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Distributive Terms, Truth, and the Port Royal Logic

2013; Taylor & Francis; Volume: 34; Issue: 2 Linguagem: Inglês

10.1080/01445340.2012.748331

1464-5149

John N. Martin,

Historical and Literary Studies

Abstract The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the truth-conditions do not require the introduction of a new concept of ‘indefinite’ term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic’s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension. Acknowledgements The author gratefully acknowledges the support of the Charles Phelps Taft Fund at the University of Cincinnati, which made this research possible. Notes 1It shall be the practice in this paper to italicize the names of concepts only if they are being defined as part of a formal definition. Expressions that are being mentioned will also be italicized. 2For the traditional operation of restriction see Buridan, Summulae 5.1.8. (2001, pp. 286, 648, 835) and Fonseca, Institytionum dialecticarum, Liber VIII, Caput 40 (1964, pp. 740–741). For restriction in the Logic see Logique et l'Art de Penser (hereafter LAP) I,6; Kremer and Moreau 2003 (hereafter KM) V (pp. 145,40); Arnauld and Nicole 1996(hereafter B) (p. 40); LAP I,7,KM V (p. 151), B (p. 45); LAP II,17, KM V (p. 248), B (p. 130); and KM V (p. 250), B (p. 131). 3Strictly speaking, on this analysis the truth-conditions are formulated in terms of identity: some S is P is true iff the indefinite extension of some S is identical to the extension of the (definite) restriction of P by that of some S. 4For Parienté’s views on supposition see Parienté Citation1985 (pp. 273–274). For the argument for indefinite restriction see page 242 and Chapters 8 and 9. On the double restriction reading see also Auroux Citation1993 (pp. 148–149, 87) and Dominicy Citation1984 (pp. 167–168). 5 Morris Citation1939. 6 LAP I,6; KM V (p. 144); and B (p. 39). 7Axiom 1. For the discussion of this and later axioms see LAP II,17, KM V (pp. 248–252), and B (pp. 130–133). 8Axiom 4. 9Axiom 5, remark, and Axiom 6. 10Axiom 6, remark. 11 LAP II,3; KM V (p. 191); and B (p. 83). 12Car lorsque le sujet d'une proposition est un terme commun qui est pris dans toute son étendue, la proposition s'appelle universelle … Et lorsque le terme commun n'est pris que selon une partie indéterminée de son étendue, à cause de qu'il est resserré par le mot indéterminé quelque, la proposition s'appelle particulière… (LAP II,3; KM V, p. 191; B pp. 83–84). 2. Le sujet d'une proposition, pris universellement ou particulièrement, est ce qui la rend universelle ou particulière. 3. L'attribut d'une proposition affirmative n'ayant jamais plus d’étendue que le sujet, est toujours considéré comme pris particulièrement: parce que ce n'est que par accident s'il est quelquefois pris généralement. 4. L'attribut d'une proposition négative est toujours pris généralement (LAP II,3, KM pp. 258–259; B, p. 139). Mai quoique cette proposition singulière soit différente de l'universelle en ce que son sujet n'est pas commun, elle s'y doit néanmoins plutôt rapporter qu’à la particulière; parce que son sujet, par cela même qu'il est singulier, est nécessairement pris dan toute son étendue, ce qui fait l'essence d'une proposition universelle, & qui la distingue de la particulière. Car il importe peu pour l'universalité d'une proposition, que l’étendue de son sujet soit grande ou petite, pourvu que quelle qu'elle soit on la prenne toute entière. Et c'est pourquoi les propositions singulières tiennent lieu d'universelles dans l'argumentation. Ainsi l'on peut réduire toutes les propositions à quatre sortes, que l'on a marquées par ces quatre voyelles A.E.I.O. pour soulager la mémoire (LAP II,3; KM V, p. 199; B p. 84) 13 LAP III,3 and B (pp. 138–142). Le moyen ne peut être pris deux fois particulièrement, mais il doit être pris au moins une fois universellement. Les termes de la conclusion ne peuvent point être pris plus universellement dans la conclusion que dans les prémisses. On ne peut rien conclure de deux propositions négatives. On ne peut prouver une conclusion négative par deux propositions affirmatives. La conclusion suit toujours la plus faible partie; c'est-à-dire, que s'il y a une des deux propositions négatives, elle doit être négative, & s'il y en a une particulière, elle doit être particulière. De deux propositions particulières il ne s'enfuit rien (KM V, pp. 259–263). 