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The Different Ways in which Logic is (said to be) Formal Catarina Dutilh Novaes
a
a
ILLC and Department of Philosophy, University of Amsterdam, Oude Turfmarkt 141-147, 1012 GC, Amsterdam, The Netherlands E-mail:
[email protected] Available online: 18 Oct 2011
To cite this article: Catarina Dutilh Novaes (2011): The Different Ways in which Logic is (said to be) Formal, History and Philosophy of Logic, 32:4, 303-332 To link to this article: http://dx.doi.org/10.1080/01445340.2011.555505
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HISTORY AND PHILOSOPHY OF LOGIC, 32 (November 2011), 303–332
The Different Ways in which Logic is (said to be) Formal Catarina Dutilh Novaes
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ILLC and Department of Philosophy, University of Amsterdam, Oude Turfmarkt 141-147, 1012 GC Amsterdam, The Netherlands
[email protected] Received 1 September 2010 Accepted 14 December 2010
Final version received 6 January 2011
What does it mean to say that logic is formal? The short answer is: it means (or can mean) several different things. In this paper, I argue that there are (at least) eight main variations of the notion of the formal that are relevant for current discussions in philosophy and logic, and that they are structured in two main clusters, namely the formal as pertaining to forms, and the formal as pertaining to rules. To the first cluster belong the formal as schematic; the formal as indifference to particulars; the formal as topic-neutrality; the formal as abstraction from intentional content; the formal as de-semantification. To the second cluster belong the formal as computable; the formal as pertaining to regulative rules; the formal as pertaining to constitutive rules. I analyze each of these eight variations, providing their historical background and raising related philosophical questions. The significance of this work of ‘conceptual archeology’ is that it may enhance clarity in debates where the notion of the formal plays a prominent role (such as debates where it is expected to play a demarcating role), but where it is oftentimes used equivocally and/or imprecisely.
1. Introduction What does it mean to say that logic is formal?1 The short answer is: it means (or can mean) several different things. Given the crucial importance of the notion of the formal in current discussions in logic, philosophy and mathematics (among other fields), it is surprising to note that the literature specifically on this concept is very scarce, if not to say virtually nonexistent. To my knowledge, the only recent study where considerable attention is paid to this notion is MacFarlane’s 2000 dissertation, aptly entitled What does it mean to say that logic is formal?, but it seems to me that his investigation is for various reasons not sufficiently exhaustive. In his study, MacFarlane is interested in the notion of the formal only insofar as it serves as a demarcating criterion for what is to count as logic, so he arguably overlooks some important aspects to the extent that they are not directly relevant for his purposes. In particular, he says nothing about the fact that the different variations seem to be organized in two main clusters, namely the formal as pertaining to forms and the formal as pertaining to rules. Moreover, some of the variations he discusses are characterized as ‘decoy’ in that they are not adequate to perform the demarcating function he is after. But the ‘decoy’ terminology suggests that these variations are of secondary importance generally speaking, which is a distorted picture of their actual (historical, practical, conceptual) significance. In other words, it seems evident that this notion deserves to be thoroughly investigated in its own right, without the pressure of any external agenda. Thus, the goal of the present investigation is to steer away from the formal as a demarcating criterion for logic and to be as general as possible so as to cover a wide range of uses of 1
This is the title of MacFarlane’s (2000) dissertation, and as will become clear throughout, I rely on MacFarlane’s work on several occasions. Nevertheless, I believe that there are important aspects in which his analysis of the notion of the formal must be complemented and/or amended, and this is indeed one of the goals of the present investigation. History and Philosophy of Logic ISSN 0144-5340 print/ISSN 1464-5149 online © 2011 Taylor & Francis http://www.tandfonline.com http://dx.doi.org/10.1080/01445340.2011.555505
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the term in connection with logic. For this end, I shall offer a taxonomy of its different uses in logical contexts; it is intended to serve as a map for further philosophical research and perhaps to contribute to the clarification of central debates within the philosophy of logic. Naturally, as with every taxonomy and every investigation having such a broad scope, not every single usage of the term ‘formal’ will necessarily fit into the schema proposed here.2 Nevertheless, if a significant number of the different usages is captured, and some of the connections between them are outlined, then the main goal of providing a bird’s eye view of this thick forest will have been attained. Another methodological clarification is necessary: I do not intend to pass judgment on what the term ‘formal’ must mean when it is used, or what the (unique) correct conception of the formal is; rather, I focus on what speakers (seem to) mean when they use this term, primarily in philosophical/logical contexts. Essentially, I am interested in the ‘grammar’ of the term ‘formal’, as Wittgenstein 1953 (Philosophical Investigations, Sections150 and 187) would put it. Currently, the term ‘formal’ covers a wide range of related but distinct concepts. In effect, there seem to be two main clusters of meanings attributed to it, each of them grouping a number of variations. The two basic senses of the formal in question are: the formal as pertaining to forms; and the formal as pertaining to (the strict application of) rules. That these are two different approaches to the general idea of being formal can also be inferred from what is usually considered to be the opposite of being formal in each case: in the case of the formal as pertaining to forms, what is not formal is usually said to be material, while in the case of the formal as pertaining to rules, what is not formal is usually said to be informal.3 I distinguish eight main variations of the notion of the formal—five pertaining to forms and three pertaining to rules. In the former case, these variations emerge from the history of applications of the form vs. matter distinction to logic, so I shall (briefly) survey these developments. The historical analysis will be essential to delineate the conceptual dissimilarities between the different variations of the formal as pertaining to forms. As for the formal as pertaining to rules, there is less of a common history underlying the different variations, but there are still important considerations to be made. For methodological reasons, I will mainly focus on the attribution of the adjective ‘formal’ to a rather narrow class of objects: arguments, logical systems, theories. However, it turns out that each of these eight variations of the formal are in fact typically attributed to different objects, which again only outlines the dissimilarities between them. A systematic discussion of the kinds of objects that can be said to be formal is certainly called for, but for reasons of space shall be left out of the present investigation.4 Such a classification of the different meanings of ‘formal’ may contribute to enhancing clarity in discussions where this notion plays a prominent role (e.g. investigations where it is expected to play a demarcating role such as MacFarlane’s), but where it is oftentimes used equivocally by different parties and/or imprecisely by one and the same person. Moreover, by means of an analysis of the semantics of the term ‘formal’, it is to be expected that one can also obtain a better understanding of the metaphysics of the property (or properties) of being formal. Indeed, I also sketch a number of philosophical issues that arise in connection 2
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Moreover, I here focus predominantly on uses of the term ‘formal’ with respect to logic, whereas the term also occupies an important place within other disciplines (mathematics, physics, sociology and economics). The taxonomy proposed here may or may not be suitable in these other cases. There are, however, interesting connections with uses of the terminology in sociology, which I discuss briefly in Section 3. The contrast between these two dichotomies is also outlined in Brun 2003, 24. See Brun 2008, which illustrates the complexity of the issue.
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with each variation. For reasons of space, these analyses cannot be exhaustive at the present occasion, but it seems to me that, without a preliminary study of the different meanings of the term ‘formal’, a proper philosophical discussion of these issues cannot even get started. Indeed, it would seem that a philosophical, conceptual examination of the notion(s) of the formal presupposes the kind of ‘semantic’ analysis that I offer here, and thus that those subsequent investigations may benefit from it. Let me mention some of the philosophical debates that might profit from the more precise understanding of the concept of the formal that shall emerge from the present analysis. One of them is the issue of the logical status of some formal systems, say second-order logic, whose logicality is viewed by many as debatable. In the case of second-order logic, since its consequence relation does not admit a complete proof procedure, this is often taken to be a sufficient reason to view it as not counting as ‘logic’ properly speaking. Presumably, given the lack of a complete proof procedure, second-order logic is not ‘formal’ and thus not logical. But of course, this line of argumentation relies on a particular (and rather specific) understanding of notion of the formal, related to the notion of a complete proof procedure. By contrast, those who claim that second-order logic is indeed formal and logical are clearly relying on a different understanding of the formal (as they do not dispute the meta-properties of the second-order consequence relation). An analysis of the different senses of ‘formal’ as presented here may illuminate the debate in that it may clarify to what extent the dispute is merely verbal (arising from an equivocation of the term ‘formal’), and to what extent there is a substantial dispute (e.g. which of the different senses of the formal should be taken to provide the right criterion for logicality). Similarly, understanding what exactly is formal about formal languages and formalisms may elucidate why their uses are so pervasive in logic and how this phenomenon relates to the formality of logic as such. These are indeed some of the long-term projects that motivate the present enterprise. Moreover, given that MacFarlane’s analysis of the different variations of the formal has been to some extent biased by his overall project (as argued above), I submit that not having particular goals or issues in mind when undertaking this work of conceptual archeology is to be recommended. In this manner, we are more likely to attain a sufficiently neutral starting point for the discussion of specific issues subsequently. 2. The formal as pertaining to forms A natural way to approach the notion of the formal is to take etymology/morphology as a starting point. The term ‘formal’ is of course the combination of the noun ‘form’ and the suffix ‘-al’, which, according to the average dictionary, means: ‘pertaining to, like, of the kind of, relating to, characterized by, belonging to’. But most importantly, it is not only a matter of etymology: as I shall argue, the notion of form as developed and applied in the philosophical tradition I am concerned with is decisive for what it means for logic to be formal for (analytic) philosophers in the twenty-first century. It all goes back to Aristotle’s metaphysical distinction between form and matter, two of the main items in his conceptual toolkit to explain the being of things.5 For Aristotle, these were concepts to be used fundamentally with respect to physics and metaphysics, not logic or language. Indeed, Aristotle himself applies the form–matter distinction to such objects only twice, in the Metaphysics 1013b19-20 and in an almost identical passage in the Physics 195a18-19. There he observes, without further elaboration, that the premises are matter for 5
Naturally, the notion of Forms (usually written with capital F) figures prominently in Plato’s doctrines, and certainly constituted a source of inspiration for Aristotle. Nevertheless, the notion of form (especially insofar as it is typically accompanied by the notion of matter) which was then to be systematically applied within logic, is surely and more immediately the Aristotelian notion.
