Applications of Logic

It is an undeniable fact that there are many logics: classical logic, modal logic, relevant logic, quantum logic, temporal logic, deontic logic, para­consistent logic, free logic, and many more (for a description of many of these logics, see Gabbay and Guenthner 2001).

Consider the distinction between pure and applied logic (da Costa 1997). We can think of a pure logic as a formal calculus that is studied primarily for its mathematical and formal features. We introduce a formal language, including connectives and quantifiers, specify rules for the behavior of the logical vocabulary, and define a concept of logical consequence for the language. We then study syntactic and semantic properties of the resulting language, and, whenever possible, prove a completeness theorem (to the effect that every logically valid statement is provable in the system). When we develop a logic in this way, we are not concerned with any applications it may have. We are simply developing a formal system.

This is entirely analogous to the notion of pure mathematics. In geo­metry, we may introduce various principles that characterize a domain of geometrical objects (such as points and curves), and we can then study these objects for their mathematical interest. In this way, we can introduce Euclidean systems - systems that satisfy the traditional assumptions about geometrical objects (in particular, in these systems the sum of the internal angles of a triangle is 180°). But we can also introduce non-Euclidean systems - systems in which some of the traditional assumptions about geometrical objects no longer hold (for example, the sum of the internal angles of a triangle is more than 180°).

Suppose, now, that we want to find out whether any of these geometrical systems, properly interpreted, apply to the physical world. We want to know what is the geometry of the universe. At this point, we leave the realm of pure mathematics and shift to applied mathematics. It is no longer enough to find out what follows (or not) from certain principles about geometrical objects. We need to develop suitable empirical interpretations of these principles that let us apply the pure geometrical systems to a messy, motley, often incongruous world. We can start with simple identifications, such as a light ray with a geometrical line, and try to work our way from there (see Putnam 1979).

Exactly the same point goes through in the case of applied logic. Logic can be applied to a huge variety of things: from electronic circuits and pieces of formal syntax through air traffic control, robotics, and computer pro­gramming to natural languages. In fact, paraconsistent logics have been applied to many of these items (see da Costa, Krause, and Bueno 2007). It is not difficult to see the motivation. For example, suppose that you are an air traffic controller operating two radars. According to one of them, there is an airplane approaching the airport from the southeast; according to the other, there is none. You have here contradictory reports, and no obvious way to decide which of the radars is correct. In situations such as this, it can be very useful to adopt a paraconsistent logic. The logic can be used to help you reason about these pieces of information without triviality; that is, without logically implying everything. This may help you determine which bits of information should be eventually rejected.

Of course, crucial among these applications of logic is the application to the vernacular (see Priest 2006b). Ultimately, what we want to find out when we apply a pure logic is the consequence relation (or the various consequence relations if logical pluralism turns out to be correct) that is (or are) used in the language we speak - although the other applications are important as well.

Now, reflection on what holds in a certain domain may give us reason to reconsider the adequacy of certain patterns of inference. For instance, as we saw, since the distributive law can be violated in quantum domains, we may adopt inferential procedures in which this law is not invoked when reasoning about quantum phenomena. This offers a certain conception of how new relations of logical consequence can be formulated, and sub­sequently applied.

In this respect, there are significant connections between the application of logic and the application of mathematics. In some cases, pure math­ematical theories that were initially developed quite independently of con­cerns with applications, turned out to be particularly useful in applied contexts. A case such as the development of non-Euclidean geometries illustrates this point. In other cases, the need for the development of suitable mathematical theories emerged from the lack of appropriate mathematical tools to describe certain empirical situations. This was the case, for instance, with the early development of the calculus by Isaac Newton and Gottfried Leibniz. The calculus was a mathematical theory that emerged, basically, from the need to articulate certain explanations in physics.

The same points go through in the case of applied logic. In certain cases, pure logics have been developed in advance of particular applications. The use of paraconsistent logic in air traffic control or in robotics was certainly not part of the original motivation to formulate this logic. In other cases, certain needs in applied contexts demanded certain relations of logical consequence. Suitable logics were then subsequently developed. As we saw, quantum logics emerged from the difficulty that the distributive law generated in the domain of quantum mechanics. In retrospect, the con­nections between applied logic and applied mathematics should not be surprising, given that ultimately both deal with inferences and with suitable descriptions of particular domains.

At this point, the monist can invoke the distinction between pure and applied logic to make sense of the perceived plurality of logics. That there are many pure logics is an undeniable fact. But the central question is to determine what is the correct logic applied to the vernacular (see Priest 2006b). When we try to answer this question, the logical monist insists, we find that there is only one. The pluralist disagrees - and the debate goes on.

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