Bolzano's consequence, relevance, and enthymemes

Historians of logic tend to view their task as the application of modern insights and symbolic techniques to old texts. Perhaps they do this on the assumption that what is good in these works must be an adumbration of what was recently done and is now well known. This holds, at any rate, for most discussions of Bolzano’s theory of logical consequence.’ In the present paper I shall reverse this procedure and comment on some problems and beliefs of contemporary logic from what I take to be Bolzano’s point of view. This will have the advantage of bringing out more forcefully than a straight exegesis what his view was and will also, I hope, put in doubt certain contemporary dogmas. I begin by applying his definition of consequence to propositional logic. Bolzano did not entertain this branch of logic, and to this extent my account is ahistorical. That it is, nonetheless, a straight extension of his theory is shown by the fact that all 23 theorems about consequence which he proves in his Theory of Science hold in this application.2 I then consider how C. I. Lewis’s so-called “independent proof” for A & --A F B fares in this system (it fails). After some comments on the proof, I show that in Bolzano-consequence premisses and conclusion share a subsentence (a necessary condition of relevance). There follows a discussion of enthymemes and a general procedure for generating the so-called “missing premiss”. At the end I sketch a taxonomy of consequence relations and briefly remark on earlier interpretations of Bolzano’s work. In using the first person plural (from now on) I mean to speak for those who think Bolzano’s approach sound, a group that includes at least Bolzano and myself.