Loading...
Please wait, while we are loading the content...
Similar Documents
Cdd : 513 . 01 the Logical System of Frege ’ S Grundgesetze : a Rational Reconstruction
| Content Provider | Semantic Scholar |
|---|---|
| Author | Cadet, Méven Panza, Marco |
| Copyright Year | 2015 |
| Abstract | This paper aims at clarifying the nature of Frege’s system of logic, as presented in the first volume of the Grundgesetze. We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic. keywords: Frege, Second-Order Logic, Grundgesetze der Arithmetik, Begriffsschrift, Frege’s notion of a Function. 1 Preliminaries and proposals The primary aim of Frege’s Grundgesetze der Arithmetik ([18]) was to set up a formal system, that he called ‘Begriffsschrift’ (FGGBS, from 1We thank, for valuable remarks, suggestions, and other sorts of support: Francesca Boccuni, Abele Lassale Casanave, Annalisa Coliva, Pieranna Garavaso, Gerhard Heinzmann, Gregory Landini, Paolo Mancosu, Dag Prawitz, Philippe de Rouilhan, Julien Ross, Marco Ruffino, Matthias Schirn, François Schmitz, Andrea Sereni, Göran Sundholm, Jamie Tappenden, Gabriele Usberti and two anonymous referees. When we quote a translation from Frege’s works, we feel free to slightly modify it, if this allows to stay closer to Frege’s original text. Manuscrito Rev. Int. Fil., Campinas, v.38, n.1, pp.5–94, jan.-jun. 2015. 6 MÉVEN CADET & MARCO PANZA now on)2, within which it should have been possible to get both a formal theory of natural numbers, and a formal theory of real numbers. Frege’s purpose was to obtain these theories by adding the appropriate explicit definitions to such a system3. To make this possible, he included in FGGBS two axioms dealing with value-ranges of functions— the infamous Basic Law V and the Basic Law VI, involving, together with extensions, Frege’s operator of definite description—and based on them to explicitly define a two-argument first-level function—the function ξ _ ζ, in Frege’s notation—, conceived in order to dispense with higher-level functions4. Frege considered these ingredients of his system to be perfectly logical in nature. Hence, he did not present his whole system as resulting from the addition of non-logical axioms and definitions to an underlying system of logic, independent both of value-ranges and of definite descriptions, provided by the remaining axioms—the Basic Laws I, IIa, IIb, III, and IV—together with a number of deductive rules. Even though this underlying system can be easily identified and is close to the one that Frege had offered some 2This system results from a revision and extension of the system previously presented in Frege’s 1879 booklet entitled in the same way ([14]). To avoid any confusion, we shall use, here, the acronym ‘FGGBS’ (for ‘Frege Grundgesetze Begriffsschrift’) to refer only to the former system. We use the whole word ‘Begriffsschrift’ (in italic) to refer to the 1879 booklet. 3Notice, however, that whereas the theory of natural numbers should have merely required appropriate definitions, since those should have been enough for warranting the existence of these numbers, that of real numbers should have also required an existence proof for domains of magnitudes (these numbers being identified by Frege with ratios of magnitudes), since their existence would have not been warranted by their definition ([38]; [12], ch. 22; [37]; [39]). 4In Frege’s parlance—which we shall also adopt here—first-level functions are those whose arguments are required to be objects, while n-level functions (n = 2, 3, . . .) are those whose arguments are required to be n − 1-level functions. As it will become clear later, Frege employed functions where we would rather employ predicates. Hence, his first-level functions often work, mutatis mutandis, as first-level predicates, and higher-level functions as higher-level predicates. Manuscrito Rev. Int. Fil., Campinas, v.38, n.1, pp.5–94, jan.-jun. 2015. THE LOGICAL SYSTEM OF FREGE’S GRUNDGESETZE 7 years earlier in his booklet Begriffsschrift5—which involves, indeed, no analogue of Basic Laws V and VI and makes, then, no mention of valueranges and of definite descriptions—, in the Grundgesetze, he made no effort to separate, within his theories of natural and real numbers, what depends on laws V and VI from what is, instead independent of them, and, thus, of value-ranges and definite descriptions. On the contrary, he pervasively used the function ξ _ ζ to avoid the appeal to higher-level functions, even when this use would have dispensed him from appealing to value-ranges and definite descriptions. These play, then, a pervasive role in Frege’s deductions, often without intrinsic necessity6. Despite this, we know that, whereas the whole FGGBS is provably inconsistent, the subsystem provided by Basic Laws I, IIa, IIb, III and IV, together with the corresponding deductive rules is not, and it is in many respects analogous to a modern system of full second-order predicate logic. This analogy goes with a number of important differences, however. The purpose of our paper is to emphasise some of these differences, by