Una delle idee centrali in buona parte della migliore filosofia nella tradizione kantiana... è stata l' enfasi sulle attività umane attraverso cui otteniamo la nostra conoscenza e sul contributo di queste attività alla struttura globale della conoscenza umana... Nonostante questo, la maggior parte degli strumenti concettuali più efficaci della filosofia più recente sono molto più adatti a trattare la struttura dell' informazione di già acquisita - la struttura delle teorie, la struttura della spiegazione, e così via - che non le attività per mezzo delle quali l' acquisiamo.

Quantificatori, giochi linguistici e argomenti trascendentali.

Mi sembra sia del tutto giustificato affermare che nella misura in cui è davvero una certezza concettuale il fatto che la nostra logica si applica a tutta l' esperienza, allora vi deve essere una connessione concettuale tra i nostri concetti logici e i modi in cui acquisiamo esperienza. Nel caso dei concetti di esistenza e universalità, che costituiscono l' argomento della teoria della quantificazione, le esperienze in questione sono quelle su cui si basa la nostra conoscenza dell' esistenza di individui. In altre parole, i quantificatori debbono necessariamente essere connessi con i processi mediante i quali acquisiamo conoscenza dell' esistenza di oggetti individuali. Ora, Kant pensava che l' unico processo di questo genere è la percezione sensibile. Tuttavia, ho già dimostrato che questo punto di vista è erroneo... Significa infatti trascurare completamente il ruolo che i nostri tentativi attivi di trovare oggetti opportuni gioca nella genesi della nostra conoscenza dell' esistenza degli individui. L' osservazione diretta può essere invece pensata come un 'caso banale' di una ricerca riuscita; perciò, i giochi linguistici di ricerca e ritrovamento non possono essere ridotti all' osservazione diretta. Dobbiamo concludere che i processi tramite i quali diventiamo consapevoli dell' esistenza di oggetti individuali in generale sono le attività di ricerca e ritrovamento, e non il processo della percezione sensibile.

I giochi linguistici della logica formale e la teoria raffigurativa del linguaggio.

1. Le 'raffigurazioni' (insiemi modello) associate a un enunciato F non sono date da F stesso. Ci si arriva soltanto partendo da F e costruendo poi 'raffigurazioni' (insiemi modello) secondo certe regole (di costruzione degli insiemi modello). Da questo punto di vista, gli enunciati non sono essi stessi raffigurazioni degli stati di cose in cui sarebbero veri, bensì istruzioni per la costruzione di un certo numero di raffigurazioni alternative.
2. Le 'raffigurazioni' (insiemi modello) coinvolte sono usualmente infinite, e, in ogni caso, un numero infinito di esse può essere ottenuto normalmente da un dato enunciato F... La nostra comprensione degli enunciati del prim' ordine deve perciò essere basata su un confronto graduale e finito tra l' enunciato F e la realtà piuttosto che sulla natura (potenzialmente) raffigurativa di F. Ciò che le regole (G) dei nostri giochi ci danno è proprio un metodo di confronto graduale tra il linguaggio e la realtà. Esse perciò mettono riparo al difetto implicito nella nostra estensione della teoria raffigurativa.

Kant on the Logic of Existence.

Following his general trascendental point of view, [Kant] maintained that our ways of reasoning about existence... must be grounded in the human activities through which we come to know the existence of individuals... What Kant was dealing with is unmistakably such logical reasoning as is now codified in the modern logic of quantification theory... He located these 'activities' in sense perception... In doing so, he was following a long tradition that goes back to Aristotle... It is only in rare cases that we can wait until the relevant individuals kindly prove their existence by showing up in our passive sense perception. Normally, we have to get up and look for them. Hence, the true basis of the logic of existence and universality lies in the human activities of seeking and finding... not in the structure of our faculty of sense perception, as Kant mistakenly claimed.

Seeking and Finding, and Game-Theoretical Semantics.

The rigth approach to the contemporary philosophy of logic and logical semantics is therefore through a study of the rule-governed activities of seeking and finding... This basic idea is to focus on the activities through which we come to know the relevant propositions, that is, the propositions whose logical form is determined by the concepts of existence and nonexistence of individuals ( plus propositional connectives ). These propositions are the ones that are studied in modern first-order logic ( quantification theory ), and the activities in question are our processes of verifying such propositions. In game-theoretical semantics, these verification processes are conceptualized as games against a recalcitrant Nature, who tries to frustrate my attempts. These games are thus two-person games. ( We can call the players Myself and Nature.)

The problem of pictorial representation will be discussed here largely by reference to one single movement, the original cubism of Picasso and Braque. The crucial importance of this movement in the history of modern art is generally recognized, but its status is still puzzling and even controversial. This puzzle centers precisely on the relation of cubist art to the idea of representation...

