Formal Methods in Philosophy: a Brief Introduction (Part II)

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the second and final part of my 'brief introduction' to formal methods in philosophy to appear in the forthcoming Bloomsbury Philosophical Methodology Reader, being edited by Joachim Horvath. (Part I is here.) In this part I present in more detail the four papers included in the formal methods section, namely Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. 

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Some of the pioneers in formal/mathematical approaches to philosophical questions had a number of interesting things to say on the issue of what counts as an adequate formalization, in particular Tarski and Carnap – hence the inclusion of pieces by each of them in the present volume. Indeed, both in his paper on truth and in his paper on logical consequence (in the 1930s), Tarski started out with an informal notion and then sought to develop an appropriate formal account of it. In the case of truth, the starting point was the correspondence conception of truth, which he claimed dated back to Aristotle. In the case of logical consequence, he was somewhat less precise and referred to the ‘common’ or ‘everyday’ notion of logical consequence.

These two conceptual starting points allowed Tarski to formulate what he described as ‘conditions of material adequacy’ for the formal accounts. He also formulated criteria of formal correctness, which pertain to the internal exactness of the formal theory. In the case of truth, the basic condition of material adequacy was the famous T-schema; in the case of logical consequence, the properties of necessary truth-preservation and of validity-preserving schematic substitution. Unsurprisingly, the formal theories he then went on to develop both passed the test of material adequacy he had formulated himself. But there is nothing particularly ad hoc about this, since the conceptual core of the notions he was after was presumably captured in these conditions, which thus could serve as conceptual ‘guides’ for the formulation of the formal theories.

Formal Methods in Philosophy: a Brief Introduction (Part I)

By Catarina Dutilh Novaes
(Cross-posted in NewAPPS)

There is a Bloomsbury Philosophical Methodology Reader in the making, being edited by Joachim Horvath (Cologne). Joachim asked me to edit the section on formal methods, which will contain four papers: Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. It will also contain a brief introduction to the topic by me, which I will post in two installments. Here is part I: comments welcome!

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Since the inception of (Western) philosophy in ancient Greece, methods of regimentation and formalization, broadly understood, have been important items in the philosopher’s toolkit (Hodges 2009). The development of syllogistic logic by Aristotle and its extensive use in centuries of philosophical tradition as a formal tool for the analysis of arguments may be viewed as the first systematic application of formal methods to philosophical questions. In medieval times, philosophers and logicians relied extensively on logical tools other than syllogistic (which remained pervasive though) in their philosophical analyses (e.g. medieval theories of supposition, which come quite close to what is now known as formal semantics). But the level of sophistication and pervasiveness of formal tools in philosophy has increased significantly since the second half of the 19^th century. (Frege is probably the first name that comes to mind in this context.)

It is commonly held that reliance on formal methods is one of the hallmarks of analytic philosophy, in contrast with other philosophical traditions. Indeed, the birth of analytic philosophy at the turn of the 20^th century was marked in particular by Russell’s methodological decision to treat philosophical questions with the then-novel formal, logical tools developed for axiomatizations of mathematics (by Frege, Peano, Dedekind etc. – see (Awodey & Reck 2002) for an overview of these developments), for example in his influential ‘On denoting’ (1905). (Notice though that, from the start, there is an equally influential strand within analytic philosophy focusing on common sense and conceptual analysis, represented by Moore – see (Dutilh Novaes & Geerdink forthcoming).) This tradition was then continued by, among others, the philosophers of the Vienna Circle, who conceived of philosophical inquiry as closely related to the natural and exact sciences in terms of methods. Tarski, Carnap, Quine, Barcan Marcus, Kripke, and Putnam are some of those who have applied formal techniques to philosophical questions. Recently, there has been renewed interest in the use of formal, mathematical tools to treat philosophical questions, in particular with the use of probabilistic, Bayesian methods (e.g. formal epistemology). (See (Papineau 2012) for an overview of the main formal frameworks used for philosophical inquiry.)

Book review: John P. Burgess' Rigor and Structure (OUP)

Rigor and Structure, Burgess tells us in the preface, was originally intended to provide for mathematical structuralism the sort of survey that A Subject with No Object (Burgess & Rosen, 1999) provided for nominalism. However, the book that Burgess has ended up writing is importantly different from his earlier work with Rosen. In large part, this is because, for Burgess, not only is mathematical structuralism true --- whereas he took nominalism to be false --- but moreover it is a ''trivial truism'', at least as a description of modern mathematics from the beginning of the twentieth century onwards (Burgess, 2015, 111). Thus, instead of providing philosophical arguments in favour of mathematical structuralism, Burgess instead devotes the first half of the book (Chapters 1 and 2) to providing an historical account of how mathematics developed into the modern discipline of which mathematical structuralism is so obviously a true description. And this is where the other component of the title enters the story. For it is Burgess' contention that modern structuralist mathematics --- which he explores in the second half of the book, that is, in Chapters 3 and 4 --- is an inevitable consequence of the long quest for rigor, which began, so far as we know, with Euclid's Elements, and was completed by work in the nineteenth and early twentieth century that led to the arithmetization of analysis, the axiomatization or arithmetic, analysis, and geometry, the formulation of non-Euclidean geometries, and the founding of modern algebraic theories, such as group theory.

Thus, in the first two chapters of Rigor and Structure, Burgess asks two questions: What is mathematical rigor? Why did mathematicians strive so hard to achieve it throughout the period just described? To answer the first question, Burgess turns initially to the pronouncements of mathematicians themselves, but he finds little that is precise enough to satisfy a philosopher there. So he turns next to Aristotle and, looking to the Posterior Analytics, extracts the following suggestion:

Mathematical rigor requires that:

• ''every new proposition must be deduced from previously established propositions'';
•  ''every new notion must be defined in terms of previously explained notions'';
• there are primitive notions from which the chain of definitions begins;
• there are primitive postulates from which the chain of deductions begins;
• ''the meaning of the primitives and the truth of the postulates must be evident''.

(Burgess, 2015, 6-7)