Monday, 5 September 2011

Van Benthem on system imprisonment

(Cross-posted at NewAPPS.)

Johan van Benthem is one of my favorite philosophers of logic (and not just because I'm ultimately an Amsterdam child!). He is completely idiosyncratic as a philosopher of logic, as he refuses to 'waste his time' with classical topics such as truth, consequence, paradoxes etc. But this is exactly what I like about what he has to say: he looks at the practices of logicians (being one himself!) and tries to make sense of what it is that we are doing when we 'do logic' in the currently established ways -- at times, adopting a rather critical stance as well. True enough, his observations are very much connected with his own research agenda, and yet they are also surprisingly general.

One of the concepts he's been talking about -- not so much in his 'official' papers, but mostly at talks, personal communication and interviews -- is the concept of system-imprisonment. (It is, however, mentioned in his 1999 'Wider Still and Wider: resetting the bounds of logic', in A. Varzi, ed., The European Review of Philosophy, CSLI Publications, Stanford, 21–44.) Here are some interesting passages:
But how good is the model of natural language provided by first-order logic? There is always a danger of substituting a model for the original reality, because of the former’s neatness and simplicity. I have written several papers over the years pointing at the insidious attractions and mind-forming habits of logical systems. Let me just mention one. The standard emphasis in formal logical systems is ‘bottom up’. We need to design a fully specified vocabulary and set of construction rules, and then produce complete constructions of formulas, their evaluation, and inferential behavior. This feature makes for explicitness and rigor, but it also leads to system imprisonment. The notions that we define are relative to formal systems. This is one of the reasons why outsiders have so much difficulty grasping logical results: there is usually some parameter relativizing the statement to some formal system, whether first-order logic or some other system. But mathematicians want results about ‘arithmetic’, not about the first-order Peano system for arithmetic, and linguists want results about ‘language’, not about formal systems that model language.
(I can't disclose the source for this quotation for now, as it is from a paper for a project I'm involved with which must remain a secret for a few more months... Anyway, the remark on mathematicians wanting results about 'arithmetic' also reminds me of the series of posts on Voevodsky and the incompleteness of arithmetic that we had a while ago.)
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.
(This is from an interview at the occasion of the Chinese translation of one of his books.)

I submit that the notions of system imprisonment and system generated problems must be taken seriously when we are using formal methods to investigate a given external target phenomenon. Oftentimes, a whole cottage industry becomes established to tackle what is taken to be a real issue, which is in fact an issue emerging from the formalism being used, not an issue pertaining to the target phenomenon itself. My favorite example here is the issue of 'free variables' in de re modal sentences, which then became seen as a real, deep metaphysical issue. In truth, it is simply an upshot of the formalism used, in particular the role of variables and the notions of bound or free variables. By adopting a different framework (as I did in a paper on Ockham's modal logic of many years ago, in the LOGICA Yearbook 2003 - pre-print version here) which does not treat quantification by means of variables, the 'issue' simply vanishes.

More generally, system imprisonment points in the direction of the epistemic limits of formal methods. Ultimately, what we prove is always relative to a given formal system, and the result lives or perishes with the epistemic reliability of the formal system itself. This does not mean that we should resign ourselves to some form of skepticism and/or relativism (Johan clearly does not!), but simply that we must bear in mind that the formal models are exactly that: models, not the real thing.

14 comments:

1. Catarina, great - that raises some very good questions.

"Ultimately, what we prove is always relative to a given formal system."

On this view, is the thing we prove a proposition or a sentence?

2. The easy answer is 'sentence' of course, but perhaps one could have a story on why, even though you operate with strings within a system, in the end what you prove is a content (relative to a given interpretation for the theory -- yet another parameter!).

But yes, it's good stuff, and while I think that a certain degree of system imprisonment cannot be avoided (it's the trade-off for some nice things like meta-theory), system-generated problems should really be closely monitored and avoided at all costs.

3. Thanks, Catarina - I'd say, on my view (I shouldn't speak for others, but I think this is what Johan is saying about arithmetic too), mathematicians prove propositions, not sentences, which are merely linguistic vehicles. One can take the sentences, focus on them, formalize them and the official inference rules, identify some basic axioms $\Delta$, etc., and then show various results, e.g., that if formula $\forall x \phi$ can be proved in $\Delta$, then $\phi(x/t)$ can be proved in $\Delta$, etc. But that's mathematical logic, not mathematics more generally. And, furthermore, these meta-claims of mathematical logic are themselves propositions (i.e., simply about formulas, etc., rather than about numbers, functions, fields, metric spaces, etc.).

On the opposite view, the mathematician is operating on strings within a system, and that seems to me a very formalist view.

To put a similar point, slightly differently, but related to your talk a few months ago at MCMP and your "External Target Phenomenon" post here, suppose $L_1$ is an interpreted natural language (or even a controlled fragment), and $L_2$ is an interpreted formalized language. It seems to me that there are various structure-preserving mappings between $L_1$ and $L_2$ (e.g., translation mappings). This is what, approximately, we mean by adequate formalization.

4. Hi Catarina,

Could you say a bit more about what problems involving de re modal sentences is sidestepped by moving to a different formal system, and how?

Thanks in advance!

Sam

5. Stu Shapiro and I have been pushing a view about the role of formalization in logic (mostly in papers, or his book, on vagueness) which takes seriously the idea that formal systems are merely models, and as such the results we prove about these formal systems don't automatically transfer to the natural languages we were originally interested in. Of course, in our work, we emphasize that they can transfer, but only with additional philosophical work. One example of this sort of thing is the objection to degree-theoretic semantics (or any precise mathematical model) for vagueness. Philosophers have objected to the use of such formal systems when attempting to understand vagueness on the grounds that they impose a kind of precision that is the very anti-thesis of the phenomenon - vagueness - that we are trying to understand. If we keep in mind that we are only building models, then the objection disappears (but is replaced with the quite difficult task of delineating exactly what aspects of the model we can, and cannot, treat as genuinely representative of the phenomenon in question).

