Answer: This is very perceptive, and it strikes a chord! I am a logician, I design and study formal systems in the standard style you describe. And nevertheless, I feel uneasy about it. It is definitely true that nobody uses formal systems with completely formal languages in some absolute sense. Even mathematicians use natural language, enriched with formal notations when they feel the need for it. And this mismatch strikes when you explain the importance of logical results to others, since the formal languages and formal systems are an impenetrable barrier. Say, Gödel’s results are about the incompleteness of formal systems of (first-order) arithmetic, whereas ordinary people, including mathematicians, thinks of ‘Arithmetic’ as an open practice – and it is much less clear what the logical results say about the latter. Likewise, traditional logic was about broad phenomena in reasoning, such as ‘negation’ or ‘quantification’, rather than what we now teach: the ladder of formal systems, from propositional logic via modal logic to first-order logic and beyond. Fortunately, logicians do not notice this problem, since they can publish formal systems and meta-theorems about them, without ever applying them to anything or explaining them to a broader audience. This ‘closed world’ can give you tenure and respect, so why worry?

Nevertheless, I am worried by what I call the ‘system imprisonment’ of modern logic. It clutters up the philosophy of logic and mathematics, replacing real issues by system-generated ones, and it isolates us from the surrounding world. I do think that formal languages and formal systems are important, and at some extreme level, they are also useful, e.g., in using computers for theorem proving or natural language processing. But I think there is a whole further area that we need to understand, viz. the interaction between formal systems and natural practice. Take mathematics itself. In the course of history, it has become a dynamic mixture of natural language and natural reasoning with formal additions: notations, proof patterns, and so on. Practizing mathematicians have recognized this, and developed hybrids of formal systems and natural language, such as De Bruyn’s ‘mathematical vernacular’, or Peter Koepke’s project on proof structure in Bonn today. More generally, philosophers like Frits Staal have emphasized the crucial role of ever new notations and their successful incorporation into existing scientific practice, in both Western and Asian traditions.

As a logician, I see an intriguing dynamics here. Mathematicians invent new notations and proof styles under pressure to become more precise, in scientific communication and face-to-face debate. Thus, more formal language enters natural language. But this happens in tandem with a reverse process. Good mathematicians are also able to step back from formalisms, and give a higher-level description of a proof in more natural language. We all feel we only really understand a proof when we can describe it at a variety of such levels. Indeed, I think that the mutual dynamics of precisation and paraphrase is an exciting form of ‘generalized proof theory’ whose study might be of benefit to everyone! And much more generally, the continual ‘successful insertion’ of cultural artefacts: notations, logical inferences, algorithms, new games, and so on is the hallmark of human behaviour. This is where logic meets reality. Even cognitive science in general would do well to understand this mixed practice.

Question by Yizhao Hu: I also have a very general question. I guess many logic students would appreciate your answer to this question. What is the best way to understand the theoretical value versus the practical value of logics? More specifically, for instance, why is completeness important to us?

Answer: I think one should pursue academic subjects because one likes them – and perhaps liking is not even enough, but there should be real love. There is the famous anecdote of Euclid who once, when asked by someone in the street for the practical value of mathematics, asked his slave to give that person some money, since he felt pity for people who go for practical value in life. How about that for an answer?

I see the main value of logic as theoretical. It is a wonderful style of thinking that provides foundations for many other sciences, and a discipline of proven value for abstract conceptual analysis. Moreover, logic is a unifying force in a fast-splintering Academia: climbing its mountains (as Fenrong once put it in a piece here in Holland) helps us see broad patterns and new connections between disciplines all across the university. That is why my own Institute for Logic, Language and Computation is the only university-wide research institute in Amsterdam. Logic works! In addition, and this has to do with falling in love, intellectually just as much as physically, logic can be beautiful, not always, but definitely in the right light at the right time of day…

Of course, and here your last example is to the point, this beauty sometimes resides in the ‘uselss’ results. Take the completeness theorem. For practical purposes, soundness is usually enough: we need to know that the inferences which we perform ar correct, so that our reasoning does not go badly astray. The reverse question, whether our system really captures all valid styles of inference, is much more a theoretical concern in the distance. But even so, what a nice concern, and what beautiful consequences, ever since Gödel’s dissertation proved completeness for first-order logic!

Now, many colleagues will give you equally strong answers on the practical applied side. It has been claimed by leading authorities that ‘logic is the calculus of computer science’, meaning that the ability to work with logical systems and methods makes you a successful engineer in information technology, software engineering, agent systems, and so on. I am sure that such uses exist, and logicians do work in industry and beyond. Likewise, it has been claimed that logic helps in computational semantics of natural language, legal reasoning, general argumentation analysis, and maybe even, that it helps you become a better public debater. (I have not noticed this phenomenon with myself yet, but there is still time…) And even police and military forces in some countries have found it useful to teach logical techniques of reasoning.

