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)

I'm looking forward to reading the Bloomsbury Philosophical Methodology Reader.

ReplyDeleteI was wondering whether there is any compendious work that presents several examples of philosophical puzzles that were (or could be) addressed through the use of formal methods in an apparently successful way.

A list (including Tarski and later developments, e.g., by Kripke) is here:

Deletehttp://m-phi.blogspot.co.uk/2012/07/list-of-achievements-of-analytic.html

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ReplyDeleteCatarina, excellent summary. Do you have a preferred citation for this---e.g. should I link to this blog?

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