It is of course indeniable that in a purely historical respect cubism did represent a step towards abstract, nonrepresentational art...

The perspective opened by the pronouncements of the most important early cubists and of their best interpreters is in fact almost diametrically opposed to what one might expect on the basis of the idea of cubists as the openers of the floodgates of abstraction. On the contrary, these pronouncements emphasize that the aim of cubists was to restore to painting the sense of concrete, solid reality which had been lost by the impressionists and by the symbolists...

Thus we have a nice paradox in our hands. Cubists were realists in some sense, but in what sense? They were trying to convey a sense of reality, but they were not doing it by depicting real objecs, at least not in the way in which they appear to our senses...

To these questions we can formulate a partial but illuminating answer in terms of an analogy between what we can find in modern art and what we can find in modern philosophy. One of the most important ideas that we can find in several different parts of recent philosophy is an emphasis on the concept of meaning. One aspect of this emphasis is a contrast between meanings and objects meant, that is to say, between on one hand the directedness or sense that enables us to refer to an object and on the other hand of this object itself. The best known form of this contrast is the distinction the German logician Gottlob Frege drew (in the case of linguistic meaning) betweeen Sinn and Bedeutung, between sense and reference as these terms are sometimes translated...

The most systematic early development of ideas of this kind is found in the phenomenology of Edmund Husserl, the main ideas of which he developed in 1901-1913, in particular between 1905 and 1913, that is to say, at the very same time at which cubism was born...

Husserl generalized Frege's distinction between sense and reference from the area of linguistic meaning to all conscious human thought-acts. Husserl was especially interested in what he called the intentionality of the acts of the human mind, that is, in the fact that they can as it were point to something beyond themselves, to be directed to it. According to Husserl, this intentionality presupposes a distinction which generalizes the Fregean distinction. Frege's references he called simply objects, and those meanings which enable an act to be directed to an object he called by the Greek word noemata (singular: noema)...

Phenomenology was conceived of by Husserl as a study of noemata. An oversimplified but non unfair brief description of the famous "phenomenological reduction" is to say that in it we disregard ("bracket") objects and focus our attention to noemata. These meaning entities are the main vehicles of human thougth, and can be reached by such a phenomenological reflexion...

After these preliminary explanations, we can give a concise answer to the question as to what it was that the cubist painters were doing. They were not painting objects. They were representing noemata, not objects. Or perhaps one should rather say that cubist paintings are a sort of concretisations of noemata. In the same way as phenomenology is supposed to be a philosophical study of noemata, cubism is the art of the noemata...

The statemens of cubists and of their most perceptive interpreters show that their art was conceptual in this very sense of dealing with our conceptions of objects, not with the external appearence of these objects. Perhaps the bluntest expression of this attitude is the statement by Picasso Gertrude Stein reported: "I do not paint things the way they look, but the way I know they are"...

The analogy between cubism and phenomenology becomes somewhat less surprising when we recall what the two movements were principally reacting against. In the case of Husserl, we have a reaction towards the sensationalism and positivism of Ernst Mach, who tried to restrict science in the last analysis to the description of sense-impression and their regularities. This is the philosophical counterpart to impressionism in painting, and of course impressionism was a part of what the cubists were reacting against. Both the impressionists and Mach were focusing their attention on our direct sense-impression, not on their objective sources. The former were trying to capture these impressions on canvas, the latter was studying their regularities. In contrast to this preoccupation both the cubists and the phenomenologists wanted to shift their emphasis "zu Sachen selbst"...

What are such noemata like according to Husserl? The following is one recent anwer. "To take an example from perception, let us consider the act of seeing a tree. When we see a tree, we do not see a collection of colored spots... distributed in a certain way; we see a tree, a material object with a back, with sides, and so forth. Part of it, for example the back, we cannot see, but we see a thing with a back. That seeing is intentional, object-directed, means that the near side of the thing is regarded only as a side of a thing, and that the thing we are seeing has other sides and features which are co-intended to the extent that the full thing is regarded as something with more than one side. The noema is the complex system of determination which unifies this multitude of features into aspects of one object."

What this means is that the noema of an object is largely independent of the way it happens to be perceived. My conception, my noema, of a tree is not tied to any particular perspective, in the sense that it comprises many things that are not perceived at a given time and perhaps cannot ever be perceived at the one and the same time. Here it might be especially helpful to think of a noema as a complex of expectations...

This is precisely what we find also in cubist painting. Just as a Husserlian noema may contain at one ande the same time expectations as to what an object or a person would look like fron many different perspectives, in the same way cubists often depicted the same subject from several different angles at one and the same time...

