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First Lecture

When I was asked to give these lectures about a year ago, I suggested the title On the Meanings of the Logical Constants and the Justifications of the Logical Laws. So that is what I shall talk about, eventually, but, first of all, I shall have to say something about, on the one hand, the things that the logical operations operate on, which we normally call propositions and propositional functions, and, on the other hand, the things that the logical laws, by which I mean the rules of inference, operate on, which we normally call assertions. We must remember that, even if a logical inference, for instance, a conjunction introduction is written

\begin{displaymath}\frac{A \hspace{.3in} B}{A \; \& \; B}\end{displaymath}

which is the way in which we would normally write it, it does not take us from the propositions A and B to the proposition $A \; \& \; B$. Rather, it takes us from the affirmation of A and the affirmation of B to the affirmation of $A \; \& \; B$, which we may make explicit, using Frege's notation, by writing it

\begin{displaymath}\frac{\vdash A \hspace{.3in} \vdash B}{\vdash A \; \& \; \vdash B}\end{displaymath}

instead. It is always made explicit in this way by Frege in his writings, and in Principia, for instance. Thus we have two kind of entities here: we have the entities that the logical operations operate on, which we call propositions, and we have those that we prove and that appear as premises and conclusion of a logical inference, which we call assertions. It turns out that, in order to clarify the meanings of the logical constants and justify the logical laws, a considerable portion of the philosophical work lies already in clarifying the notion of proposition and the notion of assertion. Accordingly, a large part of my lectures will be taken up by a philosophical analysis of these two notions.

Let us first look at the term proposition. It has its origin in the Greek $\pi
\rho \acute{o} \varsigma \alpha \sigma \iota \zeta$ used by Aristotle in the Prior Analytics, the third part of the Organon. It was translated, apparently by Cicero, into Latin propositio, which has its modern counterparts in Italian proposizions, English proposition, and German Satz. In the old, traditional use of the word proposition, propositions are the things that we prove. We talk about proposition and proof, of course, in mathematics: we put up a proposition, and let it be followed by its proof. In particular, the premises and conclusion of an inference were propositions in this old terminology. It was the standard used of the word up to the last century. An it is this use which is retained in mathematics, where a theorem is sometimes called a proposition, sometimes a theorem. Thus we have two words for the things that we prove, proposition and theorem. The word proposition, Greek $\pi
\rho \acute{o} \varsigma \alpha \sigma \iota \zeta$, comes from Aristotle and has dominated the logical tradition, whereas the word theorem, Greek $\theta \!\! \in \!\! \acute{\omega} \rho \eta \mu \alpha$, is in Euclid, I believe, and has dominated the mathematical tradition.

With Kant, something important happened, namely that the term judgement, German Urteil, came to be used instead of proposition. Perhaps one reason is that proposition, or a word with that stem, at least, simply does not exist in German: the corresponding German word would be Lehrsatz , or simply Satz. Be that as it may, what happened with Kant and the ensuing German philosophical tradition was that the word judgement came to replace the word proposition. thus, in that tradition, a proof, German Beweis, is always a proof of a judgment. In particular, the premises and conclusion of a logical inference are always called judgements. And it was the judgements, or the categorical judgements, rather, which were divided into affirmations and denials, whereas earlier it was the propositions which were so divided.