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the conclusion (of an argument), ‘in the sense of “that from which”’ (a standard formulation for the notion of material cause). But clearly, this observation does not in any way anticipate the current notion(s) of the formal as used in connection with logic in particular. And yet, it is undoubtedly the Aristotelian (and originally metaphysical) form–matter dichotomy that is at the root of modern uses of the notion. In this section, I survey the history of applications of the notions of form and matter to logic, more specifically to sentences and arguments, in order to understand how the original Aristotelian metaphysical notions have been modified to such a radical extent, while still being fundamentally at the origin of current ideas on the notion of the formal with respect to logic. Different applications of the form vs. matter distinction in logic have produced five main variations of the formal as pertaining to forms—in fact, two variations, the first one yielding two sub-variations and the second one yielding three sub-variations. I now discuss each of these five variations, indicating the historical elements behind each of them but focusing on a conceptual analysis of each. In fact, variations of the notion of the formal as pertaining to forms are all specific interpretations of a common slogan: ‘formal corresponds to abstraction from matter’.6 Accordingly, the opposite of all these variations is the notion of ‘material’. But as ‘form’ and ‘matter’ can be characterized in a variety of ways, this general slogan will yield different notions of the formal depending on what is to be considered matter.7 What is form, and formal, is what remains once matter is removed (abstracted from). (1) If the matter of an argument corresponds to a specific subclass of its terms, namely those that have a signification by themselves (i.e. ‘refer to things’), we have the schematic notion of the formal. (2) If the matter of an argument are the specific things referred to in the argument, then we have the notion of the formal as indifference to particulars. (3) If the matter of an argument is its specific subject-matter, then we have the notion of the formal as topic-neutral. (4) If the matter of an argument is its intentional content, i.e. what it talks about and its relation to objects in the world, then we have the notion of the formal as abstraction from content. (5) Finally, if the matter of an argument is thought to be its content in the sense of the meaning its terms have, then we have the notion of the formal as de-semantification (the form is then considered to be the very shape of the symbols). 2.1. The formal as variability The basic idea of the formal as schematic, as we shall see, is the idea of replacing terms with different terms in arguments, i.e. the idea of variability of terms preserving the validity of arguments. For centuries, this was the predominant view of what it means to be formal8 until, in his 1936/2002 paper, Tarski outlined the limitations of letting variation operate solely on terms. He was concerned, in particular, with the possibility of the language lacking terms for all objects; for this reason, he proposed a reformulation of the notion of the formal as variability, namely that variation should be a procedure concerning objects (objects in the world or mathematical constructions), not terms. This in turn consolidated the notion of the formal as indifference to individuals. I now discuss each of these two variations of the general idea of the formal as (validity-preserving) variability. 6
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MacFarlane (2000, 51) makes a similar remark with respect to his three main notions of the formal: ‘We can get at these three notions by construing “formal” as “independent of content or subject matter.” What does it mean to say that logic is independent of content or subject matter? That depends on what we mean by “content” or “subject matter.”’ But I think it is better to take an even more general claim as a starting point, one that is less definitive concerning how matter should be interpreted. As also noted by MacFarlane (2000, 132), matter, form and formality are all extremely plastic notions. But with at least one exception: Bolzano, in the nineteenth century, held a substitutional (variational) concept of derivability, but in his case substitution/variation concerned ideas-in-themselves, i.e. abstract entities inhabiting a third realm.
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2.1.1. The formal as schematic The variation of the notion of the formal that is historically most persistent with respect to logic and language is what can be described as the ‘schematic notion of the formal’.9 The term is coined by MacFarlane 2000, and is more felicitous than alternative widespread terminologies such as ‘variational’ and ‘substitutional’ in that it clearly indicates what is to undergo variation or be substituted: terms, linguistic objects. The idea is that the form of a sentence or argument is rendered by a schema, i.e. a construction obtained from the original sentence or argument by means of the replacement of some of its components by placeholders, often schematic letters—see Corcoran 2008 on schemata.10 What undergoes variation are the terms that are replaced by schematic letters or other placeholders/gapindicators: in turn, the schemata are seen as templates for the generation of substitutional instances, i.e. arguments generated by filling in the ‘gaps’ with arbitrary terms. Whenever a given argument (or schema) generates nothing but valid arguments as its substitutional instances, then it is usually said to be formally valid. Note, though, that one can construe schemata where more complex expressions— sentences in particular—can be replaced by placeholders, i.e. schematic letters or others. This is an old practice; the Stoics, for example, usually replaced sentences with numerals: ‘If the first then the second, but not the second; so not the first’. But all in all, be it because of the predominance of term logic up to the nineteenth century or because of the awkwardness of viewing whole sentences as ‘matter’ (whereas simple terms presumably have a straightforward relation to things),11 the schematic notion of the formal is traditionally associated with schemata where simple terms rather than propositions are replaced by schematic letters. The first occurrences of the schematic notion of the formal known to us is to be found in the oldest still extant applications of the form–matter distinction to logic, found in Alexander of Aphrodisias’ (second century AD) commentaries on Aristotle’s logical works, in particular, on the Prior Analytics, such as: The [syllogistic] figures are like a sort of common matrix: by fitting matter into them, it is possible to mould the same form in different sorts of matters. For just as things fitted into one and the same matrix differ not in form and figure but in matter, so it is with the syllogistic figures. (in Alexander of Aphrodisias 6.16–21, translation p. 48) One of Aristotle’s own analogies to explain the form vs. matter distinction is the mould/matrix analogy (Physics 195a6-8, Metaphysics 1045a26-29): bronze is the matter of both a statue and a sphere, and their forms are the shapes imprinted on them by the mould with which they are made. Moreover, Aristotle uses schematic letters extensively in both Analytics, and typically proves the invalidity of a given syllogistic mood by providing an argument displaying the given mood (we would now say: the given form), but which is obviously invalid (with true premises and false conclusion). The innovation introduced by 9
MacFarlane views this notion of the formal as a ‘decoy’ in the sense that, albeit attractive at first sight, it cannot be used for the purpose of demarcating logic, which is his main purpose. Here, however, as I am not particularly interested in the demarcational enterprise, this notion of the formal does not occupy the secondary position that it occupies in MacFarlane’s account. 10 Of course, there remains the issue of whether the logical form of a sentence is always apparent from its surface structure. As we now know very well, this is often not the case, and a considerable amount of ingenuity is usually required to uncover the ‘true’ logical form of a sentence. Moreover, sometimes there is more than just one suggestion on the table for what the correct logical form of an expression is, and it is quite controversial to claim that every sentence has a uniquely determined logical form. However, in what follows I will leave this complication aside. 11 But recall that Aristotle himself had suggested that the premises are the matter for the conclusion. Indeed, in the Arabic tradition, e.g. in Avicenna, one finds the idea that simple sentences have one of three ‘matters’, viz. necessary, impossible and contingent.
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Alexander (or whoever else Alexander was following)12 is the idea of applying the mold analogy to arguments, syllogisms in particular, in order to separate their form from their matter, while also referring to Aristotle’s own uses of schematic letters in his analysis of syllogisms.13 Typically, the form of a syllogism is associated to its figure in the tradition of the Ancient Commentators (of Aristotle) to which Alexander belongs (ranging from Alexander in the second century toAmmonius, Simplicius, Philoponus and Boethius in the sixth century—see Barnes 1990, 2007 and Lee 1984 on logical form and matter in this tradition). This implies that the terms defining the mood of a syllogism—quantifying and negating terms and the copula—were not considered to pertain to the form of a syllogism (the figure of a syllogism corresponds simply to the specific configuration of the three terms in the argument, and different moods belong to one and the same figure). Syllogistic figures are not schemata in the modern sense of the term, but syllogistic moods are, and somewhere along the way the form of arguments (syllogisms in particular) became increasingly associated to moods rather than to figures.14 The details of these developments need not detain us here, but already in the fourteenth century the schematic notion of the formal received a full-fledged formulation in the definition of formal consequence put forward by Buridan15 —a definition which is also very general in that it covers the form (and matter) of any argument/consequence, not only of syllogistic arguments: ‘Formal’consequence means that [the consequence] holds for all terms, retaining the form common to all. [. . .] a formal consequence is that which, for every proposition similar in form which might be formed, it would be a good consequence […]. (Buridan 1976, pp. 22/23, 5–9) (my emphasis) I say that, in the previous considerations (as we are speaking here of matter and form), we understand by the “matter” of the proposition or consequence the purely categorical terms, i.e. subjects and predicates, omitting the syncategorematic terms that enclose them and through which they are conjoined or negated or distributed or forced to a certain mode of supposition. All the rest, we say, pertains to the form. (Buridan 1976, I.7.2, pp. 30 (7–12)) The basic idea of forms of arguments rendered as schemata and the procedure of varying terms remained influential well beyond the scholastic period. It was often criticized as producing only sterile ‘knowledge’ (e.g. by Descartes), but uses of the term ‘formal’ to refer to such procedures based on schemata were ubiquitous.16 In the early nineteenth century, 12
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It is unclear whether it was Alexander’s original idea or whether these are only the oldest such applications of the form vs. matter distinction that have survived. ‘He uses letters in his exposition in order to indicate to us that the conclusions do not depend on the matter but on the figure, on the conjunction of the premises and on the modes. For so-and-so is deduced syllogistically not because the matter is of suchand-such a kind but because the combination is so-and-so. The letters, then, show that the conclusion will be such-and-such universally, always, and for every assumption.’ (Alexander of Aphrodisias, in Apr 53.28-54.2, translation p. 116) In two early Latin commentaries on the Prior Analytics, namely the twelfth century Anonymus Aurelianensis (Ebbesen 1981) and Robert Kilwardby’s thirteenth century commentary (Thom 2007), the form of a syllogism is said to be understood in two ways: as pertaining to its figure and as pertaining to its mood. This is crucial, as only if the form of a syllogism pertains to its mood do we obtain schematic formality as we now know it. The equivocal understanding of the form of a syllogism was apparently quite widespread in the first half of the thirteenth century (also mentioned, for example, in the anonymous Dialectica Monacensis, edited by De Rijk in volume II.II of his Logica Modernorum, p. 491) and seems to have played a crucial role for the emergence of the schematic notion of the formal. See Dutilh Novaes 2005 for further details on Buridan’s theory of consequence. Here is what Leibniz says: ‘But it must be grasped that by ‘formal arguments’ I mean not only the scholastic manner of arguing which they use in the colleges, but also any reasoning in which the conclusion is revealed by virtue of the form, with no need for anything to be added.’ (NE: IV. xvii.§ 4, 478) (emphasis added)
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the schematic notion of the formal received a sophisticated treatment by means of the notion of variation of ideas-in-themselves introduced by the philosopher, mathematician and priest Bernard Bolzano (see Siebel 2002). Bolzano may have indeed taken inspiration from mathematics, as the general procedure of considering schemata and filling in the ‘empty spaces’ designated by variables with denoting terms (numbers or letters denoting specific magnitudes) is also at the core of (algebraic) mathematics. Hence, besides the logical influence stemming from the medieval Aristotelian tradition (with which Bolzano was well acquainted), schematic formality may also be rooted in mathematical practice (but so is the formal as indifference to individuals, as we shall see). Bolzano represents a particularly important stage in the development of the schematic notion of the formal in that he may be the (indirect) source for Tarski’s use of schematic formality in his analysis of the concept of logical consequence. It has often been pointed out that there are striking similarities between Bolzano’s account of derivability and Tarski’s notion of logical consequence, and even if Tarski himself did not have direct knowledge of Bolzano’s analysis, he was certainly indirectly exposed to it via his teachers (who were students of a student of Brentano’s, who in turn heavily relied on Bolzano’s work).17 Apparently, the general idea remained ‘in the air’ for many centuries, which suggests that there is a fairly straightforward line of continuity of uses of the term ‘formal’ in the sense of schematic from the later Latin Middle Ages to Tarski, and from Tarski to us.18 The schematic notion of the formal put forward in Tarski’s ‘On the concept of logical consequence’ (Tarski 1936/2002) as a condition of material adequacy for a definition of logical consequence is essentially the one which is still widely held now. According to Tarski, a necessary condition for sentence X to be a logical consequence of the class of sentences K is:19 [. . .] If in the sentences of the class K and in the sentence X we replace the constant terms which are not general-logical terms correspondingly by arbitrary other constant terms (where we replace equiform constants everywhere by equiform constants) and in this way we obtain a new class of sentences K’ and a new sentence X’, then the sentence X’ must be true if only all sentences of the class K’ are true. (Tarski 1936/2002, p. 183/4) 17
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In a note to the English translation of his 1936 paper on logical consequence, Tarski seems to suggest that he only became aware of Bolzano’s analysis after the publication of the paper, through an article by H. Scholz. Moreover, another important difference between Bolzano’s account of derivability and more standard notions of schematic formality is that Bolzano did not work with a pre-established ‘logical term’ vs. ‘non-logical expression’ distinction; the notion of derivability is defined as relative to the set of ideas which are to undergo variation in each specific case, and there is no restriction on how this set is to be defined at each instance. The road from the Ancient Commentators to the later Latin Middle Ages is, however, much bumpier, a fact related to the virtual disappearance of Aristotle’s own writings in the earlier Latin Middle Ages. But for our purposes, what matters is that the (re)emergence of the schematic notion of the formal in the later Latin medieval period is undoubtedly an application of Aristotle’s form vs. matter distinction to logic. Moreover, a more encompassing history of the notion of the formal in logic would have to look into what Arabic philosophers such as Avicenna and Averroes, among others, have said about the form and matter of syllogisms. The Arabic authors were well acquainted with the tradition of the Ancient Commentators, but also introduced quite a few novelties of their own. For reasons of space, however, the Arabic link will be left out of the present analysis. For example: ‘When logicians say that logic is formal, they often mean that if they discover that a certain argument is valid, then all arguments of the same form as that argument will be valid too.’ (Gomez-Torrente 2000, 529) Quine also accounted for the notions of logical truth and consequence on the basis of the idea of lexical substitutions, which MacFarlane 2000 analyzes under the heading ‘grammatical formality’ (one of his decoy notions). MacFarlane distinguishes schematic formality from grammatical formality, but it seems to me that they rely essentially on the same intuition, just cashed out in slightly different ways.
