While empiricism gives us some probabilistic, mediated access to reality, our senses are frequently consistent with each other. However, we have no way of knowing if this input represents what is or simply some high-level abstract interpretation of a transformed version of what is. Both of these could be called “true” however in normal speech, so I will borrow the Buddhist doctrine of Two Truths. In very brief summary:
Absolute Truth: The underlying reality of what is.
Relative Truth: What we consistently and reliably interpret of what we sense.
Absolute truth might, for example, be nothing more than atoms interacting according to physical laws. Regardless, it is exactly the reality accepted in axiom #3. Relative truth is the world that we construct from empiricism, with tables and chairs and clouds and cats and dogs. Even if, fundamentally, there is nothing that is a dog in the underlying reality, the existence of dogs is still a relative truth, a practical concept useful for navigating the world.
The definition of knowledge is another big problem in philosophy, and is more-or-less the defining question of epistemology. Based on our axioms so far and the previous two posts, I use the following three definitions of knowledge:
True Knowledge: This is the definition of knowledge that pedants like to trot out when arguing for epistemic scepticism. “True” knowledge is knowledge of absolute truth (see above) which is impossible because of the circular trap. We have no way of knowing that the axioms we have taken are correct, thus no way of knowing anything else which we might derive from them. However, this meaning is almost never used in non-philosophical debate.
Strong Knowledge: Strong knowledge is knowledge based on “inviolate empiricism”; facts like gravity1 that consistent across a truly enormous number of observations (all relative truths, of course). Logical derivations from axioms (and from inviolate empiricism) also qualifies of course: knowledge of algebra is strong knowledge. This usage occurs outside of philosophy debates, but mostly in scientific and other formal contexts.
Weak Knowledge: Weak knowledge is knowledge based on probabilistic empiricism. It is reasonable, for example, for me to say that I “know” certain things about Shakespeare, but the chain of actual facts and observations between myself and him is quite long and tenuous. Nonetheless, I say I know these facts because they are still far and away the most probable explanation of all the various things I have experienced. This is the most common usage of knowledge in informal conversation.
(1) Yes, I know, quantum theory and relativity etc etc. What goes up must still come down.