Syncategorematic term

The distinction between categorematic and syncategorematic terms was established in ancient Greek grammar. Words that designate self-sufficient entities (i.e., nouns or adjectives) were called categorematic, and those that do not stand by themselves were dubbed syncategorematic, (i.e., prepositions, logical connectives, etc.). Priscian in his Institutiones grammaticae[5] translates the word as consignificantia. Scholastics retained the difference, which became a dissertable topic after the 13th 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.[6]

In its modern conception, syncategorematicity is seen as a formal feature, determined by the way an expression is defined or introduced in the language. In the standard semantics for propositional logic, the logical connectives are treated syncategorematically. Let us take the connective ${\displaystyle \land }$ for instance, its semantic rule is:

${\displaystyle \lVert \phi \land \psi \rVert =1}$ iff ${\displaystyle \lVert \phi \rVert =\lVert \psi \rVert =1}$

Thus, its meaning is defined when it occurs in combination with two formulas ${\displaystyle \phi }$ and ${\displaystyle \psi }$. It has no meaning when taken in isolation, i.e. ${\displaystyle \lVert \land \rVert }$ is not defined.

One could however give an equivalent categorematic interpretation using λ-abstraction: ${\displaystyle (\lambda b.(\lambda v.b(v)(b)))}$, which expects a pair of Boolean-valued arguments, i.e., arguments that are either TRUE or FALSE, defined as ${\displaystyle (\lambda x.(\lambda y.x))}$ and ${\displaystyle (\lambda x.(\lambda y.y))}$ respectively. This is an expression of type ${\displaystyle \langle \langle t,t\rangle ,t\rangle }$. 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 ${\displaystyle \land }$ function, it requires the use of ${\displaystyle \lambda }$-abstraction, in which case the ${\displaystyle \lambda }$ itself is introduced syncategorematically, thus simply moving the issue up another level of abstraction.

• Compositionality
• Generalized quantifier
• John Pagus
• Lambda calculus
• Logical connective
• Supposition theory
• William of Sherwood

• Grant, Edward, God and Reason in the Middle Ages, Cambridge University Press (July 30, 2001), ISBN 978-0-521-00337-7.