Sobre la eliminación de los contextos oblicuos

Abstract

(1) George IV was ignorant that Scott = the author of Waverley
(2) Scott = the author of Waverley
(3) George IV was ignorant that Scott = Scott
Seemingly, (1) attributes to the author of Waverley the property expressed by the predicate
(4) George IV was ignorant that Scott = ..., where the suspending points are for possible arguments. That same property is attributed in (3) to Scott, who, since (2) is true, is not other than the author of Waverley. (1) and (3) attribute then the same property to the same individual. But if (1) is true, (3) is obviously false. An analytical consequence of the notion of identity has thus been placed into doubt: The Principle of the Identity of Indiscernibles:
(5) (x = y) ⊃ (F) (Fx ≡ Fy)
II. This paradox can be traced without recurrence to particular examples whose truth or falsity is in principle capable of discussion. There are an infinite number of true sentences of the form “Scott = (ηx) Fx.” George IV must have been ignorant of some of them. Thus we can affirm the existence of two true sentences corresponding to the schemas:
(1) George IV was ignorant that Scott = (ηx) Fx,
(2) Scott = (ηx) Fx,
sentences that, in conjunction with the falsity of (3), generate a paradoxical counterexample of the Principle of the Identity of Indiscernibles.
III. The Principle of the Identity of Indiscernibles may be saved by means of the distinction made by Frege between ordinary and oblique uses of terms: in the oblique, names denote the meaning that they ordinarily express (their direct meaning). The subordinate sentence “Scott = the author of Waverley” does not denote in (1) a truth value, but the meaning as ordinarily expressed.
The doctrine of Frege-Church contains two separable elements which we may term diagnostic and remedial. The diagnostic explains the cause of the paradox: (i) “Scott” and “the author of Waverly” do not have the same direct meaning; (ii) both names appear in (1) with an oblique denotation; and therefore, (iii) they denote different entities there. The remedial consists in the construction of a language which would not be oblique. This is obtained by introducing names that would denote the meaning which others express. In this language would be the unrestricted validity of the law of Leibniz: If two names denote the same individual, they are interchangeable salva veritate.
IV. M. Furth has indicated how to construct a language without obliquity. He introduces the following notation for naming the ordinary meaning of Scott: “[Scott]l”. Thus, the name “[the author of Waverley]1” denotes the ordinary meaning of the “author of Waverley.” And according to Furth, to omit its obliquity, (l) would have to be reformulated by means of the use of “[Scott]l” and “[the author of Waverley]1” and also, by finding a predicate that allows us to affirm between [Scott]l and [the author of Waverley]1, the same relation that (1) establishes between the direct meanings of “Scott” and “the author of Waverley.” This predicate would state:
(8) George IV was ignorant that [ ]1 determines the same individual that [ ]l
Thus (1) would have to be reformulated as
(9) George IV was ignorant that [Scott]l determines the same individual that [the author of Waverley]l.
This transformation concurs with the idea that the predicate (4) does not express a property of individuals, but a property of meanings, as expressed by the predicate.
(10) George IV was ignorant [Scott]1 determines the same individual that [ ]l.
It would seem, then, that (10) allows us to respond directly to the problem of the Principle of the Identity of Indiscernibles, showing that due to the cause of obliquity, we have confused a property of meanings with that of a property of individuals, which can be avoided by the new notation.
But this procedure is not acceptable: the sentence “[Scott]1 determines the same individual that [the author of Waverley]1” normally denotes a truth value, but in (9) it appears as a name of a proposition. “[Scott]1” and “[the author of Waverley]1” are used in an obliquitous form.
It will be attempted in this paper to effectively eliminate the obliquity, but not in all of its contexts. The analysis that follows is limited to cases of obliquity relative to contexts governed by verbs of propositional attitudes and modal operators which do not contain variables bound by external quantifiers.
