Crítica, Revista Hispanoamericana de Filosofía, Volume 5, number 14, May 1971
El análisis de Platón de las relaciones y de los hechos relacionales en el Fedón
Abstract: The author points out that the exegetic tradition has maintained that Plato did not distinguish the relationship from the qualities. However, a close examination of the Phaedo has convinced him that such is an error.
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For Plato a relationship is a sequence of Forms, and a relational fact is an indivisible complex in which each Form is exemplified by an entity. Tradition has not noticed this difference between forms and facts, nor does the ordinary symbolism of the logic of the quantifiers permits one to see clearly this Platonic Notion, since the latter suggests that a relation is an indivisible entity which is exemplified by an ordered set of entities. This normal notation establishes that the relational fact — “Charles gives the book to John” — has the corresponding symbolism: Give (Charles, the book, John). Thus, schematically, this relational fact is represented by D(a,i,u).
In order to have an appropriate symbolism for the analysis of Plato, the author proposes to substitute the relational symbol ‘(D , , )’ by the ‘D1 ( ), D2 ( ), D3 ( )’, with which he succeeds in changing the simplicity of predicate ‘D’ for the complexity of ‘D1-D2-D3. Once this symbolic innovation has been introduced it gives rise to a group of semantic questions which bring along with them a new ontological concept of relation. By this way, going back to the previous example, it comes out that it is possible to represent the act of relationship “Charles gives a book to John” as follows: D1(a), D2(i), D3(u). Besides interpreting the symbol ‘D1( ), D2( ), D3( )’ as a representation of the relation give, it opens up the possibility of interpreting each of the D’s with superscript. Thus ‘D1’ may be interpreted as the condition of giver, ‘D2’ as the condition of the given and ‘D3’ as the condition of receiver. It is necessary to point out that under this notation the unity of the relation “to give” is still maintained and these three conditions are only different ways of describing this relationship.
Now it is clear what questions may be raised by this symbolic modification: What is a condition, in this sense? Is each condition an entity in itself, in such a way that the three conditions which were drawn from the relation “to give” are three distinct and separate entities? Or, are the three conditions simply three ways of conceiving or describing the same and unique give relation “to give”? Is there an ontological analysis of the multiplicity of conditions?
Within the common notion of relations, the three conditions have received an ontological explanation, that is, each condition is conceived as an abstract pair of the relation “to give” and since each one of the three positions which they can occupy in an act of relationship, the entities related by “to give”. Thus is explained the multiplicity, as well as the indivisibility of the relation. For the Platonic conception, the relations are a set of conditions, and these are the Forms of Plato. But the Forms constituting the relations cannot be exemplified independently, but only in conjunction. In this way the relational facts cannot be reduced to qualitative facts and they are distinguished by being facts constituted by a multitude of Forms, but at the same time preserving their unity.
It is immediately shown how this interpretation of Plato’s theory of relations, set forth in the Phaedo, makes possible an explanation of portions of the text which have normally remained obscure, and then proven that this theory is logically correct and therefore constitutes a genuine ontological choice.
In order to prove that this theory is logically satisfactory, it is demonstrated that a proposition which is logically true in the ordinary theory of relations is logically true in the Platonic theory. The proof of this fact is developed as follows: The characteristics of a formal Platonic language are specified, stating that the latter is like an ordinary language of the calculus of quantifiers, except that the primitive predicates are all of grade 1. Thus, there will only be one rule of formation which distinguishes the Platonic language from ordinary formal language. Assuming that the order in which the symbols signifying Forms appear is not relevant, in the formulas introduced by the previous rule, Platonic calculus includes an axiom which permits the commutability of the components. The other axioms are just like the axioms of ordinary calculus. The characteristics of an appropriate semantics for Platonic calculus are basically the same as those of ordinary semantics for atomic formulas, along with the modification brought about by the introductions of what the author calls Platonic relation structures.
Immediately the need is pointed out to prove the consistency and completeness for Platonic calculus, in relation with their corresponding Platonic models. It is simpler, however, to show the equivalency between Platonic calculus and ordinary calculus. Assuming that the difference between them only resides in their treatment of the atomic relational formulas, the problem is reduced to proving equivalency for expressions of this kind.
On the basis of Löwenheim-Skolem theorem the only thing required is the proof of a metatheorem which establishes the necessary and sufficient conditions under which a Platonic relational formula is satisfiable by an enumerable Platonic model, in terms of the satisfiability of its corresponding formula in ordinary calculus. With this metatheorem it is easy to prove that for each logically valid Platonic formula, there is a corresponding logically valid formula in ordinary calculus.
And this fact is sufficient to show that Platonic calculus adequately represents the logical structure of the related facts. Thus the difference between Plato and modern philosophy is exclusively ontological, and leaves open the problem of whether the Platonic solution is metaphysically superior with respect to the ordinary conception.