Under what condition is it right, or correct, to make a judgement, one of the form
which is certainly the most basic form of judgement, for instance? When one is faced with this question for the first time, it is tempting to answer simply that it is right to say that A is true provided that A is true, and that it is wrong to say that A is true provided that A is not true, that is, provided that A is false. In fact, this is what Aristotle says in his definition of truth in the Metaphysics. For instance, he says that it is not because you rightly say that you are white that you are white, but because you are white that what you say is correct. But a moment's reflection shows that this first answer is simply wrong. Even if every number is the sum of two prime numbers, it is wrong to me to say that unless I know it, that is, unless I have proved it. And it would have been wrong of me to say that every map can be coloured by four colours before the recent proof was given, that is, before I acquired that knowledge, either by understanding the proof myself, or by trusting its discoverers. So the condition for it to be right of me to affirm a proposition A, that is, to say that A is true, is not that Ais true, but that I know that A is true. This is a point which has been made by Dummett and, before him, by Brentano, who introduced the apt term blind judgement for a judgement which is made by someone who does not know what he is saying, although what he says is correct in the weaker sense that someone else knows it, or perhaps, that he himself gets to know it at some later time. When you are forced into answering a yes or no question, although you do not know the answer, and happen to give the right answer, right as seen by someone else, or by you yourself when you go home and look it up, then you make a blind judgement. Thus you err, although the teacher does not discover your error. Not to speak of the fact that the teacher erred more greatly by not giving you the option of giving the only answer which would have been honest, namely, that you did not know.
The preceding consideration does not depend on the particular form of judgement, in this case, A is true, that I happened to use as an example. Quite generally, the condition for it to be right of you to make a judgement is that you know it, or what amounts to the same, that it is evident to you. The notion of evidence is related to the notion of knowledge by the equation
When you say that a judgement is evident, you merely express that you have understood, comprehended, grasped, or seen it, that is, that you know it, because to have understood is to know. This is reflected in the etymology of the word evident, which comes from Latin ex, out of, from, and videre, to see, in the metaphorical sense, of course.
There is absolutely no question of a judgement being evident in itself, independently of us and our cognitive activity. That would be just as absurd as to speak of a judgement as being known, not be somebody, you or me, but in itself. To be evident is to be evident to somebody, as inevitably as to be known is to be known by somebody. That is what Brouwer meant by saying, in Consciousness, Philosophy, and Mathematics, that there are no nonexperienced truths, a basic intuitionistic tenet. This has been puzzling, because it has been understood as referring to the truth of a proposition, and clearly there are true propositions whose truth has not been experienced, that is, propositions which can be shown to be true in the future, although they have not been proved to be true now. But what Brouwer means here is not that. He does not speak about propositions and truth: he speaks about judgements and evidence, although he uses the term truth instead of the term evidence. And what he says is then perfectly right: there is no evident judgement whose evidence has not been experienced, and expeience it is what you do when you understand, comprehend, grasp, or see it. There is no evidence outside out actual or possible experience of it. The notion of evidence is by its very nature subject related, relative to the knowing subject, that is, in Kantian terminology.
As I already said, when you make, or utter, a judgement under normal circumstances, you thereby express that you know it. There is no need to make this explicit by saying,
For example, when you make a judgement of the form
under normal circumstances, by so doing, you already express that you know that A is true, without having to make this explicit by saying
or the like. A judgement made under normal circumstances claims by itself to be evident: it carries its claim of evidence automatically with it. This is a point which was made by Wittgenstein in the Tractatus by saying that Frege's Urteilsstrich, judgement stroke, is logically quite meaningless, since it merely indicates that the proposition to which it is prefixed is held true by the author, although it would perhaps have been better to say, not that it is meaningless.., but that it is superfluous, since, when you make a judgement, it is clear already from its form that you claim to know it. In speech act philosophy, this is expressed by saying that knowing is an illocutionary force: it is not an explicit part of what you say that you know it, but is is implicit in your saying of it. This is the case, not only with judgements, that is, acts of knowing, but also with other kinds of acts. For instance, if you say
it is clear from the form of your utterance that you express a wish. There is no need of making this explicit by saying,
Some languages, like Greek, use the optative mood to make it clear that an utterance expresses a wish or desire.