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The schematic notion of the formal has been immensely influential in the history of logic and philosophy, and has played a fundamental role in the development of logic as a discipline. Nevertheless, there is a wide range of philosophical difficulties associated with it, in particular the issue of establishing a (principled) partition between those terms of an argument constituting its matter and those terms of an argument constituting its form.20 The backbone of the schematic notion of the formal is of course the partition of the vocabulary into two kinds: those that are to be held fixed and those that are to be regarded as replaceable. A schema is precisely a construction with ‘gaps’ along with specifications of how these gaps can be filled in by various different objects—see Corcoran 2008. Once this partition is undertaken, it becomes quite natural to view the terms that remain fixed as constitutive of the form of a sentence/argument, while those that are to be replaced would correspond to its matter. But how to undertake this partition, that is, what criteria to use for this purpose, is of course a highly contentious matter. Where should the line separating the form from the matter of an argument be drawn? This is obviously the thorny problem of demarcating the class of logical constants, which remains to this day essentially an open problem (MacFarlane 2009). Of course, a pragmatic answer to the problem can in general be given: the class of logical constants is to be determined solely on the basis of the task at hand, as there is no fundamental, principled criterion for what is to count as a logical constant. But many believe that the pragmatic solution is no solution at all, rather it only shuns the problem altogether. And yet, as Tarski himself recognized in his 1936 paper, ‘At the foundation of our whole construction lies the division of all terms of a language into logical and extra-logical. […] however I know no objective reasons which would allow one to draw a precise dividing line between the two categories of terms’. (Tarski 1936/2002, p. 188) So while the notion of schemata as such need not involve serious philosophical difficulties, the association of schemata to the forms of sentences and arguments, yielding the schematic notion of the formal, gives rise to a variety of issues, which might arise from the metaphysical origin of the concept of forms. Appreciating the substantive, non-trivial and perhaps inappropriate association of the forms of arguments to schemata that is at the heart of the notion of the formal as schematic may represent the first step towards (dis)solving the problem of demarcating logical constants, as I argued in Dutilh Novaes forthcoming. 2.1.2. The formal as indifference to particulars21 Another pervasive notion of the formal that can be seen as having its roots in the Aristotelian tradition is the formal as indifference to (specific) particulars and their individual characteristics. Interestingly, a possible metaphysical source for the idea that attention to form entails indifference to particular individuals could have been the widespread view that individuals of the same species share the same form. An investigation of the form of those individuals would thus amount to indifference towards their individual characteristics. To my knowledge, however, the notion of the formal as indifference to particulars has never been presented in these terms in the Aristotelian tradition. What one does encounter occasionally is suggestions to the effect that the specific things referred to in an argument or sentence constitute its matter; if this is so, then abstracting form from matter in the case of arguments might amount to disregarding the peculiarities of the specific objects mentioned in the argument. In this case, an argument valid in virtue of its matter is an argument whose 20 21
I discuss these issues extensively in Dutilh Novaes forthcoming. MacFarlane’s ‘2-formality’ (2000, pp. 56–60).
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validity depends on the specific characteristics of these objects, while an argument valid in virtue of its form would be indifferent to these characteristics. The gist of the idea of anchoring the validity of certain arguments on ‘the nature of things’ as opposed to their syntactic structure (‘ex rerum natura’ as opposed to the ‘complexio’ of arguments) passed on from the Ancient Commentators to the Latin world through Boethius (and is to be found e.g. in Abelard), but it was not cast in terms of the form vs. matter distinction; rather, the underlying theoretical framework was predominantly that of the Aristotelian Topics.22 As a result, at least up to the nineteenth century the idea of indifference to particular objects was typically not associated to the notion of the formal. In contrast, this very idea is now perhaps most strongly associated to the notion of the formal by current philosophers, mathematicians and logicians (in particular, but not exclusively with respect to the demarcation of logic by its formality—see MacFarlane 2000, Ch. 6. The appearance of the notion of the formal as indifference to particulars at the central stage of discussions concerning formality illustrates the general ‘mathematical turn’ in logic that took place in the nineteenth century. In effect, it is only with the convergence of logic and mathematics that the idea of variation of objects (not terms) explicitly entered the realm of logic, and only then, in logical contexts, was it explicitly related to the notion of the formal properly speaking. Tarski’s seminal 1936 paper (which in turn relies heavily on his previous work on truth) is a likely source for the modern association between the idea of permutation of objects and the notion of the formal. After pointing out the shortcomings of a purely schematic definition of formal/logical consequence (related to the fact that a language may fail to contain names for every object), Tarski argues that a different approach is needed. Via the notion of satisfaction, he introduces the concept of a model, and then goes on to define the notion of logical consequence in terms of models.23 These few pages represent a dramatic shift of paradigm, from the predominance of schematic (term-based) formality for several centuries to the new object-based notion of the formal as indifference to particular individuals; indeed, it is eventually within the model-theoretic paradigm that the notion of the formal as indifference to particulars thrives. The shift from permutation of terms (the essence of schematic formality) to permutation of (indifference to) objects can be clearly perceived in the following passages: Since moreover it is a question here of the relation of following logically, i.e. formally,24 and therefore of a relation which has to be completely determined by the form of the sentences among which it obtains, thus following cannot depend on our knowledge of the external world, in particular on our knowledge of the objects which are spoken about in the sentences of the class K or in the sentence X, cannot be lost as a result of our replacing the names of these objects in the sentences under consideration by names of other objects. (Tarski 1936/2002, p. 183) Let us further consider an arbitrary class of sentences L and let us replace all extralogical constants occurring in the sentences of the class L by corresponding variables 22
23
24
This is a point that MacFarlane 2000 fails to stress. In his Appendix A, he includes Abelard as one of the forerunners of logical hylomorphism without further ado, but fails to note that Abelard himself does not rely on the form vs. matter distinction in his analysis of the notion of consequence/inference in any way. This is important, as it is only later, in the Latin thirteenth century, that hylomorphism is again systematically applied in logical analysis. This being said, I am prepared to concede that Abelard’s emphasis on the ‘complexio’ of an argument is indirectly related to the logical hylomorphism of the Ancient Commentators, via Boethius. At this stage, Tarski’s models are sequences of real-world objects, not set-theoretical structures. The shift occurred only in the late 1940s. For an analysis of Tarski’s notion of model in the 1936 paper, see Mancosu 2006. Note here the implication of equivalence between what is logical and what is formal.
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(equiform constants by equiform variables, non-equiform by non-equiform); we shall obtain a class of sentential functions L’. An arbitrary sequence of objects which satisfies each sentential function of the class L’ we shall call a model of the class L. (Tarski 1936/2002, p. 186) (my emphasis) Note, however, that in this article the term ‘formal’ is essentially used in its schematic sense, not in the sense of indifference to particulars. But Tarski emphasizes that respecting schematic formality is an adequacy criterion for any account of logical consequence. Another related but conceptually independent25 association of the formal to indifference to individual characteristics of particulars is to be found in Tarski’s later lecture/paper, ‘What are logical notions’ (1966/1986). In this paper, Tarski explicitly draws on Klein’s use of the concept of transformation to distinguish the notions used in different systems of geometry (Euclidean geometry, affine geometry and topology) to raise the following question: if we consider all possible one-to-one transformations of the universe onto itself, i.e. arbitrary permutations of objects, what are the notions that are still preserved? Euclidean transformations preserve proportion, collinearity and connectedness; affine transformations do not preserve proportion but do preserve collinearity and connectedness; topological transformations do not preserve proportion or collinearity but do preserve connectedness. What if one considers an increasingly larger class of transformations, which in turn will preserve fewer and fewer notions (properties)? Tarski claims that once we consider the class of all possible one-to-one transformations of the universe onto itself, i.e. once we consider all arbitrary permutations of objects, the notions that remain invariant are precisely the logical notions. (He refers to an earlier result of his with Lindenbaum, a proof that all notions that are presented as logical notions in Principia Mathematica do satisfy this criterion.) So the idea of total indifference to individual objects is presented as a solution to the problem of demarcating logic, and insofar as formality can be understood as indifference to individual characteristics of objects (something that Tarski does not explicitly say in his 1966/1986 paper, but which is forcefully argued for by Sher 2008), this approach takes care of the problem of demarcating logic and of the thesis that logic is distinctively formal in just one stroke. With the establishment of model-theory as one of the most influential approaches in logic since the 1940s, permutation invariance—which is indeed a technical version of the idea of indifference to specific objects—became increasingly seen as the very core of the notion of the formal. Indeed, one of its strengths is that it seems to offer a unification of the formal nature of logic and mathematics under a common notion—see Sher 2008. Statements of this specific conception of the formal abound; here are just some passages that clearly illustrate the pervasiveness of the permutation-invariance notion of the formal—be it under its traditional, object-based formulation or under its more recent, structuralist formulation in terms of invariance under isomorphic structures.26 [. . .] a formal system . . . is defined as a body of theorems generated by objective rules and concerning unspecified objects. (Curry and Feys 1958, p. 12) (emphasis added) 25
26
In the two cases, two kinds of indifference to particulars are referred to: one is given by quantifying over all possible interpretations of non-logical terms—logical/formal consequence—while the other is given by invariance under permutations—the logicality of logical notions. While indifference to particulars in the case of logical consequence is compatible with a pragmatic, non-principled partition of the vocabulary into logical and non-logical terms, permutation-invariance is presented by Tarski precisely as a principled way of defining the class of logical notions. I owe this point to Hannes Leitgeb. This seems like a good place to note the conceptual connection between this variation of the formal and structuralism in the philosophy of mathematics. Again, for reasons of space, I am unable to develop this point any further, but it is manifest that the notion of mathematics as characterized not by a class of special mathematical objects, but rather by the fact that it deals with pure structures, is also relevant here.
