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.

Indeed, the fact of formulating conceptual/informal but nevertheless precise desiderata is one of the philosophical strengths of Tarski’s analyses both of truth and of logical consequence. And thus, while Tarski’s classical ‘On the concept of logical consequence’ is not explicitly a methodological piece, in the process of providing a formal analysis of this concept he offers an unusually lucid discussion of how to go about when investigating philosophical concepts with formal methods.

Carnap’s most worked-out systematic analysis of what counts as adequate formalization can be found in Chapter 1 of Logical Foundations of Probability (1950), namely in his famous exposition of the notion of explication:

The task of explication consists in transforming a given more or less inexact concept into an exact one or, rather, in replacing the first by the second. We call the given concept (or the term used for it) the explicandum, and the exact concept proposed to take the place of the first (or the term proposed for it) the explicatum. (Carnap 1950, 3)

Carnap then goes on to formulate four requirements for an adequate explication: (1) similarity to the explicandum, (2) exactness, (3) fruitfulness, (4) simplicity. Exactness and simplicity seem to be purely internal criteria, going in the direction of Tarski’s criteria of formal correctness. Similarity to the explicandum seems to come quite close to what Tarski refers to as ‘conditions of material adequacy’, namely that the formal explicatum should reflect the conceptual core of the informal explicandum in question. But fruitfulness, which is both the least developed and most interesting of Carnap’s desiderata, seems to be a true novelty with respect to Tarski’s discussion in terms of material adequacy and formal correctness, and one that makes the project of formalization more complex but also significantly more interesting. Formalization thus becomes a project with pragmatic implications (Dutilh Novaes & Reck forthcoming).

One of the few recent pieces explicitly on the methodology of formalization in philosophy which goes well beyond the misguided idea that formalization amounts to translation is Hansson’s aptly entitled ‘Formalization in philosophy’ (2000), which is also included in this volume. Hansson presents a nuanced picture of formalization in philosophy, successfully avoiding both “anti-formalist and pan-formalist sentiments” by highlighting advantages and disadvantages of formalization in philosophy. Among the advantages, he mentions making implicit assumptions explicit, stimulating definitional and deductive economy, and the striving for completeness. Among the disadvantages, Hansson discusses the risks of oversimplification, false unification of concepts, false concept primitivity, implicit ontological assumptions, and enigmatic style. The conclusion is thus that, while remaining a worthwhile and potentially highly informative methodology in philosophy, formalization requires serious methodological reflection so as to minimize the risk of negative effects.

The final piece in this section was especially commissioned for the volume. We felt that it was important to include a systematic methodological discussion of uses of probabilistic frameworks in philosophy, in particular Bayesian probability theory, given how widespread such approaches have been in recent years (e.g. in formal epistemology). However, nothing in the existing literature seems to offer the systematic discussion we were after, and thus we invited Michael Titelbaum to write a contribution especially for this volume. Titelbaum’s piece offers a discussion of how formal modeling techniques can be applied to investigate normative questions, given that their typical uses in the empirical sciences are predominantly descriptive. He examines the cases of formal logic, linguistics/philosophy, and finally Bayesian epistemology, which will receive a more extensive discussion.

(See Part I for full list of references)