Sobre o 'princípio de composicionalidade'

Extraído do capítulo 5 de Thinking about Logic de Stephen Read, OUP


Como a linguagem é possível? Como é possível, a partir da aprendizagem
de um vocabulário básico e finito de uma linguagem, formar um sem
número de novos enunciados, novas proposições que expressam
pensamentos que nunca foram antes formulados? Pois isso é possível.
Apesar do vocabulário de uma linguagem ser mundo grande, como uma
olhada no dicionário revela, ele é pequeno quando comparado com o
número imenso de sentenças que compõem os livros das bibliotecas
espalhadas pelo mundo. Dentre essas sentenças, poucas são idênticas.
Dentre as sentenças que lemos, poucas são as que nós vimos antes. Como
é possível que o leitor compreenda essas sentenças? Como é possível
que o leitor as conceba e formule?

A resposta é óbvia, mas suas implicações são poderosas. Nós podemos
aprender uma linguagem porque seu vocabulário e suas regras
gramaticais são relativamente pequenas – ambas podem ser reunidas em
um pequeno número de volumes. Um dicionário de alguns volumes como o
Oxford English Dictionary contém pouco mais que o vocabulário da maior
parte dos falantes individuais do inglês. E mesmo esse dicionário
consiste de dez ou doze volumes, o que é uma pequena parte da
biblioteca onde ele está. A partir desse vocabulário, as regras
gramaticais permitem a criação de um número infinitamente grande de
sentenças. Para que possamos entender tais sentenças, os significados
das palavras individuais são combinados de acordo com a estrutura
estabelecida pela gramática. Em outras palavras, da mesma forma que
uma sentença é literalmente composta pelas palavras que ela contém, o
significado de uma sentença, a proposição, é de alguma maneira
'composto' pelos significados das palavras que ela contém. A idéia
inicial é óbvia: entendemos novas sentenças porque entendemos como
seus significados resultam dos significados das palavras que as
constituem. As implicações disso não são óbvias, e o que isso diz não
é tão claro: os significados das palavras se combinam de algum modo
para compor o significado da sentença completa, a proposição por ela
expressada.

O princípio em questão aqui é por vezes chamado de 'princípio da
composicionalidade', outras vezes de 'princípio de Frege', o grande
filósofo alemão da matemática e da linguagem do fim do século XIX. Os
dois termos cobrem aplicações bastante diferentes da idéia básica. Mas
a motivação subjacente é a mesma. De algum modo precisamos explicar a
'criatividade' da linguagem, o modo pelo qual uma criança, ao ouvir um
número finito e pequeno de enunciados, desenvolve a habilidade de
produzir e compreender um sem número de proposições que não estão
entre os dados sobre os quais tal habilidade foi desenvolvida. A
explicação é a mais simples e a mais plausível para preencher essa
lacuna, e está de acordo com a experiência pessoal, do falante de uma
linguagem, de participar de uma conversa – um conjunto de enunciados
seus e de outros falantes. Os dados iniciais e os novos enunciados
produzidos são analisados em componentes significativos, e é postulada
uma conexão entre o todo e as partes. Mas o que é essa conexão?

Aqueles que chamam essa idéia de princípio de 'composicionalidade'
estarão inclinados a interpretar essa conexão de modo bastante
literal. Eu mencionei no capítulo 1 como Russell considerou que as
proposições – significados das sentenças e objetos de crença – teriam
como constituintes particulares e universais. Assim, por exemplo, a
proposição que Sócrates é sábio teria, literalmente, Sócrates e a
sabedoria como constituintes. Para Russell, o significado de
'Sócrates' era o próprio filósofo Sócrates, em pessoa; e o significado
de 'é sábio' era o universal ou a propriedade sabedoria. Portanto, o
significado da sentença 'Sócrates é sábio' seria composto por Sócrates
e a sabedoria, do mesmo modo que a sentença é composta por sujeito e
predicado. Uma visão mais sofisticada, diferentemente, aponta para uma
dependência funcional do significado da expressão complexa em relação
aos significados das suas partes. Considere uma analogia: 4 é o
resultado do quadrado de dois, 4 = 22, mas 4 não contém literalmente o
número 2 como um constituinte, tampouco contém a função y = x2, que
recebe um número x e o eleva ao quadrado produzindo um número y.
Antes, 4 é o resultado de aplicar ao número 2 a função que eleva um
número ao quadrado. Para Frege, é desse modo que se estabelece a
conexão entre o significado de uma sentença e os significados das suas
partes. O quadro é mais complicado porque Frege distinguia o
significado da expressão dos seus componentes. Mas o princípio é
preservado: o significado de uma expressão complexa, uma sentença por
exemplo, resulta dos significados das suas partes e pode ser calculado
a partir deles. Assim, a compreensão das partes e do modo elo qual o
todo depende das partes explica a compreensão do todo.


