How many logical connectives are possible involving n simple propositions:p1, p2, . . . , pn?

Question

For 2 simple propositions, I guess we can put 4 different logical connectives (and, or, implies and iff) between 2 simple propositions p1 and p2 which gives 4 combinations. We can also put "not" in front of each simple proposition which gives 8 more combinations (e.g. not p1 or not p2), total 12 combinations for 2 simple proposition. I don't know how to go further from here? How many combinations are for 3 simple propositions? How many for n?

Answer 1

In general, there are $2^{2^n}$ such connectives possible.

This is because for each of the $2^n$ possible truth-combination of the $n$ propositions, the connective can give an output of either True or False.

So, for example, there are $2^{2^3}=2^8=256$ possible $3$-place ('ternary') connectives.

Also, notice that there are $2^{2^2}=2^4=16$ possible binary connectives, so your calculation was wrong. In fact, you didn't even get to $12$, because with $\land$, $\lor$, $\to$, and $\leftrightarrow$ (that's $4$) and their negations (that's $4$ more) you only get to $8$

The connectives you missed: connective that always gives True, the connective that returns the value of the left operand, the connective that returns the value of the right operand, the connective that works like $\leftrightarrow$, and their $4$ negations.