M. Furth does not mention the necessity of introducing the new name, “[= ]1” for denoting the direct meaning of the sign of identity. But the diadic predicate “=” as well appears in (1) with an oblique denotation. This supposition allows for the establishment of an intelligible relationship between the obliquity of the subordinate sentence and the obliquity of its component names. Within this oblique context, a subordinate sentence is not the simple name of an entity, but a combination of names that denote, by isomorphic relation, a combination of meanings.
V. As it is done in Syntax with structural descriptions, let us construct the name of a proposition by means of the simple ordered juxtaposition of the names of its component meanings. Thus, from (1) we obtain:
(11) George IV was ignorant ([Scott]l [=]l [the author of Waverley]l), where the obliquitous “that” has disappeared.
The correspondent predicate for (4) is
(12) George IV was ignorant ([Scott]1 [=]1 [ ]1).
Analogically, the binary predicate that we need is
(13) George IV was ignorant ([ ]1 [=]1 [ ]1);
(13) allows us to affirm between ([Scott]1 and [the author of Waverley]1 the same relationship established in (1). All sentences whose obliquity we wish to suppress must be interpretable as a combination of names. At times this requires a re-interpretation of its logical form. Thus, to pass from
(14) John believes that all men are mortal,
to
(15) John believes ([man]1 [⊂]1 [mortal]1), it is necessary to interpret the expression “all ... are ... “ as a diadic predicate applicable to the entities denoted by “man” and “mortal”, which can be represented by means of the sign “⊂”.
VI. In order to accept (11) and (15), it is necessary to first make explicit a rule that allows the passage from a structural name to a declarative sentence and vice versa. It is enough to stipulate, for example, that the components of the structural description must be ordered in the same way that the corresponding names in the original sentence. This problem must be separated from the “unity of the proposition” which searches for the difference between a proposition and a mere list of meanings. It is enough here that the structural description denotes univocally the proposition expressed by a declarative sentence.
VII. There remains to examine the objection that would state: It is impossible to suppress the obliquity, because any term preceded by modal operators or by verbs of propositional attitudes will automatically have an obliquitous denotation.
We said that the relative “that” allows for the formation of the name of a proposition from the name of a truth value. Thus, in the case of (14), it is necessary to have a name for the proposition which is believed. That name is “that all men are mortal”. But if to this name we attribute an obliquitous denotation, we cannot then express with it a relation of belief. The meaning of “that all men are mortal” is perhaps, following the suggestion of Church, “a certain description of a proposition by its structure and its constituents”. In consequence, if “that all men are mortal” has an obliquitous denotation in (14), then it does not denote a proposition, and either it is false, for only propositions are believed, or rather, it lacks meaning.
It is not simply the word “belief” which produces the obliquity, but it is the necessity that the second term of the relation of belief would be of an adequate type.
VIII. It could be replied that the obliquity of “([man]1 [⊂]1 [mortal]1)” in (15) reflects itself in the fact that it can restate the problem of George IV. Let us suppose that in a formalized language it is valid that:
(16) Z= [man]1
but “Z” and “[man]1” express different meanings. Then the law of Leibniz would permit us to transform (15) into
(17) John believes (Z [⊂]1 [mortal)1).
It seems that the truth of (15) is not incompatible with the falsity of (17), because John can ignore (16). But this is false. The fact that John accepts (15) and rejects (17) would only prove he does not know that “(Z [⊂]1 [mortal]1)” denotes the proposition that he had admitted to believe upon his acceptance of (15).
The situation here is opposed to that exemplified by (1) and (3). In those, the names “that Scott = the author of Waverley” and “that Scott = Scott” denote different propositions. So that the situation would be analogous, it would be necessary that “Scott” and “the author of Waverley” express the same direct meaning, in which case “that Scott = the author of Waverley” and, “that Scott = Scott”, would denote the same proposition. This shows a fact that is inferred from the premises of the doctrine: where obliquitous language requires a strict synonymy, formalized language, although it maintains the modal operators and the verbs of propositional attitudes, will be satisfied by mere extensional identity.