Ned Block distinguishes two concepts that the word 'consciousness' conflates. Phenomenal consciousness (P-consciousness) is the subjective, experiential quality of a mental state — what it is like to see red or feel pain. Access consciousness (A-consciousness) is the availability of a mental state's content for use in reasoning, report, and the rational control of behavior. Block argues these come apart: a state can be phenomenally conscious without being accessed (as suggested by partial-report experiments and Sperling's iconic memory paradigm), and a state can be access-conscious without having any distinctive phenomenal character. The 'overflow' argument holds that phenomenal experience is richer than what the cognitive access bottleneck can capture at any moment.
Start with Block's 'On a Confusion about a Function of Consciousness' (1995). Then examine the Sperling partial-report experiments and the debate between Block and cognitive-access theorists like Cohen and Dennett over whether overflow is real or an illusion of retrospective report.
From your study of qualia and the hard problem, you already know that there is something philosophically puzzling about subjective experience — about the fact that there is "something it is like" to see red or feel pain. Ned Block's distinction between phenomenal and access consciousness is an attempt to bring precision to this puzzle by separating two things the word "consciousness" has been conflating.
Phenomenal consciousness (P-consciousness) is the qualitative, subjective character of experience — the redness of red, the sharpness of a headache, the felt warmth of sun on your skin. This is the dimension that makes the hard problem hard: why does any physical process give rise to this qualitative character at all? Access consciousness (A-consciousness), by contrast, is a functional property: a mental state is A-conscious when its content is broadcast to and available for use by the cognitive systems governing reasoning, report, and the control of action. A-consciousness is, in principle, the sort of thing a cognitive science of information flow could fully describe without invoking any subjective qualities.
Block's crucial claim is that these two properties can come apart. His main empirical case draws on Sperling's classic iconic memory experiments. A subject briefly sees a 3×4 grid of letters. After it disappears, they can accurately report any row — but only one; the rest fades before it can be reported. Block argues that subjects phenomenally experienced the full grid (the full array was present in experience), but only one row at a time was *accessible* for report. Phenomenal experience "overflowed" cognitive access. Critics like Dennett and Cohen dispute this — they argue that what looks like overflow is a retrospective illusion, that subjects never really experienced the full array phenomenally. This debate remains live.
Why does the distinction matter? If Block is right, then theories that define consciousness entirely in functional terms — as availability for report and reasoning — will systematically miss the phenomenal dimension. A philosophical zombie (from your earlier study of qualia) would have full A-consciousness but no P-consciousness. A patient with certain neurological conditions might have P-consciousness without the cognitive access needed to report it. These possibilities only make sense if you hold the two concepts apart. Block's contribution is to give philosophers and cognitive scientists a cleaner vocabulary for what had been a confusing cluster of questions.
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