Material Consequence and Formal Grounding

Abstract

According to Alfred Tarski’s classical definition, logical consequence is necessary and formal. This paper focuses on the question: In what sense (if any) is material consequence a logical relation? For Tarski, material consequence has no modal force. Treating all terms (of a language with a fixed domain) as logical, he reduces logical consequence to material consequence. Thus, Tarskian material consequence seems to be a logical oxymoron designed to emphasize the importance of the distinction between logical and extra-logical terms for the definition of logical consequence. Historically, however, there have been different approaches to material consequences. This paper attempts to provide an investigation into the parallels between Tarski’s dichotomy of formal and material consequence and the modern model-theoretical approach to consequence, as well as the dichotomies of consequentia formalis and consequentia materialis in John Buridan’s logic, formal and material grounding in Bernard Bolzano’s theory of science, material and logical leading principles of reasoning in Charles S. Peirce’s classification of arguments. Firstly, I’ll focus on Buridan’s idea that a consequence is formal if it is invariant under all substitutions for its categorematic terms. I am going to suggest that not only formal consequences, but also scholastic material consequences have a modal import, e.g., the consequence is materially valid ut nunc if the antecedent cannot be true without the consequent (under the present conditions). Secondly, I’ll address Bolzano’s theory of science. According to an enduring interpretation, he inherits the substitutional concept of logical consequence while his use of various types of consequences anticipates Tarski’s model-theoretical definition of formal consequence. I’ll argue for the advantages of Bolzano’s dichotomy between formal and material grounding as an attempt of proof-theoretical approach to consequence. I’ll compare Bolzano’s formal grounding with Abelard’s perfect consequence and Peirce’s logical leading principles of reasoning. My thesis is that shifting focus from truth conditions toward grounding offers some important insights into the dynamic taxonomies of consequences and other logical relations.