Consider the pattern that we have arrived at now.
|A is true|
Here the grammatical subject I refers to the subject, self, or ego, and the grammatical predicate know to the act, which in this particular case is an act of knowing, but might as well have been an act of conjecturing, doubting, wishing, fearing, etc. Thus the predicate know indicates the modality of the act, that is, the way in which the subject relates to the object, or the particular force which is involved, in this case, the epistemic force. Observe that the function of the grammatical moods, indicative, subjunctive, imperative, and optative, is to express modalities in this sense. Finally, ``A is true'' is the judgment or, in general, the object of the act, which in this case is an object of knowledge, but might have been an object of conjecture, doubt, wish, fear, etc.
The closest possible correspondence between the analysis that I am giving and Frege's notation for a judgement
is obtained by thinking of the vertical, judgement stroke as carrying the epistemic force
and the horizontal, content stroke as expressing the affirmation
Then it is the vertical stroke which is superfluous, whereas the horizontal stroke is needed to show that the judgement has the form of an affirmation. But this can hardly be read out of Frege's own account of the assertion sign: you have to read it into his text.
What is a judgement before it has become evident, or known? That is, of the two, judgement and evident judgement, how is the first to be defined? The characteristic of a judgement in this sense is merely that it has been laid down what knowledge is expressed by it, that is, what you must know in order to have the right to make, or utter, it. And this is something which depends solely on the form of the judgement. For example, if we consider the two forms of judgement
then there is something that you must know in order to have the right to make a judgement of the first form, and there is something else which you must know, in addition, in order to have the right to make a judgement of the second form. And what you must know depends in neither case on A, but only on the form of the judgement,...is a proposition or...is true, respectively. Quite generally, I may say that a judgement in this sense, that is, a not yet known, and perhaps even unknowable, judgement, is nothing but an instance of a form of judgement, because it is for the various forms of judgement that I lay down what you must know in order to have the right to make a judgement of one of those forms. Thus, as soon as something has the form of a judgement, it is already a judgement in this sense. For example, A is a proposition is a judgement in this sense, because it has a form for which I have laid down, or rather shall lay down, what you must know in order to have the right to make a judgement of that form. I think that I may make things a bit clearer by showing again in a picture what is involved here. Let me take the first form to begin with.
Here is involved, first, an expression A, which should be a complete expression. Second, we have the form...is a proposition, which is the form of judgement. Composing these two, we arrive A is a proposition, which is a judgement in the first sense. And then, third, we have the act in which I grasp this judgement, and through which it becomes evident. Thus it is my act of grasping which is the source of the evidence. These two together, that is, the judgement and my act of grasping it, become the evident judgement. And a similar analysis can be given of a judgement of the second form.
Such a judgement has the form...is true, but what fills the open place, or hole, in the form is not an expression any longer, but a proposition. And what is a proposition? A proposition is an expression for which the previous judgement has already been grasped, because there is no question of something being true unless you have previously grasped it as a proposition. But otherwise the picture remains the same here.
Now I must consider the discussion of the notion of judgement finished and pass on to the notion of proof. Proof is a good word, because, unlike the word proposition, it has not changed its meaning. Proof apparently means the same now as it did when the Greeks discovered the notion of proof, and therefore no terminological difficulties arise. Observe that both Latin demonstratio and the corresponding words in the modern languages, like Italian dimostrazione, English demonstration, and German Berweis, are literal translations of Greek , deriving as it does from Greek , I show, which has the same meaning as Latin monstrare and German weisen.
