1 the Following Piece Appeared in the Dutch Literary Magazine “De Gids

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The following piece appeared in the Dutch literary magazine “De Gids” (http://www.literairtijdschrift-degids.nl/boekboek/show/id=59346), Amsterdam, March 2008, pp. 191 – 205.

A Stamp Full of Logic

A stamp with a logical formula Last autumn, thanks to the efforts of Dirk van Dalen, the Dutch postal service came out with their first stamp featuring a mathematician and logician. The media paid ample attention to our national genius Luitzen Egbertus Jan Brouwer (1881–1966), pitching their tones ever more highly, up to the canonization and subsequent straight ascent into the heavens in Rudy Kousbroek’s piece in our quality newspaper NRC Handelsblad.

But even so, a question remains for many. What does the cuneiform writing

|= A ! ¬A on the stamp really mean? Well, maybe we should leave the mysteries of our technical formulas intact. The motto for the exuberant bicentennial celebration of the Royal Dutch Academy of Sciences KNAW this year is “The Magic of Science”, but who knows, upon real explanation, it turns out to be black magic...

The Law of the Excluded Middle But that would be a pity for the logical formula on this stamp, which intimately touches so many Dutch lips these days. What it represents in compact form is Brouwer’s famous denial of a main principle of classical logic, namely, the Law of the Excluded Middle. This principle, known since Antiquity, says that, for every statement A, either A itself, or its negation (‘not-A’, or in notation: ‘¬A’) holds. Please note, Excluded Middle does not say that we know which of the two cases occurs, but in principle, one of the two 2 alternatives obtains. This sounds harmless and evident, and it seems hard to disagree. In this connection, one word about the notation on the stamp. Logicians use the symbol |= for validity of a statement, and the little cross-bar stands for the negation of this. But then, why not write this denial with the negation hook ¬? In order to explain that, we need to dig deeper. As in mathematics, in logic, simple notation often has deeper backgrounds.

Classical propositional logic Let us first take a look at logic the way it was when Brouwer was a student (one can see his entry in the exam book of the University of Amsterdam, the same books we are still using today to record our exams and dissertation defenses). Let’s move really fast. Ever since Aristotle and the Stoics, we know that much reasoning revolves around a small set of key notions, such as “and” (conjunction, "), “or” (disjunction, !) and “not” (negation, ¬). Who doubts that, should just travel on any Dutch train, and observe commuters. The free newspapers Metro and Spits are the most widely-read logic journals on this planet! Their Sudoku puzzles are all about logical patterns like “this square must have a digit x or y, but it cannot be x, and so it is y”, or in the notation of the stamp – with ‘A’ for the statement ‘the square has digit x’ and ‘B’ for ‘the square has digit y’:

A ! B, ¬A |= B, the conclusion B follows from the premises A ! B and ¬A.

The mathematical system of such rules, ‘classical propositional logic’, has been well- understood for centuries. It was codified by George Boole as an elegant form of algebra in his Laws of Thought of 1847. That algebra is essentially a form of binary arithmetic with two truth values 0 (‘false’) and 1 (‘true’) – an analogy already observed by Leibniz around 1700, who envisioned computing machines that would test logical validity with binary computation. That has taken a while, but since 1950, Boolean circuits have governed your computer, and they still do – whatever your newspaper science supplements may tell you about more futuristic quantum computing techniques.

The Three Main Laws But now for those ‘main laws’. Three basic principles of Boolean algebra are in fact modern mathematical versions of three classical laws of traditional logic. The first is

A = A The Law of Identity.

This does not look particularly exciting: 0 = 0 and 1 = 1, and that is that. Even an extremely quarrelsome person like Brouwer never pointed his arrows at it. Even so, we 3 will find a surprising modern reinterpretation for this principle later on. But for now, our second classical principle of logic:

¬(A " ¬A) The Law of Non-Contradiction

This principle says that no statement is true and false in the same respect. Despite some murmurings in dialectal and mystical circles (worth a story by itself), this, to, seems an unproblematic insight, and indeed: Brouwer never attacked it.

