Lassi Saario-Ramsay

The distinction between formal and material consequence was introduced into medieval logic in the fourteenth century. Authors widely adopted the new terms but disagreed on their definition. The so-called Parisian tradition regarded a formal consequence as one that was valid for any substitution of categorematic terms, whereas the so-called British tradition required that the meaning of the consequent be contained in that of the antecedent. The former criterion resembles our model-theoretic definition of logical consequence, but it was the latter that, it has been claimed, was more popular at the time. Why? I argue that the question has no answer because the contradistinction of substitution and containment does not stand up to scrutiny. I base my argument on selected texts from various fourteenth-century authors, including Walter Burley, Nicholas Drukken of Denmark, Richard Lavenham, and Peter of Mantua. Instead of two distinct criteria, one of which is favoured over the other, we find various ways of mixing the two and gradual developments towards a hybrid view. I would say that both traditions made use of a substitutional criterion and that they only disagreed on what is to be substituted and what is not, i.e. what counts as form.

Modern logic is primarily concerned with formally valid arguments, namely arguments where the conclusion is a logical consequence of the premises. Intuitively, this means that the conclusion necessarily follows from the premises in virtue of their logical form alone. Such arguments form the backbone of philosophy, mathematics, and science in general: for instance, the famous Pythagorean Theorem is a logical consequence of the axioms of Euclidean geometry. However, the task of defining the concept of logical consequence is far from straightforward. Logical consequence is the glue that holds our belief system together, but what holds the glue together? This question cuts deep into the very foundations of scientific knowledge. The standard model-theoretic definition of logical consequence, originally formulated by Alfred Tarski in 1936, has been questioned by philosophers from various perspectives over the past few decades. John Etchemendy famously argues that Tarski’s account fails to capture the pre-theoretic, intuitive concept of logical consequence. Logical pluralists suggest that there is more than one correct concept of it. On the other hand, historians have pointed out that Tarskian definitions of logical consequence had already been proposed by Parisian logicians in the 14th century, and again by Bernard Bolzano in the 19th century. It has been suggested that their proposals may have captured the intuitive concept of logical consequence even better than Tarski’s. And yet, the fact that their proposals were not widely accepted at their own time suggests that the intuitive concept itself may have changed over time. The aim of this dissertation is to analyse the continuities and discontinuities between Tarski’s account and its premodern precedents. My overall thesis is twofold. On one hand, I show that a Tarskian ‘substitutional’ criterion of logical consequence was more widespread in the 14th century than has been previously thought, and that it continued to be promoted by many prominent textbooks of logic in the early modern era until it was subsumed by the Aristotelian theory of the syllogism. On the other hand, I argue that the notion of logical form presupposed by most 14th-century authors was distinctively non-Tarskian. Back then, logical form was primarily about relationships among concepts, but it is now taken to be just about the structural properties of propositions. I suggest that Tarski is partly to blame for this transformation: his definition was successful precisely because he deliberately suppressed some philosophical intuitions to suit the needs of mathematical logicians. The dissertation consists of five articles. Article I focuses on the theories of formal consequence by the 14th-century logician William of Ockham and his student Adam Wodeham. Both distinguished between two kinds of formal consequence (consequentia formalis), the first of which anticipates Tarski’s account, but their second kinds of formal consequence differ. I show that Wodeham’s two kinds of formal consequence are based on two kinds of logical form. In Article II, I further develop and defend what I call the ‘dual form hypothesis’. I ask why the 14th-century Parisian tradition of formal consequence, as close as it is to Tarski’s definiton, was not as influential at the time as the rival British tradition. I argue that the two traditions were closer to each other than we have thought. They agreed that formal consequence involves a truth-preserving form, and only disagreed on whether that form is structural or conceptual. Article III traces the early modern reception of the Parisian definition through a large-scale analysis of Early English Books Online, which contains almost all books printed in the British empire from the 15th to the 17th century. I show that Parisian-style definitions were surprisingly widespread in textbooks of logic that were used for teaching in Oxford, Cambridge, and Dublin for a long time. In Article IV, I argue that Tarski’s definition of logical consequence is not a conceptual analysis (as Etchemendy and others have claimed) but a Carnapian explication. The point is that Tarski did not even try to capture the essence of a pre-existing everyday concept; on the contrary, he deliberately sacrificed some everyday intuitions on the altar of formal correctness. Finally, Article V takes on the challenge of formalizing the medieval British definition of formal consequence in modern logic. I try out several approaches and conclude that Gillian Russell’s recent account of logical consequence can be applied to provide a fruitful formalization of the medieval notion.

Graham Priest distinguishes between two kinds of Sorites paradoxes: standard Sorites, such as the paradox of the Heap, and non-standard Sorites, such as the Ship of Theseus. The former concerns properties of objects, whereas the latter concerns their identity conditions. Priest notes that the standard Sorites has been solved in fuzzy logic and proposes a logic of fuzzy identity to solve the non-standard Sorites in a similar way. Ideally, a definition of fuzzy identity would satisfy the fuzzy equivalents of the classical principles of identity: it should (i) be a fuzzy equivalence relation and (ii) validate a fuzzy version of the substitutivity of identicals. However, Priest’s definition fails to meet the latter requirement and only applies to a special case of the non-standard Sorites. I suggest an improvement on his logic that overcomes these shortcomings, provides the two Sorites with a unified solution, and also applies to reasoning about vague objects in general.

The distinction between formal and material consequence was introduced into medieval logic in the fourteenth century. Authors widely adopted the new terms but disagreed on their definition. The so-called Parisian tradition regarded a formal consequence as one that was valid for any substitution of categorematic terms, whereas the so-called British tradition required that the meaning of the consequent be contained in that of the antecedent. The former criterion resembles our model-theoretic definition of logical consequence, but it was the latter that, it has been claimed, was more popular at the time. Why? I argue that the question has no answer because the contradistinction of substitution and containment does not stand up to scrutiny. I base my argument on selected texts from various fourteenth-century authors, including Walter Burley, Nicholas Drukken of Denmark, Richard Lavenham, and Peter of Mantua. Instead of two distinct criteria, one of which is favoured over the other, we find various ways of mixing the two and gradual developments towards a hybrid view. I would say that both traditions made use of a substitutional criterion and that they only disagreed on what is to be substituted and what is not, i.e. what counts as form.

Our standard model-theoretic definition of logical consequence is originally based on Alfred Tarski’s (1936) semantic definition, which, in turn, is based on Rudolf Carnap’s (1934) similar definition. In recent literature, Tarski’s definition is described as a conceptual analysis of the intuitive ‘everyday’ concept of consequence or as an explication of it, but the use of these terms is loose and largely unaccounted for. I argue that the definition is not an analysis but an explication, in the Carnapian sense: the replacement of the inexact everyday concept with an exact one. Some everyday intuitions were thus brought into a precise form, others were ignored and forgotten. How exactly did the concept of logical consequence change in this process? I suggest that we could find some of the forgotten intuitions in Bernard Bolzano’s (1837) definition of ‘deducibility’, which is traditionally viewed as the main precursor of Tarski’s definition from a time before formalized languages. It turns out that Bolzano’s definition is subject to just the kind of natural features—paradoxicality of everyday language, Platonism about propositions, and dependence on the external world—that Tarski sought to tame by constructing an artificial concept for the special needs of mathematical logic.