Professor Urquhart, Also, Justifiably Argues That Frege's Grundgesetze System is a logic of Terms.

以下に記すことは、個人的な単なる memorandum です。

通説によると、Frege は文/式を名前/単称名と同一視し、そのことによって非難されてきました。この通説を述べている Frege の研究者には Dummett さんがおり、彼は Frege を批判しています。この通説に対し、異議を唱えているのが Landini さんです。彼によると、Frege とり、official な言語/論理としての Grundgesetze 体系では、文が名前と同一視されていることはなく、Frege は両者を明確に区別していると Landini さんは主張しています。Frege は文を名前と同一視していなかったという、このような通説に反する見解を保持している研究者はほとんどいないと思われるのですが、この同じ、通説に反する見解を堅持していると思われる研究者として、Alasdair Urquhart さんがいることを、以下に memo として記しておきます。


 In Frege's writings subsequent to Grundlagen, the uniqueness of sentences is no longer acknowledged. On the contrary, Frege now assimilated sentences to complex singular terms, regarding them as standing for truth-values in the same way that complex terms stand for objects of other kinds. This was a retrograde step on Frege's part, which obscured the crucial fact that the utterance of a sentence, unlike that of a complex term in general (except in special contexts, such as answering a question), can be used to effect a linguistic act, to make an assertion, give a command, etc.*1

 Unhappily, the doctrine which we considered earlier, which regards truth-values as objects and hence assimilates sentences to complex proper names, undermined the sharpness of the original perception. If sentences are merely a special case of complex proper names, if the True and the False are merely two particular objects amid a universe of objects, then, after all, there is nothing unique about sentences: whatever was thought to be special about them should be ascribed, rather, to proper names - complete expressions - in general. This was the most disastrous of the effects of the misbegotten doctrine that sentences are a species of complex name, which dominated Frege's later period: to rob him of the insight that sentences play a unique role, and that the role of almost every other linguistic expression (every expression whose contribution to meaning falls within the division of sense) consists in its part in forming sentences.*2

次に、通説に反対する Landini さんの見解を引きます。原注は省きます。

  • Gregory Landini  ''Decomposition and Analysis in Frege's Grundgesetze,'' in: History and Philosophy of Logic, vol. 17, no. 1 and 2, 1996, pp. 122-123,

2 Logic in the Grundgesetze

 In marked contrast with modern predicate logic, Frege does not begin his Begriffsschrift with a predicational form π(a1, ... , an), where π is a predicate expression and a1, ... , an are singular terms. Frege hoped to introduce the use of variables into his calculus for logic by borrowing the functional notations of mathematics and transcribing natural language sentences accordingly. To do this, he takes the True and the False as primitive objects. For instance, consider the function f, characterized as follows:

          f{\rm ( \xi )}= \left \{\begin{array}{l}{\rm the True, if  \xi  is human} \\{\rm the False, otherwise} \end{array}\right.

The value of the function f with argument Socrates is the True, because Socrates is human. The all-important point is that Socrates is not a constituent of any complex entity, state of affairs, fact, or Russellian ''proposition''. Indeed the value of the function is not a ''whole'' composed of function and argument.
 Frege's formal symbol 'f(x)' is a term, not a formula (wff), and this is so even though 'x is human' is a formula. So the formal 'f(x)' does not transcribe the natural language 'x is human'. To generate a wff in Frege's formal notation, one must add the sign '⊢'. Thus '⊢f(s)', where 's' is a constant naming Socrates, is a wff (a sentence) in the notation of the Grundgesetze. Put in a way that reflects the functional notation, '⊢f(s)' says that the value of the function f, with argument Socrates, is the True. But '⊢' is a primitive, and though our reading of '⊢f(s)' makes it appear as if the notion of '⊢' is further analysable, it is not. Accordingly, one may as well read '⊢f(s)' simply as 'Socrates is human'.
 Seeing this, we must take care never to conflate the modem predicational notation 'F(s)' which is a wff, with Frege's functional 'f(s)', which is a term. Indeed, the often repeated claim that for Frege a declarative sentence is in the category of names (terms) is quite misleading. In Frege's terminology, one distinguishes the ''content'' of a sentence from its ''assertion''. The sense of the term 'f(s)' is to be the unasserted content of the sentence 'Socrates is human'. This leads Frege to speak as if sentences are represented in the Grundgesetze as names (terms) and asserted sentences represented by employing the turnstyle. But the point is best put as we have above ''sentences'' are transcribed into Frege's ideography by use of the turnstyle[sic].

  • Gregory Landini  Frege's Notations: What They Are and How They Mean, Palgrave Macmillan, History of Analytic Philosophy Series, 2012, p. 34,

Frege's turnstile sign ''⊢'' is a sign of his object-language. Its syntax is indisputable. It attaches to a well-formed term α (open or closed) to form a wff ''⊢α''.
 This is very important, though it is uniformly missed in the literature on Frege's concept-script. The logical particles[i.e., logical constants, or logical connectives] of Frege's concept-script are quite different from those of a modern predicate language. The logical particles of modern predicate languages are flanked by wffs to form a wff. Frege's signs are flanked by terms to form terms.

 All and only terms can flanked the stroke signs[i.e., horizontal strokes] of Frege's formal language.*3

As we can see, there is unequivocal and definitive evidence that in Grundgesetze wffs are not considered to be terms and that sentences are not considered to be names (closed terms) for truth-values.*4

Landini さんの見解に対し、私はかつて詳しく説明し、簡単ながら検証を行いました。詳細は次をご覧ください。

  • 2012年7月15日、 Entry 'Why Did Frege Need Judgement Strokes and Horizontal Strokes in his Grundgesetze der Arithmetik?'

