# Syncategorematic term

In the common definition of propositional logic, examples of syncategorematic terms are the logical connectives. Let us take the connective $\land$ for instance, its semantic rule is:
So its meaning is defined when it occurs in combination with two formulas $\phi$ and $\psi$ . But it has no meaning when taken in isolation, i.e. $\lVert \land \rVert$ is not defined.
We could however define the $\land$ in a different manner, e.g., using λ-abstraction: $(\lambda b.(\lambda v.b(v)(b)))$ , which expects a pair of Boolean-valued arguments, i.e., arguments that are either TRUE or FALSE, defined as $(\lambda x.(\lambda y.x))$ and $(\lambda x.(\lambda y.y))$ respectively. This is an expression of type $\langle \langle t,t\rangle ,t\rangle$ . Its meaning is thus a binary function from pairs of entities of type truth-value to an entity of type truth-value. Under this definition it would be non-syncategorematic, or categorematic. Note that while this definition would formally define the $\land$ function, it requires the use of $\lambda$ -abstraction, in which case the $\lambda$ itself is introduced syncategorematically, thus simply moving the issue up another level of abstraction.