Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.

Does tenure review in New Zealand’s South Island give rise to rents?

Under “tenure review,” the ongoing privatization of South Island Crown pastoral leases, a pastoral lessee surrenders part of his leasehold, and acquires a freehold interest in the remainder. In order to determine whether the Crown sold the right to freehold too cheaply, we model the proportional difference between the price (per hectare) at which the Crown sold its interest to the lessee, and the prices paid to former lessees who have onsold some part of their new freeholds.tenure review; pastoral leases; New Zealand; rent seeking

Social Security and personal saving: 1971 and beyond

Feldstein (1996, 1974) reported that Social Security in the U.S.A. reduced personal saving (“saving”) in 1992 (1971) by $416 ($61) billion. I reestimate his life-cycle consumption specification using data from the latest NIPA revision, correct his calculations, and find that the implied reduction in 1992 (1971) saving is now $280 ($22) billion, 48% (16%) of actual net private saving, with a standard error of $114 ($14) billion. If structural breaks around WWII and the 1972 Social Security amendments (which raised real per capita SSW by 22%) are allowed, and the market value of Treasury debt included in the specification, the reduction in 1971 and 1992 saving attributable to Social Security is at most 0.55 times its standard error, and 12% of net private saving. I then reestimate the preferred specification of Coates and Humphreys (1999), allowing for these structural breaks and relaxing other restrictions. The implied effect of Social Security on saving is again statistically zero. Copyright Springer-Verlag Berlin Heidelberg 2003Key words: Public pensions, Social Security, personal saving, aging population, life-cycle consumption function., JEL classification: E2; E6; H3,

Getting it Right: Superannuation and Savings in the U.S.A.

Feldstein (1996) added the present value of future Social Security benefits (SSW) to the lifecycle consumption function and found that Social Security reduced private saving in 1992 by more than half. I show that this finding is significant, with a standard error ranging from 35% to 65% of actual savings in 1992. The large reduction in savings also holds when the (rejected) restrictions implied by disposable income are relaxed. But this reduction is neither robust nor significant if the sample excludes either the 1930s or the data subsequent to the 1972 legislated changes in Social Security, or if income is GNP instead of NNP

A Multicountry Characterization of the Nonstationarity of Aggregate Output.

The authors compute the scaled varlogram (the variances of kth differences scaled by the variance of first differences) of the log of annual per capita real aggregate output (GDP or GNP), as measured by (1) the long series for the United States and United Kingdom; (2) Angus Maddison's (1982) long series for twelve countries; and (3) the postwar IFS data for thirty-two countries. Simulations show that the scaled varlogram of real output is nearly always more consistent with the data being generated by parsimonious difference stationary than trend stationary univariate processes. In fact, the data reveal some "excess nonstationarity" relative to parsimonious ARIMA models. The power of the scaled varlogram to discriminate between trend stationary and difference stationary processes is somewhat greater than that of a Dickey-Fuller type F test. Copyright 1990 by Ohio State University Press.