Kant, Bolzano, and the Formality of Logic

In §12 of his 1837 magnum opus, the Wissenschaftslehre, Bolzano remarks that “In the new logic textbooks one reads almost constantly that ‘in logic one must consider not the material of thought but the mere form of thought, for which reason logic deserves the title of a purely formal science’” (WL §12, 46).1 The sentence Bolzano quotes is his own summary of others’ philosophical views; he goes on to cite Jakob, Hoffbauer, Metz, and Krug as examples of thinkers who held that logic abstracts from the matter of thought and considers only its form. Although Bolzano does not mention Kant by name here, Kant does of course hold that “pure general logic”, what Bolzano would consider logic in the traditional sense (the theory of propositions, representations, inferences, etc.), is formal. As Kant remarks in the Introduction to the 2nd edition of Kritik der reinen Vernunft , (pure general) logic is “justified in abstracting – is indeed obliged to abstract – from all objects of cognition and all of their differences; and in logic, therefore, the understanding has to do with nothing further than itself and its own form” (KrV, Bix).

Kant, Morality, and Hell

In this paper I argue that, although Kant argues that morality is independent of God (and hence, agrees with the Euthyphro), and rejects Divine Command Theory (or Theological Voluntarism), he believes that all moral duties are also the commands of God, who is a moral being, and who is morally required to punish those who transgress the moral law: "God’s justice is the precise allocation of punishments and rewards in accordance with men’s good or bad behavior." However, since we lack a strict proof of God's existence, we can still fulfill our duties from the motive of duty. if we did know that God exists, then this would undermine our pure moral motivation to do our duty, since we would have an even stronger interest in pleasing God through our good conduct. The effect of undermining our pure moral motivation would be to make us less eligible for divine reward, since God rewards us for doing our duty from the motive of duty

Using Canonical Forms for Isomorphism Reduction in Graph-based Model Checking

Graph isomorphism checking can be used in graph-based model checking to achieve symmetry reduction. Instead of one-to-one comparing the graph representations of states, canonical forms of state graphs can be computed. These canonical forms can be used to store and compare states. However, computing a canonical form for a graph is computationally expensive. Whether computing a canonical representation for states and reducing the state space is more efficient than using canonical hashcodes for states and comparing states one-to-one is not a priori clear. In this paper these approaches to isomorphism reduction are described and a preliminary comparison is presented for checking isomorphism of pairs of graphs. An existing algorithm that does not compute a canonical form performs better that tools that do for graphs that are used in graph-based model checking. Computing canonical forms seems to scale better for larger graphs

Three-Loop Predictions for the Light Higgs Mass in the MSSM

The Minimal Supersymmetric Extension of the Standard Model (MSSM) features a light Higgs boson, the mass Mh of which is predicted by the theory. Given that the LHC will be able to measure the mass of a light Higgs with great accuracy, a precise theoretical calculation of Mh yields an important test of the MSSM. In order to deliver this precision, we present three-loop radiative corrections of O(alpha_t*alpha_s^2) and provide a computer code that combines our results with corrections to Mh at lower loop orders that are available in the literature.Comment: 3 pages, 1 figure, contribution to the proceedings of the conference "Physics at LHC 2010

Radial infall of two compact objects: 2.5PN linear momentum flux and associated recoil

The loss rate of linear momentum from a binary system composed of compact objects (radially falling towards each other under mutual gravitational influence) has been investigated using the multipolar post-Minkowskian approach. The 2.5PN accurate analytical formula for the linear momentum flux is provided, in terms of the separation of the two objects, in harmonic coordinates, both for a finite and infinite initial separation. The 2.5PN formulas for the linear momentum flux are finally used to estimate the recoil velocity accumulated during a premerger phase of the binary evolution.Comment: 17 pages, 2 figure