It might seem that one of the tools for studying human thinking and human consciousness is logic. It is indubitable that logic and thought are indeed essentially interconnected; and this leads to the feeling that logic is simply a theory of how we do or should think. However, is the relationship between the two really so straightforward? Can we interpret results of logic directly as findings about ‘the structure of our thought’? Can we interpret the results of logic, e.g. Gödel’s famous incompleteness proof, as amounting to the ways we think or, as in the case of Gödel’s finding, to the limits of what we can think? Here I want to indicate that we should be more careful. To take the relationship between logic and thinking as a straightforward and unproblematic link between ‘the depicting’ and ‘the depicted’ is likely to lead us to a basically corrupted view of both logic and thought.
How do we think? It is difficult to tell; for two reasons. First, the term ‘thinking’ is notoriously ambiguous, for sometimes it is taken to mean almost everything which is going on “within our heads”, whereas sometimes it is understood as referring to some restricted parts of it, perhaps “reasoning”. Second, it is notoriously difficult to couch the workings of the mental into adequate words (which is the reason for the repeated proposals of behaviourists of all sorts to give up addressing the mental altogether in favour of addressing its observable manifestations; or, more recently, its underlying neurological machinery). However, if we do try to introspectively review what is going on within our heads, we are likely to find that it has very little to do with that which is presented in logic textbooks. The processes of our minds appear to be always at least partly picture- or model-driven, full of hardly explainable leaps and transitions which cannot be articulated in terms of logic. Only in exceptional cases do we do such things as really infer one claim from another in the way logic has it. And this concerns even such cases of thinking which we would suppose to be the most susceptible to logic, like mathematical reasoning – even there imagination, trial and error and hardly articulable leaps appear to be indispensable.
This may lead us to one of the following two conclusions. First, we may conclude that logic is not about thinking after all. However, what it is, then, that logic really is about? Second, we may keep believing that logic is about thinking, but that it has badly fallen short in its task. I am going to argue that despite appearances, it is the first answer which is closer to the truth; or at least less misleading.
To see what logic really is (and in which relationship to thought it can reasonably be seen to stand), let us turn to a man who is undoubtedly responsible for there being logic in the sense in which we now understand the word, namely to Gottlob Frege. In his famous Begriffsschrift (Nebert, Halle, 1879; English translation in van Heijenoort’s From Frege to Gödel: A Source Book from Mathematical Logic, Harvard University Press, Cambridge, Mass., 1971), where he presents a formal language which was an ingenious precursor of our ordinary language of logic -predicate calculus, Frege attempts to construct a notation which would allow articulation of mathematical (and then other) proofs and arguments in a form suitable for a foolproof assessment. A proof (or any valid argument) is, as Frege sees it, a decomposition of a link between some premises and a corresponding conclusion into a chain of steps so elementary that their validity (or, indeed, invalidity) is unmistakable. What Frege also presents in the Begriffsschrift is an inventory of the elementary valid inferential steps which could serve as a reference for mechanical proof-checking.
Note that nothing said so far concerns thinking – at least not explicitly. Patterns of inference are a matter of overt language, and their validity is a matter of the rules which govern our use of the language. (The fact that the inference from A and B to A is a valid one is a matter of the correct use of the connective and. And how it is correct to use and is a public matter.) Thinking is, of course, involved in the sense that if there were no thinking, language-using creatures, there would be no inferential patterns, but this is a far cry from logic’s being a description or norm of how to think. More relevantly, thinking also enters the scene once we realise that one can prove something merely for oneself, without speaking or writing, “before the mind’s eye”. The crucial question then is: could we take silent reasoning as parasitic upon the loud, linguistic one (i.e. could we simply say, with Wilfrid Sellars, that thinking, in this sense, is talking to oneself), or, the other way around, must we take the linguistic as parasitic upon the mental? It might seem obvious that we have to take the second answer, for language not accompanied by thought seems to be empty of content. However as Wittgenstein and others showed, the situation is far from clear-cut: simply accompanying something (in this case words) by thought can hardly grant the something a real content. Thus, although thought is surely somehow essential for language, it is hard to say that language would be simply some kind of externalisation of (pre-linguistic) thinking.