I do not want to put all this down, but for myself, I will give another answer here. It arises because people often ask me to justify why I have taught logic to generations of students. This cannot be just in order to train future research logicians, since we do not need huge amounts of these. But what I see myself as doing is this. I teach students to think in a certain abstract logical way, not as the only way they have in their repertoire, but as an ‘insertion’ in the earlier sense, or an ‘enhancement’ to what they are already able to do. This style is abstract and formal, but when used well, it can also be used to setp away from existing reality and see new patterns and options. Good logicians are creative, flexible, and can think outside of existing patterns. To me, then, ‘theory is the best investment in practice’! People with logic training can do any profession afterwards, but they do it ‘with an extra’. And this is not just self-serving speculation. I once organized a meeting on hiring students with entrepreneurs from Silicon Valley. And what they all said was this. “We want well-trained people in fundamental scientific thinking, not applied people educated to fit our current production line.” And the reason was clear. Firms today do not even know what they will be producing in some years from now, but they do know that the only invariant in this fast-changing world is theoretical insight and power of abstraction.

Question by Jiahong Guo: I have a small but very concrete question on an issue of translation. In your paper 'dynamic logic of preference upgrade', you discussed a very interesting notion, “regret”. You defined it as something that you would prefer but you know it is not the case. In Chinese, there are two possible words, “hou hui” and “yi han” to express the same meaning. Interestingly, the former term involves temporal elements. Namely one feels regretful after something happened in a way that was not expected. Inspired by this, can we add the temporal understanding into the explanation of “regret” in English? What do you think?

Answer: Finally we get to the delicate issues of translation! Indeed, in that dynamics paper, ‘regret’ was taken in an a-temporal sense in the paper you mention. That sense occurs, like in when I say I regret that, due to the Second Law of Thermodynamics, there is no never-ending energy supply. But I agree with you that many sense of regret in English are temporal, having to do with either the past (I regret the effects of my foolishness) or the future (I regret that I will have to die). Of course, in the paper, our main purpose was another one, namely to point out that regret can apply to situations that we know cannot occur – at least for people who are not out-and-out realists. Apart from ‘regret’, there must be many more cases in your translation experiences with my papers. Junwei mentioned my favourite term ‘fine-structure’, which in English means ‘detailed stucture’, but with a slight extra connotation of being ‘delicate’ and ‘good’. He said Chinese had no word with this same combination. I also enjoyed other examples in our email correspondence; these mismatches are not at all a nuisance, but rather little pleasures of comparing our two situations!

I hope we will also have a chance once to discuss your translation experiences more broadly. I realize that logic papers are special, in that they involve a lot of technical terms with fixed meanings, plus general intellectual style and jargon which is the same everywhere on the planet. But still, it would be intriguing to me to see just how logic terms translate into Chinese, and which obstacles one encounters. It has been claimed that modern logical systems are not ‘neutral’, in that they follow the subject-predicate structure of specific human languages, often Indo-European ones. On the other hand, I have never found that Chinese logic students have special difficulties with understanding modern logic, so one would want to see the abstraction level at which we communicate. (I remember this was my topic of conversation with the first student from mainland China that I ever encountered in my life, viz. Victor Guo in Stanford in 1983.) I wish I were a better expert in Chinese, so that I would get more out of this. Maybe in some years from now, I should try to translate your papers!

Question by Fenrong Liu: Jiahong’s question reminds me of the many emails exchanged between you and us translators from last October till now. We very much appreciated your patience and your kind explanations. But we have to admit that it is not an easy job to translate your work into Chinese. The general difficulty lies in the fact that it is rather a translation between two cultures than between two languages. Of course, we also have some difficulties in understanding logics. What is your impression of our questions?

Answer: Fenrong, now you raise all the hard questions. Do people really understand each other? Does language presuppose, or even determine culture, and vice versa? Is Whorf’s Thesis correct, and are we mentally locked into our language? This interview could get very long if we delved into all this! Of course,we are are not the first to raise these issues: they have been debated forever, in the case of Chinese and Western traditions, through all the centuries since our cultures first got into first-hand contacts. Frankly, I would be more interested in your own experiences as translators, and as students and researchers at the interface of Chinese and Western cultural than in my own views as a colleague and visitor from a distance.