14For example, for Rules 3–7 see William of Sherwood, Introduction to Logic III, ix (1966, p. 67); Peter of Spain, Tractatus, IV, 4 (1990, p. 40); De Rijk Citation1962–1967 (p. 45); Buridan, Summulae 5.1.8.(2001, pp. 312–313); and Fonseca, Institytionum dialecticarum, Liber VI, Caput 18 (1964, p. 382). For Rules 1 and 2 see Toletus, In Lib I posteriorum analyticorum, Cap XIX (1580, p. 202) and Fonseca, Institytionum dialecticarum, Liber VI, Caput 20 (1964, p. 386). All six rules are presented as a group in Eustace of St. Paul’s popular summary of scholastic philosophy, which was read and praised by Descartes for its clarity, Summa philosophiae quadripartita, Logia III.2.I (1609, p. 117). Letter to Mercenne, Descartes 1897–1909 (pp. 3, 251). 15See, for example, William of Sherwood, Introduction to Logic III, 9 (1966, p. 66); Peter of Spain, Tractatus IV, 13 (1990, p. 46), and de Rijk Citation1962–1967 (p. 52); and Buridan, Summulae, 5.2.2 (2001, p. 320). 16The relevant semantics for the syllogistic is set out in Part IV. For a proof of this generalization see Martin 1997, reprinted in Martin Citation2004. 17 Kneale and Kneale Citation1962 (p. 320). 18Leibniz lists seven rules, dividing the Logic’s fifth rule into two. See Lenzen Citation1990 (pp. 29–59). It should be remarked that though Leibniz (and Lenzen) presents the rule set as an ‘axiomatization’, neither his account nor the Logic’s is a true axiomatization of the valid moods. An axiom system characterizes a set of theorems as an inductive set, that is, as a set defined as the closure of a set of basic elements (axioms) under a set of construction rules (rules of inference). The Logic’s rules, on the other hand, provide a decision procedure, not an axiomatization. Curiously, the students’ mnemonic poems previous mentioned are theoretically more powerful than the Logic’s six rules because they provide not only a decision procedure but also an axiomatization. They encode how to ‘reduce’ the valid moods to Barbara and Celarent (via four ‘inference’ rules), and a ‘reduction’ is easily converted into a proof in the modern sense. See the references in Part IV for details. 19Here we shall discuss the version of ascent and descent put forward by John Buridan, not because his medieval text was known by Arnauld and Nicole, but because it is particularly clear about the logical relations at issue and because it contains all the relevant points that were to become common in the standard account from which Arnauld and Nicole abstracted their idea of distributive term. On the standard account see Corazzon. For an example of a contemporary account, see Fonseca Citation1964 1575 Liber VIII (Chapters 20–22, pp. 678–688), a text which was part of the Ratio Studiorum of 1599 for philosophy professors at Jesuit colleges of the period. 20Distributiva est secundum quam ex termino communi potest inferri quodlibet suorum suppositorum seorsum, vel etiam omnia simul copulative, secundum propositionem copulativam, ut ‘omnis homo currit’, sequitur ‘ergo Socrates currit’, ‘ergo Plato currit’, vel etiam…. Et manifestum est quod suppositio distributiva differt ab aliis suppositionibus quia terminus communis secundum eam infert quodlibet suorum singularium seorsum; aliae autem hoc non faciunt. Ideo si propositio sit vera, oportet quod sit vera pro quolibet supposito, quod non requiritur in aliis. (Summulae 4.3.6, p. 264). 21Part IV lays out the syllogistic model theory in which ⊨ is defined. For the purposes of this paper a singular term may be understood as a special case of a categorical term generally: a singular term is one that happens to supposit for a unique actual object. Thus, in the semantics of Part IV a singular term is a term that stands for a unit set. A universal affirmative with a singular term as subject is then understood as a special case of a universal affirmative, one in which the quantifier every is not explicitly expressed – such is the way it is understood in the Logic (II,2). It will follow from the semantics of Part IV that a universal affirmative with a singular term as subject and common noun as predicate is true iff the unique object in the set that is the referent of the subject is an element of the set that is the referent of the predicate, and that a proposition with either a singular term or common noun as subject and singular term as predicate is true iff the sets referred to by both terms contain one and the same individual. For perspicuity when the two terms are both singular, we shall use=to represent the copula. The connectives ∧ and ∨ here should be understood as conforming to the standard truth tables, as was the common medieval practice. 