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The usual model-theoretic notion of logical consequence is formal in precisely this sense. A sentence X is a model-theoretic logical consequence of a set of sentences K(K|MT X, for short) just in case every set-theoretic structure which is a model of all the sentences in K is also a model of X. If an argument
is such that K|MT X then any argument with the same form will be such that K’|MT X’, since the model-theoretic notion of logical consequence is intended for languages where any two sentences of the same form have as models exactly the same structures. (Gomez-Torrente 2000, p. 529)27 On my interpretation (Sher 1991 and elsewhere), the invariance-under-isomorphism criterion is a criterion of formality or structurality: isomorphic structures are formally identical; identity-up-to-isomorphism is formal identity. The basic idea is that logic is a theory of reasoning based on the formal (structural) laws governing our thinking on the one hand and reality on the other28 , and the invariance-under-isomorphism criterion says that to be formal is to treat isomorphic structures as the same structures. (Sher 2008, p. 307) In short: the notion of the formal as invariance in permutation of objects has become predominant seemingly as a consequence of the development of specific mathematical techniques in the nineteenth century and of the model-theoretic turn in logic halfway the twentieth century. Before that (as also noted by MacFarlane), indifference to particular objects was rarely explicitly associated to the notion of the formal. Why, then, should this variation be placed within the general development of the originally Aristotelian notion of the formal as pertaining to form and as opposed to matter? Essentially, because this notion of the formal can be understood as a reformulation, and thus as a continuation, of the schematic notion of the formal (as can be clearly perceived in Tarski’s logical consequence paper). The point seems to be that, instead of manipulating terms and being constrained by all kinds of linguistic limitations, one can manipulate objects directly, thus avoiding the mediation of linguistic entities. The schematic notion, in turn, is the main historical result of the application of the form vs. matter distinction in logical contexts, and thus seen formality as permutation invariance also belongs to this tradition. These two variations of the formal are not only historically but also logically related: if a given argument is formally valid in the sense of indifference to particulars, then it will also be formally valid in the sense of substitutivity of its non-logical terms by other terms. In the other direction, the implication will hold provided that the language satisfies some conditions, in particular that of having names for all objects under consideration. Nevertheless, the formal as indifference to particular objects, and more specifically as permutation invariance, also gives rise to a series of philosophical issues that often seem to go unnoticed. On the one hand, there is no obvious candidate for the role of ‘form’ here, as opposed to the identification of forms to schemata in the schematic notion of the formal. On the other hand, for permutation invariance to be successful as a procedure, a series of metaphysical assumptions related to object-identity must obtain: that objects are discrete, that they perdure, that they do not come into existence ‘out of the blue’, etc. However, it is not clear whether the partisans of permutation invariance as a criterion for logicality are 27
28
It seems to me, however, that in this paper, Gomez-Torrente does not make a sufficiently clear distinction between schematic formality, i.e. permutation of terms, and formality as indifference to particular objects, i.e. permutation of objects. The point is of course that these two properties are to be extensionally equivalent in such languages, but this does not mean that they are not intentionally different properties. Sher defends a very strong form of realism with respect to forms; she claims that there are ‘formal laws’ governing the behavior of all objects, including those in the physical world. In all cases, ‘individuals are identical to themselves, the union of non-empty properties is non-empty, etc.’ (Sher 2008, p. 307)
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aware of the fact that such substantive assumptions are being made. So again, while it is a well-established and fruitful variation of the notion of the formal, indifference to individuals and permutation invariance are concepts still awaiting further philosophical analysis.29 2.2. Formal as abstraction from subject-matter (content) The general idea of abstraction from subject-matter yields at least three clearly distinct sub-variations of the formal, corresponding to three different interpretations of ‘subjectmatter’: as abstraction from specific topics (i.e. topic-neutrality), as abstraction from (intentional) content, or as total abstraction from meaning. The first one corresponds to the notion of topic-neutrality and can (at least implicitly) be perceived in earlier developments. The second one stems from the Kantian transcendental idealist thesis that the formal laws of the understanding have no relationship whatsoever to objects. The third one is much more recent, emerging in the second half of the nineteenth century and being consolidated within Hilbert’s program. In a way, the first two variations are diametrically opposite: while the association of the formal with topic-neutrality implies that what is formal (e.g. logic) can deal with several different topics and disciplines, the formal seen as abstraction from content, in its Kantian variation, implies that what is formal cannot be about any topic at all. As for the third variation, the abstraction from content is taken to the extreme level of the symbols losing their symbolic function altogether and becoming themselves mathematical objects.
2.2.1.
Formal as topic-neutral
A recurring suggestion in the history of logic is that what is distinctive about logic is that it provides patterns of correct reasoning valid across different disciplines, that is, its topic-neutrality—a term coined by Ryle 1969. Such statements often arise when one is hard pressed to define the specific subject-matter of logic; truth and correct reasoning are the objects of all disciplines, so what is distinctive about logic? Or does it have no particular subject-matter? A classical, Aristotelian view has it that logic does not have a specific subject-matter as mathematics studies mathematical objects, biology living beings, geology minerals, etc., but rather that it is a tool to be used across disciplines, an organon. Precisely because it is not concerned with specific objects and their specificities, logic is topic-neutral and thus can (and should!) be used with respect to any (scientific) subject-matter whatsoever. Aristotle’s Posterior Analytics is a treatise on scientific method, i.e. procedures and patterns of argumentation not only valid but also actually yielding scientific knowledge—in short, Aristotle’s philosophy of science, and the methodology described there is, of course, largely based on the theory of syllogistic presented in the Prior Analytics. The view that logic (and not only with respect to syllogistic) is topic-neutral was indeed the predominant view on the nature of logic in the earlier and later Aristotelian traditions (see Barnes 2007, Ch. 5) as well as in the later Middle Ages. In both cases, logic was part of the basic academic curriculum (in the later Middle Ages it was part of the trivium together with grammar and rhetoric), as it was thought that no progress could be made in any other discipline without the appropriate logical tools. 29
Of course, there is already a wide literature on the topic. But most of the already existing analyses focus on technical examinations of whether the criterion of permutation invariance delivers the right extension for logicality. Sher’s 2008 is an attempt at a more thoroughly philosophical analysis of the concept, but many of the claims she makes concerning her realist understanding of formality and logicality seem to me to require considerably more substantiation. I discuss these issues in Dutilh Novaes unpubl.
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Of course, before formality became the distinctive trait of logic with Kant (as convincingly argued by MacFarlane 2000),30 the topic-neutrality of logic did not imply that being formal amounted to being topic-neutral or vice versa. However, there were several suggestions to the effect that the topic-neutrality of logic was somehow related to its (schematic) formality.31 With Kant, as we shall see, logic loses entirely its ability to produce (scientific) knowledge: Kant explicitly rejects the role of organon traditionally assigned to logic. For Kant (general), logic is beyond being topic-neutral: it has no topic at all, as it only defines the formal conditions of pure thought. In the Kantian sense, ‘formal’ is not what is topic-neutral, and thus can be applied to a variety of topics, but rather what cannot have any content at all. But despite the great influence of the Kantian characterization of logic as formal in subsequent developments, this particular aspect of his characterization did not prevail. Indeed, later authors such as Frege will reject Kant’s dismissal of logic as a tool to produce knowledge, and the general idea that topic-neutrality as a key feature of logic is to be found in several influential authors of the twentieth century.32 Frege 1885, sketches two notions of the formal, and clearly rejects one of them in favor of the other. The notion he rejects was the one defended by his contemporaries Heine and Thomae (more on the latter below), according to which the symbols of mathematics are to be viewed as meaningless, and this is what constitutes a formal approach to mathematics. Instead, he defends the idea of the universality of the concept of number (it can be applied to entities of any kind), and connects this ‘general applicability’ of arithmetic to its formal nature. He is thereby also rejecting the Kantian conception of arithmetic as non-formal precisely because it is contentual: for Frege, arithmetic (among other formal theories) is both formal and contentual, as later spelled out in more detail in the Grundgesetze der Arithmetik (Frege 1903), as it is general and yet pertains to specific objects, numbers. Hence, this particular aspect of Frege’s notion of the formal in these texts echoes the traditional conception of the formal as ‘topic-neutral’ and ‘universally applicable’. Now, if logic is seen as essentially topic-neutral and as essentially formal, the association of the formal to topic-neutrality follows in a rather straightforward way. In fact, it would seem that logic is topic-neutral because it is formal, i.e. because it deals with forms and patterns that are valid irrespective of the specific subject-matter being dealt with. A valid argument-form will be valid for all substitutional instances of its material terms (according to the schematic notion of the formal), regardless of the discipline or topic dealt with. In other words, the topic-neutrality variation of the formal follows from its schematic variation. Similarly, topic-neutrality is entailed by indifference to individual characteristics of objects: if logic is indifferent to specificities of objects, then a fortiori it is indifferent to specificities of objects related to their being objects of investigation of specific domains. Indeed, it is inspired by the topic-neutrality of (formal) logic that Husserl introduced the concept of ‘formal ontology’ in his Logical Investigations (see Smith 2005). Husserl says of logical sequences that they ‘may be so generalized, so purely conceived, as to be free of all essential relation to some concretely limited field of knowledge.’ (Husserl 1973, I A19/65) Accordingly, a formal ontology would deal with ontological objects and relations that are also topic-neutral: 30 31
32
See also Hodges 1999 on how ‘formal logic’ became a set phrase in English logic only in the nineteenth century. For example: ‘For conversions—and in general deductions in the figures—do not depend, as I have already said, on the peculiarities of the matter (which is different in different cases), but on the nature of the figures themselves. That is why proofs in their case are universal’. (Alexander of Aphrodisias, in Apr 35.6-9, translation in p. 91, my emphasis) ‘logic’ . . . [is] . . . the name of a discipline which analyzes the meaning of the concepts common to all the sciences, and establishes the general laws governing the concepts. (Tarski 1959, p. xi.)
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Husserl’s conception of the science of logic as relating also to formal-ontological categories such as object, state of affairs, unity, plurality, and so on, is not an arbitrary one. These concepts are, like the concepts of formal logic, able to form complex structures in non-arbitrary, law-governed ways, and they, too, are independent of the peculiarity of any material of knowledge. This means that in formal ontology, as in formal logic, we are able to grasp the properties of given structures in such a way as to establish in one go the properties of all formally similar structures. (Smith 2005, p. 300) The Husserlian notion of formal ontology inaugurated a tradition of uses of the concept of the formal as applied to philosophical concepts outside logic, and more generally to other disciplines.33 For example, commenting on the formal vs. material dichotomy with respect to concepts, Correia and Keller 2004 say: The most widespread characterization invokes the concept of topic-neutrality. A formal concept, it is said, is a topic-neutral concept, i.e. a concept under which can fall objects of any kind. Objecthood, existence and identity, for instance, are arguably topic-neutral. (Correia and Keller 2004, p. 276) But the idea that logic is formal in the sense that it is topic-neutral has been challenged in recent decades by the development of ‘logics’ of specific subject-matters, such as modal logic, epistemic logic, deontic logic, tense logic, etc., much to the dissatisfaction of ‘purists’ such as Kneale: With them [the developers of such ‘topic-bound’logics] ‘logic’is no longer the name of a science concerned with the principles of inference common to all studies, but rather a name for any collection of rules in accordance with which we may argue in some context. […] it seems clear that this way of talking involves abandonment of the notion that logic is concerned with form as opposed to subject matter. (Kneale 1956, p. 238) The purists seem to have lost the battle, though, and in view of the development of several logics dealing with specific topics, the equation ‘formal = topic-neutral’ can no longer be taken for granted. So either these logics are not formal, or topic-neutrality is not all there is to being formal. Of course, the latter alternative seems the more plausible one; nevertheless, the association of formality to topic-neutrality remains pervasive, in particular outside the scope of logic—such as in e.g. ‘formal ontology’. 2.2.2.