If you want to have a first approximation to the notion of proof, a first definition of what a proof is, the strange thing is that you cannot look it up in any modern textbook of logic, because what you get out of the standard textbooks of modern logic is the definition of what a formal proof is, at best with a careful discussion clarifying what a formal proof in the sense of this definition is not what we ordinarily call a proof in mathematics. That is, you get a formal proof defined as a finite sequence of formulas, each one of them being an immediate consequence of some of the preceding ones, where the notion of immediate consequence, in turn, is defined by saying that a formula is an immediate consequence of some other formulas if there is an instance of one of the figures, called rules of inference, which has the other formulas as premises and the formula itself as conclusion. Now, that is not a what a real proof is. That is why you have the warning epithet formal in front of it, and do not simply say proof.
What is a proof in the original sense of the word? The ordinary dictionary definition says, with slight variations, that a proof is that which establishes the truth of a statement. Thus a proof is that which makes a mathematical statement, or enunciation, into a theorem, or proposition, in the old sense of the word which is retained in mathematics. Now, remember that I have reserved the term true for true propositions, in the modern sense of the word, and that the things that we prove are, in my terminology, judgements. Moreover, to avoid terminological confusion, judgements qualify as evident rather than true. Hence, translated into the terminology that I have decided upon, the dictionary definition becomes simply,
Accepting this, that is, that the proof of a judgement is that which makes it evident, we might just as well say that the proof of a judgement is the evidence for it. Thus proof is the same as evidence. Combining this with the outcome of the previous discussion of the notion of evidence, which was that it is the act of understanding, comprehending, grasping, or seeing a judgement which confers evidence on it, the inevitable conclusion is that the proof of a judgement is the very act of grasping it. Thus a proof is, not an object, but an act. This is what Brouwer wanted to stress by saying that a proof is a mental construction, because what is mental, or psychic, is precisely our acts, and the word construction, as used by Brouwer, is but a synonym for proof. Thus he might just as well have said that the proof of a judgement is the act of proving, or grasping, it. And the act is primarily the act as it is being performed. Only secondarily, and irrevocably, does it become the act that has been performed.
As is often the case, it might have been better to start with the verb rather than the noun, in this case, with the verb to prove rather than with the noun proof. If a proof is what makes a judgement evident, then, clearly, to prove a judgement is to make it evident, or known. To prove something to yourself is simply to get to know it. And to prove something to someone else is to try to get him, or her, to know it. Hence
This means that prove is but another synonym for understand, comprehend, grasp, or see. And, passing to the perfect tense,
We also speak of acquiring and possessing knowledge. To possess knowledge is the same as to have acquired it, just as to know something is the same as to have understood, comprehended, grasped, or seen it. Thus the relation between the plain verb to know and the venerable expressions to acquire and to possess knowledge is given by the two equations,
On the other hand, the verb to prove and the noun proof are related by the two similar equations,
It is now manifest, from these equations, that proof and knowledge are the same. Thus, if proof theory is construed, not in Hilbert's sense, as metamathematics, but simply as the study of proofs in the original sense of the word, then proof theory is the same as theory of knowledge, which, in turn, is the same as logic in the original sense of the word, as the study of reasoning, or proof, not as metamathematics.
Remember that the proof of a judgement is the very act of knowing it. If this act is atomic, or indivisible, then the proof is said to be immediate. Otherwise, that is, if the proof consists of a whole sequence, or chain, or atomic actions, it is mediate. And, since proof and knowledge are the same, the attributes immediate and mediate apply equally well to knowledge. In logic, we are no doubt more used to saying of inferences, rather than proofs, that they are immediate or mediate, as the case may be. But that makes no difference, because inference and proof are the same. It does not matter, for instance, whether we say rules of inference or proof rules, as has become the custom in programming. And, to take another example, it does not matter whether we say that a mediate proof is a chain of immediate inferences or a chain of immediate proofs. The notion of formal proof that I have referred to in the beginning of my discussion of the notion of proof has been arrived at by formalistically interpreting what you mean by an immediate inference, by forgetting about the difference between a judgement and a proposition, and, finally, by interpreting the notion of proposition formalistically, that it, by replacing it by the notion of formula. But a real proof is and remains what it has always been, namely, that which makes a judgement evident, or, simply, the evidence for it. Thus, if we do not have the notion of evidence, we do not have the notion of proof. That is why the notion of proof has fared so badly in those branches of philosophy where the notion of evidence has fallen into disrepute.