suggesting a reconstruction of such a subsystem, which is intended to both follow the modern usual standards of exposition, and to be faithful to Frege’s conceptions, especially in what makes them contrast with other conceptions, nowadays quite standard and on which modern systems of predicate logic are based. This double intent makes our purpose distinct from that of a number of other presentations and critical discussions of Frege’s conceptions and systems of logic (which the reader can find, for example, in [42], [12], [5], [35], [1], [25], [31]). The adoption of modern standards of exposition apart, there are, however, some important differences between our reconstruction and Frege’s original presentation. One of them pertains to our focusing on a fragment of Frege’s whole system, to which he does not seem to have been willing to assign a dis5Cf. footnote 2, above. Though there are many relevant differences between the two systems, we avoid this matter here, and rather limit our attention to the (more mature) system of the Grundgesetze, namely FGGBS. 6On the eliminability of the appeal to value-ranges from many of Frege’s deductions in the Grundgesetze, cf. [24], pp. 581-584, and [25], 6.1. Notice, however, that eliminating value-ranges would often require the appeal to functions of third or even higher level, which results in formulas that have not a plain rendering within a second-order predicate language. Manuscrito Rev. Int. Fil., Campinas, v.38, n.1, pp.5–94, jan.-jun. 2015. 8 MÉVEN CADET & MARCO PANZA tinct role7. This goes together with our distinction between a propositional and a second-order system. Moreover this distinction does not correspond to any distinction that Frege would have emphasised in the Grundgesetze. Though many of the arguments and considerations included in the first part of this treatise ([18], I.1-I.52)8 are inescapably propositional in nature, he was, in fact, unconcerned, both in this part, and in the successive ones, with a clear identification of a propositional fragment of his whole system. Comparing the Grundgesetze with the Begriffsschrift could certainly offer another picture, since the system offered in the latter does not only involve, as we have said above, neither value-ranges nor definite descriptions, but also includes a propositional fragment which is much more easily separable from the whole system. Still, it seems important to us to notice that in the Grundgesetze, to which we shall here limit our attention, things go otherwise. Another often mentioned radical difference between Frege’s conception of logic and our own depends on his not making a clear separation between the syntax of his system and the meaning to be attributed to the symbols involved in it. As it has been often remarked, this results in his interweaving syntactical and semantic aspects (at least with respect to the usual modern conception of the distinction between syntax and semantics). Though our reconstruction will largely reflect this attitude, we shall not insist on this well-known feature of Frege’s exposition. This 7Notice, however, that in the Foreword to the Grundgesetze, Frege remarks that a “dispute [Streit]” about the logical nature of his derivations “can arise only concerning my basic law of value-range (V)” and continues as follows: “I take it to be purely logical. At any rate, the place is hereby marked where there has to be a decision ” ([18], Vorwort, pp. VII; [22], p. VI1). This seems to suggest that Frege deemed questionable the logical nature of Basic Law V (and VI), or, at least, considered it less certain than that of his other laws. 8This part is plainly titled ‘Exposition [Darlegung] of the Begriffsschrift’. It contains an informal presentation of Frege’s system, namely its language, its Basic Laws (or axioms), its deductive rules, and its basic (explicit) definitions. The remaining part of the Grundgesetze (with the exception of II.55-II.164, offering a critical discussion of some alternative definitions of real numbers and advancing the desiderata that a suitable definition should have met, according to Frege) are devoted to prove theorems within this very system. Manuscrito Rev. Int. Fil., Campinas, v.38, n.1, pp.5–94, jan.-jun. 2015. THE LOGICAL SYSTEM OF FREGE’S GRUNDGESETZE 9 is only an aspect of a more general attitude with regard to the nature and role both of logic and of formal languages, which we shall not focus on, though we hope that the discussion on this matter, quite intense at present, could benefit from the reconstruction we offer. In fact, our reconstruction is so conceived as to make its syntactical ingredients easily identifiable, so as to allow to separate them from the semantic ones. Doing this would result in a rendering of the purely syntactical features of Frege’s system. It is noteworthy that this rendering would stand on its own. This shows that Frege’s interweaving syntactical and semantic aspects does not result in |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/download/8641949/9447 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