(なぜ Landini さんが Frege の Grundgesetze の syntax を正しく剔出できたのでしょうか。よくはわかりませんが、Principia の syntax をきちんと突き止めようと苦心された経験が Landini さんにはあったことから、その時に得た経験値を生かして、Grundgesetze でもきちんと syntax を確定しようと冷静かつ客観的に原文を精査されたのだろうと推測します。)

そして、Landini さんと同じ見解を有していると思われる Urquhart さんの文章を引用します。

  • Alasdair Urquhart  ''The Unnameable,'' in Achille C. Varzi and William Seager eds., Truth and Values: Essays for Hans Herzberger, University of Calgary Press, Canadian Journal of Philosophy Supplementary Volume 34, 2011, pp. 125-126,

4. The syntax of Frege's logic

The world of logical entities in Frege's system is dark and obscure, illuminated only by the flash of occasional metaphors. However, this world is the ontological reflection of a world of logical syntax that is clear, explicit, and easily describable. Hence, to form a picture of Frege's entities themselves, we must inevitably start from syntax.
 Frege's logic is essentially a logic of terms; the notion of assertion is concentrated entirely in the judgment stroke (Frege 1893[Grundgesetze, Band I], 1967[its partial English translation], 38). The basic category is that of objects, which we shall denote by the sign O. Terms with no free variables are of type O and correspond to the linguistic category of proper names. Terms with free variables are considered as names of functions. Frege includes in his system certain primitive function names, for example, the name of the identity function 'ξ = ζ', and the name of the conditional function. Both of these terms are terms of the type (O, OO), that is to say, in Frege's terminology, names of first-level functions of two arguments. In other words, they are functions that take two objects as arguments and whose value is an object. Frege also allows functional variables intended to range over first-level and second-level functions. The rules for the formation of terms do not allow a functional variable standing on its own to be a term; if (for example) a first-level variable 'f' occurs in a term, it can only occur in expressions of the form f(t), where is a term of type O (Frege 1993[sic. Correctly, 1893], §21; Frege 1967, 73). Thus the most general type of a term in Frege's logic is

          (μ1, ..., μk, φ1, ..., φl, O, ..., OO),

where μ1, ..., μk is a list of types of second-level functions and φ1, ..., φl is a list of types of first-level functions.

Urquhart さんの論文末尾、参考文献表には Landini さんの文献は上がっていません。Urquhart さんの論文のどこを見ても Landini さんへの言及はありません。ですので、Urquhart さんは独自に上記の引用文にあるような見解に至ったのでしょう。

Frege は文を名前と見なしていたという通説に対する反論者が Landini さんだけだと少し心細いですが、Urquhart さんも反論者に加わって来れば、幾分心細さが和らぎます。

Landini さんも Urquhart さんも、ご両人とも Frege の専門家ではないところが面白いですね。Frege の専門家であれば、Dummett さんの意見に思わず従ってしまうところですが、お二人とも Frege の専門家でなかったためか、虚心坦懐に Grundgesetze を読んで、実際にそこから読み取れるもののみに従ったのかもしれません。

Urquhart さんの上記の論文は短いものながら、とても興味深いです。以前に拝読し、その時、Urquhart さんが、非常に珍しくも、(おそらく) 正しく Grundgesetze の syntax を捉えていることに驚きました。通説に流されず、きちんと正確に Grundgesetze を見ておられることに感嘆致しました。私にとってはこれだけでもすごいと思ってしまいます。

上記の Urquhart さんの論文では Frege が Russell Paradox に足をすくわれた原因が検討されています。詳しい話はできないのですが、足をすくわれた遠因に、対象と概念、対象と関数とは、まったく異なる存在者であるとする Frege の見解があったと Urquhart さんは考えておられます。この Frege の見解に対する疑義としては、the concept horse paradox が彼の存命中に提示されており、Urquhart さんもこの paradox について論文中で言及されています。Urquhart さんがおっしゃるように、Russell Paradox の遠因に、Frege による対象と概念/関数の区別があったとするならば、Urquhart さんは明示的には述べておられませんが、the concept horse paradox が Frege に提起された際、このことは、自身の見解に何か重大な過誤があるかもしれないということに Frege が気が付く数少ない chance だったのではなかろうか、と個人的には何となく思いました。The concept horse is not a concept. このことを、「自然言語が悪いのだ」と言うのではなく、ここで自身の何かがおかしいと感じたならば、Frege は軌道修正できていたかもしれません。その時、どこへ向かって行ったかは、私には全然想像できませんが…。


もう一つ、the concept horse paradox が提起されたのと大体同じ時期に、Frege が過誤に気が付く chance があったかもしれません。それは Cantor による Frege の Grundlagen への書評です。これを Frege が読んだ時、その時にも、もしかすると軌道修正の chance が訪れていたのかもしれませんが、この辺りのことについては、例えば、次を参照。

  • Charles Parsons  ''Postscript to Essay 5 [''Some Remarks on Frege's Conception of Extension''],'' in his From Kant to Husserl: Selected Essays, Harvard University Press, 2012, pp. 133-135.

ただし、私自身としては、この Parsons さんの文章をちらっと読んだだけなので、内容を十分理解しているとは言えないことを、ここに正直に記しておきます。


*1:There is no difference between the quotation from the Duckworth edition and the one from the Harvard UP edition.

*2:Dummett, Frege: Philosophy of Language, p. 196. Again, there is no difference between the quotation from the Duckworth edition and the one from the Harvard UP edition.

*3:Landini, Frege's Notations, p. 34.

*4:Landini, Frege's Notations, p. 36.