Formal as total abstraction from (intentional) content34
As already mentioned, Kant rejects the characterization of logic as topic-neutral, and thus the association of formality to topic-neutrality, as he rejects any relation whatsoever between (general) logic and objects. His transcendental logic might be thought to be formal in the sense of indifference to particular objects and of topic-neutrality, as it concerns the a priori relation to (bare) objects, but in his general logic even this a priori relation to objects is to play no role whatsoever. Kant emphasizes that logic is formal in the sense of total abstraction from content, and ‘content’ here is to be understood as any intentional relation 33
34
‘Formal disciplines are set apart from ‘regional’ or ‘material’ disciplines in that they are independent of the peculiarities of any given field of knowledge.’ (Smith & Smith 1995, p. 28) This is MacFarlane’s ‘3-formality’ (2000, pp. 60–65).
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to objects in the physical realm.35 In other words, the total abstraction from content (matter) that characterizes (general) logic as formal for Kant entails that logic cannot in any way talk about the (physical) world. If it is about anything at all, it is about the pure forms of thought in the understanding, and this is why it is formal. This is clearly a distinctively Kantian, idealist interpretation of the nature of logic and of the notion of the formal. But we can establish the principles (rules) governing the pure forms of the understanding, and this is what logic is about. Be that as it may, the idea that logic has no intentional content is (to my knowledge) virtually absent in the history of philosophy prior to Kant.36 As already mentioned, according to the ancient and medieval Aristotelian traditions, the forms of logic were meant to be applied in all scientific contexts (logic’s topic-neutrality), which implies that, even if indirectly (by means of applications), there is always some sort of content in logical investigations. Similarly, but in a pejorative sense, Descartes accuses logic of only dealing with forms and thus of disregarding intuition and a proper grasp of content. For Descartes, clearly the forms of logic are not in the mind. One of the aspects of the Kantian revolution consisted precisely in subverting the bad image that forms had had in the early modern period (with the exception of Leibniz and a few others), while at the same time internalizing these forms in the understanding.37 Kantian forms are in no way forms as schemata, but rather the very conditions of possibility of thought. Moreover, Kant’s idealism implies that these forms have no connection with the physical world whatsoever. General logic. . . abstracts from all content of knowledge, that is, from all relation of knowledge to the object, and considers only the logical form in the relation of any knowledge to other knowledge; that is, it treats of the form of thought in general. (KrV:A55/B79) Kant’s idealistic interpretation of logic is (or so it seems to me) the obvious source for the widespread view that logic is strictly an aprioristic matter and lacks any empirical content, in sum that it is entirely exiled from the physical world on account of its formality.38 This is exactly what Kant seems to mean when he says that logic is formal, i.e. that it deals with the pure forms of the understanding. The heart of the matter here is the (metaphysical) claim that these forms are absolutely not in the world. It is according to such a view of forms that formal investigations, i.e. investigations pertaining to these ethereal forms, are purely aprioristic and devoid of any empirical content. It is thus chiefly in this sense that the notion of the formal is associated to the notion of a priori, strictly non-empirical knowledge. By the same token, logic is in no way to be held accountable towards the physical world or towards metaphysics, as these elements purportedly play no role in the determination of the valid principles of logic. According to such views, in a world with fundamentally different metaphysical properties, ‘our’ valid principles and rules of logic are just as valid. Indeed, from this angle, the relation between logic and the world is perfectly symmetrical: logic cannot tell us anything about how the world is, but the world cannot in any way determine how logic ought to be. MacFarlane 2000, p. 3.3 claims that four distinguished twentieth century authors (Schlick, Einstein, Carnap and Nagel) explicitly held this view of logic, in particular insofar 35
36
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‘Since for Kant the content of a concept depends on its relation to an object (A69/B94, A139/B178, A155/B194-5, A239/B298, JL:Section 2), these formulations are equivalent (A59/B83, A63/B87)’. (MacFarlane 2000, p. 61 fn. 8) A possible exception would be the recurrent suggestion in the Aristotelian tradition that logic deals with concepts of ‘second intention’ (concepts of concepts, not of things), an idea that seems to have originated in Avicenna. See Longuenesse 1998, 147 et passim on Kant’s internalization of forms. But one must not assume that, whenever Kant uses the term ‘formal’, there is no connection whatsoever with objects. In the moral domain, for example, a categorical imperative is formal and yet it has a bearing on the world.
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as logic is formal. But even if not all philosophers and logicians are entirely explicit about or even aware of such traces of ‘neo-Kantianism’, it seems fair to say that the received view still has it that logic is purely aprioristic. This is so precisely because they are formal, i.e. because they deal with these forms that do not inhabit the physical world.39
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2.2.3. The formal as de-semantification40 The third, perhaps most radical variation of the notion of the formal as abstraction from content is the pervasive view that the abstraction in question concerns abstraction from all meaning whatsoever, i.e. that it amounts to what we could refer to as a process desemantification41 of some portions of (written) language. On this view, to be purely formal amounts to manipulating symbols as blueprints with no meaning at all, as pure mathematical objects and thus no longer as signs properly speaking. The forms in such cases are, presumably, the very shapes of the symbols involved—no more, no less. When one speaks of ‘formal systems’, ‘formal theories’, etc., de-semantification is typically an important element.42 This variation of the notion of the formal is the most recent one among those considered so far. Indeed, I have never come across any pre-nineteenth century suggestion to the effect that, when symbols are treated as non-symbols, i.e. as inscriptions without meaning, the approach is then ‘formal’. In fact, in medieval theories of supposition, the significative, meaningful kinds of supposition (personal and material) were taken to be formal precisely insofar as they were significative, given that signification was associated to the notion of form. In contrast, when terms were considered in their materiality only, thus not in their meaningfulness, then material supposition occurred.43 So interestingly, we have exact opposite uses of the terminology in medieval theories of supposition and in the modern conception of the formal as ‘meaningless’. Historically, this variation of the notion of the formal has its roots in the nineteenth century and was later consolidated within Hilbert’s program for the foundation of mathematics in 39
40
41 42
43
Taking up a suggestion put forward by Quine in ‘Two dogmas of empiricism’, Putnam 1979, reprint argued that the failure of distributivity of conjunction over disjunction constituted an empirical motivation to revise classical logic. Putnam’s original paper provoked heated reactions (for example, Dummett 1976), and he himself came to admit later on that his original position was untenable (see Maudlin 2005). Perhaps the failure of Putnam’s attempt served to discredit even further the already notso-popular view that logic is not aprioristic, but in truth attempts to argue for the purely aprioristic nature of logic (e.g. Field 1996, 1998) have not been entirely successful either (in particular, it is not at all obvious how the aprioristic nature of logic can be reconciled with falibilism and the possibility of revision in logic). For a recent attempt to argue for the non-aprioricity of logic, see Bueno and Colyvan 2003. This is MacFarlane’s ‘syntactic formality’ (2000, Section 2.1), one of his ‘decoy’ variations. I find his terminology here misleading, as a syntactic approach does not necessarily take a stance concerning the meaningfulness or lack thereof of the language in question. I owe the term to Krämer 2003. Nevertheless, there is a crucial step from ‘formal objects’ understood as meaningless inscriptions to the inductive generation of a potentially infinite collection thereof by means of recursive rules. The latter aspect pertains to the notion of the formal as computable, to be discussed in the next section. I owe this point to Wilfried Sieg. See Dutilh Novaes 2011 for an account of medieval theories of supposition. Note that there are many other uses of the term ‘formal’ that have been important in the history of philosophy at one point or another, but which have by now disappeared from the analytic philosopher’s terminological toolkit. Uses of ‘formal’ closer to the metaphysical origin of the terminology, in particular in connection with the Aristotelian notion of formal causes, were abundant in the Aristotelian tradition, but they are now mostly of historical interest. So even though my main interest here is the history of the different variations of the formal that are now still pervasive in philosophical circles, my focus on these variations should not be interpreted as ignorance of these uses which are no longer current.
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the 1920s.44 To be sure, Hilbert was not the first to suggest focus on mathematical symbols in their materiality, as inscriptions: already in the Grundlagen der Arithmetik (Frege 1884) and later in ‘Über formale Theorien der Arithmetik’ (1885/6) Frege criticizes this approach, in particular, as defended by his colleague in Jena, Carl Johannes Thomae.45 Responding to Frege’s criticism, in the second edition (1890—the first edition had been published in 1880) of his textbook Elementare Theorie der analytischen Funktionen einer komplexen Veränderlichen, Thomae writes:
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For the formal conception, arithmetic is a game with signs which one may call empty; by this one wants to say that (in the game of calculation) they have no other content than that which has been attributed to them concerning their behavior with respect to certain rules of combination (rules of the game).46 (emphasis added) However, it was only within Hilbert’s program in the 1920s that this approach acquired the status of a truly significant approach to mathematics and that this particular sense of the term ‘formal’ became widespread, especially once the approach became known as ‘formalism’.47 The gist of this notion of the formal is aptly captured in the following passage by Bernays: The typically mathematical character of the theory of provability reveals itself especially clearly, through the role of the logical symbolism. The symbolism is here the means for carrying out the formal abstraction. The transition from the point of view of logical content to the formal one takes place when one ignores the original meaning of the logical symbols and makes the symbols themselves representatives of formal objects and connections. For example, if the hypothetical relation ‘If A then B’ is represented symbolically by A→B Then the transition to the formal standpoint consists in abstracting from all meaning of the symbol → and taking the connection by means of the ‘sign’ → itself as the object to be considered. To be sure one has here a specialization in terms of figures instead of the original specialization in terms of content; this, however, is harmless insofar as it is easily recognized as an accidental feature. Mathematical thought uses the symbolic figure to carry out the formal abstraction. (Bernays 1998, p. 248). A key term in this passage is ‘formal abstraction’: abstraction is of course a crucial concept in mathematics, and within the Aristotelian mathematical tradition abstraction typically corresponds to abstraction of form from matter. Here, however, abstraction corresponds to ignoring specifically the meaning (content) of signs; as the inverted commas used by Bernays indicate, ‘signs’ thus viewed are no longer signs once they become the objects to 44
45
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‘To Hilbert is due […] the emphasis that strict formalization of a theory involves the total abstraction from the meaning, the result being called a formal system or formalism (or sometimes a formal theory or formal mathematics).’ (Kleene 1951, p. 61/2.) In the nineteenth century English mathematics, the idea of manipulating ‘meaningless symbols’ was also present, in particular, with Peacock and Babbage. (I owe this point to W. Hodges.) Translation taken from Thomae’s biography, available at http://www-groups.dcs.st-and.ac.uk/∼history/Biographies/ Thomae.html. The term ‘formal’ as used here is very probably reminiscent of the Kantian idea of the formal as abstraction from content discussed in the previous section. The term ‘formalism’ was initially introduced as a pejorative characterization of Hilbert’s position by Brouwer. Moreover, Hilbert’s views are significantly more complex than the mere idea of ‘treating mathematical symbols as meaningless’.