* * *

Sobre a natureza da lógica

In this section we want to show that a formal system with a normative character designed to deal with epistemological contradictions finds its place in the very nature of logic. We start by calling attention to the fact that there is a perennial philosophical question about the nature of logic, namely, whether the main character of logic is epistemological, ontological, or linguistic. We emphasize the epistemological character of intuitionistic logic, which is in clear opposition to the realist (and, we claim, ontological) view of logic found in Frege’s works. We argued in section 2 that at least some contradictions that appear in scientific theories are epistemological. Our main argument here depends on the duality between the rejection of explosion and the rejection of excluded middle, both of which may be motivated by epistemological reasons. From this perspective, the rejection of excluded middle does not mean that both A and ¬A may be false. Dually, the rejection of explosion does not mean that both A and ¬A may be true.

In building formal systems we deal with several logical principles, and it may justly be asked what these are principles about. This is a central issue in philosophy of logic. Here we follow Popper (1963, pp. 206ff), who presents the problem in a very clear way. The question is whether the rules of logic are:

(I.a) laws of thought in the sense that they describe how we actually think;

(I.b) laws of thought in the sense that they are normative laws, i.e., laws that tell us how we should think;

(II) the most general laws of nature, i.e., laws that apply to any kind of object;

(III) laws of certain descriptive languages. 

We thus have three basic options, which are not mutually exclusive: the laws of logic have (I) epistemological, (II) ontological, or (III) linguistic character. With respect to (I), they may be (I.a) or descriptive (I.b) normative. Let us illustrate the issue with some examples.

Aristotle, defending the principle of non-contradiction, makes it clear that it is a principle about reality, “the most certain principle of all things” (Metaphysics 1005b11). Worth mentioning also is the well-known passage, “the same attribute cannot at the same time belong and not belong to the same subject in the same respect” (Metaphysics 1005b19-21), which is a claim about objects and their properties. 

On the other hand, a very illustrative example of the epistemological side of logic can be found in the so-called logic of Port-Royal, where we read:

Logic is the art of conducting reasoning well in knowing things, as much to instruct ourselves about them as to instruct others.

This art consists in reflections that have been made on the four principal operations of mind: conceiving, judging, reasoning, and ordering.

(…) [T]his art does not consist in finding the means to perform these operations, since nature alone furnishes them in giving us reason, but in reflecting on what nature makes us do, which serves three purposes.

The first is to assure us that we are using reason well …

The second is to reveal and explain more easily the errors or defects that can occur in mental operations.

The third purpose is to make us better acquainted with the nature of the mind by reflecting on its actions. (Arnauld, A. & Nicole 1662, (1996) p. 23)

Logic is conceived as having a normative character. So far so good. But logic is also conceived as a tool for analyzing mental processes of reasoning. This analysis, when further extended by different approaches to logical consequence, as is now done by some non-classical logics, shows that there can be different standards of correct reasoning in different situations. This aspect of logic, however, has been relegated to secondary status by Frege’s attack on psychologism. Frege wanted to eliminate everything subjective from logic. For Frege, laws of logic cannot be obtained from concrete reasoning practices. Basically, his argument is the following. From the assumption that truth is not relative, it follows that the basic criterion for an inference to be correct, namely, the preservation of truth, should be the same for everyone. When different people make different inferences, we must have a criterion for deciding which one is correct. Combined with Frege’s well-known Platonism, the result is a conception of logic that emphasizes the ontological (and realist) aspect of classical logic. 

Our conception of the laws of logic is necessarily decisive for our treatment of the science of logic, and that conception in turn is connected with our understanding of the word ‘true’. It will be granted by all at the outset that the laws of logic ought to be guiding principles for thought in the attainment of truth, yet this is only too easily forgotten, and here what is fatal is the double meaning of the word ‘law’. In one sense a law asserts what is; in the other it prescribes what ought to be. Only in the latter sense can the laws of logic be called ‘laws of thought’(…) If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace (Frege 1893, (1964) p. 13).

(…) [O]ne can very well speak of laws of thought too. But there is an imminent danger here of mixing different things up. Perhaps the expression "law of thought" is interpreted by analogy with "law of nature" and the generalization of thinking as a mental occurrence is meant by it. A law of thought in this sense would be a psychological law. And so one might come to believe that logic deals with the mental process of thinking and the psychological laws in accordance with which it takes place. This would be a misunderstanding of the task of logic, for truth has not been given the place which is its due here (Frege 1918, (1997) p. 325).

For Frege, logic is normative, but in a secondary sense. Along with truths of arithmetic, the logical relations between propositions are already given, eternal. This is not surprising at all. Since he wanted to prove that arithmetic could be obtained from purely logical principles, truths of arithmetic would inherit, so to speak, the realistic character of the logical principles from which they were obtained. Logic thus has an ontological character; it is part of reality, as are mathematical objects.

It is very interesting to contrast Frege’s realism with Brouwer’s intuitionism, whose basic ideas can be found for the first time in his doctoral thesis, written at the very beginning of twentieth century. The approaches are quite opposed.

Mathematics can deal with no other matter than that which it has itself constructed. In the preceding pages it has been shown for the fundamental parts of mathematics how they can be built up from units of perception (Brouwer 1907, (1975) p. 51).

