Logical Truth and Meaning

Abstract

W.V.O. Quine has considered a certain concept as illegitimate because for its characterization it is necessary to use the notion of synonymy. This paper tries to show that if one accepts this objection, one equally has to reject the concept of logical truth since in its definition the notion of synonymy also has to be employed. Afterwards some difficulties and conclusions derived from the use of the notion of synonymy in the characterization of ‘logical truth’ are also analyzed.
1. A classical criticism of the notion of analyticity
In ‘Two Dogmas of Empiricism’ Quine rejects the distinction between analytic and synthetic statements backing his attitude in a series of objections he formulates against the notion of analyticity. Here we are specially interested in singling out one of the characterizations criticized by Quine. According to it, an analytical statement is either a logical truth (analytical statement of first class) or else it is a statement which can become a logical truth by substituting synonyms for synonyms (analytical statement of second class). Quine’s criticism consists of pointing out that in the above definition the concept of synonymy, which is as obscure as that of analyticity, is made use of. Quine does not believe that the concept of synonymy can be characterized in a clear and independent way; hence he thinks that the concept of synonymy as well as that of analyticity (as just defined) both have to be taken as illegitimate. The synonymy here discussed is cognitive synonymy (identity of cognitive meaning), and Quine points out that this is the concept relevant to the discussion. Besides this it can be observed that this synonymy is one between terms and not between statements (this synonymy we will call ‘global synonymy’). But the kind of synonymy considered is not essential to the discussion since Quine takes both of them to be suspect. In fact the notions of analyticity, synonymy between terms and global synonymy are all suspect for Quine and, as he shows in his paper, interdefinable. Hence if one could give a clear characterization of any of them, all three would get the status of legitimate concepts.
To conclude this section, it is well to remember the core of Quine’s objection considered here: if in characterizing a concept one makes an essential use of the notion of synonymy (either between terms or global), such a concept cannot be taken as legitimate.
2. The notion of logical truth
We will now give a characterization of logical truth as it is usually expressed in logic texts: it is said that a statement is a logical truth if and only if it has at least one logical form such that all of its substitution instances are true.
In this definition we are essentially using five notions: statement, logical form, to have (a logical form), to be a substitution instance (of a logical form) and true.
The third and fourth notions are basically identical, hence we will study in what follows just the notions of logical form and to be a substitution instance (of a logical form), since for our purposes we can assume that the first and last notions are reasonably clear.
3. Logical Form
For reasons of space we will treat this subject in a very schematic way leaving aside several problems belonging to it. In what follows, by ‘logical form’ we will only mean that belonging to statements, leaving aside, e.g., that of arguments. It can be said that logical forms are certain symbolic schemes consisting of logical signs we have, e.g., ‘˙’, ‘v’, ‘⊃’, etc., which are used to stand for certain logical expressions of ordinary language like ‘and’, ‘or’, ‘if…then’, etc. These logical signs have an unambigous meaning as opposed to the ambiguity found in every-day speech. Schematic letters are constituents of logical symbolism which do not have a fixed meaning but do have, associated with different types of them, certain linguistic categories, like statement, proper name, monadic predicate, etc. as examples, ‘p’, ‘q’, ‘r’, are associated with the category of statement, ’F’, ‘G’, ‘H’, with that of monadic predicate. Auxiliary signs are those constituents of logical symbolism used as punctuation marks (parentheses would be examples of these).
We can state now the following definition: a logical form is an expression built up only by means of logical signs, schematic letters and auxiliary signs and such that it becomes a statement if the schematic letters which occur in it are substituted by linguistic expressions of the proper semantic categories.
4. Substitution instances of a logical form
We are going to distinguish between two types of substitution instances, i.e., direct and semidirect substitution instances.
We will say that p is a direct instance of F (where F is a logical form) if p is obtained from F by a replacement of expressions of the associated categories by schematic letters in F, where any occurrence of the same schematic letter in F is replaced by the same associated expressions. In the general case, direct substitution instances of a form F are not statements in ordinary language, since logical signs (if any) which occur in F are not substituted at all.
