(Cross-posted at M-Phi.)
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.
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.
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.