Jaakko Hintikka on the analytic/synthetic distinction in logic and arithmetic

What kind of judgement is required to establish an arithmetic truth like 2+2=4 or 5+7=12?

One way to divide up kinds of judgment is into two kinds: analytic and synthetic. Broadly, analytic judgements are true in virtue of their definitions, while synthetic judgements are true in virtue of their application to the empirical world.

Frege, and many 20th century logical positivists defended a view that these kinds of truths only needed "analytic" reasoning to be established. Conversely, Kant - whose writings popularised the analytic/synthetic distinction chose 5+7=12 as a paradigm case of a synthetic (though still a priori) truth.

While it does seem defensible that arithmetic statements like these are not merely true in virtue of their definitions (as Kant says, pace Frege/Russell/Whitehead, nowhere is it contained in the definition of 5 or the definition of 7 that adding the two together should make 12). It is strange to say that we learn them from experience, for example by seeing many cases of such an addition and learning some kind of general law. Kant's position is not anything like this, rather he thinks that we can figure out truths like these in abstract by reasoning about the limits of possible experiences we might have.

My understanding of how this kind of reasoning is supposed to work for Kant is that it relies on our general processes for generating images from ideas of numbers¹. Once you prove that some mathematical truth holds based on this kind of reasoning, you've proved that it holds for any possible case - and this is why Kant was motivated to write about these kinds of judgement in the first place - while they are synthetic they are also universal, necessary, and a priori. Frege seems to have been especially concerned that if this were the foundation of all mathematics it might imply that mathematical truths are not, in fact, universally or necessarily true precisely because things like "our general processes for generating images form ideas" seemed to him to be too dependent on contingent limitations of human minds.

As a result Frege, and after him Russell and Whitehead, tried to remove the need for this kind of reasoning by trying to use only the rules of logic in order to provide a firm foundation for mathematics. Doing this required introducing increasingly more powerful logics². One concern you might raise with these logics is that the reasoning that they capture as they get more powerful might go beyond pure "analytic" reasoning, in the sense of "being true only in virtue of definitions". (though I think this kind of concern is only motivated if you are sympathetic to Kant's original paradigm examples (facts about Euclidean triangles, 5+7=12) in the first place.

The logician and philosopher Jaakko Hintikka raised exactly this worry, and I recently read an in depth review of his approach by the contemporary logician and philosopher Costanza Larese.

Hintikka's approach was proof theoretic, that is: he was interested in giving a list of permissible proof-rules such that any logical derivation you can make using those rules is an analytic judgement, and any logical derivation that goes beyond them is a synthetic one. The particular proof rule which Hintikka disallows from analytic reasoning is a proof rule named existential instantiation.

Hintikka received some criticism from Kant scholars for this approach, and while I agree that the focus on existential instantiation is less motivated by Kant's writings than Hintikka makes it out to be, I think the general approach seems right for capturing what Kant actually meant by an "analytic" judgement, especially when it comes to mathematics. One reason to believe this is that one of Kant's students, Johann Schultz, was already pushing back on attempts by Leibniz to reduce arithmetic to logic. In my blog post on this topic, I spell out what role I think Schultz/Kant were attributing to the imagination in these kinds of synthetic judgement, and it does seem to have a lot to do with our ability to posit new, temporary, objects for the purpose of proving some general rule:

My current understanding of Kant's view is that to judge that 2 + 2 = 4 you must firstly construct in imagination 2 objects and then 2 more objects beside them. Secondly you count the total as 4. Lastly you observe that nothing in this process depended on the particular objects involved, but only on the general procedure for being able to generate two things in the imagination. This means that this judgement holds necessarily and universally for any case of two and two things. I think he also perhaps thought you could do this by working with these imaginative procedures in other ways - which is how it is still possible to do arithmetic on unimaginably large numbers.

A related paper, coauthored by Larese, on "Depth-Bounded Natural Deduction" follows up on Hintikka's work and generalises it beyond existential instantiation by disallowing, from analytic judgements, any proof steps which require imagining new virtual objects³ not already introduced in the axioms, even if these objects are eliminated/discharged later in the proof. The effect of this is that, "analytic" truths end up being a fairly narrow set of truths that you can establish by tautology or very basic Aristotelian-style syllogism.

This is about as far as I have thought this topic through. My main take-away is that, at least in mathematics, some useful notion of analytic vs synthetic judgements can be made rigorous.

Update: even if you make this distinction, you might still want to further divide the class of synthetic judgements into those which are computationally tractable and those which are not, shortly after publishing this I wrote a follow up post on this question which you can read here.