Logic, Metalogic, Metaphysics

Introduction

To assert that “there is no subject, there is only calculation”¹ is, at the very least, to overlook the subject making that judgment; it is undoubtedly to conflate calculation with the result of calculation through an unwarranted identification, and perhaps to disregard the distinction between law and rule, between language and metalanguage, between logic and metalogic.

Distinguishing Between Law and Rule

Whereas a law operates, a rule establishes the conditions and processes of operation:

Whereas a law is expressed within a calculus, a rule stands outside the calculus over which it presides. The former belongs to the object language, the latter to the metalanguage: it is a metalogical statement that says something about the statements of logic. Rules of commutation, for example, speak of equivalences, whereas the corresponding laws are equivalences; rules mention a connective that laws use.²

A logical law may be defined as “a set of formulas representing a sequence of propositions in a true discourse” (that is, independently of the truth or falsity of the individual propositions), and a logical rule as “an element of metalogic determining which expressions are admissible in logical discourse (formation rules) and which inferences are legitimate (rules of proof)”:

The laws of logical discourse can be transposed into metalogical rules: thus, a logical law of implication may correspond to a rule of proof.³

Distinguishing Between Language and Metalanguage

A language—a logical language—cannot be constructed by means of itself; one must be able to speak about it and explain it. Hence the distinction between object language and metalanguage, or even between object language and observer language⁴, established by modern logic⁵.

Indeed, within a linguistic system, there may be things that it cannot express⁶, or different levels of meaning⁷, and, in particular, a risk of confusion in cases of autonymous usage⁸.

Thus, in the study of modern logic, at least two linguistic systems are distinguished: the symbolic language (object language), and the language in which the logician sets out its properties and explains its structure: the metalanguage, which is in fact a natural language fulfilling this function. If, in linguistics, one wishes to distinguish between non-scientific and scientific metalanguages, the scientific metalanguage must itself be specified, as an object language, by a meta-metalanguage of the natural-language type⁹.

Algirdas Julien Greimas (1917–1992) showed that four levels of language were “necessary to achieve a genuinely scientific semantics, capable of describing any signifying system, whatever form it may take: the object language, the descriptive language, the methodological language, and the epistemological language”¹⁰.

Metalogic or Metaphysics?

To inquire into the foundations of logic leads directly to the “first principles”: non-contradiction¹¹, identity, and the excluded middle¹².

If these principles are regarded as self-evident and as belonging to the very structure of the thinking human mind, the ontological order is introduced; if they are placed within a higher-order logic, namely metalogic, the question is merely deferred. Metaphysics and metalogic thus have the same material object—these first principles—but continue to differ in their formal object¹³: metaphysics studies these first principles insofar as they reflect the structure of being (the ontological order), whereas metalogic studies them insofar as they reflect the structure of thought (the logical order)¹⁴.

If metaphysical questions are inaccessible to logic, this is because, for logic, the true essence of things gives way to an operational essence “whose practical content commands consensus”¹⁵.

Conversely, “the metaphysician may require logic to serve as an effective instrument for his science.” Nevertheless, while “classical logic is particularly well suited to the exposition and resolution of metaphysical problems,” the same cannot be said of modern logic, given its quantitative reductionism; to confuse the second and third degrees of abstraction would be to “commit a serious error”¹⁶.

Yet, since there can be no metaphysics without metalogic through which it can express itself, nor metalogic without metaphysics to provide it with first principles, we are led to consider that “at the origin there is language [natural language] as ‘metalanguage’”¹⁷. It is the inevitable connection between concept and word that “explains why every linguistic theory either begins from a metaphysical a priori or leads to questions of a metaphysical order”¹⁸.