The term judgement also has a long history. It is the Greek $\kappa \rho
\acute{\iota} \sigma \iota \zeta$, translated into Latin judicium, Italian giudizio, English judgement, and German Urteil. Now, since it has as long a history as the word proposition, these two were also previously used in parallel. the traditional way of relating the notions of judgement and proposition was by saying that proposition is the verbal expression of a judgement. This is, as far as I know, how the notions of proposition and judgement were related during the scholastic period, and it is something which is repeated it the Port Royal Logic, for instance. You still find it repeated by Brentano in this century. Now, this means that, when, in German philosophy beginning with Kant, what was previously called a proposition came to be called a judgement, the term judgement acquired a double meaning. It came to be used, on the one hand, for the act of judging, just as before, and, on the other hand, it came to be used instead of the old proposition. Of course, when you say that a proposition is the verbal expression of a judgement, you mean by judgement the act of judging, the mental act of judging in scholastic terms, and the proposition is the verbal expression b means of which you make the mental judgement public, so to say. That is, I think, how one thought about it. Thus, with Kant, the term judgement became ambiguous between the act of judging and that which is judged, or the judgement made, if you prefer. German has here the excellent expression gefälltes Urteil which has no good counterpart in English.


  the act of judging that which is judged
old tradition judgement proposition
Kant Urteil ( sakt) ( gefälltes) Urteil

This ambiguity is not harmful, and sometimes it is even convenient, because, after all, it is a kind of ambiguity that the word judgement shares with other nouns of action. If you take the word propositioin, for instance, it is just as ambiguous between the act of propounding and that which is propounded. Or, if you take the word affirmation, it is ambiguous between the act of affirming and that which is affirmed, and so on.

It should be clear, from what I said in the beginning, that there is a difference between what we now call a proposition and a proposition in the old sense. In order to trace the emergence of the modern notion of proposition, I first have to consider the division of propositions in the old sense into affirmations and denials. Thus the propositions, or the categorical propositions, rather, were divided into affirmations and denials.

(categorical) propositions
$\overbrace{ \hspace{1in}}$
affirmation            denial

And not only were the categorical propositions so divided: the very definition of a categorical proposition was that a categorical proposition is an affirmation or a denial. Correlatively, to judge was traditionally, by which I mean before Kant, defined as to combine or separate ideas in the mind, that is, to affirm or deny. Those were the traditional definitions of the notions of propositions and judgement.

The notions of affirmation and denial have fortunately remained stable, like the notion of proof, and are therefore easy to use without ambiguity. Both derive from Aristotle. Affirmation is Greek $\kappa \alpha \tau \acute{\alpha}
\phi \alpha \sigma \iota \zeta$, Latin affirmatio, Italian affermazione, and German Bejahung, whereas denial is Greek $\alpha \pi
\acute{o} \phi \alpha \sigma \iota \zeta$, Latin negatio, Italian negazione, and German Verneinung. In Aristotelian logic, an affirmation was defined as a proposition in which something, called the predicate, is affirmed or something else, called the subject, and a denial was defined as a proposition in which the predicate is denied of the subject. Now, this is something that we have certainly abandoned in modern logic. Neither do we take categorical judgments to have subject-predicate form, nor do we treat affirmation and denial symmetrically. It seems to have been Bolzano who took the crucial step of replacing the Aristotelian forms of judgement by the single form

A is, A is true, or A holds.

In this, he was followed by Brentano, who also introduced the opposite form

A is not, or A is false.

and Frege. And through Frege's influence, the whole of modern logic has come to be based on the single form of judgement, or assertion, A is true.

Once this step was taken, the question arose, What sort of thing is it that is affirmed in an affirmation and denied in a denial? that is, What sort of thing is the A here? The isolation of this concept belongs to the, if I may so call it, objectivistically oriented branch of German philosophy in the last century. By that, I mean the tradition which you may delimit by mentioning the names of, say Bolzano, Lotze, Frege, Brentano, and the Brentano disciples Stumpf, Meinong, and Husserl, although, with Husserl, I think one should say that the split between the objectivistic and the Kantian branches of German philosophy is finally overcome. The isolation of this concept was a step which was entirely necessary for the development of modern logic. Modern logic simply would not work unless we had this concept, because it is on the things that fall under it that the logical operations operate.