6. Roy, saying formalization is modelling is fine with me. In teaching, I usually say: formalization is a translation mapping (into a formalized language) that approximately models the logical consequence relation between the propositions expressed in natural language.

But as you say, the thing is:

"but is replaced with the quite difficult task of delineating exactly what aspects of the model we can, and cannot, treat as genuinely representative of the phenomenon in question"

Indeed. If M is the "model" (what is a model?) and T is the "target phenomenon" (what exactly is that?), then what is their relation and how are properties of T inferred from those of M? Even distinguishing between the "model" and the "phenomenon" is not easy here.

Suppose I formalize:

(i) Anyone who draws a circle draws an ellipse

as,

(ii) $\forall x(\exists y(Cy \wedge Dxy) \rightarrow \exists y(Ey \wedge Dxy))$

I think the question is: what is being modelled? What has to be preserved under this translation?
Maybe this is the logical form of (i); or maybe how (i) fits into an overall pattern of natural language implications; or maybe how it fits into a pattern of inferences; possibly other candidates.

And there is an intermediate between the natural language sentence and the formalized string, that Quine calls "regimentation", and we usually call translating into "loglish". For example,

(iii) For any person $x$, if $x$ draws a circle, then $x$ draws an ellipse.

Presumably, (iii) is be thought of as synonymous with (i)? (expressing the same proposition).

7. @Sam

The idea is that if you have a term logic without variables, the whole thing becomes a matter of scope (whether the modal operator ranges over the whole sentence or only over a part of it, namely the predicate), so it's not about the modality being attributed to a sentence or to a thing. As I said, I have a paper on this but somehow I can't find it on the internet anymore (it used to be there). If I have the time I'ii upload it to my academia page later today.

8. Hi Catarina,

Thanks. If you do get time to upload it, that would be great.

Best,
Sam

9. @Sam, The paper has been uploaded and the link can be found in the text above.

10. @Roy and @Jeff,

As I think I mentioned many times, I believe that the view that formal languages are models of so-called natural languages is misguided. I of course like the bit of them being *models*, but I reject the claim that natural languages is what they are models *of*. Just yesterday I wrote the section of my book where I criticize this position (referring to Roy and Stewart indeed!). But rather than giving you my arguments against it, let me quote Johan again (again, from the paper whose source shall remain unnamed for now). I agree with most of the points he makes in this passage:

"The idea that logical languages may, or may not, be good ‘models’ for a given natural language is still in line with the idea that some fixed reality is given, which logical systems are then supposed to model. In that sense, the very term ‘semantics’ is conservative, because the given natural language becomes the yardstick for judging the quality of the logical system. But a logical system is not just a model for natural language, or some reasoning practice. It is also a tool for independent conceptual analysis of some cognitive activity, which can bring to light important features that are not encoded in natural language as we have it. In that sense, logical systems can also be much more radical in their thrust, helping us, as Marx advocated, to move away from interpreting the world to changing it. I find nothing repugnant to the idea that we might want to change natural language because of logical considerations. In fact, I often worry about the adjective “natural”, which often hides a sort of unthinking admiration for what is supposed to be pristine ‘nature’. In reality, ‘natural language’ is about as natural as a Dutch polder or a Chinese lake: constantly tinkered with by humans. What also helps in seeing this broader conceptual horizon is the fact that a language is first and foremost a practice, not the algebra with operators that Montague saw as its core."

11. Hi Catarina,
But then the question becomes: what is a "formal system" a model of? My attitude is based on my background in physics. We have a physical system P. We define a simplified and approximate mathematical model M. There are transfer principles between M and P.

Here is the sort of physics problem I used to do as a physics undergrad in classical mechanics.
A small ball slides down from the top of a hemispheric dome. It leaves contact with the dome at a certain point. I can define a mathematical model of this and compute the angle. I can also do an experiment, which confirms the prediction. Since the model is idealized, I can de-idealize in various ways.

I just googled and this problem is solved here:

http://minerva.union.edu/labrakes/Work_and_Energy_Problems_Solutions.pdf

But I can't quite see what the object of the modelling is on your approach.

(I agree with bits, and disagree with bits, of what Johan says here. I'd comment that Russell says in HWP somewhere that Theses on Feuerbach contained the earliest version of the pragmatist theory of truth.)

Cheers,

Jeff

12. Jeff, quick answer (I should really be turning off the computer now!): formal systems are models of whatever their target phenomena are, in each case, very much like in physics. As you may recall, in my talk in Munich last June I said that syllogistic, for example, is a model (for a restricted language) of the truth-preservation definition of validity at the beginning of the Prior Analytics. When I was doing formalizations of medieval logical theories, the target phenomenon was the theory in question, understood as a conceptual structure, *not* its expression in actual text. Of course, I could only apprehend this conceptual structure through the text, but once this was done, I could throw the ladder away and go straight from concepts to formalism.
I've been writing on all this just this week; if you are interested, I can send you the chapter as soon as it is more or less ready for consumption, and perhaps we can discuss it next week when I'm in Munich -- you WILL be there, I suppose! :)

13. I'm in Edinburgh! Not sure if I can make the MCMP thing in Munich for various reasons. (One being that I want to spend more time with my family before the teaching begins.)

Yes, I'd really like to read the chapter (and the earlier paper, but I'm not on academia ...).

14. Hi Jeff, I'm running around a bit in the next couple of days, but I'll send you the paper and the chapter soon.
It will be a shame if you are not in Munich next week! But it's for a good cause, one I'm very sympathetic to :)