Abstract
The medieval distinction between categorematic and syncategorematic words is usually given as the distinction between words which have signification or meaning in isolation from other words (such as nouns, pronouns, verbs) and those which have signification only when combined with other words (such as conjunctions, quantifiers, and articles). Some words, however, are classified as both categorematic and syncategorematic. One such word is Latin infinita ‘infinite’. Because infinita can be either categorematic or syncategorematic, it is possible to form sophisms (logical puzzles or paradoxes) using infinita whose solutions turn on the distinction between categorematic and syncategorematic uses of infinita. As a result, medieval logicians were interested in identifying correct logical rules governing the categorematic and syncategorematic uses of the term. In this paper, we look at 13th–15th-century logical discussions of infinita used syncategorematically and categorematically. We also relate the distinction to other medieval distinctions with which it has often been conflated in modern times, and show how and where these conflations go wrong.
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Notes
It is clear, then, that Murdoch and Thijssen’s assertion that the syn/categorematic distinction arose in the 14th century (Murdoch and Thijssen 2001, p. 129) is simply incorrect.
Medieval authors vary as to whether they take infinita or infinitum as the basic form; we follow Sherwood and use infinita simply because a choice has to be made.
We should note that discussions of correct uses of infinita are not restricted to treatises on logic; they also occur in treatises on natural philosophy (cf. footnote 19). However, since in this paper we are interested in explicit discussions of rules for reasoning with infinita, we have restricted our attention to logical texts.
Though while Priscian was indeed strongly influential on the medieval development of the distinction in logical contexts, it would be a mistake to take his grammatical definition as representative of the logical definition (Kretzmann 1982, pp. 212–213).
Terminus syncategorematicus dicitur qui, significative acceptus, non potest esse subiectum aut praedicatum, aut pars subiecti aut pars praedicati distributi, propositionis categoricae (Paul of Venice 1979, p. 6).
Dicuntur sincathegoreumatice, quasi: consignificative, idest: cum aliis significative scilicet cum cathegoreumaticae; non quia de se nichil significant, sed quia habent significationem non finitam sed infinitam, cuius finitationem trahunt ab adjunctis (Kretzmann 1982, p. 214, fn. 12). For the dating of Henry’s treatise, see (Braakhuis (1981), p. 135).
Qui tam per se quam cum alio positus habet significatum proprium seu determinatum...qui tam per se quam cum alio positus habet significatum sed indeterminatum (Paul of Pergola 1961, p. 7). Translations not otherwise footnoted are my own.
Qui nec per se, nec cum alio positus aliquid significat (Paul of Pergola 1961, p. 7).
Dictiones sincategorematice significant res aliquas. Sed non significant res subicibiles vel predicabiles. Ergo significant res que sunt dispositiones subicibilium vel predicabilium (Peter of Spain 1992, p. 38).
“The consequence is invalid because in the one proposition the term ‘infinite’ supposits categorematically, in the other syncategorematically” (Heytesbury 1988, p. 422), Et sic de talibus non valet consequentia: Quia in una propositione iste terminus infinitum supponit cathegorematice, in alia sincathegorematice (Heytesbury 1494a, fol. 3v).
Terminus categorematicus est signum, tam implicite quam explicite simplex, de communi lege, non extremorum aliqualiter unitivum, sed alterius a se et suo consimili per se in notitiam deductivum (Paul of Venice 1979, p. 2).
Terminus syncategorematicus est signum officii executivum, nullius a se et suo consimili sine nova impositione per se significativum (Paul of Venice 1979, p. 4).
This is not a new suggestion; it occurs in, e.g., (Murdoch and Thijssen (2001), p. 129) and (Braakhuis (1981), p. 143), albeit not explicitly in the latter. However, we feel quite strongly that this suggestion has not been taken up as widely as it should be, and thus wish to stress this point, which comes up again in the next section.
An almost identical wording appears in (Moore 2012, pp. 108–09).
