In this paper: https://www.aclweb.org/anthology/J/J16/J16-2006.pdf, the author breaks the Pointwise Mutual Information of a phrase up into several components:
They use the ∧ notation to denote conjunction. But what does that actually mean?
What is the difference between the p(a,c) joint probability and the p(a∧c)? In logic form, aren't they the same?
Yes, when $a, b, c$ are all events, structures like $\mathsf P(a,b,c)$ are normally interpreted as the probability of the conjunction of the listed events; that is it is indeed identical to $\mathsf P(a\wedge b\wedge c)$ .
$$\mathsf P(a,c) = \mathsf P(a\wedge c)\\\mathsf P(a,b,c) = \mathsf P(a\wedge b\wedge c)$$
It is just a matter of style which an author may use. The former is often preferred merely because the typesetting is more compact. However, it seems that this author would rather be more explicit about the conjunction.
NB: It is more usual to use set operators ($\cap,\cup$), but not exactly uncommon to use logical connectives ($\wedge,\vee$).