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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...
and
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
, an expression is syncategorematic if it lacks a
denotation In linguistics and philosophy, the denotation of an expression is its literal meaning. For instance, the English word "warm" denotes the property of being warm. Denotation is contrasted with other aspects of meaning including connotation. For inst ...
but can nonetheless affect the denotation of a larger expression which contains it. Syncategorematic expressions are contrasted with categorematic expressions, which have their own denotations. For example, consider the following rules for interpreting the
plus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
. The first rule is syncategorematic since it gives an interpretation for expressions containing the plus sign but does not give an interpretation for the plus sign itself. On the other hand, the second rule does give an interpretation for the plus sign itself, so it is categorematic. # ''Syncategorematic'': For any numeral symbols "n" and "m", the expression "n + m" denotes the sum of the numbers denoted by "n" and "m". # ''Categorematic'': The plus sign "+" denotes the operation of addition. Syncategorematicity was a topic of research in
medieval philosophy Medieval philosophy is the philosophy that existed through the Middle Ages, the period roughly extending from the fall of the Western Roman Empire in the 5th century until after the Renaissance in the 13th and 14th centuries. Medieval philosophy ...
since syncategorematic expressions cannot stand for any of
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
's
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
despite their role in forming
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the ...
s. Medieval logicians and grammarians thought that quantifiers and
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s were necessarily syncategorematic. Contemporary research in formal semantics has shown that categorematic definitions can be given for these expressions in which they denote generalized quantifiers, but it remains an open question whether syncategorematicity plays any role in
natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
. Both categorematic and syncategorematic definitions are commonly used in contemporary
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...
and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
.


Ancient and medieval conception

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 Priscianus Caesariensis (), commonly known as Priscian ( or ), was a Latin grammarian and the author of the ''Institutes of Grammar'', which was the standard textbook for the study of Latin during the Middle Ages. It also provided the raw mater ...
in his ''Institutiones grammaticae'' translates the word as ''consignificantia''. Scholastics retained the difference, which became a dissertable topic after the 13th century revival of logic.
William of Sherwood William of Sherwood or William Sherwood (Latin: ''Guillielmus de Shireswode''; ), with numerous variant spellings, was a medieval English scholastic philosopher, logician, and teacher. Little is known of his life, but he is thought to have studie ...
, a representative of terminism, wrote a treatise called ''Syncategoremata''. Later his pupil,
Peter of Spain __NOTOC__ Peter of Hispania ( la, Petrus Hispanus; Portuguese and es, Pedro Hispano; century) was the author of the ', later known as the ', an important medieval university textbook on Aristotelian logic. As the Latin ''Hispania'' was considere ...
, produced a similar work entitled ''Syncategoreumata''.Peter of Spain
''Stanford Encyclopedia of Philosophy'' online


Modern conception

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 Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comp ...
for
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
, the logical connectives are treated syncategorematically. 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 Thus, its meaning is defined when it occurs in combination with two formulas \phi and \psi. It has no meaning when taken in isolation, i.e. \lVert \land \rVert is not defined. One could however give an equivalent categorematic interpretation 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 In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progra ...
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.


See also

*
Compositionality In semantics, mathematical logic and related disciplines, the principle of compositionality is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. ...
* Generalized quantifier * John Pagus *
Lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...
*
Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
*
Supposition theory Supposition theory was a branch of medieval logic that was probably aimed at giving accounts of issues similar to modern accounts of reference, plurality, tense, and modality, within an Aristotelian context. Philosophers such as John Buridan ...
*
William of Sherwood William of Sherwood or William Sherwood (Latin: ''Guillielmus de Shireswode''; ), with numerous variant spellings, was a medieval English scholastic philosopher, logician, and teacher. Little is known of his life, but he is thought to have studie ...


Notes


References

* Grant, Edward, ''God and Reason in the Middle Ages'', Cambridge University Press (July 30, 2001), . {{DEFAULTSORT:Syncategorematic Term Logic Semantics Philosophy of language Medieval philosophy Term logic