22Commentary: Dico tamen quod in suppositione determinata non oportet veritatem esse pro uno solo supposito, immo aliquando est vera pro quolibet, sed hoc requiritur et sufficit quod sit vera pro aliquo uno. Unde notandum est statim quod duae sunt condiciones suppositionis determinatae alicujus termini communis. Prima est quod ex quolibet supposito illius termini possit inferri terminus communis remanentibus aliis in propositione positis. Verbi gratia, quia in ista ‘homo currit’ iste terminus ‘homo’ supponit determinate, ideo sequitur ‘Socrates currit; ergo homo currit’, ‘Plato currit; ergo homo currit’, et sic de quolibet alio singulari contento sub ‘homine’. Secunda condicio est quod ex termino communi sic supponente possint inferri omnia singularia disjunctive, secundum propositionem disjunctivam; verbi gratia, sequitur ‘homo currit; ergo Socrates currit vel Plato currit vel Johannes currit…’ et sic de aliis (Summulae 4.3.5, p. 263). 23Sed confusa tantum est secundum quam non sequitur aliquod singularium seorsum retentis aliis in propositione positis, nec sequuntur singularia disjunctive, secundum propositionem disjunctivam, licet forte sequantur secundum propositionem de disjuncto extremo….\ Suppositio autem confusa tantum differt a suppositione determinata quia secundum suppositionem confusam non inferuntur ex termino communi singularia secundum propositionem disjunctivam, quod bene fit secundum suppositionem determinatam. Verbi gratia, in ista propositione ‘omnis homo est animal’ … et sic de aliis, quia prima est vera et omnes aliae sunt falsae (Summulae 4.3.6, p. 264). 24Strictly, the syllogistic's existential presupposition should also be expressed here and below, by an additional conjunct ∃xSx. This additional condition is made explicit in the more careful formulation of Part IV. 25See Ashworth Citation1973 and Priest and Read Citation1980. 26As explained in an earlier note, strictly speaking, in syllogistic syntax s i =p j is the universal affirmative every s i is p j and s i ≠p j is its contradictory, the particular negative some s i is not p j . 27The terminology is from Carnap Citation1947. 29For a more precise statement of the proof-theoretic system (both in axiomatic or natural deduction form) and the completeness theorem, see Martin Citation1997. 30See Keenan and Westerståhl Citation1997. 31Apart from the discussion of truth-conditions in which the wording describing restriction closely follows these formulations, the only passage in which Arnauld and Nicole describe ‘indeterminate’ quantifier restriction is this: Or cette restriction ou resserrement de l'idée générale quant à son étendue, se peut faire en deux manières.La première est, par une autre idée distincte & déterminée qu'on y joint, comme lorsqu’à l'idée générale du triangle, qui est le triangle rectangle, je joins celle d'avoir un angle droit: ce qui resserre cette idée à une seule espèce de triangle, qui est le triangle rectangle.L'autre en y joignant seulement une idée indistincte & indéterminée de partie; comme quand je dis, quelque triangle: on dit alors que le terme commun devient particulier, parce qu'il ne s’étend plus qu’à une partie des sujets auxquels il s’étendoit auparavant; sans que néanmoins on ait déterminé quelle est cette partie à laquelle on l'a resserré. (LAP I,6; KM V, p. 145; B p. 40) Parienté interprets the last paragraph of this text as introducing a second and new operation of ‘indefinite’ restriction. What the text as a whole is saying, however, in the terms just defined, is that although the quantifiers in both universal and particular affirmatives are subject conservative, which is a concept defined in terms of standard restriction alone, the subject of the universal proposition is universal and therefore (the relevant quantity of) the predicate (in this case, at least one) is true of (i.e. is identical to) each of the entire restricted class, but the subject of the particular is particular and therefore (the relevant quantity of) the predicate (in this case at, least one) is true of (i.e. identical to) at least one of the restricted class. The introduction of a second notion of restriction is gratuitous. 32More precisely, in the terminology of the Logic, ASP is true iff the extension of the predicate restricted by that of the subject is identical to that of the subject. 33For a full defense of the existential reading of the semantics of the Logic, see Martin Citation2011, 2012.