Gregory Wheeler has written an eloquent new defense of formal philosophy.

Quotes:

...formal epistemology is an interdisciplinary research program that includes work by philosophers, mathematicians, computer scientists, statisticians, psychologists, operations researchers, and economists which aims to give mathematical and sometimes computational representations of, along with sound strategies for reasoning about, knowledge, belief, judgment and decision making.

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Why... bother being so formal? [Rich] Thomason, commenting on philosophers who view formal methods as a distraction to real philosophical advancement, observed that the only real advantage that we have over the great philosophers of the past are the new methods that we have at our disposal. Probability. First-order logic. Calculus. The number zero. It is hard to imagine improving on Aristotle without resorting to methods that were simply unavailable to him. Knowing just this much about history, a better question is this: why limit your options?

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The problem with aspiring to counterexample-proof philosophy without taking into account either formal or empirical constraints is that the exercise can quickly devolve into a battle of wits rather than a battle of ideas. And the problem is only compounded by pseudo-formal philosophy — the unfortunate practice of using formal logic informally — because this encourages philosophers to describe rather than define the fundamental operations of their theories. Memories are ‘accessed in the right way’; justified beliefs are ‘based’ on one’s ‘evidence’; coherent beliefs ‘hang together’. But, like a bump in a rug carefully pushed from one corner of a crowded room to another, this reliance on pseudo-formalisms to avoid any and all counterexamples inevitably means that the hard, unsolved philosophical problems are artfully avoided rather than addressed. At its worst, rampant counterexample avoidance turns philosophy into little more than a performance art.

But, one way to arrest this slide is by constraining epistemological theories by a combination of empirical evidence and formal models. For if you replace those fudged terms with a formal model, or a provably correct algorithm, and hem in imagination by known empirical constraints, then if a theory is successful in explaining a range of cases, that hard won success can be weighed against the theory’s failings. In other words, if we set aspirations for epistemology higher than conceptual analysis, that will open more room to judge success and failure than the all-or-nothing stakes of counterexample avoidance.

See also: An Overview of Formal Epistemology.