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be considered in themselves, precisely because they are de-semanticized. Significant for our purposes is the fact that the process of abstraction as well as the result obtained (a ‘purely formal system’) are both qualified as ‘formal’. On the one hand, the formal as de-semantification may be seen as a radical variation of the general ‘abstraction from content’ Kantian motto; in effect, Kantian ideas had considerable influence over Hilbert and Bernays (Bernays 1998), Kant is mentioned several times).48 But on the other hand, there is a sense in which the formal as de-semantification can also be seen as stemming from the schematic notion of the formal, as described in the following passage by Tarski: Already at an earlier stage in the development of the deductive method we were, in the construction of a mathematical discipline, supposed to disregard the meanings of all expressions specific to this discipline, and we were to behave as if the places of these expressions were taken by variables void of any independent meaning. But, at least, to the logical concepts we were permitted to ascribe their customary meanings. […] Now, however, the meanings of all expressions encountered in the given discipline are to be disregarded without exception, and we are supposed to behave in the task of constructing a deductive theory as if its sentences were configurations of signs void of any content. (Tarski 1959, p. 134) At any rate, it is patent that the notion of the formal as de-semantification is best understood against the general background of the formal as pertaining to form and as abstraction from matter as I have been describing it. To be sure, the adjective ‘formal’ used in this sense also qualifies a certain methodology, that is to say a specific manner of proceeding—the blind, mechanical manipulation of symbols—which will be discussed in more detail in the next section. But it is important to distinguish the formal as de-semantification from the formal as computable,49 as it is possible to proceed ‘formally’, i.e. mechanically, even without the preliminary step of ‘formal abstraction’ from meaning, as is attested by logical systems that have favorable computational properties but which operate with interpreted languages. Similarly, it is quite a conceptual step from treating signs as meaningless to the purely computational manipulation of signs (see below). With respect to the notion of the formal as de-semantification, again quite a few philosophical difficulties arise. Forms here are simply the shapes of the ‘signs’ considered as meaningless blueprints, as pure inscriptions: they are mere physical occurrences. But of course, one must give an account of what is to count as identity—equiformity—in these cases: are A and A the same object? What about A, using a different font? Is every token occurrence to be considered as having a form of its own, or do different tokens share the same form? These are again philosophical issues requiring further attention, but which are all too often overlooked. 2.3. Conclusion These are, I claim, the five main variations of the formal as pertaining to forms. Historically, the formal as schematic is the oldest and most pervasive among them, stemming essentially from the Ancient Commentators’ application of the Aristotelian form vs. matter distinction to syllogisms. The other four variations all emerge from it, having been to a greater or lesser extent influential at different times, but at any rate it is clear that these five 48
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See also fn. 10 of MacFarlane 2000, p. 142 for additional historical connections between the program of ‘rigorization’ of mathematics and a general Kantian background, and Section 2.1 of Zach 2003 on the influence of Kant over Hilbert and Bernays. Something which, to my mind, MacFarlane fails to do.
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variations are all still relevant for current conceptions of what it means to say that logic is formal. 3. The formal as pertaining to rules After the examination of the cluster of variations of the notion of the formal as pertaining to forms, I shall now focus on the cluster pertaining to rules, which is, as such, predominantly normative. Unlike the variations pertaining to forms, the rule-related variations of the terminology do not share (or at least not to the same extent) a long and complex philosophical history. Rather, although the normative element was implicitly present in the developments described in the previous section, systematic terminological connections between the notion of the formal and the phenomenon of rules and rule-following are rather recent. The main aspect of dissimilarity between the formal as pertaining to rules and the formal as pertaining to form is thus a normative import and an emphasis on doings and actions. As pertaining to forms, the formal is essentially related to what something is; as pertaining to rules, the formal is essentially related to what something (or someone) does, and is generally characterized by a strict adherence to (usually explicitly formulated) rules, norms or laws. Its opposite, ‘informal’, amounts to a lax attitude towards rules—either unsystematic application or outright noncompliance. Significantly, the normative import of these variations of the formal is clearly related to the (putative) normative role of logic, an important topic which has received renewed attention recently (see Field 2009, for example). The three variations to be examined are: (1) the formal as algorithmic or computable—a variation of the formal particularly pervasive in much of twentieth and twenty-first century work in (the philosophy of) logic; (2) the formal as pertaining to regulative rules and conventions; (3) the formal as pertaining to constitutive rules, and, in particular, the Kantian view of logic as defining the constitutive forms of thought. (For the regulative vs. constitutive distinction, I will rely essentially on Searle’s formulation of it.) Clearly, variations (2) and (3) have an intrinsic normative component: in such cases, the rules in question are thought to distinguish right from wrong; correct from incorrect; what is permitted from what is forbidden; obligations from entitlements and prohibitions. The formal then concerns (strict) adherence to the salient norms, or even the system of norms itself. In variation (1), however, the normative component is less clearly perceived: when speaking of the formal as computable, it is not a matter of adjudicating between right and wrong but rather of implementing instructions and procedures in a strict, blind manner. In other words, the notion of the formal as computable is not explicitly deontic/normative because it does not involve the idea of choices being made, i.e. of deliberation and voluntary compliance or non-compliance to rules; in the case of formal as computable, actions cannot but follow the pre-established paths. Nevertheless, again it is a matter of strict application of (explicitly stated) rules/instructions, which justifies that these three variations be treated in the same cluster. 3.1. The formal as computable Uses of the term ‘formal’ as synonymous with ‘computable’ are extremely pervasive in the logical developments of the last eight decades; for example, a ‘formal system’ is often defined as a recursively axiomatized theory. But such uses appear to be a relatively recent phenomenon in philosophy and logic: unlike the variations of the formal as pertaining to forms, which have an old and distinguished philosophical pedigree, ‘formal’ in the sense of computable seems to have become widely used only in the twentieth century. Moreover, it can be argued that, according to a more precise use of the terminology, the extension of ‘computable’ is actually a proper sub-class of the extension of ‘formal’ in this very sense, as there are, for example, systems where one operates ‘formally’, i.e. according to explicitly
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stated rules, but which do not display full computability in that they are not recursively axiomatizable. Nevertheless, uses the term ‘formal’ as a synonymous with ‘computable’ have become current practice.50 It is also in this sense that we usually speak of a ‘formal language’,51 i.e. a language with strict formation rules such that its syntax is absolutely exact—an idea that dates back to Carnap’s seminal The Logical Syntax of Language. In the 1930s, growing interest in the phenomenon of computability span from Hilbert’s formalistic program in connection with the ‘decision problem’(Entscheidungsproblem), and subsequently in light of the incompleteness theorems. In effect, in the 1930s the concept of computability/effective calculability was perhaps the most widely discussed topic in logic and the foundations of mathematics, to a great extent as an aftermath of the blow inflicted by Gödel’s incompleteness theorems to Hilbert’s program.52 But there was not only conceptual continuity; there was also transmission of the terminology. Seemingly, the first uses of the term ‘formal’ in the sense of computable were directly influenced by the sense of ‘formal’ as de-semantification just discussed. Turing himself does not use the term ‘formal’ in this sense in his 1936 paper, but many of the logicians involved in the development of the concept of computation—such as Kleene and Gödel—explicitly used the term ‘formal’ as synonymous with computable, in particular, in the phrase ‘formal system’. Curry remarks that the notion of a ‘formal system’ [. . .] is quite a different notion from that of a postulate system, as naively conceived a half century ago. In the older conception a mathematical theory consisted of a set of postulates and their logical consequences. The trouble with this idea is that no one knows exactly what a logical consequence is [...]. In the modern conception this vague and subjective notion is replaced by the objective one of derivability according to explicitly stated rules. (Curry 1957, p. 1) (emphasis added) These explicitly stated rules should allow for a mechanical, blind implementation, and it is in this very sense that the system is said to be formal. Similarly, commenting on the problem of decidability for mathematical theorems, Gödel says: The first part of the problem has been solved in a perfectly satisfactory way, the solution consisting in the so-called ‘formalization’ of mathematics, which means that a perfectly precise language has been invented, by which it is possible to express any mathematical proposition by a formula. Some of these formulas are taken as axioms, and then certain rules of inference are laid down which allow one to pass from the axioms to new formulas and thus to deduce more and more propositions, the outstanding feature of the rules of inference being that they are purely formal, i.e. refer only to the outward structure of the formulas, not to their meanings, so that they could be applied by someone who knew nothing about mathematics, or by a machine. (Gödel 1995, p. 45) (emphasis added) This passage by Gödel illustrates the tight connection between the Hilbertian ‘formal’ view of symbols as pure shapes and the concept of the formal as computable: once the total abstraction from meaning is undertaken, the different transformations and manipulations of symbols may become a purely mechanical endeavor. Nevertheless, it is in fact a dramatic step (something that is not made sufficiently explicit by Gödel’s phrase ‘so that’) from the idea of treating (mathematical or otherwise) symbols as meaningless (an idea already 50
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E.g. the widespread view according to which second-order logic is not ‘logic’ properly speaking because its set of validities is not recursively enumerable, as discussed in the introduction above. But also in the sense of the formal as de-semantification. Sieg 2008 offers a detailed account of how the modern notion of computability emerged from attempts to solve the decision problem. See also Smith 2009, Ch.. 33.
