The truth functional hypothesis does not imply the liars paradox

The truth-functional hypothesis states that indicative conditional sentences and the material implication have the same truth conditions. Haze (2011) has rejected this hypothesis. He claims that a self-referential conditional, coupled with a plausible assumption about its truth-values and the assumption that the truth-functional hypothesis is true, lead to a liar’s paradox. Given that neither the self-referential conditional nor the assumption about its truth-values are problematic, the culprit of the paradox must be the truth-functional hypothesis. Therefore, we should reject it. In this paper I argue that, contrary to what Haze thinks, the truth-functional hypothesis is not to blame. In fact, no liar’s paradox emerges when the truth-functional hypothesis is true; it emerges only if it is false

The spectrum of a vertex model and related spin one chain sitting in a genus five curve

We derive the transfer matrix eigenvalues of a three-state vertex model whose weights are based on a $\mathrm{R}$-matrix not of difference form with spectral parameters lying on a genus five curve. We have shown that the basic building blocks for both the transfer matrix eigenvalues and Bethe equations can be expressed in terms of meromorphic functions on an elliptic curve. We discuss the properties of an underlying spin one chain originated from a particular choice of the $\mathrm{R}$-matrix second spectral parameter. We present numerical and analytical evidences that the respective low-energy excitations can be gapped or massless depending on the strength of the interaction coupling. In the massive phase we provide analytical and numerical evidences in favor of an exact expression for the lowest energy gap. We point out that the critical point separating these two distinct physical regimes coincides with the one in which the weights geometry degenerate into union of genus one curves.Comment: 22 pages, 12 figure

Integrable Vertex Models with General Twists

We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions. Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method. Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the $B_n$, $D_n$ and $C_n$ Lie algebras are presented. The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.Comment: 9 pages, Proceedings of Recent Progress in Solvable Models: RIMS Project, Kyoto, Japan, 20-23 July 200