The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement (veridical justification) with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic (gives rise to logical truths and consequences), logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic.

A Martin-Löf random sequence is an infinite binary sequence with the property that every initial segment σ has prefix-free Kolmogorov complexity K(σ) at least ∣σ∣ − c, for some constant c ϵ ω. Informally, initial segments of Martin-Löf randoms are highly complex in the sense that they are not compressible by more than a constant number of bits. However, all Martin-Löf randoms necessarily have contiguous substrings of arbitrarily low complexity. If we demand that all substrings of a sequence be uniformly complex, then we arrive at the notion of shift-complex sequences. In this paper, we collect some of the existing results on these sequences and contribute two new ones. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is zero. We strengthen this result by proving that the Martin-Löf random sequences that do not compute shift-complex sequences are exactly the incomplete ones, in other words, the ones that do not compute the halting problem. In order to do so, we make use of the characterization by Franklin and Ng of the class of incomplete Martin-Löf randoms via a notion of randomness called difference randomness. Turning to the power of shift-complex sequences as oracles, we show that there are shift-complex sequences that do not compute Martin-Löf random (or even Kurtz random) sequences.