Jacques Rivière also expressed effectively the deeper reasons for the overthrow of perspective and consistent lighting by the cubists. Lighting, he says, " is the sign of a particular instant... It has the effect of profoundly altering the forms themselves... It can therefore be said that ligthing prevents things from appearing as they are". Likewise, "perspective is as accidental as lighting... It indicates not the situation of the object, but of the spectator"...

In the last couple of decades another kind of meaning analysis has developed which addresses itself to similar problems from a somewhat different-looking vantage-point. This approach is often referred to by the metaphysically loaded term 'possible-worlds semantics'. In reality its basic ideas are not very far from Frege and Husserl. Frege said that in his notion of Sinn or sense more is involved than the reference. It includes also the way in which the reference is given (die Art des Gegenbenseins). Now possible-worlds semantics arises when it is realized that all such talk of 'ways of being given' is functional, that the only reasonable way of understanding Frege's statement in the last analysis is to interpret the sense or Sinn as the function which gives us the reference, by means of which we as it were can find this reference. This is already one of the main ideas of possible-worlds semantics. It is not so far from Husserl, either, for what Husserl was interested in was precisely those 'vehicles of directedness', the noemata, which enable us to intend or refer to objects...

But what are the arguments of these functions? What are they allowed to depend on?... At the abstract level on which possible-worlds semantics is moving, the only general answer to the argument question is therefore: the whole possible world in question. It is for this reason that possible-worlds semantics say that meanings are functions from possible worlds to extensions or references. Given a term and a world, they yield as a value the object the term refers to in that world...

One problem we face is due to the fact that one cannot grasp a meaning function as the abstract entity it is in the eyes of a logician. For him, a function is a class, usually an infinite class, of pairs of correlated arguments and values. Such a class is not usually apprehended directly, but only by means of some 'algorithm' or 'recipe' for obtaining the value from the argument. This already complicates the situation, and partly explains why it was so hard for Frege and Husserl to reach the abstract interpretation of meanings as functions just outlined.

Curiously enough, cubists were sometimes accused of a 'mistake' which is parallel to the mistake of considering meaning functions in the abstract, as mere classes of pairs of arguments and values. Jacques Rivière once claimed that the "first mistake" of the cubists was to try to represent an object from all the different angles at one and the same time. This is of course not the same thing as representing it in all possible situations, but the analogy is nevertheless amusing...

An especially important group of questions that arises here concerns the constant functions which in different circumstances or situations (in different 'possible worlds') pick out the same object or individual. Such meaning functions define what it is to be one and the same individual. In the Husserlian jargon, they constitute the objects we think and speak about. Questions of this sort are what philosophers call questions of individuation. In thinking of them, it may be of some illustrative help to think first of the different possible worlds as existing separately and then think of the embodiments of one and the same individual in them as being connected by a 'line', the world line of that individual. (David Kaplan has coined a nice pun and called them trans world heir lines or TWA's.) They are as it were graphical representations of the meaning functions which tell us how find the same object or individual in different possible situations...

Now possible-worlds semantics allows for several different answers to the question as to how such individuation takes place. The easier answer would be that we recognize objects by their identical looks both in pictures and in real life. However, according to the cubists this answer just does not work. For we saw that they were not depicting objects and individuals by means of their appearances or looks, but by means of the properties they are known or thought to have. But what properties? Both in art and in philosophy one's first idea is to say here: by means of certain especially important properties. Philosophers call them essential properties... Braque mention as his most important representational means "volume, line, mass, weigth", which are all among philosophers' primary qualities. However, it seems to me that neither in art nor in philosophy is essentialism the essential thing...

In the realm of the logical and the philosophical individuation problem, it is the insight... that the individuation principles are not absolute in the sense that they are not fixed by Logic or Nature or by some equally inescapable power. The world lines are in principle "drawn and described" by ourselves. This does not mean of course that they are arbitrary. They may be grounded in our innate nature and also in the laws of nature. The point is that all this still in principle leaves room for more than one way individual objects are 'constituted', for more than one individuation method...

This specific doctrine gains some additional interest from being o kind of natural sequel to a most important overall change in philosophers' and logicians' attitudes to the logic of our own language. The two contrasting conceptions concerning it have been labelled by van Heijenoort 'logic as language' and 'logic as calculus'... In the conception of 'logic as language' we have a view to the effect that our language and its logic is the inescapable medium of all communication. We can modify it by small steps, but we just cannot seriously contemplate an altogether different system of logical rules and of representational relationships between language and reality. We just cannot get outside our language according to this view. This type of conception was represented by Frege...

