Catarina Dutilh Novaes

The Different Ways in which Logic is (said to be) Formal

What does it mean to say that logic is formal? The short answer is: it means (or can mean) several different things. In this paper, I argue that there are (at least) eight main variations of the notion of the formal that are relevant for current discussions in philosophy and logic, and that they are structured in two main clusters, namely the formal as pertaining to forms, and the formal as pertaining to rules. To the first cluster belong the formal as schematic; the formal as indifference to particulars; the formal as topic-neutrality; the formal as abstraction from intentional content; the formal as de-semantification. To the second cluster belong the formal as computable; the formal as pertaining to regulative rules; the formal as pertaining to constitutive rules. I analyze each of these eight variations, providing their historical background and raising related philosophical questions. The significance of this work of ‘conceptual archeology’ is that it may enhance clarity in debates where the notion o...What does it mean to say that logic is formal? The short answer is: it means (or can mean) several different things. In this paper, I argue that there are (at least) eight main variations of the notion of the formal that are relevant for current discussions in philosophy and logic, and that they are structured in two main clusters, namely the formal as pertaining to forms, and the formal as pertaining to rules. To the first cluster belong the formal as schematic; the formal as indifference to particulars; the formal as topic-neutrality; the formal as abstraction from intentional content; the formal as de-semantification. To the second cluster belong the formal as computable; the formal as pertaining to regulative rules; the formal as pertaining to constitutive rules. I analyze each of these eight variations, providing their historical background and raising related philosophical questions. The significance of this work of ‘conceptual archeology’ is that it may enhance clarity in debates where the notion of the formal plays a prominent role (such as debates where it is expected to play a demarcating role), but where it is oftentimes used equivocally and/or imprecisely.

Buridan's Consequentia: Consequence and Inference Within a Token-Based Semantics

I examine the theory of consequentia of the medieval logician, John Buridan. Buridan advocates a strict commitment to what we now call proposition-tokens as the bearers of truth-value. The analysis of Buridans theory shows that, within a token-based semantics, amendments to the usual notions of inference and consequence are made necessary, since pragmatic elements disrupt the semantic behaviour of propositions. In my reconstruction of Buridans theory, I use some of the apparatus of modern two-dimensional semantics, such as two-dimensional matrices and the distinction between the context of formation and the context of evaluation of utterances.

Reassessing logical hylomorphism and the demarcation of logical constants

The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds in particular of the vexed issue of demarcating the class of logical constants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as ‘logical hylomorphism as we know it’—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions.The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds in particular of the vexed issue of demarcating the class of logical constants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as ‘logical hylomorphism as we know it’—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions.

Form and Matter in Later Latin Medieval Logic : The Cases of Suppositio and Consequentia

The paper investigates the influence of Aristotelian metaphysics, especially of the concepts of form and matter, in the development of Latin medieval logical theories, in particular theories of consequence and theories of supposition. More specifically, it presents the main lines in the development of the concepts of formal/material consequence and formal/material supposition in the 13th and 14th centuries. It is suggested that the growing influence of Aristotelian metaphysics in the 13th century is an important element in the development of these logical theories; nevertheless, the early stages of these theories pre-date the rediscovery of Aristotles metaphysics in the 13th century. Indeed, other relevant channels of transmission of the metaphysical concepts of form and matter to logical contexts are discussed: the Prior Analytics commentaries of the ancient commentators, Arabic texts, and the neo-Platonic metaphysics of the grammarian tradition stemming from Priscian.

Ralph Strode's obligationes : The Return of Consistency and the Epistemic Turn

In what follows, I analyze Ralph Strodes treatise on obligations. I have used a hitherto unpublished edition of the text (based on 14 manuscripts) made by Prof. E.J. Ashworth. I first give a brief description of Strodes text, which is all the more necessary given that it is not available to the average reader; I also offer a reconstruction of the rules proposed by Strode, following the style of reconstruction used in my analysis of Burleys and Swynesheds rules elsewhere—that is, essentially based on the idea that obligationes can be viewed as logical games. In the second part, I address Strodes explicit arguments contra Swyneshed. In the third part, I discuss Strodes epistemic and pragmatic approach to obligationes.

An Intensional Interpretation of Ockham's Theory of Supposition

According to a widespread view in medieval scholarship, theories of supposition are the medieval counterparts of theories of reference, and are thus essentially extensional theories. I propose an alternative interpretation: theories of supposition are theories of properties of terms, but whose aim is to allow for the interpretation of sentences. This holds especially of Ockham’s supposition theory, which is the main object of analysis in this paper. In particular, I argue for my intensional interpretation of his theory on the basis of two key-phrases in his Summa Logicae: ‘denotatur’ and ‘propositio est distinguenda’. Finally, I offer a reconstruction of his theory as a set of instructions to be carried out in order to generate the possible readings of (certain) sentences.

Validity, the Squeezing Argument and Alternative Semantic Systems: the Case of Aristotelian Syllogistic

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran’s deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel’s famous ‘squeezing argument’; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure.

Medieval obligationes as a theory of discursive commitment management

In earlier work, I have presented an interpretation of Obligationes as logical games of consistency maintenance; this interpretation has some advantages, in particular that of capturing the multi-agent, goal-oriented, rule-governed nature of the enterprise by means of the game analogy. But it has as its main limitation the fact that it does not provide a satisfactory account of the deontic aspect of the framework—i.e. of what being obliged to a certain statement consists in. In order to remedy this shortcoming, this paper argues for the more encompassing thesis that Obligationes can be viewed as a theory of rational dialogical practices, in particular concerning the management of one’s discursive commitments. The main inspiration for this interpretation comes from R. Brandom’s inferentialist, normative pragmatism, but more generally, it relies heavily on a dialogical conception of logic. I argue that it offers a more compelling account of several aspects of the obligational framework, in particular the role of doubting responses and the ubiquity of obligational vocabulary in the context of scientific discourse in the medieval tradition.

The Buridanian Account of Inferential Relations between Doubly Quantified Propositions: a Proof of Soundness

On the basis of passages from John Buridans 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 Buridans logic’. In the reconstruction, she focused on the syntactical elements of Buridans 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 Buridans 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 vi...On the basis of passages from John Buridans 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 Buridans logic’. In the reconstruction, she focused on the syntactical elements of Buridans 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 Buridans 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 Buridans logic can be said to be sound.

Theory of Supposition vs. Theory of Fallacies in Ockham

I propose to examine the issue of whether the ancient tradition in logic continued to be developed in the later medieval period from the vantage point of the relations between two specific groups of theories, namely the medieval theories of supposition and the (originally) ancient theories of fallacies. More specifically, I examine whether supposition theories absorbed and replaced theories of fallacies, or whether the latter continued to exist, with respect to one particular author, William of Ockham. I compare different parts of Ockhams Summa Logicae , namely III-4 (on fallacies), and the final chapters of part I and first chapters of part II (on supposition). I conclude that there is overlap of conceptual apparatus and of goals (concerning propositions that must be distinguished) in Ockhams theories of supposition and of fallacies, but that the respective conceptual apparatuses also present substantial dissimilarities. Hence, theories of supposition are better seen as an addition to the general logical framework that medieval authors had inherited from ancient times, rather than the replacement of an ancient tradition by a medieval one. Indeed, supposition theories and fallacy theories had different tasks to fulfil, and in this sense both had their place in fourteenth century logic.