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!


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 19th 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 20th 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.)

In recent years, many of the discussions of what counts as a good formalization seem to assume that the core problem is how to translate portions of ordinary language into formal languages such as first-order predicate calculus or other logical systems (Baumgartner & Lampert 2008; Brun 2008; Cook 2002; Sainsbury 2001). However, it can be argued that using formal methods to treat philosophical issues goes much beyond a mere process of translation from one language into another (Dutilh Novaes, chap. 3). Indeed, it involves a fair amount of conceptual analysis, as illustrated by Tarski’s work on truth, logical consequence, and logical constants, or by Carnap’s applications of the concept of explication.

The application of formal methods to philosophical questions has (had) its friends as well as its foes. A number of critics (the later Wittgenstein in Philosophical Investigations, Strawson (1963), Ryle (1954), Kripke (1976) himself) have voiced concerns regarding the suitability of treating philosophical questions with formal methods, resisting what some of them view as a regrettable form of scientism. Strawson notes:

…it seems prima facie evident that to offer formal explanations of key terms of scientific theories to one who seeks philosophical illumination of essential concepts of non-scientific discourse, is to do something utterly irrelevant—is a sheer misunderstanding, like offering a text-book on physiology to someone who says (with a sigh) that he wished he understood the workings of the human heart.(Strawson 1963, 504–505)

Thus, according to these critics, scientific (formal) methods cannot simply be transposed to philosophical inquiry without substantive contentual loss or even outright ‘change of subject’, simply because science and philosophy are two radically different domains of inquiry (see (Dutilh Novaes & Geerdink forthcoming) for a discussion of the Carnap vs. Strawson debate). Moreover, formal approaches run the risk of only making more obscure what is otherwise already clear and accessible.

These concerns are not to be taken lightly, and the proponents of formal methods for philosophical inquiry must engage in serious methodological reflection on what justifies the application of such methods in philosophy. Equally importantly, they must formulate criteria of what makes for a good formalization; what does it take for a formal account of a philosophical issue or concept to be adequate? In other words, they must produce clearly formulated conditions of adequacy for formal analyses in philosophy.

In particular, the ultimate goal of these analyses is arguably to provide new insight into the phenomena being analyzed, and so the question becomes: what does it take for such formal analyses to be illuminating and informative? One example of a success story is Lewis’ (1976) triviality result against the tenability of the idea that the probability of a conditional is given by the corresponding conditional probability; it was was described as ‘a bombshell’, given how utterly surprising (and thus informative) it was.

Conversely, what are the risks and pitfalls involved in applying formal methods to philosophical questions? One of the risks can be described as ‘system imprisonment’: what guarantee do we have that the ‘discoveries’ being made are not merely artifacts of the system, rather than tracking independent facts about the target phenomenon? (Dutilh Novaes 2012, chap. 3) Another difficulty is ensuring that there are suitable bridge principles properly connecting the informal object of formalization – its target phenomenon – to the formal framework. As any form of analysis, formalization must tread the thin line between staying close enough to the target phenomenon so that it is truly a formalization of the phenomenon, while also revealing something about the phenomenon that informal examination cannot easily reveal so as to be truly informative (a version of the so-called ‘paradox of analysis’).


  • Awodey, Steve & Reck, Erich H. (2002). Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic. History and Philosophy of Logic 23 (1):1-30.
  • Baumgartner, Michael & Lampert, Timm (2008). Adequate formalization. Synthese 164 (1):93-115.
  • Brun, Georg (2008). Formalization and the objects of logic. Erkenntnis 69 (1):1 - 30.
  • Carnap, Rudolf (1950).  Logical Foundations of Probability. University of Chicago Press.
  • Cook, Roy T. (2002). Vagueness and mathematical precision. Mind 111 (442):225-247.
  • Dutilh Novaes, Catarina (2012). Formal Languages in Logic – A philosophical and cognitive Analysis. Cambridge University Press.
  • Dutilh Novaes, Catarina & Geerdink, Leon (forthcoming). The dissonant origins of analytic philosophy: common sense in philosophical methodology. In S. Lapointe & C. Pincock (eds.), Innovations in the History of Analytical Philosophy. Basingstoke: Palgrave Macmillan.
  • Dutilh Novaes, Catarina & Reck, Erich (forthcoming). Carnapian explication, formalisms as cognitive tools, and the paradox of adequate formalization. Synthese.
  • Frege, Gottlob (1997). Begriffsschrift. A formula language of pure thought modelled on that of arithmetic. In Michael Beaney (ed.), The Frege Reader. Blackwell.
  • Hansson, Sven Ove (2000). Formalization in philosophy. Bulletin of Symbolic Logic 6 (2):162-175.
  • Hodges, Wilfrid (2009). Traditional logic, modern logic and natural language. Journal of Philosophical Logic 38 (6):589 - 606.
  • Kripke, Saul (1976). Is there a problem about substitutional quantification? In G. Evans & J. McDowell (eds.), Truth and Meaning: Essays in Semantics, Clarendon Press, 325-420.
  • Lewis, D. (1976). Probability of Conditionals and Conditional Probabilities. Philosophical Review, 85: 297-315.
  • Russell, Bertrand (1905). On denoting. Mind 14 (56):479-493.
  • Papineau, David (2012). Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets. Oxford University Press.
  • Ryle, Gilbert (1954). Formal and informal logic. In Dilemmas, the Tarner Lectures 1953. Cambridge, CUP, 111-129.
  • Sainsbury, R. M. (2001). Logical Forms: An Introduction to Philosophical Logic. Blackwell Publishers.
  • Strawson, P. F. (1963). Carnap’s views on constructed systems versus natural languages in analytic philosophy. In Schilpp, Paul Arthur, ed. The Philosophy of Rudolf Carnap. Chicago, Open Court, 503–518.
  • Tarski, Alfred (2002). On the concept of following logically. History and Philosophy of Logic 23 (3): 155-196.


  1. Great post and formal method is easy-way to resolve your philosophy subject and i read your article its great help us how to use formal method for resolve the problem thanks for sharing sentence rephrasing tool .


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