Alzubaidy, "Deduction in Monadic Algebra", Journal of mathematical sciences,18, 1-5 (2007). [6] K.H. AlzubaidyOn "Algebraization of Polyadic Logic", Journal of mathematical sciences,18 , 15-18 (2007). [7] P.R. Halmos, "Lectures on Boolean Algebra" . D.Van Nostran, Princeton (1963). [8] A.G.Hamilton, "Logic for Mathematicians". Cambridge University Press, Cambridge (1988).
ISBN: 978-988-19253-5-0 WCE 2014 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)