This new concept, which simply did not exist before the last century, was variously called. And, since it was something that one had not met before, on had difficulties with what one should call it. Among the terms that were used, I think the least committing one is German urteilsinhalt, content of a judgement, by which I mean that which is affirmed in an affirmation and denied in a denial. Bolzano, who was the first to introduce this concept, called it proposition in itself, German Satz an sich. Frege also grappled with this terminological problem. In Begriffsschrift, he called it judgeable content, German beurteilbarer Inhalt. Later on, corresponding to his threefold division into expression, sense and reference, in the case of this kind of entity,m what was the expression, he called sentence, German Satz, what was the sense, he called thought, German Gedanke, and what was the reference, he called truth value, German Wahrheitswert. So the question arises, What should I choose here? Should I choose sentence, thought, or truth value? The closest possible correspondence is achieved, I think, if I choose Gedanke, that is, thought, for late Frege. this is confirmed by the fact that, in his very late logical investigations, he called the logical operations the Gedankengefüge. Thus judgeable content is early Frege and thought is late Frege. We also have the term state of affairs, German Sachverhalt, which was introduced by Stumpf and used by Wittgenstein in the Tractatus. And, finally, we have the term objective, German Objektiv, which was the term used by Meinong. Maybe there were other terms as well in circulation, but these are the ones that come immediately to my mind.

Now, Russell used the term proposition for this new notion, which has become the standard term in Anglo-Saxon philosophy and in modern logic. And, since he decided to use the word proposition in this new sense, he had to use another word for the things that we prove and that figure as premises and conclusion of a logical inference. His choice was to translate Frege's Urteil, not by judgement, as one would expect, but by assertion. And why, one may ask, did he choose the word assertion rather than translate Urteil literally by judgement? I think it was to avoid any association with Kantian philosophy, because Urteil was after all the central notion of logic as it was done in the Kantian tradition. For instance, in his transcendental logic, which forms part of the Kritik der reinen Vernunft, Kant arrives at his categories by analyzing the various forms that a judgement may have. That was his clue to the discovery of all pure concepts of reason, as he called it. Thus, in Russell's hands, Frege's Urteil came to be called assertion, and the combination of Frege's Urteilsstrich, judgement stroke, and Inhaltsstrich, content stroke, came to be called the assertion sign.

Observe now where we have arrived through this development, namely, at a notion of proposition which is entirely different, or different, at least, from the old one, that is, from the Greek $\pi \rho \acute{o} \tau \alpha \sigma \iota
\zeta$ and the Latin propositio. To repeat, the things that we prove, in particular, the premises and conclusion of a logical inference, are no longer propositions in Russell's terminology, but assertions. Conversely, the things that we combine by means of the logical operations, the connectives and the quantifiers, are not propositions in the old sense, that is, what Russell calls assertions, but what he calls propositions. And, as I said in the very beginning, the rule of conjunction introduction, for instance, really allows us to affirm A & B, having affirmed A and having affirmed B,

\begin{displaymath}\frac{\vdash A \hspace{.3in} \vdash B}{\vdash A \; \& \; B}\end{displaymath}

It is another matter, of course, that we may adopt conventions that allow us to suppress the assertion sign, if it becomes too tedious to write it out. Conceptually, it would nevertheless be there, whether I write it as above or

\begin{displaymath}\frac{A \; \mbox{true} \hspace{.3in} B \; \mbox{true}}{A \; \& \; B \; \mbox{true}}\end{displaymath}

as I think that I shall do in the following.

So far, I have made no attempt at defining the notions of judgement, or assertion, and proposition. I have merely wanted to give a preliminary hint at the difference between the two by showing how the terminology has evolved.

