SEMANTIC TABLEAUX

My interest in mathematical logic has come about as a result of three personal interests: As an undergraduate physics major, I found that the semi-rote application of formulas when calculating solutions to pre-formulated problems caused a great deal of what was being presupposed to be hidden, assumptions which I was impatient to understand. The habitual, trained application of the formulas and the mathematics needed to transform and apply formalized physical laws in the context of applied problems masked the most basic foundations that were presupposed. I sensed that these most basic presuppositional foundations were not of central concern to most physicists, and I questioned then, as I do now, whether their study is explicitly encouraged even in the most purely theoretical physics. Largely for this reason, I was attracted by the philosophy of science. While engaged in its study, I discovered that working with the formal systems of mathematical logic offered the kind and depth of thorough and clear understanding, reaching to the very foundations of the subject, that I was searching for.

My second interest in mathematical logic resulted from a desire to see philosophers employ stricter, more rigorous, more objectively accountable reasoning than I saw exhibited by my professors of philosophy, fellow students, and by most philosophers whose works I read.

My third interest came about in the context of my own teaching. I became interested in the possibility of improving the reasoning skills and perhaps even the cognitive intelligence of my students no matter what course I taught, whether it was a basic historical introduction to philosophy, phenomenology, philosophy of mathematics, philosophy of behavioral science, epistemology, courses in mathematical logic, philosophical uses of modern logics, computer programming, or other classes.

In the context of my skill-focused teaching, I observed that when students become skilled in the use of a system of natural deduction, as contrasted with an axiomatic approach to mathematical logic, their ability to reason in wider areas of interest, beyond the framework of mathematical logic, noticeably improves. They become more disciplined in their thinking, more accustomed to following strict rules, and come to appreciate the advantages that careful, reflectively monitored, rule-directed mental activity brings.

Elsewhere, I have described how my teaching of mathematical logic led to the development of a learning game, VALIDITY. Much like the generalizable skills acquired through the study of systems of natural deduction, semantic tableaux are a natural complement.

For readers interested in semantic tableaux, here is a link to a paper that considers the model-theoretic character of proofs and disproofs by means of counterexample constructions. The paper contrasts the two approaches to semantic tableaux proposed by Beth and Lambert-van Fraassen, and takes note of the fact that Beth’s original approach had not benefitted by a precisely formulated rule of closure for detecting tableaux sequences terminating in contradiction. A technique for determining closure is therefore proposed to remedy this deficiency.