On Logic. Inquiries into the Justification of Deduction
Arthur Nieuwendijk

Abstract:
This dissertation is about logic. It investigates three different ways of viewing logic—three different ways of answering the same question: How can logical principles be justified? It considers an ontological perspective in the fourth chapter, a transcendental perspective in the third chapter, and a semantic perspective in the first two chapters.
In the first two chapters, the principle of bivalence is central. According to this principle, every statement is either true or false. This is a semantic principle because it says something, albeit in a very general sense, about the meaning of statements. The correctness of this principle is challenged by a philosophical movement known as anti-realism. Michael Dummett is the most prominent advocate of this movement. Characteristic of anti-realism is not only the rejection of the principle of bivalence but especially the logical consequences attached to that rejection. Anti-realism posits that, given that the principle of bivalence is untenable, the logical principle of the excluded middle must also be rejected. According to this principle, for every statement p, the sentence $p \lor \neg p$ is universally valid. This principle forms an essential element of classical propositional and predicate logic; its rejection therefore implies a rejection of these logics.
Chapter 1 goes into detail regarding the considerations on the basis of which an anti-realist, following Dummett, rejects the principle of bivalence as incorrect. The guiding thread is the meta-question: "How can the question 'How is it possible to criticize generally accepted logical principles?' be answered?" The traditional view is that a justification of logical principles is possible on the basis of a prior metaphysical investigation. Dummett argues that it is precisely this view that stands in the way of an adequate answer to the question of justification. Opposing the traditional view, Dummett states that metaphysics does not precede logic, but that, conversely, questions regarding metaphysics should be answered on the basis of a prior theory-of-meaning investigation. Seen from this perspective, the fundamental question is: What is the correct theoretical model of meaning? Dummett does not answer this question directly. He limits himself to an explication of the conditions a model of meaning must satisfy to be acceptable. Chapter 1 further examines the considerations underlying these conditions, as well as the thesis that, given these conditions, a semantic model endorsing the principle of bivalence cannot be accepted.
Chapter 2 connects with a well-known counterexample to the anti-realist thesis that the rejection of the principle of bivalence entails that the principle of the excluded middle is also untenable. This chapter shows that an anti-realist does not necessarily have to accept this counterexample. However, the goal of this chapter is not so much to defend the anti-realist position as to show that the anti-realist defense ultimately relates to the assumption that certain conceptual distinctions are so important that they must be expressible in language. Chapter 2 concerns the distinction between direct and indirect evidence. It is argued that, given the rejection of the principle of bivalence, the principle of the excluded middle can only be derived if the distinction between direct and indirect evidence cannot be expressed in the logical language.
The anti-realist argument against classical logic, as discussed step-by-step in Chapters 1 and 2, is negative in nature. That is, it attempts to convince us that classical logic cannot be justified. Chapter 3 investigates whether a positive answer can also be given to the question: What is the correct logic? According to Dummett, this is possible through an analysis of the notion of a proof. This analysis must provide insight into the minimum we must be able to do to obtain something like a logical proof. An important starting point is that a proof is considered the result of a certain way of using language. The basic idea is that among the many ways we can use logical connectives, there are a few without which we could not arrive at a logical proof. These "minimal" forms of use are considered constitutive for the possibility of a logical proof. The following criterion then seems obvious: the interpretation of a logical connective, as well as the logic induced by this interpretation, is justified without further ado if this interpretation corresponds to a "minimal" form of use of the connective in question. This approach relies heavily on the assumption that it is possible to unambiguously establish what counts as a logical proof solely in terms of use. In this chapter, however, it is argued that this assumption is incorrect: what we consider a logical proof is not solely dependent on the way we use language. The conclusion to be drawn is that it is not possible, in the manner Dummett advocates, to determine what the correct logic is independently of how the world is.
Even if it is, to say the least, doubtful whether it is possible to justify a logic independently of how the world is, it is indeed possible to justify the choice for a logic by understanding it against the background of philosophical views concerning reality. This is illustrated in Chapter 4 through the question of the interpretation of the logic developed by Navyany={a}ya, an Indian philosophical school with a rich history. It is argued that when this logic is considered in light of the ontological starting points endorsed by Navyany={a}ya, an interpretation and assessment of Navyany={a}ya logic in terms of classical predicate logic is completely inadequate.