Non-Contradiction also occurs in other cultures where logic arose independently. For instance, the Mohist tradition in China around 500 B.C. reads it as a rule of conversation: contradictions between discussion partners should not be accepted, but resolved. Culture jumps are natural to a logician. Already Leibniz saw China as a serious dialogue partner, and even sent (unsolicited) advice to Rome about religious contacts.

The Mohist also subscribed to our third main law, which we have seen already:

Law of Excluded Middle A ! ¬A

According to Boole this compound statement always gets value 1. If A has value 0, then ¬A gets value 1, and using addition, the disjunction gets value 1. If A has value 1, things end up likewise. There is no third option. Here, too, the Mohist interpretation is more oriented toward conversation. Participants in a dialogue should eventually choose one of the statements A or ¬A, though maybe not right at the start. This point will return.

This third main law has been debated since Greek Antiquity. Aristotle himself thought it dubious for statements about the future. Whether there will be a sea-battle to-morrow is not yet true or false right now, otherwise, the future would be fixed deterministically.

By the way, Greek stamps about Aristotle exists in many sizes and denominations:

And other cultures know how to honour their logicians, too. Yet more stamps circulate all over the Islamic world of the great scholar Avicenna (active around the year 1000) – though they mainly celebrate his medical, rather than his logical innovations:

Brouwer’s objection: from truth to provability But now, what was Brouwer’s problem with A ! ¬A? That the Dutch stamp does not postulate a valid principle, but an objection to other people’s views, is quite in style. As Pim Levelt, the President of our Royal Academy KNAW, once said during a ceremony at the Blaricum cemetery where Brouwer lies buried, “This tombstone was only realized after a long history of conflicts – and that, ladies and gentlemen, seems to me entirely in the spirit of the deceased”. But Brouwer’s doubts concerning Excluded Middle did emanate from his positive ‘intuitionist’ view of mathematical truth. This revolves around the activity of a mathematician who constructs objects, and proves their properties constructively. On this view, made explicit by Heyting in 1929, but also occurring with Russian contemporaries like Kolmogorov, proofs for complex assertions can be constructed systematically from proofs for simpler component assertions. For the Boolean key notions, this works as follows. A proof for a conjunction A"B consists of a pair of proofs, one for A and one for B. A proof for a disjunction A!B is either a proof for A or one for B. And a proof for a negation ¬A is a refutation of A, or more precisely, an effective method transforming each proof of A into one for a manifest contradiction.

Read in this way, Excluded Middle undergoes a dramatic character change. In this new interpretation, it says that each mathematical statement A is either provable or refutable. And this, of course, is not plausible, and in fact, it is false. Ever since Gödel’s Theorems of 1931 we know that ‘incompleteness’ is the norm in mathematics: some statements are neither provable nor refutable in the most common theories of arithmetic, analysis, or set theory (always assuming that these theories are consistent). So far, Kurt Gödel, often 5 considered the greatest logician since Aristotle, has not received more than a limited- edition stamp for collectors in his native country of Austria:

Even so, Brouwer’s objection was by no means absolute. In particular, he agreed that many mathematical settings do admit of reasoning with Excluded Middle, provided they are ‘simple’ enough – as is the case, in particular, with finite structures. But the extrapolation to infinite structures was problematic on his view.

But are we still talking about the same thing? At this stage, you might object at once that the above objection does not affect classical Excluded Middle at all. For, a major shift has taken place, from speaking about truth to speaking about provability. So, Brouwer did not fight classical logic on its own territory, he changed the agenda. Making this distinction is the ‘polder solution’ reflecting age-old Dutch habits of compromise, probably favoured by most professional logicians. Classical logic is right about truth, but Brouwer got things right about provability. If you are quarrelsome, and are spoiling for a fight, then you have got to twist some ideological screws. Thus, defenders of philosophical ‘verificationism’ claim that provability, or more general ‘verifiability’ is the only sensible notion of truth. Then classical logic becomes a system devoid of sense, at best ‘correct for its own cult members’. But in reality, though verificationism has some famous adherents, it is the smaller philosophical church.