A theological perspective may then, in principle, be unitive or synthetic (as compared with a scientific approach, which is necessarily analytical in the first instance). Following St. Augustine, it would maintain that

natural language is merely the reflection of the eternal Word, creator of both the external and the internal world, according to the formula “causa essendi, ratio intelligendi, ordo vivendi”¹⁹ (De Civitate Dei, VIII, 4). The Word is at the origin of things, just as it is at the origin of the intellect that knows things, and also at the origin of the word (verbum) that expresses things and, in a sense, creates them for others.²⁰

Although “perfectly explanatory,” a theological explanation is not widely acceptable and, quite rightly, is entirely inadmissible in science. Thus Aristotle, while establishing a connection between words, thoughts, and things, regards language as a human “invention”:

Spoken words are merely images of the affections of the soul²¹; and writing is merely an image of the words expressed in speech. Just as writing is not identical among all human beings, neither are languages the same. But the affections of the soul, of which words are the immediate signs, are identical in all human beings, just as the things of which these affections are faithful representations²² are also the same for all.²³

From a more strictly philosophical perspective, it seems to us that we can only acknowledge that principles constitute the very nature of intelligence²⁴.

Indeed, the principles of reason (such as the principle of non-contradiction) possess a self-evidence that becomes apparent only when they are disregarded or when the consequences of denying them are considered. They cannot be demonstrated—for otherwise they would no longer be principles—but constitute an internal requirement of reason, a requirement to which reason assents “through a genuine intellectual intuition, whose necessity is properly ‘irrefutable.’”

These principles are therefore “meta-logical, or meta-rational, insofar as logic and reason designate the order of purely discursive knowledge, that is, knowledge which is purely mediated (and therefore demonstrative).”

They can consequently be grasped in their true meta-logical nature only by philosophy, which transcends logic (without contradicting it). Thus:

• “Logic is nothing other than the set of intellectual operations through which the human mind subordinates itself to principles in its cognitive activity.”
• “Philosophy is not subordinate to principles; […] they think themselves within it and are connatural to it; […] intelligence knows them implicitly in knowing itself.”²⁵

This is why “these natural structures of intelligence cannot be regarded in a Kantian manner, as though they conditioned intelligence a priori. On the contrary, they are perfectly transparent to it; they are intelligence itself, they are the Logos.”

For whatever belongs to the order of intelligence is necessarily meaning, Logos. Intelligence cannot speak of unintelligible principles which it would obey without understanding them, nor can it judge that its understanding of them would be purely illusory.

Principles are indeed the reflection of the structures of the intellect in objective knowledge, we grant this; but these principles are also those of the Logos in itself, necessary and purely intelligible, for this is the only proposition that intelligence can intelligibly maintain—which implies that there can be no essential heterogeneity between our intelligence and the Logos.²⁶

