Otávio Bueno (Miami) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Inconsistent scientific Theories: A Framework". Abstract: Four important issues need to be considered when inconsistent scientific theories are under discussion: (1) To begin with, are there–and can there be–such things as inconsistent scientific theories? On standard conceptions of the structure of scientific theories, such as the semantic and the syntactic approaches (Suppe [1989], and van Fraassen [1980]), there is simply no room for such theories, given the classical underpinnings of these views. In fact, both the syntactic and the semantic approaches assume that the underlying logic is classical, and as is well known, in classical logic everything follows from an inconsistent theory. Despite this fact, it seems undeniable that inconsistent scientific theories have been entertained–or, at least, stumbled upon–throughout the history of science. So, it looks as though we need to make room for them. (2) But once some room is made for inconsistent scientific theories, how exactly should they be accommodated? In particular, it seems crucial that we are able to understand the styles of reasoning that involve inconsistencies; that is, the various ways in which scientists and mathematicians reason from inconsistent assumptions without deriving everything from them. It is tempting, of course, to adopt a paraconsistent logic to model some of the reasoning styles in question (see da Costa and French [2003], da Costa, Krause, and Bueno [2007], and da Costa, Bueno, and French [1998]). This is certainly a possibility. However, actual scientific practice is not typically done using paraconsistent logic. And if our goal is to understand that practice in its own terms, rather than to produce a parallel discourse about that practice that somehow justifies the adequacy of the latter by invoking tools that are foreign to it, an entirely different strategy is called for. (3) What are the sources of the inconsistencies in scientific theories? Do such inconsistencies emerge from empirical reasons, from conceptual reasons, from both, or by sheer mistake? By identifying the various sources in question, we can handle and assess the significance of the inconsistencies in a better way. Perhaps some inconsistencies are more important, troublesome, or heuristically fruitful than others—and this should be part of their assessment. (4) Several scientific theories become inconsistent due to the mathematical framework they assume. For example, the theories may refer to infinitesimals, as the latter were originally formulated in the early versions of the calculus (see Robinson [1974] and Bell [2005]), the theories may invoke Dirac’s delta function (Dirac [1958]), or some other arguably inconsistent mathematical framework. The issue then arises as to how we should deal with inconsistent applied mathematical theories. What is the status of these theories? Which commitments do they bring? Are we committed to the existence of inconsistent objects if we use such theories in explaining the phenomena? Can an inconsistent scientific theory ever be indispensable? Questions of this sort need to be answered so that we can make sense of the role of inconsistent theories in applications. (For an insightful discussion, see Colyvan [2009].) In this paper, I examine these four issues, and develop a framework–in terms of partial mappings (Bueno, French and Ladyman [2002], and Bueno [2006]), and the inferential conception of the application of mathematics (Bueno and Colyvan [forthcoming])–to represent and interpret inconsistent theories in science. Along the way, I illustrate how the framework can be used to make sense of various allegedly inconsistent theories, from the early formulations of the calculus through Dirac’s delta function and Bohr’s atomic model (Bohr [1913]).