Two conceptions of criteria of adequacy for formalization: Tarski and Carnap

(Cross-posted at NewAPPS)

Formal/mathematical philosophy is a well-established approach within philosophical inquiry, having its friends as well as its foes. Now, even though I am very much a formal-approaches-enthusiast, I believe that fundamental methodological questions tend not to receive as much attention as they deserve within this tradition. In particular, a key question which is unfortunately not asked often enough is: what counts as a ‘good’ formalization? How do we know that a given proposed formalization is adequate, so that the insights provided by it are indeed insights about the target phenomenon in question? In recent years, the question of what counts as adequate formalization seems to be for the most part a ‘Swiss obsession’, with the thought-provoking work of Georg Brun, and Michael Baumgartner & Timm Lampert. But even these authors seem to me to restrict the question to a limited notion of formalization, as translation of pieces of natural language into some formalism. (I argued in chapter 3 of my book Formal Languages in Logic that this is not the best way to think about formalization.)

However, some of the pioneers in formal/mathematical approaches to philosophical questions did pay at least some attention to the issue of what counts as an adequate formalization. In this post, I want to discuss how Tarski and Carnap approached the issue, hoping to convince more ‘formal philosophers’ to go back to these questions. (I also find the ‘squeezing argument’ framework developed by Kreisel particularly illuminating, but will leave it out for now, for reasons of space.)

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 framework for 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. (I’ve argued elsewhere that this is a problematic idea.)

These two conceptual starting points allowed him 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. (In my SEP entry on medieval theories of consequence, I’ve done a bit of conceptual genealogy to unearth where these two conditions for logical consequence came from.)

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 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.

Carnap’s 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 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 me to come very 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 me to be a true novelty with respect to Tarski’s discussion in terms of material adequacy and formal correctness, and one that makes the whole thing significantly more complicated but also significantly more interesting.

As I’ve argued in a talk at the Carnap on Logic conference in Munich last year, Carnap’s concepts of similarity and fruitfulness are in fact somewhat in tension with one another, in the sense that a formalization (explication) can be viewed as all the more fruitful if it reveals aspects of the informal concept which were not visible ‘to the naked eye’. So what Carnap added to the Tarskian framework is the idea that the goal of a formalization is not only to capture exactly what is already explicit in the informal concept in question. To be sure, Carnap himself does not say that much about what he understands under fruitfulness, and seems to focus in particular on the explicatum’s capacity to generate ‘many universal statements’. But it seems to me that the Carnapian notion of the fruitfulness of a formalization can be developed in other interesting directions, in particular in the more epistemological/cognitive direction of formalization as a tool for discovery. (I’m supposed to write a paper for the special issue ensuing from the Carnap workshop, and the plan is to develop these ideas more fully.)

Be that as it may, I hope to have made it clear that both Tarski and Carnap offer excellent starting points for (much-needed) sustained discussions of what counts as adequate formalization, and more generally of the methodological aspects of applying formal/mathematical methods to philosophical questions.

UPDATE: I want to add plugs to two excellent books on Carnap and explication: Carnap's Ideal of Explication and Naturalism (ed. Pierre Wagner) and A. Carus' monograph Carnap and Twentieth-Century Thought: Explication as Enlightenment


  1. Thanks for this. Part of what I like about researching histories is finding shadowy old paths covered in briars that weren't originally explored (in addition to re-treading the ancestral path and plotting its trajectory on a map).
    The tension you mention strikes me as the type which is required for a violin concerto, but which snaps the string when too strong. An astute mathematician knows to plant trees that bear fruit, but one can never know which tree comes from which seed. Sometimes they bear lemons, but still the wood can be used to build a house. (God, I hope not wandering too close to the border of metaphor and nonsense)
    When I attempt an explication a la Carnap (which reminds me much of Frege), it seems I have a number of possible routes. The most straight forward is of a somewhat discrete unpacking of a box of christmas ornaments. Once they're all laid out, I can decide which to decorate the tree with. This is useful for math proofs, where terms are often just shorthand or compositions of shorthands. For example, saying that a function is continuous can be expanded to a statement of the Bolzano-Weierstrass definition when in the context of real analysis. The statement "a holomorphic function is analytic" is an example of a 'composition,' or nested shorthand, which can be unpacked piece by piece.
    Another route is the one I find amenable to analysis of ordinary language, and is a continuous unpacking, more like a 'zooming in.' The explicandum is first as if it were far away and visible as a meaningful, if somewhat mysterious, whole. Embarking on the explication brings one closer to it gradually, revealing increasing levels of detail until I'm satisfied ("closer" is somehow not quite right; it's a falling in, a reaching out, an assimilation). It may conclude swiftly, almost automatically, or I may grasp around blindly and in vain at ever expanding swirls of images and sounds. Complicated, emotional concepts tend to take this route. In a way, the concept is already "exact," but it's exactness is not perceived.
    The last approach that comes to mind, and the most difficult, is a modular-constructive approach, wherein tenuously related concepts may be explored in hopes of finding similarities. Small pieces that might be useful are glued together and wired up in various ways, and one tests to see how the resulting device works at each stage, sizing up new additions and assessing how it's progressing towards the goal. This is useful when one seeks a causal narrative of an observed phenemona.
    Perhaps these are special cases of a more general process. The first may be a limiting case of the second at least, which has resulted in a somewhat algebraic conception on my part (with the explitata a collection of factors, whose product is the explicandum). But I can feel my feet leaving the ground now, so that's where I'll stop. Thanks for the post(s), by the way. I agree that this needs to be talked about. Accounts of formalist principles generally hide so much 'under the hood.' -P


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