H. Sheffer. A set of five independent postulates for Boolean algebras, with application to logical constants. Notre Dame J. Formal Logic, 10:266--70, 1913.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....successes in Boolean algebra. Many of these successes focus on equational systems in terms of the Sheffer stroke. In 1913, Sheffer presented the following three axiom basis for Boolean algebra in terms of a binary connective now known as the Sheffer stroke, or NAND, that is, xjy = x y [Sheffer, 1913]: x x) x x) x Sheffer 1 x (y (y y) x x Sheffer 2 (x (y z) x (y z) y y) x) z z) x) Sheffer 3 Using a new technique called proof sketches a method for systematically generating sequences of clauses that provide valuable guidance and ....

H. Sheffer. A set of five independent postulates for Boolean algebras, with application to logical constants. Notre Dame J. Formal Logic, 10:266--70, 1913.

....v9.tex; 2 04 2001; 16:02; p.1 2 Less well known is the following (equivalent) 2 basis due to Meredith in 1968 [10, p. 221] x 0 y) 0 x = x (Meredith 1 ) x 0 y) 0 (z y) y (z x) Meredith 2 ) Boolean algebra can be axiomatized with other connectives, and in 1913, Sheffer [13] presented the following 3 basis for Boolean algebra in terms of a binary connective now known as the Sheffer stroke, or NAND, that is, xjy = x 0 y 0 . xjx)j(xjx) x (Sheffer 1 ) xj(yj(yjy) xjx (Sheffer 2 ) xj(yjz) j(xj(yjz) yjy)jx)j( zjz)jx) Sheffer 3 ) Meredith [9] simplified ....

Sheffer, H.: 1913, `A set of five independent postulates for Boolean algebras, with application to logical constants'. Trans. AMS 14(4), 481--488.

....v6.tex; 16 10 2000; 13:16; p.1 2 Less well known is the following (equivalent) 2 basis due to Meredith in 1968 [8, p. 221] x 0 y) 0 x = x (Meredith 1 ) x 0 y) 0 (z y) y (z x) Meredith 2 ) Boolean algebra can be axiomatized with other connectives, and in 1913, Sheffer [11] presented the following 3 basis for Boolean algebra in terms of a binary connective now known as the Sheffer stroke, or NAND, that is, xjy = x 0 y 0 . xjx)j(xjx) x (Sheffer 1 ) xj(yj(yjy) xjx (Sheffer 2 ) xj(yjz) j(xj(yjz) yjy)jx)j( zjz)jx) Sheffer 3 ) Meredith [7] simplified ....

Sheffer, H.: 1913, `A set of five independent postulates for Boolean algebras, with application to logical constants'. Trans. AMS 14(4), 481--488.

....approach was to first prove the theorem in TV and then to successively eliminate (from the input) the axioms of TV until only the axioms of MV remained. The value of proof sketches is not limited to condensed detachment. Example 6.3, Open Questions from Boolean Algebra. In 1913, Henry Sheffer [13] presented a 3 axiom basis for Boolean algebra using the Sheffer stroke . f(f(x; x) f(x; x) x (Sheffer 1) f(x; f(y; f(y; y) f(x; x) Sheffer 2) f(f(x; f(y; z) f(x; f(y; z) f(f(f(y; y) x) f(f(z; z) x) Sheffer 3) More recently, a number of simplifications ( abridgements ) of ....

Sheffer, H., "A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants," Trans. American Mathematical Society, vol. 14 (1913), pp. 481--488.

....by the number of axioms in a system or by the lengths of the axioms in a system. In this report, we show that a short Sheffer stroke identity is a single axiom for Boolean algebra. The Sheffer stroke j can be interpreted as the NOR operation, xjy = x 0 y 0 . In 1913, Sheffer [5] presented the following three axiom equational basis (3 basis) for Boolean algebra. xjx)j(xjx) x (Sheffer 1) xj(yj(yjy) xjx (Sheffer 2) xj(yjz) j(xj(yjz) yjy)jx)j( zjz)jx) Sheffer 3) Partially supported by National Science Foundation grant no. CDA 9503064. y Supported by the ....

H. Sheffer. A set of five independent postulates for Boolean algebras, with application to logical constants. Trans. AMS, 14(4):481--488, 1913.

....than using other complete bases. It is interesting to observe that the fact that NAND offers the most suitable basis for Boolean network simulation within DNA computation continues the traditional use of this basis as a fundamental component within new technologies. Thus, from the work of Sheffer [17] that established the completeness of NAND with respect to propositional logic, through classical gate level design techniques [8] and, continuing, in the present day, with VLSI technologies both in nMOS [11] and CMOS [20, pp. 9 10] The simulation proceeds as follows. An n input, m output ....

H.M. Sheffer. A set of five independent postulates for boolean algebras, with application to logical constants. Trans. Amer. Math. Soc., 14:481--488, 1913.

....of the 2 basis presented in this article led directly to the proof of previously unknown shortest single axioms for Boolean algebra in terms of the Sheffer stroke [4] Specifically, the single axioms were first proved by deriving equations 26a and Commutativity. 2. Background In 1913, Sheffer [7] presented the following 3 basis for Boolean algebra in terms of the Sheffer stroke. x j x) j (x j x) x (Sheffer 1) x j (y j (y j y) x j x) Sheffer 2) x j (y j z) j (x j (y j z) y j y) j x) j ( z j z) j x) Sheffer 3) More recently, a number of equivalent simplifications ....

Sheffer, H.: A set of five independent postulates for Boolean algebras, with application to logical constants, Trans. Amer. Math. Soc. 14(4) (1913), 481--488.

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H.M. Sheffer, A set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants, Am. Math. Soc. Tr. nr. 14, 1913,

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