To motivate my next step, consider any of the usual inference rules of the propositional or predicate calculus. Let me take the rule of disjunction introduction this time, for some change,

\begin{displaymath}\frac{A}{A \vee}\end{displaymath}

or, writing out the affirmation,

\begin{displaymath}\frac{A \mbox{true}}{A \vee B \mbox{true}}\end{displaymath}

Now, what do the variables A and B range over in a rule like this? That is, what are you allowed to insert into the places indicated by these variables? The standard answer to this question, by someone who has received the now current logical education, would be to say that A and B range over arbitrary formulas of the language that you are considering. Thus, if the language is first order arithmetic, say, then A and B should be arithmetical formulas. When you start thinking about this answer, you will see that there is something strange about it, namely its language dependence. Because it is clearly irrelevant for the validity of the rule whether A and B are arithmetical formulas, corresponding to the the language of first order arithmetic, or whether they contain, say, predicates defined by transfinite, or generalized induction. The unary predicate expressing that a natural number encodes an ordinal of the constructive second number class, for instance, is certainly not expressible in first order arithmetic, and there is no reason at all why A and B should not be allowed to contain that predicate. Or, surely,f or the validity of the rule, A and B might just as well be set theoretical formulas, supposing that we have given such a clear sense to them that we clearly recognize that they express propositions. Thus what is important for the validity of the rule is merely that A and B are propositions, that is, that the expressions which we insert into the places indicated by the variables A and B express propositions. It seems, then, that the deficiency of the first answer, by which I mean the answer that Aand B should range over formulas, is eliminate by saying that the variables A and B should range over propositions instead of formulas. And this is entirely natural, because, after all, the notion of formula, as given by the usual inductive definition, is nothing but the formalistic substitute for the notion of proposition: when you divest a proposition in some language of all sense, what remains is the mere formula. But then, supposing we agree that the natural way to of the first difficulty is to say that A and B should range over arbitrary propositions, another difficulty arises, because, whereas the notion of formula is a syntactic notion, a formula being defined as an expression that can be formed by means of certain formation rules, the notion of proposition is a semantic notion, which means that the rule is not longer completely formal in the strict sense of formal logic. That a rule of inference is completely formal means precisely that there must be no semantic conditions involved in the rule: it may only put conditions on the forms of the premises and conclusion. The only way out of this second difficulty seems to be to say that, really, the rule has not one but three premises, so that, if we were to write them all out, it would read

\begin{displaymath}\frac{A \mbox{ prop} \hspace{.1in} B \mbox{ prop} \hspace{.1in} A \mbox{ true}}{A \vee B \mbox{ true}}\end{displaymath}

that is, from A and B being propositions and from the truth of A, we are allowed to conclude the truth of $A \vee B$. Here I am using

\begin{displaymath}A \mbox{ prop}\end{displaymath}

as an abbreviated way of saying that

\begin{displaymath}A \mbox{ is a proposition.}\end{displaymath}

Now the complete formality of the rule has been restored. Indeed, for the variables A and B, as they occur in this rule, we may substitute anything we want, and, by anything, I mean any expressions. Or, to be more precise, if we categorize the expressions, as Frege did, into complete, or saturated, expressions and incomplete, unsaturated, or functional, expressions, then we should say that we may substitute for A and B any complete expressions we want, because propositions are always expressed by complete expressions, not by functional expressions. Thus A and B now range over arbitrary complete expressions. Of course, there would be needed here an analysis of what is understood by an expression, but that is something which I shall not go into it in these lectures, in the belief that it is a comparatively trivial matter, as compared with explaining the notions of proposition and judgement. An expression in the most general sense of the word is nothing but a form, that is, something that we can passively recognize as the same in its manifold occurrences and actively reproduce in many copies. But I think that I shall have to rely here upon an agreement that we have such a general notion of expression, which is formal in character, so that the rule can now count as a formal rule.

