Page 271 - Encyclopedia of Philosophy of Language

In informal mathematics, to have shown of an arbi- trary triangle that its interior angles add up to 180° is to establish that all triangles have interior angles adding up to 180°. This was once commonly held to show that in addition to individual triangles, there are 'arbitrary' triangles. In traditional logic, there was a time when the grammatical similarity between sen- tences like 'John owns a donkey' and 'every farmer owns a donkey' was taken to show that the phrase 'every farmer' denotes an entity called the 'arbitrary farmer.' This view entailed that, in addition to indi- vidual objects, there are arbitrary objects. By the prin- ciple of 'generic attribution,' an arbitrary object has those properties common to the individual objects in its range. (For an overview of the history of arbitrary objects see Barth 1974.)
1. Generic Attribution
The notion of an arbitrary object has fallen into total disrepute, because of fundamental problems con- cerning the principle of generic attribution. In its informal formulation, the principle of generic attri- bution leads in a straightforward way to con- tradictions for 'complex properties.' Take an arbitrary triangle. Then it is oblique or right-angled, since each individual triangle is either right-angled or oblique. But it is not oblique, since some individual triangle is not oblique; and it is not right-angled since some individual triangle is not right-angled. Therefore it is oblique or right-angled, yet not right-angled and not oblique. A contradiction. These problems have brought many logicians to the conclusion that arbi- trary objects belong to the 'dark ages of logic' (Lewis
1972).
2. Fine'sTheory
In a series of articles resulting in the book Reasoning with Arbitrary Objects, K. Fine (1985b) has set out to reinstate arbitrary objects, by formulating a coherent account of the principle of generic attribution, and constructing formal models for interpreting languages with constants denoting arbitrary objects. Fine argues
convincingly that there are various areas of research where the introduction of arbitrary objects is well motivated. In linguistics, for instance, there are cases where reference to arbitrary objects seems most natu- ral. Consider the following text:
Every farmer owns a donkey. He beats it regularly. (1)
In Discourse Representation Theory, pronouns are taken to refer to objects that have in some sense been introduced by the previous text. But admittingonly individuals in our ontology, this leads to problems; for to what individualfarmer and individual donkey can the pronouns 'he' and 'it' be said to refer? It seems natural to have them refer respectively to the arbitrary farmer and the arbitrary donkey he owns.
The heart of Fine's theory of arbitrary objects con- sists of a reformulation of the principle of generic attribution. According to Fine, the argument showing that the notion of an arbitrary object leads to con- tradictions for complex properties, depends upon the failure to distinguish two basically different for- mulations of this principle: one is merely a rule of equivalence and is stated in the material mode; the other is a rule of truth and is stated in the formal mode.
To formulate the two versions of the principle, let a be the name of an arbitrary object a; and let i be a variable that ranges over the individuals in the range of a. The equivalence formulation of the principle of generic attribution then takes the form: 0(a) = V/</>(/). Given this formulation, contradictions can be derived. For, let R(i) (O(i)) be the statement that triangle / is rectangle (oblique). Because V/(/?(z) v 0(/)) it follows that R(a) v O(a) for arbitrary triangle a. But because —tfiR(i) and —tfiO(i) it follows that —Jt(a) A —,0(0), and we have arrived at a contradiction. The truth formulation of the principle takes the form: Thesen- tence </>(a) is true if the sentence V/$(/) is true. In this version of the principle, the argumentation leading to a contradiction is blocked, and a coherent formulation is reached. But there is a price to pay; in general, formulas containing names for arbitrary objects don't decompose truth-functionally.
SECTION VI Language and Logic
Arbitrary Objects W. P. M. Meyer Viol
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