Psychological Eudaimonism and Interpretation in Greek Ethics

Plato extends a bold, confident, and surprising empirical challenge. It is implicitly a claim about the psychological — more specifically motivational — economies of human beings, asserting that within each such economy there is a desire to live well. Call this claim ‘psychological eudaimonism’ (‘PE’). Further, the context makes clear that Plato thinks that this desire dominates in those who have it. In other words, the desire to live well can reliably be counted on (when accompanied with correct beliefs about the role of morality or virtue in living well) to move people be virtuous. As we will argue, this general claim appears in not only Plato but Aristotle and the Stoics as well. But it is one we might wonder about, in three ways. First, we might wonder about its warrant. After all, the claim is universal in scope; yet it is about a highly contingent fact about the motivational propensities of individual human organisms, and there is abundant variability in the individual forms human nature takes. What grounds could the ancients have for their confidence that there are no outliers (assuming, as we do, that they do not merely misspeak in framing general claims as universal ones)? Second, we might wonder about its truth. For were it true, it would entail something remarkable about the nature of rationality that we (post-)moderns would be wise to heed. And third, we might wonder about its relationship with normative eudaimonism. By ‘normative eudaimonism’ (‘NE’) we mean the claim that we have conclusive reason to act in ways that conduce to our own eudaimonia. As we will show, the key to these three questions is the first. If we consider what justification the ancients have for their claim, we can see why that claim must be true. Moreover, as we will also show, it must be true because of the nature of practical rationality as the ancients understood it — that is, in terms of normative eudaimonism. We will show this by marshalling unexpected resources: Donald Davidson’s work in understanding how we interpret others and in so doing make sense of them as rational beings. If we couple Davidson’s account of interpretation with the eudaimonist structure of practical rationality essential to these ancient ethical theories, psychological eudaimonism is a consequence. The paper proceeds as follows. In Section I, we lay out the textual basis for ascribing PE to Plato, Aristotle, and the Stoics. In Section II, we introduce Davidson’s account of interpretation. This allows us to appropriate that account in Section III to the particular purposes of normative eudaimonism, to support the claim that we must ascribe the desire to live well to those whom we would see as rational. Finally, in Section IV we consider challenges to this strategy

The complexity of counting locally maximal satisfying assignments of Boolean CSPs

We investigate the computational complexity of the problem of counting the maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over the Boolean domain {0,1}. A satisfying assignment is maximal if any new assignment which is obtained from it by changing a 0 to a 1 is unsatisfying. For each constraint language Gamma, #MaximalCSP(Gamma) denotes the problem of counting the maximal satisfying assignments, given an input CSP with constraints in Gamma. We give a complexity dichotomy for the problem of exactly counting the maximal satisfying assignments and a complexity trichotomy for the problem of approximately counting them. Relative to the problem #CSP(Gamma), which is the problem of counting all satisfying assignments, the maximal version can sometimes be easier but never harder. This finding contrasts with the recent discovery that approximately counting maximal independent sets in a bipartite graph is harder (under the usual complexity-theoretic assumptions) than counting all independent sets.Comment: V2 adds contextual material relating the results obtained here to earlier work in a different but related setting. The technical content is unchanged. V3 (this version) incorporates minor revisions. The title has been changed to better reflect what is novel in this work. This version has been accepted for publication in Theoretical Computer Science. 19 page

Embedding a Forest in a Graph

For \math{p\ge 1}, we prove that every forest with \math{p} trees whose sizes are $a_1,..., a_p$ can be embedded in any graph containing at least $\sum_{i=1}^p (a_i + 1)$ vertices and having a minimum degree at least $\sum_{i=1}^p a_i$.Comment: Working paper, submitte

The Complexity of Approximately Counting Tree Homomorphisms

We study two computational problems, parameterised by a fixed tree H. #HomsTo(H) is the problem of counting homomorphisms from an input graph G to H. #WHomsTo(H) is the problem of counting weighted homomorphisms to H, given an input graph G and a weight function for each vertex v of G. Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in #P. We give a complete trichotomy for #WHomsTo(H). If H is a star then #WHomsTo(H) is in FP. If H is not a star but it does not contain a certain induced subgraph J_3 then #WHomsTo(H) is equivalent under approximation-preserving (AP) reductions to #BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class #RHPi_1 under AP-reductions. Finally, if H contains an induced J_3 then #WHomsTo(H) is equivalent under AP-reductions to #SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, #WHomsTo(H) is complete for #P under AP-reductions. The results are similar for #HomsTo(H) except that a rich structure emerges if H contains an induced J_3. We show that there are trees H for which #HomsTo(H) is #SAT-equivalent (disproving a plausible conjecture of Kelk). There is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model. In particular, we show that for a family of graphs J_q, parameterised by a positive integer q, the problem #HomsTo(H) is AP-interreducible with the problem of approximating the partition function of the q-state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of #HomsTo(J_q)

Enhancing the Engineering Curriculum: Defining Discovery Learning at Marquette University

This paper summarizes the results of our investigation into the feasibility of increasing the level of discovery learning in the College of Engineering (COE) at Marquette University. We review the education literature, document examples of discovery learning currently practiced in the COE and other schools, and propose a Marquette COE-specific definition of discovery learn-ing. Based on our assessment of the benefits, costs, and tradeoffs associated with increasing the level of discovery learning, we pre-sent several recommendations and identify resources required for implementation. These recommendations may be helpful in enhancing engineering education at other schools

Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org