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Substitutivity SUBSTITUTIVITY CARLOS EMILIO GARCÍA DUQUE pUNIVERSI hDAD DE CALDAS RESUMEN ABSTRACT En este artículo, procuro hacer un examen In this paper, I attempt a careful examination cuidadoso del principio de sustitutividad, of the principle of substitutivity, sometimes algunas veces llamado ‘la ley de Leibniz’, referred to as ‘Leibniz’ Law’, or ‘Leibniz’ o ‘el principio leibniciano de identidad Principle of Identity of the Indiscernibles’. de los indiscernibles’. Tras ofrecer una After giving a precise and independent formulación precisa e independiente de formulation of these three principles, I take estos tres principios, me ocupo de ciertos issue with several alleged counterexamples supuestos contraejemplos al principio de to the principle of substitutivity and sustitutividad y discuto, por separado, las discuss, separately, the supposed failures of supuestas fallas que se pueden atribuir substitutivity attributable to the paradoxes a las paradojas de la intensionalidad, las of intensionality; difficulties involving the dificultades implícitas en las condiciones truth conditions of modal expressions and de verdad de expresiones modales y los belief contexts. I contend that the alleged contextos epistémicos. Sostengo que los counterexamples hardly account for the supuestos contraejemplos a duras penas meaning Leibniz himself originally ascribed se ajustan al sentido original que el mismo to the principle. Furthermore, I shall argue Leibniz le quiso dar al principio. Además, that a restricted version of Substitutivity argumentaré que una versión restringida may be true, and urge additional attempts de la sustitutividad puede ser verdadera to rescue this principle, under the conviction y recomiendo que se hagan esfuerzos that an appropriate dismissal of the attacks adicionales para salvar el principio, bajo on it, would give us better grounds to la convicción de que el rechazo adecuado understand Leibniz’ claims on identity de los ataques en su contra nos permitiría comprender mejor los planteamientos de Leibniz sobre la identidad. PALABRAS CLAVE KEY WORDS Principio de sustitutividad, ley de Leibniz, The principle of substitutivity, Leibniz, principio leibniciano de identidad de los ‘Law Leibniz’, Principle of Identity of the indiscernibles, salva veritate, intensionalidad, Indiscernibles, salva veritate, intensionality, contextos epistémicos, identidad, identidad belief context, identity, concept-identity. de conceptos. p hRecibido el 12 de Diciembre de 2006 y aprobado el 11 de Abril de 2007. Carlos Emilio García Duque THE PRINCIPLE OF SUBSTITUTIVITY In literature on identity, there is a frequent discussion of a principle sometimes referred to as 'Leibniz' Law', 'Leibniz' Principle of Identity of the Indiscernibles' or 'the Principle of Substitutivity'. Such variety of labels suggests that there must be a close relationship among the purported principles so designated. Letting issues of ambiguity aside, it transpires that the three labels just mentioned do not refer to the same concept. Scholars like Ishiguro, for example, have pointed out that the general understanding of 'Leibniz' law' aims to the Principle of Indiscernibility of Identicals, while the Principle of Substitutivity is a principle which defines identity of concepts. The disagreements about what could be the correct way to understand each of the three principles contrast with the almost universal agreement about the falsity of the Principle of Substitutivity. It is my interest in this paper to determine whether the standard arguments against the Principle of Substitutivity do justice to the meaning Leibniz himself originally ascribed to it. To accomplish this task, I shall examine a formulation of the Principle of Substitutivity and I shall try to spell out the arguments that show its falsity. Then I will argue that a restricted version of this Principle is true, and urge additional attempts to salvage it, under the conviction that, if such a project becomes successful, then one would be in a better position to interpret Leibniz' claims on identity. I To begin with, it will be useful to see what the three principles alluded to are exactly. Unfortunately, Leibniz did not arrive at a singular and definitive formulation of any of the principles (it is even dubious that he regarded them as strictly different principles), since he discussed the matter in several places. However, there are key features in each of them, which enable us to distinguish a formulation from another. In Leibniz’ own words the principles can be stated as follow: (A) “Those are the same of which one can be substituted for the other without loss of truth, such as triangle and trilateral, quadrangle and quadrilateral.” This is what has been called ‘Leibniz’ law’, or the principle of Indiscernibility of Identicals. It says that If A and B are identical then everything that is true of A is true of B, or in a more formal way that: (A = B) → (φ) (φA ≡ φB). 