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developed, for example, by Thomae, as we saw) to the rule-governed inductive generation of a potentially infinite collection of (‘meaningless’) formulae, which could then be used to express mathematical propositions, as described by Gödel. This step could only be taken once the notion of computability was sufficiently mature, which was clearly not the case prior to the developments in the 1930s.53 It is also important to note that algorithmic procedures are used on two distinct levels within a formal system, both of which are mentioned in Gödel’s passage. On one level, the formulae of the language are inductively generated by means of the application of its rules of syntax (rules of formation) to its basic terms, generating a potentially infinite collection of formulae. On another level, the rules of inference (rules of transformation) defined for the formal system can be ‘mechanically’ applied to the formulae in order to establish what follows from what within the system—and, in particular, what its theorems are. They consist in the formulae obtained by the application of the rules of inference to axioms, in first instance, and then to the formulae obtained from the axioms, in a procedure that can be iterated as many times as one wishes. So a formal system is ‘formal’ both in the sense of how its language is generated and in the sense of how inferences are drawn within the system.54 One might expect that Frege would have used the term ‘formal’in the sense of computable. However, as I read him, Frege was not particularly concerned with the mechanical move from premises to conclusion as such; rather, his main concern seemed to be that every inferential step be made thoroughly explicit, i.e. that there be no gaps in the expression of an argument (see, for example, the preface of the Begriffsschrift or his critique of Dedekind’s lack of explicitness concerning inferential steps in the Grundgesetze). Frege’s goal was epistemological clarity, not ‘mechanical reasoning’ as such, although the requirement that no hidden contentual considerations be incorporated into the application of rules can be read as the demand for a blind, ‘mechanical’ application of rules (see Sieg 1994, p. 74/5).55 Another likely precursor of uses of ‘formal’ in the sense of computable is Leibniz, given the crucial role of both the notion of calculability and of the terminology of ‘form’ and ‘formal’ in his investigations. However, as far as I could establish, he does not use the term ‘formal’ distinctively in the sense of computable or calculable. So if the notion of the formal as computable is not to be found (in any case not fully developed) in Leibniz, then it is probably not to be found in any other author prior to the 1930s. (To my knowledge, it is also not to be found in the tradition of the ‘algebra of logic’ stemming from the works of Boole).56 The question of what it means to say that something is formal can thus be (partially) answered rather straightforwardly with to the notion of computability—effectively calculability—as this is indeed one important sense in which the term is used. But of course, 53
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Although Gödel is not usually seen as having made significant contributions to the topic of computability, he exerted a significant influence on some of the founders of the field such as Church, Kleene and Rosser—see Zach 2006. Note that, in this sense, a system can be ‘formal’ even if it is not decidable, i.e. even if it cannot prove or disprove all the formulae of its underlying language. Moreover: ‘I here want to consider two views, both of which bear the name ‘formal theory’. I shall agree with the first; the second I shall attempt to confute (Thomae’s). The first has it that all arithmetic propositions can be derived from definitions alone using purely logical means, and consequently that they also must be derived in this way’. (Frege 1885, p. 94, transl. p. 141) By ‘purely logical means’ Frege surely meant the rules put forward in the Begriffsschrift, and in this sense it can be said that Frege was already relying in a proto-form of the formal as computable. Naturally, this is also a clear formulation of Frege’s logicism. Interestingly, though, the Oxford English Dictionary lists a circa 1390 occurrence of the term ‘formal’ in connection with algorithms, in John Gower’s Confessio Amantis: ‘The wise man accompteth After the formal proprete Of algorismes a, be, ce.’ (John Gower, Conf. III 89)
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this is only a relative answer, and a rather uninformative one if not accompanied by a discussion of what it is to be computable, i.e. by a proper understanding of the concept of computable as in ‘not requiring any insight or ingenuity’. What must be kept in mind is that, prior to the formulation of technical notions such as that of a Turing machine, general recursive functions and lambda-definability, the notion of computability is originally an intuitive, informal notion.57 ‘Computable’ can be used as synonymous with ‘effectively calculable’ and ‘mechanical’, but these are also informal notions as such, so again these relative definitions only take us around in circles. So what are then the core elements of the intuitive, informal notion of being computable? A recurrent and to some extent illuminating clarification of the concept of effective calculation or computation is based on the notion of a mechanical procedure. What is a mechanical procedure, i.e. how do machines in general operate? The basic intuition seems to be that they implement previously determined actions in a sequence, step by step, in a completely predictable way, thus reaching the desired outcome. The mechanism of a (wellfunctioning) clock, for example, goes from state to state by means of the previously defined (within the mechanism itself) operations of its parts. To compute is thus in first instance an action undertaken by a human agent, but who acts as if she was a machine, following pre-determined patterns (as specified for example by means of an algorithm). To compute is to reason as machines would if they could reason: mechanical thinking.58 Note that even Turing, when introducing the notion of (what we now call) a Turing machine in his 1936 paper, did not refer to machines properly speaking. The agent operating the ‘machine’, i.e. undertaking the actions determined by the table of instructions, is actually a human being, to whom he refers to as the ‘computer’. The human computer simply follows strictly the instructions contained in the table of instructions of a given Turing machine, thus not adding any insight of his/her own.59 Now, just as a ‘Turing machine’ is (originally) not a machine properly speaking, as it is operated by a human being, the description of a ‘mechanical procedure’as a procedure where every step is completely determined does not describe the operations of actual machines. As any owner of a car, a washing machine or even a computer knows, machines often do unpredictable things, i.e. they act differently at different occasions even when the same buttons seem to be pushed: same input, different outputs. So clearly, when speaking of a ‘mechanical procedure’, it is not so much actual machines that we have in mind, but rather the ideal of a machine, i.e. the abstract machine which always behaves according to plan, that is according to the instructions contained in it. The ideal machine is aptly described in Section 193 of Wittgenstein’s Philosophical Investigations; he reminds us that actual machines do not (usually) behave ‘mechanically’, especially but not exclusively due to hardware limitations and constraints. Thus, what occurs in a calculation/computation is the passage from one initial state (which can be the premises, or more generally the initial state of information) to a final state (the conclusion, the result of the calculation), by means of successive ‘small’ passages from state to state. It is a successive transformation of enumerable, finite states, yielding 57
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But note that this does not mean that the gap between the informal notion of computability and its technical counterparts cannot be bridged somehow. Smith 2009, Ch. 34 argues convincingly that a variation of Kreisel’s ‘squeezing argument’ can be used here. Personally, I think that Turing’s analysis is not only formally correct; but also conceptually insightful, and the idea of a Turing machine truly captures the conceptual core of the notion of computability. Of course, now that computers are omnipresent, the expression ‘mechanical thinking’ does not sound particularly absurd, but one must appreciate that this is a very recent development. Recall that Descartes considered animals to be ‘machine-like’ precisely because, unlike human beings, they do not have a soul and thus cannot think. Thus described, the actions of the human computer also make it patent that the variation of ‘formal’ as computable does have traces of a normative component after all.
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a particular output from a given input. The heart of the matter is defining the rules or instructions allowing for the passages from a state to the next state in such a way that any agent would effect the passage in exactly the same way. Thus seen, a computation has a few fundamental properties such as: it progresses through discrete, very ‘small’ individual steps; it is a dynamic process, taking place in time and with a clear temporal direction; every step is completely determined,60 i.e. the instructions are applied blindly, requiring no insight or ingenuity. Indeed, Turing’s great accomplishment was to provide a precise, technical formulation to the informal idea that a calculating procedure should involve no ingenuity or insight. Note also that such mechanical procedures can be implemented even if they will not be able to deliver all the answers within a system (e.g. whether a given formula is a theorem of the system); that is, a system can be formal in the sense of ‘mechanical’ applications of its rules of transformation and yet be undecidable, and in such cases the term ‘computable’ in its more precise sense should not be used. For our purposes, what matters is that procedures displaying these characteristics— passages from one state to another by means of determined actions, not requiring any insight but determined by explicit rules or instructions—are usually described as formal procedures. But an important question is whether the converse holds. That a procedure displaying the characteristics described here can be said to be formal appears to be an irreproachable use of the terminology; but are we prepared to say that only the procedures satisfying these conditions can be said to be formal? Or is the term ‘formal’ to be used also in circumstances where the procedures in question cannot be fully automated? In particular, mathematics is the formal science par excellence; but can the whole of mathematics be reduced to strictly mechanical procedures? Arguably not. Hence, there seems to be more to formality than just mechanical reasoning—that is, if mathematics is to be seen as formal at all.61 So it would seem that we are actually dealing with three different but nested classes of phenomena: the computable is a proper sub-class of the mechanical, which in turn is a proper sub-class of the formal. 3.2. The formal as pertaining to regulative rules After dealing with the variation of the formal as computable, which does concern rules and instructions but which is not explicitly normative, we now enter the domain of the formal as explicitly normative, i.e. as concerning rules determining what ought to be done. For this purpose, it shall prove convenient to rely on a popular (albeit controversial) taxonomy of rules into two kinds: regulative vs. constitutive rules, relying on Searle’s formulation of the distinction.62 For my purposes, what is important is that, if there is such a distinction of kinds of rules (or simply if it is generally thought that there is such a distinction), and if the notion of the formal is often used as pertaining to rules, then it is to be expected that there shall be (at least) two distinctive variations of the formal as pertaining to rules, 60
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Of course, there are now theories of non-deterministic computation. But here I shall focus on the classical notion of computability, as it is arguably the important one in order to understand the concept of formal as computable. Note that the term ‘formal’ is also often used simply as synonymous with ‘mathematical’, such as in the phrase ‘formal epistemology’, which refers to an approach to epistemology based on a specific mathematical theory, namely the Bayesian probabilistic framework. ‘I distinguish between two sorts of rules: Some regulate antecedently existing forms of behavior; for example, the rules of etiquette regulate interpersonal relationships, but these relationships exist independently of the rules of etiquette. Some rules, on the other hand, do not merely regulate but create or define new forms of behavior. The rules of football, for example, do not merely regulate the game of football but as it were create the possibility of or define that activity. The activity of playing football is constituted by acting in accordance with these rules; football has no existence apart from these rules. I call the latter kind of rules constitutive rules and the former kind regulative rules. Regulative rules regulate a pre-existing activity, an activity whose existence is logically independent of the rules. Constitutive rules constitute (and also regulate) an activity the existence of which is logically dependent on the rules’ (Searle 1971, p. 41)
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indeed accompanying the constitutive vs. regulative distinction. Even in non-philosophical contexts, where (presumably) no explicit awareness of the constitutive vs. regulative distinction is present, one also finds different uses of the term ‘formal’ essentially following this dichotomy: contrast, for example, the phrase ‘formal conditions of employment’ (the formal as constitutive) to ‘formal education’ or ‘dress code: formal’ (the formal as regulative). Moreover, the constitutive vs. regulative distinction has some important metaphysical implications: constitutive rules have logical and temporal priority over (or at least simultaneity with) the phenomena they constitute, while regulative rules come about only after the establishment of the phenomena they regulate. Regulative rules are inductive and backward-looking, i.e. they generalize from previous events and experiences; constitutive rules, in contrast, can be said to ‘look forward’. It is important to note though that these qualifications—constitutive, regulative—are better seen as relative to a given phenomenon and thus not as absolute qualifications. A rule is not constitutive or regulative an sich, but rather with respect to a particular phenomenon or activity. For example, the rules and conventions constituting the institution of money (one of Searle’s favorite examples) are constitutive of this particular social construction—money—but they can be viewed as regulative of a broader phenomenon, namely the exchanges and interactions related to the trade of goods and values in a given community. Usually, constitutive rules come to create institutions which are nevertheless embedded in larger contexts (typically filling in a particular niche in these contexts, that is, meeting particular needs felt within the larger contexts), and with respect to the larger contexts they tend to be essentially regulative, not constitutive. The formal as pertaining to regulative rules is at first sight the least ‘philosophical’ of the eight variations under consideration here. Indeed, uses of the term in everyday life generally fall under this variation, such as in ‘a formal person’, ‘dress-code: formal’, ‘formal education’, etc. These uses all pertain to the rigorous (possibly excessive) observance of recognized (but not necessarily explicitly formulated) regulative rules, norms or laws. It is frequently related to the notions of ceremony and ritual, i.e. situations defined by strict rulefollowing. These everyday uses of ‘formal’ can be traced back to several centuries ago, as attested by the examples provided in the Oxford English Dictionary, and one might think that there are no interesting theoretical/philosophical implications in this variation. It seems to me, though, that this is not the case. For example, uses of ‘formal’ as strict compliance with regulative rules permeate much of the discussions on formality in sociology and philosophy of law. Moreover, the ontological status of such rules is directly related to debates opposing naturalistic vs. normativistic accounts in a variety of philosophical topics. Finally, insofar as language use is a rule-governed activity, it is important to raise the question of the status of rules within the phenomenon of language broadly understood; this may shed light on the issue of the ‘formality’ and ‘informality’ of different kinds of languages.63 Note that systems of regulative rules are in fact products of processes of abstraction very similar to the abstraction of form from matter according to the Aristotelian account of substance as a hylomorphic compound. We have seen that a very general trait of different uses of the notion of the formal as pertaining to forms is the idea of abstraction from matter, and that the different variations of the formal can be understood as generated by different interpretations of the notions of form and matter. Well, a similar pattern is in place here, and again it is a mechanism of ‘making it explicit’ (in this case, making implicit norms explicit). The difference is that, when speaking of the formal as pertaining to forms, I focused on arguments viewed as ontologically independent entities, while in the case of the formal as pertaining to rules in general, and to regulative rules in particular, we are dealing 63
For example: what kind of formality did Montague have in mind when he claimed that English (and presumably French, Spanish, Japanese, Arabic etc.) is a formal language?
