(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.