# How many truth functions are there?

The textbook says that given $$n$$ atomic wff, there are $$2^{2^n}$$ truth functions. Is it correct to say that since the set of all the atomic wff is denumerable, there are $$2^{2^{\aleph_0}}$$ truth functions (that is, truth functions are uncountable)?

• If you define "truth function" in a certain way, then yes.
– Karl
Mar 2 at 6:53
• The text book refers to a standard definition of truthfunction for two-valued propositional logic. Mar 2 at 7:28

The set of all truth assignments can be identified with the set of all infinite sequences of $$1$$'s and $$0$$'s, since a truth assignment gives a truth value to a countable set of propositional variables.
This in turn can be identified with the set of all subsets of $$\mathbf{N}$$, since we can think of a sequence of $$1$$'s and $$0$$'s as a characteristic function of natural numbers, i.e. a number $$n$$ is in the subset iff the value of the sequence at the $$n$$-th place is $$1$$.
The set of subsets of natural numbers has cardinality $$2^{\aleph_0}$$.