96 Discusiones Filosóficas. Año 8 pNº 11,Enero–Diciembre, 2007. pp. 95 - 108 h SUBSTITUTIVITY (B) “Those terms of which one can be substituted for the other without affecting truth are identical.” This corresponds to the Principle of Identity of Indiscernibles. It can be stated by saying that: If everything that is true of A is true of B, and vice versa, and hence if there is no discernible difference between A and B, then A is identical with B: (φ)(φA ≡ φB) → (A = B). (C) “Two terms are the same if one can be substituted for the other without altering the truth of any statement. If we have A and B and A enters into some true proposition, and the substitutions of B for A wherever it appears, results in a new proposition which is likewise true, and if this can be done for every such proposition, then A and B are said to be the same; and conversely, if A and B are the same, they can be substituted for one another as I have said.”1 This is what is properly called ‘the Principle of Substitutivity’ or ‘the Salva Veritate Principle’. Notice that (B) is the converse of (A), and that slightly different formulations of both (A) and (B) can be found in or derived from (C). On the other hand, many have contended that the three formulations are infected by a confusion of use/mention. What can be substituted for one another are words, not things as Leibniz seems to suggest, and what can be true or false are the propositions expressed by the sentences in which those words figure. A more precise formulation of the notions conveyed by (A) through (C) is offered by Richard Cartwright in his essay “Identity and Substitutivity”.2 I will use his formulation for the remaining of the paper: (D) for all expressions α and β, α = β expresses a true proposition if and only if, for all sentences S and S’, if S’ is like S save for containing and occurrence of β where S contains an occurrence of α, then S expresses a true proposition only if S’ does also. but this is just the conjunction of: 1 The first two formulations come from Leibniz’ Logical Papers (pages, 84 and 52), as quoted by Hide Ishiguro in her book: Leibniz’s Philosophy of Logic and Language. New York, Cornell University Press, 1990. p. 19. The third formulation comes from Richard Cartwright’s paper: “Identity and Substitutivity.” In: Identity and Individuation. Ed., by Milton K. Munitz, NYU Press, 1971. 2 I agree with Cartwright’s main conclusions and my own views are strongly influenced by his arguments. I do not, however, share his evaluation about the success of the counterexamples to the Principle of Substitutivity. p h 97 Carlos Emilio García Duque (E) for all expressions α and β, α = β expresses a true proposition if substitution of β for α is truth preserving with (F) for all expressions α and β, α = β expresses a true proposition only if substitution of β for α is truth preserving. In recent literature, mostly references to (F) are considered as references to “the Principle of Substitutivity”. On the other hand, there appears to be widespread agreement on the charge that (F) is simply false; and the reason to justify such charge is that there are counterexamples to it. The main arguments to this effect come from the paradoxes of intensionality, difficulties involving the truth conditions of modal expressions and failures in belief contexts. In my discussion, I will stick to the standard examples which are already familiar to the reader. Let us tackle one at a time. It has been argued that while the sentence ‘Giorgione was so-called because of his size’ (S1) is true, the sentence ‘Barbarelli was so-called because of his size’ (S2) is false, and yet Giorgione = Barbarelli. Therefore, S1 and S2 appear to be a pair of sentences that falsify (F). For (F) says that if the proposition ‘α = β’ is true, then substitution of β for α should be truth preserving. Under the supposition that there is no discussion regarding the truth of ‘α = β’, and according to (F), we should expect that if S1 and S2 only differ in that S2 contains an occurrence of β, where S1 contains an occurrence of α, then if S1 is true, S2 would be true also, but S2 seems to be false. Let me spell out the core of this argument. Since the Principle of Substitutivity, as expressed in (F), contains no restrictions whatsoever it is supposed to cover all occurrences of all expressions. The name ‘Giorgione’ was attributed to someone because of his size, but the same person was also called ‘Barbarelli’, so while the proposition ‘Barbarelli was called ‘Giorgione’ because of his size’ seems perfectly appropriate, the proposition ‘Barbarelli was called ‘Barbarelli’ because of his size’ seems not. It looks like we have a legitimate counterexample to (F). However, as it has been established by Cartwright, the contention that S2 is false (on the grounds that there is a property that Giorgione has but Barbarelli lacks) seems to be incoherent.
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