In contrast to this, the conception of 'logic as calculus' allows us as it were to step back and have a detached view from the outside of the whole of our own language, as if it were an artificial calculus to be freely interpreted. The possibility of such bird's-eye view includes the possibility of allowing changes in the representational relationships between our language and the reality. In technical jargon, it allows the development of a model theory of our language, for the basic idea of this theory (sometimes also called logical semantics) is just to study a language by letting its interpretation vary systematically... It is no accident that the development of logical semantics and of model theory has been one of the most important large-scale changes in modern logic and in contemporery philosophy. The growth of possible-worlds semantics is one aspect of this development....

Now here we have another analogy with cubism which in my view is even more important than the analogy between phenomenology and cubism... What the cubists did was to substitute for the traditional illusionistic and naturalistic system of representation others which were at least partly created freely by the painter himself. This is the explanation for the apparent abstractness of many cubistic pictures. It is not that these pictures do not represent. Rather, they represent their objects by means of conventions, by means of a 'language' which the spectator has not yet learned. Once one realizes this 'key', one finds much more method in the apparent madness of cubist paintings.

There are in fact clear-cut representational principles underlying the pictures of the cubists. For instance, in analytic cubism the painter as it were split the surface of the object into small facets and then turned most of them into one and the same plane represented in the picture. On other occasions other rules were followed... It is also fascinating and instructive to see how Picasso takes one set of representational conventions, for instance the ones Velasquez had used, and transposes them into his own system, or, more accurately, into one of the many systems he is using...

That one of the main novelties of the cubism was the rejection of naturalistic representation is of course a common place. What is not appreciated equally often are the realistic motives of this rejection... In his inimitable ironic way, Picasso emphasized this point in one of his revealing discussions with Kahnweiler. They had been discussing Michelangelo's 'Last Judgement' and Picasso had emphasized the impossibility of painting such a picture "honestly" and without presenting mere "charm" by using naturalistic means. "Look at Michelangelo's 'Last Judgement'", Picasso said. "Is it truly worked out? Is it truly worked out rigth down to the nostrils of Christ's nose? Of course not! It is just charm, decoration. When you reduce all that to its basic elements there is enough left to make a nice tie. To produce a big picture like that would need the help of every painter since the beginning of time and even so it would not be finished yet"...

We can conclude with Kahnweiler: "I began to see what he was leading up to. He meant that Cubism was the only honest painting and that honest painting could be conceived only in the form of language with invented signs and no attempt to imitation." Picasso assented to this and reiterated, "Cubism is the only real painting"...

Cubists emphasized the freedom of choice here and especially the resulting relativity of representational conventions. This feature of the cubist 'philosophy' and its reasons are summed up very well by Robert Rosenblum who writes: "The inevitable conclusion is that a work of art presents a complex interchange between artifice and reality. A picture depends upon external reality, but the Cubists means of recording this reality - unlike the means devised by the Renaissance - are not absolute but relative. One pictorial language is no more 'real' than another...". Yet it is far from obvious that this relativity is absolute. Are there limits to what a painter can do on his own and yet be understood by others? It is clear that in the apparent freedom which an artist enjoys in choosing his own language we have one of the reasons why cubism has in practice led artists toward greater and greater abstraction. It is also clear that in this direction we soon run into the aesthetic counterpart of Wittgestein's famous problem of the impossibility of completely private languages...

When cubists renounced the appearances of objects for conceptual reality, some of their early interpreters compared this move to the idealistic philosophy of Immanuel Kant. This is misleading, for the representational freedom which the cubists asserted goes much beyond the innate forms of sense-perception which Kant emphasized and which are grounded not in artistic choice but in immutable human nature...

If one allows oneself full freedom in varying the representational relationship between language and reality or between pictures and reality, one will come up with surprising results. In logic, a technique sometimes used in model theory is a nice instance of this. It was apparently first used by Leon Henkin in his important completeness proofs around 1950, and has been frequently employed since. It consists in interpreting certain sets of formulae as literally speaking of themselves, as being the very reality they speak about. This idea of making as it were a chunk of language into an independent reality of its own has a neat counterpart in the artistic repertoire of the cubists and their followers. For that is precisely what the technique of collage amounts to: you insert an actual object, a chunk of reality, into your picture. Thus the so-called model sets and other sets of formulae one can use in the Henkin technique are precisely logicians' collages, we may perhaps say...

Indeed, what the cubists did was even more ambivalent than this. The very first collage on the history of modern art, Picasso's 'Still Life with Cane Chair' of 1912, is enough to show this. There Picasso represents the bottom of the chair, not by painting it, but by pasting a real object on the canvas. However, the irony is heightened by the fact that this 'real' object is itself 'false': it is not a piece of real caning but only a ready-made piece of wax cloth made to look like caning. The nice self-irony of this trick illustrates strikingly cubists' emphasis on the relativity of all representation...