Epistemological reductionism is, broadly speaking, an attempt to answer sceptical worries concerning epistemic "access". For example, how are we to have representational epistemic access to:
- states of affairs (e.g., future ones or long past ones),
- mathematicalia (e.g., infinite sets),
- moral properties (e.g., the property of being morally obliged somehow),
- possibilities (e.g., a possible world in which there are $\aleph_0$ members of the Beatles)
- the structure of space and time (e.g., the fine-grained topology of space below the Planck scale),
- causal connections (e.g., the connection between the magnetic field and force on a nearby electron),
1. If $p$, then it is knowable that $p$.Each of these reductionist proposals attempts to "close the gap" between the world and the mind. For example, if $p$, then rational inquiry would yield an epistemic warrant for $p$. This is the core assumption of Semantic Anti-Realism: that each truth is knowable. (A similar view was advocated by Kant, Peirce and Dummett.)
2. If a term $t$ has a value, then that value can be computed/constructed.
3. If a term $t$ has a value, then that value has been physically tokened.
4. If $P$ is a proof of $\phi$, then someone (or some community) grasps and accepts $P$.
However, Descartes, Hume, Russell and Popper all argued, in their own way, that these epistemic "gaps" cannot be closed. (Descartes went on to try and close the gap by a complicated argument, set out in his Meditations, involving God.) For the possibility of the obtaining of a state of affairs, of which we are non-cognizant cannot, at least not with certainty, be ruled out.
That said, such a conclusion does not imply that one ought to be a sceptic. Human cognition, which I assume is neurophysiologically much like primate cognition (and in some respects like all animal cognition), presumably functions reasonably well in acquiring representational states which count as knowledge. Unfortunately, little is understood on this important topic in cognitive psychology, mainly because it is incredibly unclear what these representational states are. It merely says that we can't rule out sceptical scenarios.