Now, if we stick to our previous decision to call what we prove, in particular, the premises and conclusion of a logical inference, by the word judgement, or assertion, the outcome of the proceeding considerations is that we are faced with a new form of judgement. After all, A prop and B prop have now become premises of the rule of disjunction introduction. Hence, if premises are always judgements,

\begin{displaymath}A \mbox{ is a proposition.}\end{displaymath}

must count as a form of judgement. This immediately implies that the traditional definition of the act of judging as an affirming or denying and of the judgement, that is, the proposition in the terminology then used, as an affirmation or denial has to be rejected because A prop is certainly neither an affirmation not a denial. Or, rather, we are faced with the choice of either keeping the old definition of judgement as an affirmation or a denial, in which case we would have to invent a new term for the things that we prove and that figure as premises and conclusion of a logical inference, or else abandoning the old definition of judgement, widening it so as to make room for A is a proposition as a new form of judgement. I have chosen the latter alternative, well aware that, in so doing, I am using the word judgement in a new way.

Having rejected the traditional definition of a judgement as an affirmation or a denial, by what should we replace it? How should we now delimit the notion of judgement, so that A is proposition, A is true, and A is false all become judgements? And there are other forms of judgement as well, which we shall meet in due course. Now the question, What is a judgement? is no small question, because the notion of judgement is just about the first of all the notions of logic, the one that has to be explained before all the others, before even the notions of proposition and truth, for instance. There is therefore an intimate relation between the answer to the question what a judgement is and the very question what logic itself is. I shall start by giving a very simple answer, which is essentially right: after some elaboration, at least, I hope that we shall have a sufficiently clear understanding of it. And the definition would simply be that, when understood as an act of judging, a judgement is nothing but an act of knowing, and, when understood as that which is judged, it is a piece or, more solemnly, an object of knowledge.


the act of judging that which is judged
the act of knowing the object of knowledge

Thus, first of all, we have the ambiguity of the term judgement between the act of judging and that which is judged. What I say is that an act of judging is essentially nothing but an act of knowing, so that to judge is the same as to know, and that which is judged is a piece, or an object, of knowledge. Unfortunately, the English language has no counterpart of Ger. eine Erkenntnis, a knowledge.

This new definition of the notion of judgement, so central to logic, should be attributed in the first place to Kant, I think, although it may be difficult to find him ever explicitly saying that the act of judging is the same as the act of knowing, and that what is judged is the object of knowledge. Nevertheless, it is behind all of Kant's analysis of the notion of judgement that to judge amounts to the same as to know. It was he who broke with the traditional, Aristotelian definition of a judgement as an affirmation or a denial. Explicitly, the notions of judgement and knowledge were related by Bolzano, who simply defined knowledge as evident judgement. Thus, for him, the order of priority was the reverse: knowledge was defined in terms of judgement rather than the other way round. The important thing to realize is of course that to judge and to know, and, correlatively, judgement and knowledge, are essentially the same. And, when the relation between judgement, or assertion, if you prefer, and knowledge is understood in this way, logic itself is naturally understood as the theory of knowledge, that is, of demonstrative knowledge, Aristotle's $ \acute{\epsilon} \pi \iota \sigma \tau \acute{\eta} \mu
\eta~\acute{\alpha} \pi o \delta \epsilon \iota \kappa \tau \iota \kappa
\acute{\eta}$. Thus logic studies, from an objective point of view, our pieces of knowledge as they are organized in demonstrative science, or, if you think about it from the act point of view, it studies our acts of judging, or knowing, and how they are interrelated.

As I said a moment ago, this is only a first approximation, because it would actually have been better if I had not said that an act of judging is an act of knowing, but if I had said that it is an act of, and here there are many words that you may use, either understanding, or comprehending, or grasping, or seeing, in the metaphorical sense of the word see in which it is synonymous with understand. I would prefer this formulation, because the relation between the verb to know and the verb to understand, comprehend, grasp, or see, is given by the equation

to know = to have understood, comprehended, grasped, seen

which has the converse

to understand, comprehend, grasp, see = to get to know.