We will say that p is a semidirect substitution instance of F if p is a direct substitution instance of F and the logical signs in F have been replaced by ordinary language expressions which in that context have the same sense. At this point it can be thought that we are using a certain non-problematic notion of synonymy, that which holds between two expressions because one of them is explicitly introduced as an abbreviation of the other. But this is not so since, e.g., ‘˙’ is not used as just an abbreviation of ‘and’; ‘˙’ gets a precise definition by means of truth tables within logical symbolism. And once a meaning has been assigned to it by these means one cannot stipulate that it is an abbreviation of an expression which has not been defined this way — such a stipulation would be a new assignation of meaning. Nor can one argue that ‘and’ is an abbreviation of ‘˙’; so if there is any synonymy between ‘and’ and ‘˙’ this is so not because of any stipulation, but because the truth table associated to ‘˙’ constitutes an adequate clarification of a certain usual sense of ‘and’. Hence we can now conclude this point by saying that in an adequate definition of ‘semidirect substitution instance’ one has to employ a notion of synonymy of the type considered problematic by Quine.
The direct and semidirect substitution instances of a certain logical form F do not exhaust the set of its substitution instances; as an example, we can consider the following statement:
(1) John and Peter are Argentinian
which can become a semidirect substitution instance of the form ‘p ˙ q’ only by paraphrasing (1) into, say
(1′) John is Argentinian and Peter is Argentinian.
But to do this, we must have a notion of “appropriate reformulation”; following Quine, we will call “paraphrase of p” a reformulation of p in which appropriate logical notation is made use of. Now we have to ask for the conditions which have to be satisfied by p′ for it to be an adequate paraphrase of p. This we will do in the following section; for now we will offer a general definition of substitution instance assuming we know what an adequate paraphrase is:
p is a substitution instance of F = df
(i) p is a direct substitution instance of F, or
(ii) there is a p′ such that p′ is an adequate paraphrase of p and p′ is a direct substitution instance of F.
5. The requirements for the paraphrase. Synonymy
There are many difficulties present in giving the requirements which a statement p′ must fulfil to be an adequate paraphrase of a statement p. Some logicians (among them Copi and Quine) have proposed criteria which are implicitly used for this purpose and they seem to point out that the way to find an adequate paraphrase of p is, first of all, to understand the meaning of p and then find a statement written in logical notation which conveys that same meaning. We might then conclude that a requirement which p′ ought to satisfy to be an adequate paraphrase of p is that of being synonymous to p. This leads us to the following criterion of adequate paraphrase:
p′ is an adequate paraphrase of p iff:
(i) p′ is a direct substitution instance of a logical form which contains special logical signs
(ii) p′ is (cognitively) synonymous to p.
We have reached this criterion by noting that the only general norm which in practice seems to guide the search for an adequate paraphrase is the vague indication of finding a statement synonymous to the given one, but written in logical notation. Now I will present another argument independent of the above reasons, directed to show that it is convenient to take synonymy as a necessary requirement for an adequate paraphrase.
Let us consider the following statement:
(E) Every logical truth is true.
(E) seems to convey something implicit in the very same concept of logical truth. Hence we can say that an adequate characterization of the concept of logical truth has to have (E) as a consequence.
Let us call this last statement (C), and we will take it as a minimal adequacy criterion which any acceptable definition of logical truth has to satisfy if this concept is understood in the usual way. Naturally, (C) is taken as a necessary but not as a sufficient condition of adequacy.
Now let us consider q, a statement which is a substitution instance of F, a logical form true under any interpretation, and let q be neither a direct nor a semidirect substitution instance of F. Now we must remember that for this to be the case (cf. Section 4), there has to be a statement q′ an adequate paraphrase of q which is also a direct substitution instance of F. From what we have said, q is a logical truth (cf. Section 2). How, then, are we to transfer our confidence in the truth of q′ to q? If q and q′ were synonymous, there would be no problem at all.
We usually accept that q is true because a paraphrase of it is true; when investigating if a statement is a logical truth or not, we accept paraphrases of it which, in our judgment, say the same thing. However, if we do not require the adequate paraphrase q′ to be synonymous to q, then it seems that we cannot guarantee that every logical truth is true.
Naturally if there were a requirement other than synonymy which could guarantee the truth of a statement when its adequate paraphrase is true, we could drop the requirement of synonymy. It is difficult, however, to see what this requirement would be. We cannot appeal to logical equivalence; this in itself is a case of logical truth and therefore cannot be used in the definition of adequate paraphrase, since we would create a vicious circle.
From what we have said, we have the following three alternatives from which to choose:
(1) There is no guarantee that every logical truth is true.
(2) There is always a cognitive synonymy relation between a statement p and its adequate paraphrase p′.
(3) There is always, between p and its adequate paraphrase p′, a relation R which is different from cognitive synonymy and logical equivalence and which guarantees that if p′ is true, then p is also true.