We also speak of a judgement being immediately and mediately evident, respectively. Which of the two is the case depends of course on the proof which constitutes the evidence for the judgement. If the proof is immediate, then the judgement is said to be immediately evident. And an immediately evident judgement is what we call an axiom. Thus an axiom is a judgement which is evident by itself, not by virtue of some previously proved judgements, but by itself, that is, a self-evident judgement, as one has always said. That is, always before the notion of evidence became disreputed, in which case the notion of axiom and the notion of proof simply became deflated: we cannot make sense of the notion of axiom and the notion of proof without access to the notion of evidence. If, on the other hand, the proof which constitutes the evidence for a judgement is a mediate one, so that the judgement is evident, not by itself, by only by virtue of some previously proved judgements, then the judgement is said to be mediately evident. And a mediately evident judgement is what we call a theorem, as opposed to an axiom. Thus an evident judgement, that is, a proposition in the old sense of the word which is retained in mathematics, is either an axiom or a theorem.
Instead of applying the attributes immediate and mediate to proof, or knowledge, I might have chosen to speak of intuitive and discursive proof, or knowledge, respectively. That would have implied no difference of sense. The proof of an axiom can only be intuitive, which is to say that an axiom has to be grasped immediately, in a single act. The word discursive, on the other hand, comes from Latin discurrere, to run to and fro. Thus a discursive proof is one which runs, from premises to conclusion, in several steps. It is the opposite of an intuitive proof, which brings you to the conclusion immediately, in a single step. When one says that the immediate propositions in the old sense of the word proposition, that is, the immediately evident judgements in my terminology, are unprovable, what is meant is of course only that they cannot be proved discursively. Their proofs have to rest intuitive. This seems to be all that I have to say about the notion of proof at the moment, so let me pass on to the next item on the agenda, the forms of judgement and their semantical explanations.
The forms of judgement have to be displayed in a table, simply, and the corresponding semantical explanations have to be given, one for each of those forms. A form of judgement is essentially just what is called a category, not in the sense of category theory, but in the logical, or philosophical, sense of the word. Thus I have to say what my forms of judgement, or categories, are, and, for each one of those forms, I have to explain what you must know in order to have the right to make a judgement of that form. By the way, the forms of judgement have to be introduced in a specific order. Actually, not only the forms of judgement, but all the notions that I am undertaking to explain here have to come in a specific order. Thus, for instance, the notion of judgement has to come before the notion of proposition, and the notion of logical consequence has to be dealt with before explaining the notion of implication. There is an absolute rigidity in this order. The notion of proof, for instance, has to come precisely where I have put it here, because it is needed in some other explanations further on, where it is presupposed already. Revealing this rigid order, thereby arriving eventually at the concepts which have to be explained prior to all other concepts, turns out to be surprisingly difficult: you seem to arrive at the very first concepts last of all. I do not know what it should best be called, maybe the order of conceptual priority, one concept being conceptually prior to another concept if it has to be explained before the other concept can be explained.
Let us now consider the first form of judgement,
or, as I shall continue to abbreviate it,
What I have just displayed to you is a linguistic form, and I hope that you can recognize it. What you cannot see from the form, and which I therefore proceed to explain to you, is of course its meaning, that is, what knowledge is expressed by, or embodied in, a judgement of this form. The question that I am going to answer is, in ontological terms,
This is the usual Socratic way of formulating questions of this sort. Or I could ask, in more knowledge theoretical terminology,
of, if you prefer,
or, this may be varied endlessly,
These various ways of posing essentially the same question reflect roughly the historical development, from a more ontological to a more knowledge theoretical way of posing, and answering, questions of this sort, finally ending up with something which is more linguistic in nature, having to do with form and meaning.
