Conditional Counting of Qualia

(Abstract accepted to Athens conference by GCAS.)

Two well-known prisoner schemes represent two different logics:
1) Prisoner’s Dilemma (PD) in Game Theory
2) Three Prisoners (TP) examined by Jacques Lacan.
Their difference concerns the dialectic of Quantity and Quality: In PD, qualities have a fixed arrangement and quantities are pre-determined as Game Rules; whereas in TP, quantities emerge only through the conditional counting of qualia. PD counts “score” under a fixed structure; whereas TP counts conditional existences. Their contexts rarely intersect: in Google, PD yields 507000 results, TP yields 345000 results, PD+TP together yields only 1310 results. There is a clear logical gap. Here, we devise a formalization to represent both as two modes of counting.

Structural counting consists of usual set theory representations:
0 = {}
1 = {0} = {{}}
2 = {0,1} = {0,{0}} = {{},{{}}}
3 = {0,1,2} = {0,{0},{0,{0}}} = {{},{{}},{{},{{}}}}
Increments: n ∪ {n} = n+1

Conditional numbers are obtained by “opening” and “dotting” them:
0′ = ·
1′ = · {·}
2′ = · {·} { · {·} }
3′ = · {·} { · {·} } { · {·} { · {·} } }
Increments: n’ n = (n+1)’

Their “openness” indicates that the counting is yet incomplete, that it consists of suppositions. Their “dots” represent their indeterminacy by distributing the possibilities. Each of these conditional numbers signifies a different qualitative situation:
0′ is unicity: it signifies the “less than nothing” of Slavoj Žižek’s famous book.
1′ is an independent singular existence: it signifies “less than one”, to be or not to be.
2′ is a dilemma: it signifies “less than two”, this or that or else.
3′ is “less than three”, and the counting goes on to infinity.

In this paper, we describe the formal procedure of conditional counting and use it to represent TP, thereby resolving its “paradoxical” status with respect to the accepted “canonical example” status of PD.

I think I found a relevant parapraxis by Joan Copjec. In ITINW (2002) when she wants to refer to Lacan’s “Three Prisoners”, she instead mentions “Prisoner’s Dilemma”, which is a separate thing in Game Theory. I had described these concepts’ difference here:


Throughout his teaching, Lacan referred several times to “The Prisoner’s Dilemma,” a logical puzzle out of which he essayed to compose a kind of “Group Psychology beyond the Ego.” * In this puzzle, each of three prisoners has either a black or a white circle attached to his back. In other words, the game begins with each player placed behind a “veil of ignorance.” Operating with the knowledge that the prison warden has five circles, three white and two black, to work with— that is, knowing that two circles have been withdrawn from play—each prisoner must determine from the evidence of the circles he is able to see on the backs of the other two whether he is himself a black or a white. The prisoner who figures this out first will be freed.

* Jacques Lacan, “Logical Time and the Assertion of Anticipated Certitude,” Newsletter of the Freudian Field, vol. 2 (1988).

Joan Copjec 2002 Imagine There Is No Woman: Ethics And Sublimation, p.184


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