Intuitionist logic: the constructive alternative However all this may be, a mere objection to someone else’s logic is hardly a ground for scientific greatness. What makes Brouwer’s insights so remarkable is really the positive alternative that turned out to lie behind his general views of mathematics, a system known today as intuitionist logic. The proof interpretation of the logical notions turned out to validate a system of laws of its own which combines consistency with beauty. For instance, the Law of Non-Contradiction still holds. Assuming that we have both a proof and a refutation for A is untenable, and this observation itself is a proof for ¬(A " ¬A). 6

And many other useful principles remain correct, be it with a new proof-theoretic flavour. In particular, all inferences needed to solve your Sudoku are intuitionistically correct, so that may be a relief when you are traveling home on the train. What does happen is that classical reasoning principles often ‘split’ into variants, some intuitionistically correct, others not. A beautiful example is the famous method of ‘proof by contradiction’. In classical logic, one can establish an assertion ¬A by first assuming A, and then deriving a contradiction. This much-used pattern is totally unproblematic to an intuitionist. But classical logic also has a more mysterious variant: prove an assertion A by deriving a contradiction from its negation ¬A. This second method is intuitionistically unacceptable: ‘refuting refutability’ is not the same as positive proof. The former road does establish something, but is only the double negation ¬¬A. Connected to this is the fact that intuitionists reject the classically valid equivalence between the two assertions ¬¬A and A. In general, they find the first weaker than the second.

From proof to information Over time, it has turned out possible to interpret intuitionistic logic in new ways, not directly related to proof, with Brouwer’s original observations again high-lighting some interesting phenomena. Evert Beth, Brouwer’s colleague in the 1950s at the University of Amsterdam, interpreted intuitionism in terms of semantic models of growing information stages, from poorer to richer, where statements are always interpreted at stages of the process. A mathematician, or a rational agent in general, gradually acquires more information, moving upward in this order. And now we see a peaceful coexistence between two logical systems. At final stages of the process, where all information is in, classical logic holds. But in intermediate preliminary stages, where we normally find ourselves in life, classical laws may fail. Let us call a negation not-A true at a stage of the model when A has been refuted right now already: no further richer stage makes A true. Under this interpretation, it is easy to see that Excluded Middle will fail in general. Take an extremely simple information model with two stages, an initial one where we do not have A yet, followed by a final stage where we do have it:

A 1 2

At the first stage 1, A fails by assumption, but neither does not-A hold there according to the reading of negation that we just gave, since we do get A at stage 2 after all. Thus, the disjunctive statement A ! ¬A does not hold at stage 1.

A bit of feeling for finesse is required here. In Beth’s semantics, and intuitionist logic in general, there is a delicate difference between saying that ‘A does not hold’ in some situation, and saying that ‘not-A holds’ in that situation. But you really saw that already on the stamp. Brouwer’s objection is formulated with a little ‘bar’, standing for the first, weak negation ‘not’. It is not formulated with the intuitionist’s own strong negation ¬. That would even be impossible, as Brouwer considered the strong negation of Excluded Middle contradictory – and hence the following principle is intuitionistically valid:

¬¬(A ! ¬A)

But is not this just Excluded Middle, modulo some contraction of a redundancy, viz. the two negations? It looks that way, but: as we already saw, the classical law ¬¬A # A is intuitionistically invalid… The distinction is easily visualized in Beth’s models: ¬¬A unpacks to the assertion that for each stage, there is a later stage where A is made true (A is ‘inevitable’), but this is not the same as saying that A is true right here, right now. You may know who is the person in front of you, but you ‘have not been introduced yet’. For instance, in stage 1 of the above picture, ¬¬A is true, but A is not.

Here we get to a fateful parting of the ways. To many, the preceding is insufferable scholastics and juggling with negations – but to others, a new style of thinking opens up, revealing subtle distinctions that are suppressed in classical logic. In any case, it seems fair to say that intuitionist logic provides a coherent and delicate description of constructive reasoning in a context of proof, information, and computation. It has not managed to edge out classical logic as a norm, let alone as a discipline, but it has found its own place in the foundations of mathematics and also computer science. That place is not as vast as some modern partisan of Brouwer would like you to believe, but is also not as tiny as many standard logic text books would have it, who either ignore intuitionism, or just accord it some minor air-play as ‘the revolution that failed’.