Footnotes

1. Alain Chauve, “La logique et la vérité,” Philopsis, January 24, 2010 (online), p. 30.[↩]
2. Robert Blanché (1898–1975), Introduction à la logique contemporaine, Paris: A. Colin, 1957; cited by F. Chenique, Comprendre la logique moderne, Paris: Dunod, 1974, p. 1.[↩]
3. Chenique, ibidem.[↩]
4. “We regard our study of this language and its logic [object language], including our use of logic in that study, as belonging to another language, which we call the observer language. We may also speak of object logic and observer logic,” Stephen Cole Kleene (1909–1994), Logique mathématique (Mathematical Logic, 1967), trans. J. Largeault, Paris: Armand Colin, 1971, p. 11.[↩]
5. Beginning with Bertrand Russell’s Theory of Types.[↩]
6. “In my introduction to the Tractatus, I suggested that, although in any language there are things which it cannot express, it is always possible to construct a language of a higher order in which these things can be said. There will still be things which cannot be said in the new language, but which can be expressed in the next language, and so on ad infinitum. This suggestion, which was then new, has since become a commonplace of logic”; Bertrand Russell, Histoire de mes idées philosophiques, 1959, trans. G. Auclair, Paris: Gallimard, 1961, p. 141.[↩]
7. See the paradoxes of the Barber, the Librarian, Brouwer, etc.[↩]
8. A word refers to itself as a word: in “‘rose’ has four letters,” it is the word, not the flower, that is being referred to; hence the use of quotation marks in such cases, in French as in English.[↩]
9. One cannot escape the level of natural language, which gives an intuitive operational meaning to the meaning established by any given axiom; Chenique, … Logique moderne, p. 7. Algirdas Greimas (1917–1992) considers it impossible in practice—and perhaps even in principle—to describe the vast semantic universes encompassed by natural languages, and maintains that only closed systems, or “semantic micro-universes,” can be studied; Sémantique structurale. Recherche de méthode, Paris: Larousse, 1966, p. 141.[↩]
10. Greimas, op. cit., 1966, pp. 16–22.[↩]
11. The principle of non-contradiction is the first in the logical order, grounding the others, which derive from it their certainty of truth.[↩]
12. Without excluding the principle of the excluded middle, the notion of the included middle is one of the foundations of Stéphane Lupasco’s (1900–1988) Dynamic Logic of the Contradictory, a general logic encompassing classical logic and a logic of “energy” intended to account for the Three Matters (Paris: Julliard, 1960): macrophysical matter-energy, living matter, and psychic matter. Applied to dynamic processes, through the notions of potentialization (in the sense of potential in physics or biology) and the corresponding antithetical actualization, the principle of non-contradiction takes on a new aspect. Although this principle may have been weakened in the so-called Dutch (Brouwer) or Polish (Łukasiewicz, Tarski, etc.) schools, whose logics introduce intermediate values between true and false, these logics remain static; Lupasco’s contradictory potentialization-actualization is intrinsically dynamic and, perhaps one might put it this way, constitutes an “expanded” logic: ontologico-logical or logico-ontological.[↩]
13. On the notions of material and formal objects, see Métaphysique du paradoxe, Vol. 1, Appendix 2, pp. 239–240.[↩]
14. Chenique, … Logique classique, op. cit., pp. 107–108.[↩]
15. The designation “operational essence” is proposed by François Chenique in connection with the notion of an “operational concept,” which is necessary and sufficient for logic; unlike the classical concept derived from Aristotle’s philosophy, it is simply “a general idea accepted by everyone, whatever its deeper nature may be,” … Logique classique, pp. 58–59.[↩]
16. Ibid., p. 58, n. 4.[↩]
17. Ibid., p. 108.[↩]
18. Ibid., p. 60. Particularly in the human sciences, the difficulty of completely separating the various domains—logic being compelled to use natural language and spontaneous logic to expound its logical system, while philosophy necessarily relies on logic, even if only spontaneous logic, to express itself—is illustrated by antithetical doctrines: for example, traditionalism (which regards ideas and words as a primordial revelation) and structuralism, which is vigorously anti-metaphysical (ibid.).[↩]
19. “Sicut Deus est causa essendi, ita est ratio intelligendi et ordo vivendi,” St. Bonaventure (1221?–1274), De Scientia Christi, q. 4, fund. 24 – V, 19b. Cf. St. Bonaventure, Questions disputées sur le savoir chez le Christ, trans. H. Wéber, Paris: O.E.I.L., 1985.[↩]
20. Ibid., p. 108.[↩]
21. “To describe words as images of the affections of the soul is to attribute an entirely human origin to language. Greek antiquity never resolved this question otherwise, nor did it seek to explore it in greater depth,” comments Barthélemy Saint-Hilaire.[↩]
22. In his introduction to his translation of Thomas Aquinas’s Commentary on Aristotle’s Posterior Analytics, op. cit., Guy-François Delaporte notes that “representation” does not adequately render the Latin similitudo, which he relates to semel (“once”) and which must be understood in reference to Aquinas’s three types of unity: identity (substantial unity), equality (quantitative unity), and similitude (qualitative unity), namely, a single quality present in two distinct subjects; pp. 22–23.[↩]
23. Hermeneia, or On Interpretation (De Interpretatione), ch. I, §§ 2–3 (16a), trans. Barthélemy Saint-Hilaire, Logique d’Aristote, Paris: Ladrange, 1844, Vol. I, p. 147.[↩]
24. Here we draw on certain elements presented in our Jean Borella, La Révolution métaphysique…, op. cit., Part IV, Introduction, Article 3, pp. 277–278.[↩]
25. Jean Borella, La crise du symbolisme religieux, op. cit., p. 281.[↩]
26. Ibidem, pp. 281–283.[↩]