But the resolving of this deep issue is not what we are after now; the only thing we want to point out here is that if logic is an articulation of our thinking in virtue of being an articulation of our linguistic activities, then it must be an articulation of this very specific and perhaps a rather rare species of thinking which can be safely identified with speaking to oneself. And what is important, is that nothing in logic, at least if we construe the term in the way Frege did, really hangs on our really entertaining this kind of thinking; for, as Frege stressed, it is important to distinguish between “the psychological way of arousal” of a statement within our mind and the “fullest way of carrying out its proof”. This means that logic is not concerned with how it happens that one comes to believe a statement, but rather with how the statement could be derived from obvious truths by a chain of obviously truth-preserving steps. And this also means that if we understand the term ‘thinking’ so narrowly as to apply to the actual processes going on within our minds/brains, then logic is not about thinking (at least not in any direct sense).
This means that a logical proof of a statement is neither a record of the activities of the mind or the brain of her who comes to believe the statement, nor a prescription of how these activities should proceed in order for her to come to do so. What a proof really is is a demonstration of the truth of the statement (indeed the Latin word for proof is demonstratio): everybody who sees the proof should be able to acknowledge the truth of the conclusion once she accepts the truth of the premises. (From the other side, it seems to be just the existence of a proof, of a publicly performable and persuasive demonstration, what is required if something, which is not self-evident, is to be commonly accepted as true.)
The only obvious connection between a proof and thinking thus appears to consist in the fact that if something be a proof, then whoever sees it should not fail to see the truth of that which is proved. So perhaps we could take the proof as a description of one possible way of arriving at the proved truth (that way, which is “canonical” and hence especially safe)? However, if we take thinking in the normal sense of the term, viz. as addressing goings-on within the head, even this seems to be dubious: what really happens in the head of one who sees a proof might differ wildly from person to person.
I think it is necessary to clarify this, for the concept of logic is being constantly overloaded. Frege’s notion was, as we saw, quite simple and perhaps unexciting: It aimed at classification and structuring of our sentences in terms of those valid inferences in which they participated. (Perhaps I should immediately stress that simple and unexciting means neither trivial, nor irrelevant, nor unimportant.) Such was, as it seems, also the notion of logic developed by the first great logician of our world, Aristotle. But between and after Aristotle and Frege, a number of philosophers have wanted to have logic as something much more encompassing. They have wanted to use logic to directly address the ultimate structure of our world (‘logic as metaphysics’, or as the late Wittgenstein was fond or putting it, “logic as ultraphysics”), or of our thought (“logic as transcendental psychology”).
I think that the lesson taught us by Frege is that logic can yield interesting results only if we do not overload it. I am convinced that the very reason why Frege did manage to accomplish that which many philosophers before him only dreamed of (i.e. to develop a symbolic language making correctness of reasoning transparent) is that his original aims were incomparably less ambitious than those of his precursors (e.g. Leibniz): what he was after was not a universal language to perspicuously formalise and consequently solve all our problems, but only a tool to clearly articulate mathematical proofs.
Now what I want to suggest is that the interpretations of the results of formal logic as being directly about the nature of human thinking, which I think are misleading, are just the outgrowths of such an overloaded notion of logic. The most common cases in point are the various interpretations of the most exciting logical result of this century, Gödel’s incompleteness theorem, which make the theorem into a revelation of various breath-taking facts about the way we think. Perhaps the most well known statement to this effect is Roger Penrose’s employment of Gödel’s result for his analysis of the concept of human consciousness and his resulting claim that the upshot of Gödel’s theorem is that human thinking is unalgorithmic (see Penrose’s The Emperor’s New Mind, Vintage, London, 1990).
What does the famous incompletness theorem consist in? Gödel proved that no matter how many axioms of arithmetic we accept, there will inevitably be a certain proposition, G, which will be neither provable, nor refutable. The reason is that G says, in effect, “I am unprovable”, and that hence both its provability and its refutability leads to a contradiction; for if it were provable and thereby true, it would have to be unprovable, and if it were refutable, it would have not to be unprovable, i.e. be provable. Moreover, G is obviously true, for it says of itself that it is unprovable, and it is unprovable. Hence there exists a formula which is true, but not provable. Gödel also pointed out that this result transfers to any language which is nontrivial enough to be able to spell out elementary arithmetic.
