Reductive Analysis

In Mathematics we often create or consider different structures and properties that can be reductively understood in terms of others. For example, the definition of finite set in Set theory can be formally understand as the property that holds when every injective function on the set is bijective, and when it satisfies a well order. Mathematics thrives on being able to build these sort of links between difference branches. We can connect algebra and topology by understanding fundamental groups.

Given the similarity between Mathematics and Philosophy, we might hope that we can employ the same sort of notions in resolving Philosophical problems. If we could reduce a philosophically difficult concept into a series of other conditions that we do understand, we can reductively understand the original concept. This approach is extremely tempting to apply when we seem to deal with certain topics that have an intuitive basis and seem to imply other notions.

For example, it seems that we have an intuitive understanding of what it means to Know. We link certain ideas with knowledge; truth, belief, infallibility, mental states, abilities. It seems reasonable to suggest that we could attempt to understand Knowledge by trying to see how these other concepts relate to how we use or apply the concept of knowledge. Certainly it seems plausible to think that if I know that $P$, then it must be the case that I believe that $P$, or that it is true that $P$. To think this is to suggest that they are necessary conditions for knowledge, which implies an important logical relationship. The thought might go that if we could find these different necessary conditions for knowledge then perhaps we would be able to find sufficient conditions for knowledge. From here, we might be able to even develop a reductive analysis of Knowledge by turning it into a series of conjunctive properties. That is, we'd be be able to say that "I know that $P \iff \phi$".

In this way, we'd in theory be able to take our concept of knowledge that we are struggling with, and simply check whether it satisfies some different conditions in $\phi$, that we presumably do understand. As one can see, finding such a reductive analysis for these concepts would be incredibly helpful as it would allow us to quickly pinpoint the salient conditions. I personally think however that as tempting as this is, reductive analysis is a dangerous road to go down.

Unlike in Mathematics where we are bound by certain rules and are able to at completely independent of reality, Philosophy lies in a strange position where it lacks these axiomatic rules to stand upon. The only thing it plausibly has is the basis of informal logical argument, and even this might lie on shaky ground.  This isn't to think that no concepts can be reduced - A bachelor is reductively analysable to unmarried adult male. I think however that philosophical concepts that have a reductive analysis are in fact in a rarity, and that it is the wrong approach to search for this analysis. Mathematics perhaps can be build upon a series of necessary and sufficient conditions because it uses formal logic as a foundation. There does not seem to be an obvious reason why the same would apply to philosophical notions that can often depend on heuristic notions.

Perhaps one is aware of the attempts to reductively analyse the concept of Knowledge. An initial attempt was to postulate the tripartite analysis; to think that one has knowledge if one has a justified true belief. Whilst we might intuitively think that this is plausible, there appear to be a number of issues in our way. The first is that such a claim doesn't seem to be analytic in that it doesn't just follow from the definition of Knowledge. So it seems that we are unable to verify that this claim is true via pure logic. In particular one might think that such approach is circular given our task is to understand the notion of Knowledge (which would then fuel development of a definition). Thus without this foundation, it seems that we must use other tools that depend on informal reasoning and argument to assess the validity of our reduction of Knowledge

One common way that Philosophers do this is by considering different thought experiments that is meant to elicit different intuitions. In response to our Tripartite analysis, we might consider the case of a broken clock, with its hands stuck at 2pm. Now when we look at the Clock it says it is 2pm, and presumably this justifies us in thinking that it is 2pm. Imagine however that the instance we look at the clock, it actually is 2pm. It seems now that we have satisfied our three conditions (justified true belief), yet it seems rather odd to think that we actually know that it is 2pm. Plausibly this suggests that our reductive analysis is missing something. Although since our method of verification is not purely logical, such an assertion is also not direct. It seems to rely on an element of judgement, and it is not at all obvious how we ought to assess this.

Regardless, such a thought experiment seems to suggest that the notion of knowledge can not be reduced to these three conditions. Yet this does not deny that such conditions are irrelevant, even if they might not be logically related. One might think that it is possible to construct further examples that would challenge our intuitions that knowledge relies on belief or even truth. What is rather interesting is that we seem regardless justified in suggesting that notions of truth, belief and justification are still pertinent to Knowledge. In stark contrast with Mathematics, counter examples do not seem to entirely render invalid different relations. It seems that we can consider different concepts and say that they are related to others without them having any logical relationship in terms of necessary or sufficient conditions

Thus I think that our attempts to understand Philosophy via reduction is missing something foundational about concepts that lack necessary connection via logical stability. Many concepts miss out on generating if and only if connections, yet our study of such notions still seem to capture important details of ideas, perhaps even their intrinsic nature. The way they are linked is just not via a logic that an be universally processed in the way that Mathematics can. It is different, and I think we're missing something in our philosophical investigation if we fail to respect this.