Catarina Dutilh Novaes

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s :69–90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus second-order logic divide. The descriptive approach is illustrated by Dedekind’s ‘discovery’ of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Frege’s Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects.

A range of formal models of human reasoning have been proposed in a number of fields such as philosophy, logic, artificial intelligence, computer science, psychology, cognitive science, etc.: various logics, probabilistic systems, belief revision systems, neural networks, among others. Now, it seems reasonable to require that formal models of human reasoning be empirically adequate if they are to be viewed as models of the phenomena in question. How are formal models of human reasoning typically put to empirical test? One way to do so is to isolate a number of key principles of the system, and design experiments to gauge the extent to which participants do or do not follow them in reasoning tasks. Another way is to take relevant existing results and check whether a particular formal model predicts these results. The present investigation provides an illustration of the second kind of empirical testing by comparing two formal models for reasoning – namely the non-monotonic logic known as preferential logic; and a particular version of belief revision theories, screened belief revision – against the reasoning phenomenon known as belief bias in the psychology of reasoning literature: human reasoners typically seek to maintain the beliefs they already hold, and conversely to reject contradicting incoming information. The conclusion of our analysis will be that screened belief revision is more empirically adequate with respect to belief bias than preferential logic and non-monotonic logics in general, as what participants seem to be doing is above all a form of belief management on the basis of background knowledge. The upshot is thus that, while it may offer valuable insights into the nature of human reasoning, preferential logic is ultimately inadequate as a formal model of the phenomena in question.

This thesis is on the history and philosophy of logic and semantics. Logic can be described as the ‘science of reasoning’, as it deals primarily with correct patterns of reasoning. However, logic as a discipline has undergone dramatic changes in the last two centuries: while for ancient and medieval philosophers it belonged essentially to the realm of language studies, it has currently become a sub-branch of mathematics. This thesis attempts to establish a dialogue between the modern and the medieval traditions in logic, by means of ‘translations’ of the medieval logical theories into the modern framework of symbolic logic, i.e. formalizations. One of its conclusions is that, when properly understood within their own framework, the interest of medieval logical theories for modern investigations go beyond mere historical interest, but that a thorough conceptual analysis of such theories must be undertaken in order to avoid conceptual misprojections. While such translations of medieval into modern logic have been attempted before, the approach presented here is innovative in that attention is paid to the similarities as well as to the dissimilarities between the two traditions, and to what can be learned from the medieval masters for modern investigations in logic and semantics.

On the basis of passages from John Buridan's Summula Suppositionibus and Sophismata, E. Karger has reconstructed what could be called the 'Buridanian theory of inferential relations between doubly quantified propositions', presented in her 1993 article 'A theory of immediate inference contained in Buridan's logic'. In the reconstruction, she focused on the syntactical elements of Buridan's theory of modes of personal supposition to extract patterns of formally valid inferences between members of a certain class of basic categorical propositions. The present study aims at offering semantic corroboration 'a proof of soundness' to the inferential relations syntactically identified by E. Karger, by means of the analysis of Buridan's semantic definitions of the modes of personal supposition. The semantic analysis is done with the help of some modern logical concepts, in particular that of the model. In effect, the relations of inference syntactically established are shown to hold also from a semantic point of view, which means thus that this fragment of Buridan's logic can be said to be sound.

We investigate the relations among Brandom’s three dimensions of semantic inferential articulation, namely, incompatibility entailments, committive consequences, and permissive consequences. In his unpublished manuscript “Conceptual Content and Discursive Practice” Brandom argues that (1) incompatibility entailment implies committive consequence, and that (2) committive consequence in turn implies permissive consequence. We criticize this hierarchy both on internal and external grounds. Firstly, we prove that, using Brandom’s own definitions, the reverse of (1) also holds, and that the reverse of (2) may hold (but the proof relies on substantive assumptions). This suggests that there are no three different notions of inference emerging from Brandom’s definitions, but at most two, and perhaps even just one. Secondly, this result puts into question the connections between the three inferential relations and the familiar notions of deduction and induction