The reason why the first answer needs elaboration is that you may use know in English both in the sense of having understood and in the sense of getting to understand. Now, the first of the preceding two equations brings to expression something which is deeply rooted in the Indo-European languages. For instance, Gr. $o \vec{\iota} \delta \alpha$, I know, is the perfect form of the verb whose present form is Gr $\epsilon \acute{\iota} \delta \omega$, I see. Thus to know is to have seen merely by the way the verb has been formed in Greek. It is entirely similar in Latin. You have Lat. nosco, I get to know, which has present form, and Lat. novi, I know, which has perfect form. So, in these and other languages, the verb to know has present sense but perfect form. And the reason for the perfect form is that to know is to have seen. Observe also the two metaphors for the act of understanding which you seem to have in one form or the other in all European languages: the metaphor or seeing, first of all, which was so much used by the Greeks, and which we still use, for instance, when saying that we see that there are infinitely many prime numbers, and, secondly, the metaphor of grasping, which you also find in the verb to comprehend, derived as it is from Lat. prehendere to seize. The same metaphor is found in Ger fassen and begreifen, and I am sure that you also have it in Italian. (Chorus. Afferare!) Of course, these are two metaphors that we use for this particular act of the mind: the mental act of understanding is certainly as different from the perceptual act of seeing something as from the physical act of grasping something.

Is a judgement a judgement already before it is grasped, that is, becomes known, or does it become a judgement only through our act of judging? And, in the latter case, what should we call a judgement before it has been judged, that is, has become known? For example, if you let G be the proposition that every even number is the sum of two prime numbers, and then look at

G is true

is it a judgement, or is it not a judgement? Clearly, in one sense, it is, and, in another sense, it is not. It is not a judgement in the sense that it is not known, that is, that it has not been proved, or grasped. But, in another sense, it is a judgement, namely, in the sense that G is true makes perfectly good sense, because G is a proposition which we all understand, and, presumably, we understand what it means for a proposition to be true. The distinction I am hinting at is the distinction which was traditionally made between an enunciation and a proposition. Enunciation is not a word of much currency in English, but I think that its Italian counterpart has fared better. The origin is the Gr. $\acute{\alpha} \pi \acute{o} \phi \alpha \nu \sigma \iota
\varsigma$ as it appears in De Interpretatione, the second part of the Organon. It has been translated into Lat. enuntiatio, It. enunciato, and Ger. Aussage. An enunciation is what a proposition, in the old sense of the word, is before it has been proved, or become known. Thus it is a proposition stripped of its epistemic force. For example, in this traditional terminology, which would be fine if it were still living, G is true is a perfectly good enunciation, but it is not a proposition, not yet at least. But now that we have lost the term proposition in its old sense, having decided to use it in the sense in which it started to be used by Russell and is now used in Anglo-Saxon philosophy and modern logic, I think we must admit that we have also lost the traditional distinction between an enunciation and a proposition. Of course, we still have the option of keeping the term enunciation, but it is no longer natural. Instead, since I have decided to replace the term proposition in its old sense, as that which we prove and which may appear as premise or conclusion of a logical inference, by the term judgement, as it has been used in German philosophy from Kant and onwards, it seems better, when there is a need of making the distinction between an enunciation and a proposition, that is, between a judgement before and after it has been proved, or become known, to speak of a judgement and an evident judgement, respectively. This is a well-established usage in the writings of Bolzano, Brentano, and Husserl, that is, within the objectivistically oriented branch of German philosophy that I mentioned earlier. If we adopt this terminology, then we are faced with a fourfold table, which I shall end by writing up.

judgement     proposition
evident judgement     true proposition


Thus, correlated with the distinction between judgement and proposition, there is the distinction between evidence of a judgement and truth of a proposition.

So far, I have said very little about the notions of proposition and truth. The essence of what I have said is merely that to judge is the same as to know, so that an evident judgment is the same as a piece, or an object, of knowledge, in agreement with Bolzano's definition of knowledge as evident judgement. Tomorrow's lecture will have to be taken up by an attempt to clarify the notion of evidence and the notions of proposition and truth.

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Joan Lockwood