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with a different kind of phenomenon. As already remarked, the notion of the formal as pertaining to rules typically concerns what something (someone) does, not what something (someone) is. So now we are dealing with phenomena characterized by actions (in general, human actions), most saliently (but not exclusively) social inter-actions. The ‘substance’ in question is no longer the individual, but the very collection of certain (human) practices and actions, in a way constituting an entity of its own.64 One can then speak of an abstraction of the ‘form’ of such a social entity, understood as the collection of the general patterns, regularities, rules and norms underlying the practices and actions constituting the social entity in question. Such an abstraction should yield a system of rules detached from their actual implementations, and it is precisely such a system that is often said to be formal. Relatedly, strict observance of patterns underlying social practices is also often described as ‘formal’, even if the actual process of abstraction and explicit formulation of such rules has not taken place. Hence, a formal person is not only one following explicitly formulated rules of etiquette, but also one that follows implicit rules and norms more strictly than most other people. Admittedly, uses of the term ‘formal’ as pertaining to regulative rules in (analytic) philosophy are quite scarce, except in the philosophy of law, where they seem to be rather frequent.65 But it is worth noting that, in sociology, the formality terminology is widely used, both in the sense of pertaining to regulative rules and in the sense of pertaining to constitutive rules; in fact, some of the debates on the formal in sociology seem to revolve around the failure to recognize both kinds of rules as yielding different kinds of formality.66 So, while it may not have received extensive attention from philosophers under this very terminology (Searle, for one, does not use the term ‘formal’ in this sense), the notion of the formal as pertaining to regulative rules encompasses an array of conceptual implications that, again, would deserve further attention. 3.3. The formal as pertaining to constitutive rules67 Above, I briefly mentioned that the constitutive-regulative distinction is not unanimously accepted. But I take it that, while there are good reasons to believe that the distinction is certainly not a sharp, clear-cut one, it is still convenient as a means to explain certain differences between kinds of rules that have an impact on different understandings of the term ‘formal’. Indeed, in this section, I shall discuss uses of the term ‘formal’ that are quite different from those just discussed (even though they also pertain to norms and rulefollowing) precisely because they seem to pertain to a different kind of rules, which can be described as constitutive rules. Constitutive rules are typically seen as conditions of 64
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The ontology of social phenomena is of course an interesting and complex topic, and for reasons of space we cannot go very deep into the discussion. There is a whole research area dedicated to it, known as ‘social ontology’: generally characterized, social ontology deals with philosophical questions concerning the nature, existence and identity of social facts, entities, properties and actions. Unsurprisingly, Searle has been very active in this field. Here is one such use: ‘…we are concerned primarily with the law in the sense that it is facts even though these facts have a normative skin or formal side. The law is dynamic and deals with the development of life itself. Its formal divisions are incident to its substantive divisions.’ (Sayre, Philosophy of Law, p. 48.) In a review of Stinchcombe’s (2001) When Formality Works, Richard Scott seems to propose precisely this diagnosis of the disagreement between Stinchcombe and the so-called ‘new Institutionalists’, such as Meyer (a locus classicus being Meyer and Rowan 1977): ‘By contrast [to Stinchcombe], the new institutionalists stress that collective beliefs and rules act to constitute actors and actions. While for Stinchcombe the formalisms are distinct from the actors and behavior they were crafted to regulate, […] for Meyer and other social constructionists, the formalisms give rise to these actors and behaviors. They do not reflect reality; they create it’. (Scott 2002; p. 636, emphasis added) MacFarlane’s ‘1-formality’ (2000, pp. 52–56) belongs to this variation, but does not exhaust it.
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possibility for the existence of whatever phenomenon (entity, fact) they constitute, and this conception yields yet another variation of the notion of the formal. In sociology, the notion of the formal is extensively employed to discuss the constitutive, creative import of certain kinds of rules, namely the rules that we philosophers call constitutive rules—see, for example, Meyer and Rowan 1977. In philosophy, however, the term ‘formal’ is typically not used in connection with the relation between constitutive rules and (social) institutions—Searle, for example, does not use the term when discussing issues pertaining to social ontology such as collective intentionality and institutions. For our purposes, the most salient uses of the term ‘formal’ as pertaining to constitutive rules or norms concern the Kantian thesis of the existence of laws of thought as constitutive of the understanding as such. This means that the Kantian analysis of the formal as pertaining to forms of thought as such has (at least) two distinctive components, which must not be conflated: the idea that the laws of thought are completely dissociated from the physical world (the variation of the formal discussed in Section 2.2.2), and the idea that the laws of thought have a normative import. Kant holds both views, but it is perfectly possible to hold either one of them and not the other. As noted by MacFarlane 2000, Ch. 4, Kant himself does not use the term ‘formal’ to qualify the laws and norms of thought as such; rather, he uses the term ‘general’ in this sense, as in e.g. ‘general logic’. He does use the expression ‘the mere form of thought as such’ (JL:13), as opposed to its matter, and from ‘the form of thought’ to these norms of thought being formal it is but a small step. In effect, many works in logic of the nineteenth century (in particular, those influenced by Kant) use the term ‘formal’ to refer to the norms that are constitutive for thought, i.e. the ‘laws of thought’.68 Still now, the term ‘formal’ is often used in connection with logic in the sense of outlining its normative component, i.e. the (presumed) fact that the laws of logic are constitutive of correct reasoning insofar as they capture the forms of thought as such.69 Such an emphasis on the normative component of reasoning is one of Kant’s main contributions to the history of ideas; while the view that there are proper and improper moves when it comes to reasoning (in particular in logic) had been implicit in ancient, medieval as well as early-modern philosophy (e.g. Descartes’ Rules for the Direction of the Mind), it was Kant who first brought it to the fore and developed it thoroughly. MacFarlane 2000, p. 53 offers a concise and illustrative collage of passages from Kant expressing these very ideas: The laws of logic, by contrast, are defined as the “… necessary laws of the understanding and of reason in general, or what is one and the same, of the mere form of thought as such …” (JL:13). By “necessary laws of the understanding,” Kant means “… those [laws] without which no use of the understanding would be possible at all …” (JL:12), that is, the norms constitutive of thought. Similarly, in the first Critique he says that general logic “… contains the absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding” (KrV:A52/B76). What does it mean to say that the laws of general logic are constitutive of thought as such? Does it mean that it is impossible to think in any other way, and thus that every occurrence of thinking is, per definition, correct? Not so; it is possible to form a thought that breaks one 68
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One example: Poland 1892, The Laws of Thought or Formal Logic – A brief, comprehensive treatise on the laws and methods of correct thinking. Echoes of this idea can be found for example in charges against dialethism and paraconsistency to the effect that denying the law of non-contradiction entails that thinking can make no sense without it, as the law of non-contradiction would be constitutive of thought as such.
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or more of the ‘laws of though’ but which still counts as a thought, albeit an improper one. So (according to the Kantian view) the formal (i.e. constitutive) laws of thought are binding in the sense that any thinker must be responsible towards them, and must thus acknowledge their authority over him/her. Should he/she refuse to do so, then indeed his/her ‘thoughts’ can no longer be viewed as thoughts properly speaking, as they are no more than erratic ruminations. Now, the idea of laws of thought as normative, constitutive constraints on our cognitive activities as human beings does not seem to be entirely implausible; what seems contentious is rather the idea that nothing else can count as thought (human or otherwise) unless it is accountable towards these very laws. Moreover, it is far from obvious that the ‘laws of logic’ as we know them—that is, on a traditional conception of logic—capture indeed the cognitive architecture of human beings. Of course, Kant’s project is overtly normative as opposed to descriptive; he specifically rejected Wolff’s psychological approach to logic (MacFarlane 2000, p. 4.2.1). Nevertheless, even if one is sympathetic to the Kantian idea of constitutive rules of though, the exact content of such rules still remains to be determined, and it is by no means uncontroversial that they should coincide with the rules of logic as traditionally construed.70 But the tradition following Kant in the 19th century (e.g. Boole) simply took the rules of logic to coincide with the laws of thought, an idea that is still quite widespread and which reinforces the association of the concept of the formal to the idea of normativity and constitutive rules. 4. Conclusion The taxonomy of uses of the term ‘formal’ presented here might be viewed as having merely historical rather than genuine philosophical interest, belonging at best to the realm of the history of ideas. But it seems to me that such an assessment would disregard not only the practical advantages of a regimentation of the rather chaotic myriad of uses of this notion in current (philosophy of) logic. It would also fail to appreciate the deeper philosophical understanding of the very property (or properties!) of being formal that may ultimately emerge from the analysis of the semantics of the term ‘formal’ undertaken here. The least we can expect from philosophers is that they use their terms with the appropriate clarity, rigor and systematicity, but they seem to be failing this requirement quite miserably in their multiple and often imprecise uses of the term ‘formal’. So it is to be hoped that the present study can contribute to minimizing this terminological chaos by providing a conceptual analysis of such an important notion. Moreover, the taxonomy presented here may help elucidate some of the most pressing debates within the philosophy of logic. For example, the project that initially motivated the present investigation is that of attaining a better grasp of the very concept of ‘formal languages’, and of the role they play within the practices of logicians. Now, a natural question to be raised is in which of the different senses of formality formal languages are indeed formal. I believe that other pressing issues can be analogously addressed by means of the taxonomy presented here, and thus that it may represent a solid platform on which central discussions within the philosophy of logic can be conducted. Acknowledgements I am grateful to Wilfrid Hodges, Wilfried Sieg, Hannes Leitgeb and two anonymous referees for comments on earlier drafts.
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See, for example, Harman 1986. Moreover, a case could be made for the claim that this is ultimately at least partially an empirical question, even if the normative import is still maintained.
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Appendix: A visual representation of the relations between the different variations Formal as schematic
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Formal as pertaining to forms
Formal as abstraction (from subject-matter)
Formal as variability Formal as indifference to particular objects Formal as topic-neutral
Formal as abstraction from intentional content Formal as desemantification Formal as pertaining to rules
Formal as computable Formal as pertaining to regulative rules
Formal as pertaining to constitutive rules
