Multiplicative Gains, Non-ergodic Utility, and the Just One More Paradox (with Supplemental Information)

Interest in multiplicative vs. additive returns on bets has been revived by Peters, who proposes ergodicity and added noise are useful in understanding utility preferences. Peters requires a Monte Carlo simulation to demonstrate empirically a supposed paradox that arithmetic expectation is inappropriate for multiplicative gain distribution forecasting. Here I formalize the r operator notation, which significantly simplifies multiplicative problems, as an extension of the arithmetic group's Δ/d discrete and continuous operators into the multiplicative semigroup. I show how the annihilating (absorbing) element of the multiplicative semigroup at 0, not +/-∞, may be used to conveniently represent nonlinear sequence occurrences, such as running out of money, without the need for special computer rules outside the mathematics. I use this to solve Peters' expected-value paradox elegantly, without ergodicities nor noise. But Peters misses the real paradox, called “Just One More”: the outcome of an advantageous additive gamble is identical to the outcome of a similar disadvantageous multiplicative gamble, after one trial; hence, by induction, an agent will keep playing. I propose games “Hero or Heroin” and “American Roulette” to highlight this paradox. This may help in explaining addiction. The Supplement contains further visualizations and arguments against the need and applicability of ergodics for utility. The results contribute to the understanding of repeated multiplicative gambles with annihilating states, and their utility.

On the Curious Calculi of Wittgenstein and Spencer Brown

In his Tractatus, Wittgenstein sets out what he calls his N-operator notation which can be used to calculate whether an expression is a tautology. In his Laws of Form, George Spencer Brown offers what he calls a “primary algebra” for such calculation. Both systems are perplexing. But comparing two blurry images can reduce noise, producing a focus. This paper reveals that Spencer Brown independently rediscovered the quantifier-free part of the N-operator calculus. The comparison sheds a flood light on each and from the letters of correspondence we shall find that Russell, as one might have surmised, was a catalyst for both.

TRACTARIAN FIRST-ORDER LOGIC: IDENTITY AND THE N-OPERATOR

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503

Dynamics of Gyroelastic Continua

This paper introduces the idea of distributed gyricity, in which each volume element of a continuum possesses an infinitesimal quantity of stored angular momentum. The continuum is also assumed to be linear-elastic. Using operator notation, a partial differential equation is derived that governs the small displacements of this gyroelastic continuum. Gyroelastic vibration modes are derived and used as basis functions in terms of which the general motion can be expressed. A discretized approximation is also developed using the method of Rayleigh-Ritz. The paper concludes with a numerical example of gyroelastic modes.