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.
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ARISTOTLE. Metaphysics. The Complete Works of
Aristotle, Oxford
University
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
__________. “The Thought”, 1918. The Frege Reader. Oxford
HEYTING, A. Intuitionism: an Introduction. London
HUNTER, G. Metalogic. University of California
NICKLES, T. ‘From Copernicus to Ptolemy: inconsistency
and method’ in Inconsistency in Science
(Ed. J. Meheus). Dordrecht
POPPER, K. Conjectures and Refutations, New York
PRIEST, G. In
Contradiction. Oxford University
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.
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