Joel Martinez

The tree-based data structure of △-tree for propositional formulas is introduced in an improved and optimised form. The △-trees allow a compact representation for negation normal forms as well as for a number of reduction strategies in order to consider only those occurrences of literals which are relevant for the satisfiability of the input formula. These reduction strategies are divided into two subsets (meaning- and satisfiability-preserving transformations) and can be used to decrease the size of a negation normal form A at (at most) quadratic cost. The reduction strategies are aimed at decreasing the number of required branchings and, therefore, these strategies allow to limit the size of the search space for the SAT problem

The tree-based data structure of -tree for propositional formulas is introduced in an improved and optimised form. The -trees allow a compact representation for negation normal forms as well as for a number of reduction strategies in order to consider only those occurrences of literals which are relevant for the satisfiability of the input formula. These reduction strategies are divided into two subsets (meaning- and satisfiability-preserving transformations) and can be used to decrease the size of a negation normal form A at (at most) quadratic cost. The reduction strategies are aimed at decreasing the number of required branchings and, therefore, these strategies allow to limit the size of the search space for the SAT problem.

In this paper, I argue against the claim that in Plato's Republic the most important distinguishing feature between the philosopher and non-philosopher is that the philosopher has knowledge while the non-philosopher has, at best, true opinion. This claim is, in fact, inconsistent with statements Plato makes in later books of the Republic. I submit that the important distinction Plato makes concerns the type of knowledge possessed by the philosopher-ruler. As a result, we need to amend widely held scholarly interpretations of important passages in the Republic ; most notably the passages containing the Sun, Line, and Cave. I consider the views of a number of important scholars and suggest a proposal that avoids this inconsistency with the text. An important consequence of my argument is that Philosophers are indeed not the only ones with knowledge in the Kallipolis.

Virtue ethicists argue that modern ethical theories aim to give direct guidance about particular situations at the cost of offering artificial or narrow accounts of ethics. In contrast, virtue ethical theories guide action indirectly by helping one understand the virtues—but the theory will not provide answers as to what to do in particular instances. Recently, this had led many to think that virtue ethical theories are self-effacing the way some claim consequentialist and deontological theories are. In this paper I defend virtue ethics against the charge of self-effacement. I distinguish between modestly self-effacing theories, immodestly self-effacing theories and theories that recommend indirect guidance. Though all self-effacing theories are indirect, not all indirect theories are self-effacing. I argue that virtue ethics is not self-effacing, but rather indirectly action-guiding. The response I articulate draws on the distinctive virtue ethical mode of action-guidance: namely, that thinking hard about virtue and what kind of person one aims to be offers the kind of guidance we want (or should want) as we face practical moral problems.