But since you ask, maybe I will say this. I do think language, reasoning, and culture are deeply tied up. But many discussions about it seem fruitless to me, since one does not acknowledge some facts, and make some practical distinctions. First, given the relatively short period since our shared ancestors walked out of Africa, I would expect no very deep biological difference in neural-cognitive structure – though cultural artefacts like alphabet versus character script will no doubt influence some short-term processing facts. Now of course, culture provides a context where these basic skills are exercised, and clearly, when social life and educational organization are different, tendencies of interpretation and behaviour may well be different. But then I would say that all cultures face the same problems of arranging a just society and a reasonable life, so how much variation do we expect in stable solutions? I am often amazed at how similar situations are when you look at the invariances, rather than the accidental coding of behaviour. Chinese are supposed to be obsessed with ‘face’, but I can assure you the same is true in the West: it is just coded differently. Also, terms like ‘culture’ are so uniform that hey hide the enormous diversity between classes. Compare a Dutch intellectual to a Chinese farmer, and you will see worlds apart. But my guess is the same is true when you compare that farmer to a Chinese intellectual. Next, Chinese and Western intellectuals also share worldwide identities which make their personality no longer ‘mono-cultural’. Well, no conclusion here.

Personally, I am most struck by a simple empirical observation. All through world history, cultures have had contacts, mixed, taken over ideas, and so on. Good ideas – and alas, also bad ones – cross almost without effort, even though it sometimes may involve 'creative misunderstanding' (but that also happens between people inside the same culture). On a more specific note: consider the LORI lecture by Professor Zhang and Fenrong on Mohist Logic. I was struck by the amazing similarities with the Greek tradition, minds so far apart geographically seemed to have run on parallel tracks.

Once again, this is not an answer. I should learn from you, not the other way around.

Question by Jiahong Guo: Maybe this is a question that many of us want to ask, but I am saying it now. We have talked a lot so far on modal logic, but your research is not only on modal logic. Just to name a few, you have many works on spatial logic, logic of time, logic and cognition, and so on. Can you say something on the trend of logic in general in the 21st century?

Answer: My own interests are indeed much broader than Modal Logic, and over the years I have worked in several other areas. This is hard to avoid when you get older! Modal logic is my favourite methodology, though I do not see it as exclusive: I have also worked with first-order logic, and even type theory. And indeed, I have spent ten years of my life in the syntax and semantics of natural language, and other large chunks of time on methodology of science, connections with cognitive science, and recently, logic and games. I hope our translation Book series will eventually also show the full spectrum of these activities.

You ask for more general views on logic. I find it hard to repeat these here, since I feel I have produced a stream of programmatic papers and interviews on these, addressed to many different audiences. Maybe far too many already! My most recent lecture for the DMLPS Congress in Beijing will give you a fair impression. Another reason for a little modesty is that my views keep evolving, so one year’s truth can be another year’s falsity. For instance, our institute ILLC is a very different place today from when we founded it, and I cannot say that I foresaw or planned all of this.

So let me stick to just one slogan, which has already been mentioned several times in this interview. I think that modern logic may be on the threshold of changing its traditional agenda to become a broad theory of intelligent interaction, at the interface of the humanities, exact sciences, and social sciences, and I see my work as helping history along a little bit here – in a gentle non-violent persuasive mode.

Question by Fenrong Liu: I just got a notification that the Science Press, a prestigious Chinese publisher in Beijing, is willing to publish our book. The book will be titled as “A Door to Logic: Selected Works by Johan van Benthem-- Modal Logic”, this is the first Volume, as we planned. The publisher is also happy to publish two further volumes, namely “Logic and Natural Language” (volume 2) and “Logic and Philosophy” (volume 3). You have done much work already, especially for Volume 1, writing a preface for the book, and more importantly, new introductions to each section. I believe that readers will benefit a lot from these new additions. Would you like to take this chance to say a few words to your future Chinese readers?

Answer: So the wish in the previous answer has been granted already. China is really a place where dreams come true very fast! My word to readers is just this.

I hope that my papers indeed open a ‘Door to Logic’, an old discipline dating bach some 2500 years to Ancient Greece and India, but also a vibrant and diverse modern area with many great minds, lots of different opinions, and great panoramic points of view on scientific life. I think you might appreciate learning what my discipline has to offer, but equally well, logic will no doubt benefit from joining forces with Chinese readers, and increasingly also, original Chinese contributors to the field, who will bring their own insights and add new themes. I also think that such contacts are important in shaping a common intellectual universe where we can all participate as equals, and agree and disagree in a civilized and productive manner. The benefits of that, of course, go far beyond any special academic discipline.