The words of your mathematical demonstration merely accompany a mathematical construction that is effected without words …

While thus mathematics is independent of logic, logic does depend upon mathematics: in the first place intuitive logical reasoning is that special kind of mathematical reasoning which remains if, considering mathematical structures, one restricts oneself to relations of whole and part (Brouwer 1907, (1975)  p. 73-74).

            It is remarkable that Brouwer’s doctoral thesis (1907) was written between the two above-quoted works by Frege (1893 and 1919). Brouwer, like Frege himself, is primarily interested in mathematics. For Brouwer, however, the truths of mathematics are not discovered but rather constructed. Mathematics is not a part of logic, as Frege wanted to prove. Quite the contrary, logic is abstracted from mathematics. It is, so to speak, a description of human reasoning in constructing correct mathematical proofs. Mathematics is a product of the human mind, mental constructions that do not depend on language or logic. The raw material for these constructions is the intuition of time (this is the meaning of the phrase ‘built up from units of perception’).

These ideas are reflected in intuitionistic logic, which was formalized by Heyting (1956). Excluded middle is rejected precisely because mathematical objects are considered mental constructions. Accordingly, to assert an instance of excluded middle related to an unsolved mathematical problem (for instance, Goldbach’s conjecture), would be a commitment to a Platonic realm of abstract objects, an idea rejected by Brouwer and his followers.

            With respect to the linguistic aspects of logic, we shall make just a few comments. According to one widespread opinion, a linguistic conception of logic prevailed during the last century. From this perspective, logic has to do above all with the structure and functioning of certain languages. Indeed, sometimes logic is defined as a mathematical study of formal languages. There is no consensus for this view, however, and it is likely that it is not prevalent today. Even though we cannot completely separate the linguistic from the epistemological aspects – i.e., separate language from thought –, we endorse the view that logic is primarily a theory about reality and thought, and that the linguistic aspect is secondary. 

            At first sight, it might seem that Frege’s conception according to which there is only one logic, that is, only one account of logical consequence, is correct. Indeed, for Frege, Russell, and Quine, the logic is classical logic. From a realist point of view, this fits well with the perspective of the empirical sciences: excluded middle and bivalence have a strong appeal. Ultimately, reality will decide between A and not A, which is the same as deciding between the truth and falsity of A.

The identification of an intuitionistic notion of provability with truth was not successful. As is shown by Raatikainen (2004), in the works of Brouwer and Heyting we find some attempts to formulate an explanation of the notion of truth in terms of provability, but all of them produce counterintuitive results.

On the other hand, the basic intuitionistic argument that rejects a supersensible realm of abstract objects is philosophically motivated – and it is notable that this argument usually seems rather convincing to students of philosophy. As Velleman & Alexander (2002, pp. 91ff) put it, realism seems to be compelling when we consider a proposition like every star has at least one planet orbiting it. However, when we pass from this example to Goldbach’s conjecture, the situation changes quite a bit. In the former case, it is very reasonable to say that reality is one way or the other; but if we say with regard to the latter case that ‘the world of mathematical numbers’ is one way or the other, there is a question to be faced: where is this world?

What is the moral to be taken from this? That classical and intuitionistic logic are not talking about the same thing. The former is connected to reality through a realist notion of truth; the latter is not about truth, but rather about reasoning. In our view, assertability based on the intuitionistic notion of constructive proof is what is expressed by intuitionistic logic.

ARNAULD, A.; NICOLE, P. Logic or the Art of Thinking. Cambridge University Press, 1996.

ARISTOTLE. Metaphysics. The Complete Works of Aristotle, Oxford University Press, 1996.

BROUWER, L.E.J. “On the Foundations of Mathematics” 1907. Collected Works vol. I. (ed. A. Heyting), North-Holland Publishing Company, 1975.

FREGE, G. The Basic Laws of Arithmetic, 1893. Transl. M. Furth. University of California Press, 1964.

__________. “The Thought”, 1918. The Frege Reader. Oxford: Blackwell Publishers, 1997.

HEYTING, A. Intuitionism: an Introduction. London: North-Holland Publishing Company, 1956.

HUNTER, G. Metalogic. University of California Press, 1973.

NICKLES, T. ‘From Copernicus to Ptolemy: inconsistency and method’ in Inconsistency in Science (Ed. J. Meheus). Dordrecht: Springer, 2002.

POPPER, K. Conjectures and Refutations, New York: Harper, 1963. 

PRIEST, G. In Contradiction. Oxford University Press, 2006.

RAATIKAINEN, P. “Conceptions of Truth in Intuitionism”. History and Philosophy of Logic, 25: 131–145, 2004.

VELLEMAN, D.J.; ALEXANDER GEORGE, A. Philosophies of Mathematics. Oxford: Blackwell Publishers, 2002.

Extraído de Carnielli & Rodrigues, TOWARDS A PHILOSOPHICAL UNDERSTANDING OF THE LOGICS OF FORMAL INCONSISTENCY.

http://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8641922