Now, one particular answer to this question, however it be formulated, is that a proposition is something that is true of false, or, to use Aristotle's formulation, that has truth or falsity in it. Here we have to be careful, however, because what I am going to explain is what a proposition in the modern sense is, whereas what Aristotle explained was what an enunciation, being the translation of Greek , is. And it was this explanation that he phrased by saying that an enunciation is something that has truth or falsity in it. What he meant by this was that it is an expression which has a form of speech such that, when you utter it, you say something, whether truly or falsely. This is certainly not how we now interpret the definition of a proposition as something which is true or false, but is is nevertheless correct that it echoes Aristotle's formulation, especially in its symmetric treatment of truth and falsity.
An elaboration of the definition of a proposition as something that is true or false is to say that a proposition is a truth value, the true or the false, and hence that a declarative sentence is an expression which denotes a truth value, or is the name of a truth value. This was the explanation adopted by Frege in his later writings. If a proposition is conceived in this way, that is, simply as a truth value, then there is no difficulty in justifying the laws of the classical propositional calculus and the laws of quantification over finite, explicitly listed, domains. The trouble arises when you come to the laws for forming quantified propositions, the quantifiers not being restricted to finite domains. That is, the trouble is to make the two laws
evident when propositions are conceived as nothing but truth values. To my mind, at least, they simply fail to be evident. And I need not be ashamed of the reference to myself in this connection: as I said in my discussion of the notion of evidence, it is by its very nature subject related. Others must make up their minds whether these laws are really evident to them when they conceive of propositions simply as truth values. Although we have had this notion of proposition and these laws for forming quantified propositions for such a long time, we still have no satisfactory explanations which serve to make them evident on this conception of the notion of proposition. It does not help to restrict the quantifiers, that is, to consider instead the laws
unless we restrict the quantifiers so severely as to take the set A here to be a finite set, that is, to be given by a list of its elements. The, of course, there is no trouble with these rules. But, as soon as A is the set of natural numbers, say, you have the full trouble already. Since, as I said earlier, the law of the excluded middle, indeed, all the laws of the classical propositional calculus, are doubtlessly valid on this conception of the notion of proposition, this means that the rejection of the law of excluded middle is implicitly also a rejection of the conception of a proposition as something which is true or false. Hence the rejection of this notion of proposition is something which belongs to Brouwer. On the other hand, he did not say explicitly by what it should be replaced. Not even the wellknown papers by Kolmogorov and Heyting, in which the formal laws of intuitionistic logic were formulated for the first time, contain any attempt at explaining the notion of proposition in terms of which these laws become evident. It appears only in some later papers by Heyting and Kolmogorov from the early thirties. In the first of these, written by Heyting in 1930, he suggested that we should think about a proposition as a problem, French problème, or expectation, French attente. And, in the wellknown paper of the following year, which appeared in Erkenntnis, he used the terms expectation, German Erwartung and intention, German Intention. Thus he suggested that one should think of a proposition as a problem, or as an expectation, or as an intention. And, another year later, there appeared a second paper by Kolmogorov, in which he observed that the laws of the intuitionistic propositional calculus become evident upon thinking of the propositional variables as ranging over problems, or tasks. The term he actually used was German Aufgabe. On the other hand, he explicitly said that he did not want to equate the notion of proposition with the notion of problem and, correlatively, the notion of truth of a proposition with the notion of solvability of a problem. He merely proposed the interpretation of propositions as problems, or tasks, as an alternative interpretation, validating the laws of the intuitionistic propositional calculus.