The story goes on: intuitionism and dialogue games Intuitionism is often associated with the foundations of mathematics, and pure proofs of extra-terrestial precision, leaving no room for human weakness and conflict. But ideas have their own evolutionary dynamics, and we can equally well position Brouwer’s work 8 in a very different logical tradition going back to Antiquity, viz. that of logical patterns in dialogue and debate. And really, that may be an appropriate perspective for mathematics as well. A proof is not, as many people seem to think, a ‘conversation stopper’, but rather the ultimate attempt at clarity toward others, and hence at intersubjective communication. And so our story of logical laws takes a turn toward argumentation and games.

As early as the mid 1950s, this turn was made by the German logician Paul Lorenzen, who was looking for an underpinning of the logical laws in our daily practice of argumentation and interaction. His remarkable idea was that the logical core operations “or”, “and” en “not” will then function as a sort of ‘switches’, not just in a Boolean computer, but also in a discussion. When I defend that A ! B, then you can hold me to this, and I have to choose eventually which of the two I will defend. Thus, a disjunction offers a choice to its defender – and likewise, a conjunction A " B offers a choice to the attacker: since the defender of a conjunction is committed to both its parts. In this manner, an argumentation game unrolls between the defender of an assertion and its attacker, and the account just given also immediately explains in an attractive ‘dynamic’ manner why conjunction en disjunction behave so analogously from a formal standpoint. They are really the same act, namely choice, but performed by different players. Moreover, I a dialogue, interesting interactions arise by means of the third item of Boolean algebra: logical negation. This triggers a role switch: defending ¬A is attacking A, and vice versa. Indeed, being able to ‘put yourself in another person’s place’ has been called the most essential human cognitive achievement. In this way, logic comes to describe the structure of rational interaction between conversation partners.

In this setting, Lorenzen called a logical inference valid if the defender of the conclusion has a winning strategy: that is, a rule for playing which will always lead her to win the game against any defender of the premises. Here is a simple example with the earlier Sudoku rule. When you defend a conclusion B against someone defending premises A!B and ¬A, then first attack that disjunction, forcing him to choose. If his answer is B, then you win at once – and if his choice is A, then you can now safely attack A, since he has just placed himself in the shameful conversational position of ‘self-contradiction’: A, ¬A. General winning strategies may of course be much more complex that this simple gambit: after all, successful argumentation is a highly non-trivial human ability.

Here is the connection to our stamp: we are still on its trail... What Lorenzen observed is that Excluded Middle is not plausible in this game setting. The defender of A ! ¬A need not have a general strategy telling her infallibly which of the two disjuncts to choose, and then win. In fact, the logical validities for which winning strategies do exist in Lorenzen’s dialogue games are precisely those of Brouwer’s intuitionistic logic! It has to be said, though, that in the meantime, dialogue games have been found that do match classical logic, and the difference is actually an instructive point of additional procedure. Their debating rules allow the defender of a disjunction to revise her initial choice later on. In this light, intuitionistic logic is that of an immediate righteousness, while classical logic treats us more humanely. Lorenzen’s preference was clearly for the first. He was an intimidating person, and when I once had to accompany him as a young assistant professor for a lunch at Groningen university, I broke my knife in terror.

Logic and solving games Lorenzen’s work has had little impact on the foundations of mathematics, and it ended largely as a source of inspiration for general ‘argumentation theory’, also in The Netherlands, the very homeland of intuitionism. But the idea that logic and games have intrinsic ties has persisted, and is even coming to the fore much more these days. Important influence here have come from informatics (sometimes called ‘computer science’) , a fundamental discipline where logic still supplies foundations for the study of processes involving information and computation. And modern computation, you know it from the email and internet in your life, has long ceased to be a matter of just Boolean circuits in a lonesome computing device, but rather a complex interactive information- processing phenomenon spread out over large networks of ‘agents’.