On the same page he expands on this rough approximation, and attributes the distinction to Peter of Spain, whom he identifies with Pope John XXI. Both of these points deserve comment; current scholarship believes that Peter of Spain is not the same Petrus Hispanus who became Pope, and, as noted above, the distinction is not original to Peter but rather derives from the grammatical distinction in Priscian. The tradition of tracts on syncategoremes also predates Peter; the early 12th C Dialectica Monacensis refers to an as-yet-unidentified treatise on syncategorematic terms (Braakhuis 1981, p. 136).
For Bacon, a noun of the second intention is any noun which “does not signify a thing itself, but a concept that an intellect places over a thing, e.g., ‘genus’, ‘species’, and the like” (Bacon 2009, p. 128). As examples, he also gives ‘proposition’ and ‘syllogism’.
Infinitus, infinita, infinitum fit sensus compositus quando tenentur categorematice, divisus vero quando tenentur syncategorematice. Et tenentur categorematice quando subsequitur verbum principale vel cum praecedit aliquod determinabile, syncategorematice vero quando praecedit nullo determinabili praecedente (Paul of Pergola 1961, p. 152).
Knuuttila and Lehtinen also appear to do the same, but immediately afterwards express puzzlement as to how the two authors they consider, Richard Kilvington and Gregory of Rimini, could maintain such a conflation [11, p. 312].
The importance of being able to reason about infinity was especially recognized in the 14th century, when infinity and continuity made up a significant portion of discussions in natural philosophy (Murdoch 1982, p. 564).
Prima regula ista, a sensu composito ad sensum divisum et e contra non valet argumentum (Paul of Pergola 1961, p. 154).
Acceptio termini per se sive pro re sua, vel pro aliquo supposito contempto sub re sua vel pro aliquibus suppositis contemptis sub re sua (Franco 1971, p. 206).
Quando dictio supponit pro aliquo uno (William of Sherwood 1995, p. 136).
See also (Knuuttila and Inkeri Lehtinen 1979, p. 310).
“Signa autem universalia sunt ut ‘omnis’, ‘quilibet’, ‘quisque’, ‘quicumque’, ‘unumquodque’, ‘ubique’, ‘semper’, ‘qualislibet’, ‘quantumlibet’, ‘infinita’, ‘totus’, ‘ambo’, ‘uterque’, ‘nullus’, ‘neuter’, ‘nunquam’, ‘nusquam’, ‘nihil’, et huiusmodi” (de Libera 1986, p. 232, Sect. 108). My thanks to Ana María Mora Márquez for providing me with this citation. See also (Geach (1967), p. 41).
Utrum haec syncategoremata ‘omnis’, ‘nullus’, ‘quilibet’, ‘uterque’, et similia possunt esse partes subiecti vel praedicati propositionis significative sumpti (Paul of Venice 1979, p. 10).
Quod ly .omnis. et similia possunt esse partes subiecti vel praedicati significative sumpti (Paul of Venice 1979, p. 10).
Aliqua syncategoremata bene possunt esse partes subiecti vel praedicati, sed ly .omnis. non (Paul of Venice 1979, p. 10).
Non fieret praecises de praedicato subiectum (Paul of Venice 1979, p. 12).
In conversione simplici requiritur quod termini in conversa et convertente supponant praecises pro eodem vel pro eisdem, et eodem modo (Paul of Venice 1979, p. 34).
Nec omnis universalis negative est convertibilis simpliciter (Paul of Venice 1979, p. 34).
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The author would like to thank the audience of the FotFS VIII conference in Cambridge, September 2013, and two anonymous referees whose comments helped improve the paper.
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Uckelman, S.L. The logic of categorematic and syncategorematic infinity. Synthese 192, 2361–2377 (2015). https://doi.org/10.1007/s11229-015-0670-z
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DOI: https://doi.org/10.1007/s11229-015-0670-z