Returning now to the form of judgement
the semantical explanation which goes together with it is this, and here I am using the knowledge theoretical formulation, that to know a proposition, which may be replaced, if you want, by problem, expectation, or intention, you must know what counts as a verification, solution, fulfillment, or realization of it. Here verification matches with proposition, solution with problem, fulfillment with expectation as well as with intention, and realization with intention. Realization is the term introduced by Kleene, but here I am of course not using it in his sense: Kleene's realizability interpretation is a nonstandard, or nonintended, interpretation of intuitionistic logic and arithmetic. The terminology of intention and fulfillment was taken over by Heyting from Husserl, via Oskar Becker, apparently. There is a long chapter in the sixth, and last, of his Logische Untersuchungen which bears the title Bedeutungsintention und Bedeutungserfüllung, and it is these two terms, intention and fulfillment, German Erfüllung, that Heyting applied in his analysis of the notions of proposition and truth. And he did not just take the terms from Husserl: if you observe how Husserl used these terms, you will see that they were appropriately applied by Heyting. Finally, verification seems to be the perfect term to use together with proposition, coming as it does from Latin verus, true, and facere, to make. So to verify is to make true, and verification is the act, or process, or verifying something. For a long time, I tried to avoid using the term verification, because it immediately gives rise to discussions about how the present account of the notions of proposition and truth is related to the verificationism that was discussed so much in the thirties. But, fortunately, this is fifty years ago now, and, since we have a word which lends itself perfectly to expressing what needs to be expressed, I shall simply use it, without wanting to get into discussion about how the present semantical theory is related to the verificationism of the logical positivists.
What would an example be? If you take a proposition like,
to know that proposition, you must know what counts as a verification of it, which in this case would be the direct seeing of the shining sun. Or, if you take the proposition,
then it would be a direct thermometer reading. What is more interesting, of course, is what the corresponding explanations look like for the logical operations, which I shall come to in my last lecture.
Coupled with the preceding explanation of what a proposition is, is the following explanation of what a truth is, that is, of what it means for a
p to be true. Assume first that
and, because of the omnipresence of the epistemic force, I am really asking you to assume that you know, that is, have grasped, that A is a proposition. On that assumption, I shall explain to you what a judgement of the form
means, that is, what you must know in order to have the right to make a judgement of this form. And the explanation would be that, to know that a proposition is true, a problem is solvable, an expectation is fulfillable, or an intention is realizable, you must know how to verify, solve, fulfill, or realize it, respectively. Thus this explanation equates truth with verifiability, solvability, fulfillability, or realizability. The important point to observe here is the change from is in A is true to can in A can be verified, or A is verifiable. Thus what is expressed in terms of being in the first formulation really has the modal character of possibility.
Now, as I said earlier in this lecture, to know a judgement is the same as to possess a proof of it, and to know a judgement of the particular form A is true is the same as to know how, or be able, to verify the proposition A. Thus knowledge of a judgement of this form is knowledge how in Ryle's terminology. On the other hand, to know how to do something is the same as to possess a way, or method, of doing it. This is reflected in the etymology of the word method, which is derived from Greek , after, and , way. Taking all into account, we arrive at the conclusion that a proof that a proposition Ais true is the same as a method of verifying, solving, fulfilling, or realizing A. This is the explanation for the frequent appearance of the word method in Heyting's explanations of the meanings of the logical constants. In connection with the word method, notice the tendency of our language towards hypostatization. I can do perfectly well without the concept of method in my semantical explanations: it is quite sufficient for me to have access to the expression know how, or knowledge how. But it is in the nature of our language that, when we know how to do something, we say that we possess a method of doing it.
Summing up, I have now explained the two forms of categorical judgement,
respectively, and they are the only forms of categorical judgement that I shall have occasion to consider. Observe that knowledge of a judgement of the second form is knowledge how, more precisely, knowledge how to verify A, whereas knowledge of a judgement of the first form is knowledge of a problem, expectation, or intention, which is knowledge what to do, simply. Here I am introducing knowledge what as a counterpart of Ryle's knowledge how. So the difference between these two kinds of knowledge is the difference between knowledge what to do and knowledge how to do it. And, of course, there can be no question of knowing how to do something before you know what it is that is to be done. The difference between the two kinds of knowledge is a categorical one, and, as you see, what Ryle calls knowledge that, namely, knowledge that a proposition is true, is equated with knowledge how on this analysis. Thus the distinction between knowledge how and knowledge that evaporates on the intuitionistic analysis of the notion of truth.