Syncategorematic term

The distinction between categorematic and syncategorematic terms was established in ancient Greek grammar. Words which design self sufficient entities e.g. nouns or adjectives were called categorematic and those which fail to stand by themselves were dubbed syncategorematic, e.g. prepositions, logical connectives, etc. Priscian in his Institutiones grammatice translates the word by 'consignificantia'. Scholastics retained the difference which became a dissertable topic after the XIIIth century revival of logic. William of Sherwood, a representative of terminism, wrote a treatise called Syncategoremata. Later his pupil, Peter of Spain, produced a similar work entitled Syncategoreumata

In propositional calculus, a syncategorematic term is a term that has no individual meaning (a term with an individual meaning is called categorematic). Whether a term is syncategorematic or not is determined by the way it is defined or introduced in the language.

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:

$\lVert \phi \land \psi \rVert =1$ iff $\lVert \phi \rVert =\lVert \psi \rVert =1$

So its meaning is defined when it occurs in combination with two formulas $\phi$ and $\psi$. But is 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 $\lambda$-abstraction: $\lambda x,\lambda y.x\land y$. 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.