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[2] 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.[3]

The word "alone" can be used both as categorematic term or a syncategorematic in respect of the Most Holy Trinity. According to the St Thomas Aquinas' Summa theologiae, the unity of the three divine persons, who is God, can be solely predicated with categorematical terms, e.g. in the proposition "God alone is eternal". Differently from the unity of God, the three divine persons can be predicated with syncategorematic terms: for example, if the sentence "God alone creates" (categorematic) is true, the sentence "God the Father alone creates" is false given that also God the Son creates (the word "alone" used as a syncategorematic term).[4] In other words, the John 1's creation is an example of a twofold use of the word "alone": as a categorematic term in respect of the unity of God, and also as a syncategorematic term in respect of two or all of the three divine persons.

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 ${\displaystyle \land }$ for instance, its semantic rule is:

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

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

We could however define the ${\displaystyle \land }$ in a different manner, e.g., 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.

• John Pagus
• William of Sherwood
• Supposition theory

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