The "Tau" of Science - How to Measure, Study, and Integrate Quantitative and Qualitative Knowledge

Scientists' ability to integrate diverse forms of evidence and evaluate how well they can explain and predict phenomena, in other words, $\textit{to know how much they know}$, struggles to keep pace with technological innovation. Central to the challenge of extracting knowledge from data is the need to develop a metric of knowledge itself. A candidate metric of knowledge, $K$, was recently proposed by the author. This essay further advances and integrates that proposal, by developing a methodology to measure its key variable, symbolized with the Greek letter $\tau$ ("tau"). It will be shown how a $\tau$ can represent the description of any phenomenon, any theory to explain it, and any methodology to study it, allowing the knowledge about that phenomenon to be measured with $K$. To illustrate potential applications, the essay calculates $\tau$ and $K$ values of: logical syllogisms and proofs, mathematical calculations, empirical quantitative knowledge, statistical model selection problems, including how to correct for "forking paths" and "P-hacking" biases, randomised controlled experiments, reproducibility and replicability, qualitative analyses via process tracing, and mixed quantitative and qualitative evidence. Whilst preliminary in many respects, these results suggest that $K$ theory offers a meaningful understanding of knowledge, which makes testable metascientific predictions, and which may be used to analyse and integrate qualitative and quantitative evidence to tackle complex problems.