(1), however, violates our criterion (C) above, and we know of no relation such as the one described in (3). Besides, we have no assurance that if such an R were found, this relation would be devoid of all the problems attached to the cognitive synonymy requirement. For these reasons, we have chosen the alternative (2). We have noted before that the definition of adequate paraphrase has a bearing on the definition of logical truth, if our proposal is accepted, the concept of synonymy is involved in the characterization of that of logical truth.
6. Possible difficulties with the proposed approach
6.1. The first difficulty brought about by the use of synonymy in the definition of logical truth is the one based on the above-mentioned objection by Quine: we lack definitions and criteria of synonymy which are sufficiently clear. Besides, the notion of synonymy is terribly vague and there are many cases in which it is unclear whether the relation holds or not.
An answer to this objection can be given on the following counts:
(a) There are pragmatic reasons to retain the concept. If a concept has some clear cases of application and is useful in some scientific or philosophical contexts, it may be worthwhile to retain it even if it is vague and lacks a precise definition.
(b) It can be pointed out that the type of synonymy which links a statement p with its adequate paraphrase p′ is fortunately clearer than other types of synonymy.
(c) It can be said that any time the synonymy between p and p′ is completely contextual, it is a relation which makes two expressions interchangeable in a given context (without any alteration in its global meaning), in spite of the fact that those expressions do not act in the same way in any other context.
(d) Finally it should be added that the notion of global synonymy does not entirely lack a general criterion of application.
From what we have said, we do not mean to sidestep the difficulties of application that the concept of synonymy presents; we would like, rather, to single out some of the reasons why the status of concept of synonymy is not so despairing nor its applications so difficult as to banish it from our semantic vocabulary.
6.2. The criterion of adequate paraphrase offered before might present some difficulties with respect to logics with stronger ontological commitments than first-order functional calculus. We might start, say, from the statement ‘All men are mortal’ and obtain as its paraphrase the statement ‘^x Hx  ^x Mx’, this statement being interpreted as one about two classes, ^x Hx and ^x Mx. It can, however, be doubted whether the original statement says anything about classes: it would then be reasonable to conclude that the given paraphrase is not synonymous to the original statement. But since the concept of logical truth is mostly used in relation to first-order logic, we will not go into too many details about these problems.
6.3. Up to this point, we have discussed mainly whether synonymy is a necessary condition for an adequate paraphrase. We have said nothing about its also being a sufficient condition, but I do not think it is so, because in this case there would be no difference between first and second class analytical statements (cf. Section 1). That is, ‘logical truth’ and ‘analytical statement’ become synonymous, if the former expression is defined as in the present paper and the latter as in Quine’s paper. In any case, we will not pursue this point since our aim in this paper is to show that synonymy is a necessary condition for an adequate paraphrase and that it has to be used in the definition of ‘logical truth’.
7. Some consequences of the previous analysis
We will conclude this paper by indicating some methodological and epistemological consequences of our previous analysis.
7.1. Logical form and meaning. It is sometimes thought that considerations about the meaning of a statement are not required for an analysis of its logical form. But if our previous analysis is correct, we have shown that this last statement is partly wrong since we do have to make some analysis of the meaning of a statement to determine what its logical form is.
7.2. Criteria of admissibility of semantic concepts. In this paper I have tried to show that Quine’s criteria for admitting semantic concepts as legitimate are too restrictive and, as a result, they eliminate very important logical and semantic concepts such as that of logical truth as it is normally used.
7.3. Paraphrases and logical hypotheses. Gregorio Klimovsky has argued that there is no methodological gap between logic and the other sciences and so he thinks that (interpreted) logical systems are hypothetical-deductive systems. I want to show that the argument in this paper does also suggest the presence of hypotheses in the heart of logic.
If my previous analysis is adequate, the statements which show the way to paraphrase certain linguistic uses state relations of synonymy. It is possible to be mistaken about synonymies which are not the result of explicit stipulations; for this reason statement of the last type mentioned turn out to be true hypotheses about ordinary language. This point is important, since when the logician determines that a given statement is a logical truth, he backs up his argument not only with the principles and methods at his disposal to ascertain if a given logical form is a logical law, but also with some assumptions which allow him to paraphrase the statement in such a way that it shows certain logical form. Hence, he could erroneously attribute the character of logical truth to a statement which is not so, if his hypotheses about paraphrase are wrong, even if the principles he uses to analyze logical forms were infallible.

(José A. Robles)