Now this contact with informatics has many facets today, most of them actually running via classical logic. And even more to the point, in the games arena, Excluded Middle has some of its finest hours! To see this, let us go back in time to a contemporary of Brouwer, the German mathematician Ernst Zermelo, one of the founding fathers of modern set theory. Zermelo was interested in the following question: “Who has the best starting position in the game of Chess?” (Great mathematicians have a broad range of interests.) In 1913 Zermelo proved the following theorem, now named after him. In a finite game for two players with only two possible outcomes ‘Win’ or ‘Lose’, where each run of the game must end after finitely many steps, one of the to players must have a winning strategy. We give the attractive mathematical proof here:

“Suppose that the game lasts for only 1 move, and Player I starts. Then either I has a move that makes her win (and her winning strategy I “choose such a winning move”), or all initial moves lead to loss for I, and in that case, it is Player II who has a winning strategy (‘just wait, and win’). When a game lasts for two rounds, we just repeat the given reasoning. Either the starting player has a move taking her to a position where she has a winning strategy, and then she has a winning strategy in the whole game – or all moves of the starting player lead to a position with a winning strategy for the other player, and then the latter player has a winning strategy. And so on for longer games.”

This line of reasoning can even be used to write a mechanical algorithm that traverses the tree of all possible moves in a finite game, and determines node-by-node which player has the winning strategy there. This algorithm is the basis for all further sophisticated methods that solve games today using computers, a form of ‘meta top sport’ where our Dutch IKAT/MICC group in Maastricht is a major international player.

Of course Chess has a third outcome option, viz. a draw. Zermelo’s theorem says in this case that either one of the players in Chess has a winning strategy, or the other has a ‘non-losing strategy’ ensuring she always wins or draws. This result was rediscovered independently by the Dutch world champion in Chess Max Euwe in 1929. As it happens, almost a century after Zermelo’s result, we still do not know which of the two options holds for Chess: the complete game tree is just too large to comb through. In that sense, even the finite Excluded Middle is an idealization, which need not lead to explicit knowledge. But the clock is ticking for Chess. Recently, for the board game” Checkers”, an American variant of our Dutch ‘Damspel’, 15 years of continuous computer labour yielded the Zermelo answer: the starting player has a non-losing strategy.

If one thinks for a moment about Zermelo’s argument as sketched above, the pivotal role of Excluded Middle will be clear, and indeed, the power and elegance of this principle. Of course, intuitionists would not object: the games considered were finite. For infinite games, matters quickly get more delicate, and we will now follow our thread into some other disciplines. Logical themes often ’cross’ from one academic domain to another, and that nomadic ‘walkabout’ behaviour is precisely what makes them so fascinating.

Non-determined games, set theory, and game theory A game in which one of the two players has a winning strategy is called determined. Now, are all games determined? With this simple question, we are right in the 11 foundations of set theory. Examples have been found of infinite non-determined games, but their construction turned out to depend strongly on the mathematical axioms one assumes for sets, in particular, the famous ‘Axiom of Choice’. Therefore, in the 1960s, it has been proposed to turn the tables’, and just postulate that all games are determined. This Axiom of Determinacy might be viewed as ‘Excluded Middle run wild’, but then a gallop with beautiful mathematical consequences. There is a broad consensus today that set theory needs new axioms, but much less: which ones, and Determinacy is just one option. In any case, it may be said that games are an important source of intuitions here.

But set theory still is not game theory in the usual economic sense. The latter discipline describes more refined preferences that players have concerning the outcomes of a game, far beyond winning and losing. Accordingly, mathematical game theory emphasizes ‘equilibria’ between strategies chosen by players, in which no one can improve his pay- off by unilaterally deviating. The recent movie “A Beautiful Mind” tells a little about this equilibrium theory, and a lot about the tragic life of its main founder John Nash. Indeed, Nash equilibria describe interactions between players far beyond Zermelo’s framework, and are essential to the study of realistic economic and social behaviour. But even in this richer area, lively contacts are growing between logic and game theory, sometimes under the slogan of ‘intelligent interaction’.

We seem to have strayed far from our stamp by now. But even in this broader field, we meet with Brouwer. The other Brouwer, to be sure, pioneer in topology, a branch of mathematics emerging in the 20th century as the most abstract study of spatial structures. Brouwer’s fame with mathematicians largely rests on his Fixedpoint Theorem, a fundamental result with a wide range of surprising applications. It says roughly that all continuous functions on suitable topological spaces have ‘fixed-points’: points x where the function value f(x) equals x itself. Now, game-theoretic equilibria and topological fixed-points turn out closely related, and – as was already observed by John von Neumann, important game-theoretic results follow from Brouwer’s Theorem. Incidentally, in his proof for that theorem, Brouwer did not impose any intuitionistic restrictions, and indeed, strictly speaking, the Fixedpoint Theorem is even intuitionistically unprovable. (Discussion about constructive variants continues…) So, Brouwer preached against Excluded Middle, but he was no stranger to sin.

