Badiou talks of joy, happiness, and beatitude in various places, so one is entitled to wonder what difference, if any, exists between these concepts. As Badiou is using these terms in his recent book « Métaphysique du bonheur réel » (2015), « happiness » is the generic term. It can be broken down into the affects of « enthusiasm » in politics, « beatitude » in the sciences, « pleasure » in the arts, and « joy » in love (page 40).
Badiou discusses these affects only briefly, affirming
« As we have seen in the preceding chapters « happiness » is a synthetic word for several affects tied to distinct truth procedures. In LOGICS OF WORLDS (2006), I indicate explicitly, for the first time, that the participation of an individual in a truth is signaled by an affect, and that for each type of truth there is a different affect » (page 57, my translation).
We should bear in mind that later in the book Badiou declares that we must abstract from his terminology, and that he finds that Deleuze’s concept of sense corresponds quite closely to his own concept of truths.
Secondly, I think we must reject Badiou’s obsession with magic numbers such as the FOUR truth procedures. Deleuze and Guattari recognise many more semiotic régimes, and we should see their book WHAT IS PHILOSOPHY? as a pedagogical simplification of their full system.
Another interesting development is Badiou’s application of this typology of philosophical affects to his own works. He claims that in BEING AND EVENT the predominant affect is the beatitude that comes from scientific comprehension:
Beatitude is the name of the happiness provided by being as being, when it is taken up in the writing of its purity (59).
In contrast, the predominant affects in LOGICS OF WORLDS are the pleasure and joy arising from the exploration of the logic of appearing, as it manifests in the relations between bodies and truths within specific worlds:
The fact that the question of the body of truths is central obviously explains that pleasure (of the formalised sensible) and joy (of the other, of the sexed Two as sovereign in the world) are at this level the most clearly explored forms of happiness (60).
Badiou’s terminology is far from satisfactory, but at least it has the merit of spelling out certain distinctions and sketching a useful typology of affects, even if we choose to apply it differently.
I think Deleuze would treat « joy » as a more generic term than « happiness ». « Pleasure » seems very insufficient to describe the artistic affects, especially given the importance of the sublime in modern art. But Deleuze also told us not to be too attached to particular words, and to be ready to change them. Deleuze exemplified this attitude in his aborted dialogue with Foucault, where he tried to show that what he and Guattari meant by « desire » corresponds to what Foucault refers to with the expression « bodies and their pleasures ».
Note: I am grateful to a discussion with Anna Powell that helped me clarify my ideas on this point.