For those abreast of the jargon (those who aren’t can skip this paragraph), let me briefly recapitulate the proof: First, Gödel showed that there can be established a one-one correspondence between formulae of the language of formal arithmetic and natural numbers, and that in virtue of this, arithmetical formulae, which are about numbers, can be interpreted as being about arithmetical formulae. Second, he demonstrated that there is an arithmetical predicate which holds of a number iff it is a number of a formula which is provable; i.e. that what this predicate says, in effect, is ‘is provable’. Third, he showed that there is a formula G which is equivalent to the negation of the formula which arises out of the application of the provability predicate to the number of G, hence which says, in effect, of itself that it is unprovable.
The situation can be seen also in terms of the Liar paradox (see F. Moorcroft’s paper in The Philosophers’ Magazine, Winter 1997, p. 63): Tarski pointed out that the Liar paradox necessarily occurs in a language once the language fulfils the following three conditions. (i) It contains negation. (ii) It contains names of its own sentences. (iii) It contains its own truth predicate. Now what Gödel showed was that arithmetic does contain names of its own sentences (namely numerals); and that it contains its own provability predicate; so as it surely does contain negation, it is bound to contain a statement analogous to the Liar one with provability in place of truth. This statement is not paradoxical in the sense in which the Liar statement is, but it is bound to be unprovable as well as irrefutable.
Now what Penrose says about Gödel’s proof represents a fairly common way of interpreting this extraordinary result, a way which consists in seeing it as describing, or at least constraining, the manner a human being can think: “Mathematical truth is not something that we ascertain merely by use of an algorithm. … We must ‘see’ the truth of a mathematical argument to be convinced of its validity. This ‘seeing’ is the very essence of consciousness. … When we convince ourselves of the validity of the Gödel’s theorem … we reveal the very non-algorithmic nature of the ‘seeing’ process itself.” (pp. 540-541). However, it is obvious that such interpretations quite unscrupulously transgress a boundary which the previous paragraphs indicated were important: the boundary between thinking as a matter of activity of our brain (hence a matter of the “causal order”, which can be studied by physics, psychology and the rest of natural science), and “thinking” as a matter of ideal, abstract proofs (a matter of the “logical order”, to be studied by logic). This means that from the Fregean viewpoint, they contain a basic conflation.
If we see the situation from the Fregean standpoint, then we have to conclude that, first, Gödel’s result does not tell us anything about what is happening in the head of those who ascertain the truth of G or any other proposition. This result even does not tell us what should be happening in their heads in order to ascertain the truth of G. The only thing it really tells us is that that there does not exist a certain kind of demonstration of the truth G.
However, we have stated above that if we are to accept the truth of a sentence, then we need a proof; and now it seems that G is both unprovable and commonly accepted as true. Is this not a contradiction, which calls for a deus ex machina solution as the one urged by Penrose (who claims that it is consciousness which makes us see the truth even without the proof)? I think not, for the simple reason, that G is not undemonstrable – after all, we have demonstrated, and in this sense proved, its truth four paragraphs before.
To be more rigorous – how do we know that G is true? We know it for we know that G says of itself that it is unprovable and that it indeed is unprovable. We know that G says of itself that it is unprovable because Gödel demonstrated to us that G is equivalent (within the axiomatic system of Peano arithmetic) to the proposition ‘G is not provable’; and we know that G is not provable because we know that its provability implies its unprovability (for if G would be provable, it would have to be true and hence what it says would have to be the case). This seems to be as clear a demonstration as a demonstration could be. Its only peculiar feature is that it requires mixing reasoning within the system of arithmetic and reasoning about it. That we cannot prove G otherwise is a fact astonishing for a mathematical logician, but there is no reason why this feature should make the demonstration wrong – after all, we do demonstrate the truth of G to any reasonable human being (perhaps with some background in mathematics) in this way.
Hence we might do better if we do not see logic as too intimately connected with thinking, and we might also do better if we do not see Gödel and co. as direct theoreticians of thought or consciousness. This is not to say that their results are irrelevant for considerations of how we do or should think; it is to warn that the ways in which they can be so relevant are not straightforward and potentially delusive.