Logic, interactive computation, and linear game algebra Our long story of Brouwer’s intuitionism, classical logic, informatics, set theory, and game theory has one last surprising twist in store. Excluded Middle remains ever active as a source for new insights. To get to the edge of current research, we first return to the basic notions of logic itself, which are still a matter of debate today. In games, dialogue, and argumentation, interaction between two or more persons is the central feature, creating a complex process taking place over time. But then our earlier interpretation of the logical disjunction ‘A or B’ as a choice for one of the sub-games A, B seems rather drastic. For, that choice needs to be made ‘globally’ at the beginning, when we know nothing yet about what happened in the course of those sub-games. For this reason, it has been proposed to add a second logical form of choice. Now, A, B are both played in parallel, while the player who has to choose may decide ‘locally’, at each of his turns, in which of the two games, the next move is to be played. This delayed choice is reminiscent of what we saw earlier with Lorenzen dialogues for classical logic, and with some historical good-will, also with the Mohists in China. A much-cited, and often rediscovered example gain comes from Chess. Here is a method you can use to beat Bobby Fisher, also known as the ‘Copy Cat’ strategy:

“You play two chess games simultaneously, once as White and once as Black. You let Fisher open as White, then you copy his move into the other game as White, and you stay there, forcing him to respond as Black with an answer to your (i.e., his own) opening move. Next you copy this answer in the other game (you are the one with the power to switch), etcetera. In this way, both games will get exactly the same sequence of moves, and you must win one of them (or draw in both).”

The corresponding notion of ‘parallel disjunction’ A + B was proposed around 1970 as a new logical operation by the American mathematician Andreas Blass, triggered by a study of Lorenzen dialogues. As we have seen already, in games, classical Excluded Middle need not always hold for the initial choice disjunction !. To have a winning strategy, the starting player in

A ! ¬A must chose right at the star, and therefore, she needs to have a winning strategy in A of ¬A separately. But this was exactly what failed in non-determined games A (unless we 13 postulate the Axiom of Determinacy). But the Copy Cat argument does not assume at all that the game is determined, and thus, it shows that quite generally, the logical principle

A + ¬A is valid without any restrictions! In other words, ‘delayed’ or ‘interactive’ choice A + B does satisfy Excluded Middle. Therefore, the validity of this principle depends on how we read the disjunction, and for that, there are different legitimate options. In this way, we suddenly see a new richer world of logical operations with their own laws. Observations like this have led to a much richer form of game algebra, with more operators that those of either classical or intuitionistic logic. Important contributions have been made in recent years by Jean-Yves Girard and Samson Abramsky, and the area continues to flourish. Thus, Boolean algebra turns out to be just one corner of a beautiful general theory of complex interactive behaviour.

This modern view of logic as a rational interaction between many actors is radical, and too much to swallow for most colleagues. But it does give interesting new meanings to the classical laws of logic that we started our story with. We saw this for the flamboyant Excluded Middle, but it is equally true for inconspicuous domestic principles like the earlier Law of Identity. From the commonplace that A = A this becomes the Copy Cat rule ‘Make your A into my A”, and then it suddenly becomes a key to interaction. This perspective makes old conflicts between defenders and critics of Excluded Middle somewhat obsolete. With the distinction between ! and + and further game operators, the whole debate about ‘competing logics’ becomes one notch more refined.

A series of stamps? It will be abundantly clear that Dutch TNT Post is not done with logic yet. The Brouwer stamp merely honours an objection to classical logic. It seems reasonable to me, at the very least, to also add a positive statement, in the form of a valid intuitionistic law:

|= ¬¬(A ! ¬A)

But most attractive of all would be a series of three stamps, culminating in Excluded Middle in its modern game-based interpretation:

|= ¬¬(A ! ¬A) |= A + ¬A

¬¬( The choice of the values for those three staAmps, I gladly leave to our Postal Service.

¬ ! A) Conclusion This is about all that can be said about Brouwer and Excluded Middle. Or not.