‘My Future Son is Possibly Alive’. Existential Presupposition and Empty Terms in Abelard's Modal Logic

Abstract

The aim of this paper is to investigate the problem of existential import in Abelard's modal logic, and to ask whether the system of logical relationships that he proposes for modal propositions maintains its validity when some of the terms included in these propositions are empty. In the following, I first argue that, just as in the case of non-modal propositions, Abelard interprets modal propositions as having existential import, so that it is a necessary condition for the truth of propositions like ‘It is possible for my son to be alive’ or ‘it is necessary that all men are animals’ that their subjects’ referents exist. Then, I present the schemata of inferences that Abelard proposes to describe the logical behaviour of de rebus modal propositions. I argue that these systems of relations are valid only as long as all the terms contained in the formulas have an existing referent. I also claim that Abelard was aware of this difficulty (at least in the Logica Ingredientibus), and, accordingly, he explicitly decided to restrict the validity of his modal system to propositions that do not contain empty terms.

ORCID

Irene Binini http://orcid.org/0000-0001-5031-1710

Notes

1 The English formulation “it is (not) possible for an/some/every S (not) to be P’ aims to capture the semantical ambiguity of the corresponding Latin construction, where the modal term has a nominal form and is followed by an accusative-infinitive clause. When not specified otherwise, all English propositions of this form should be taken as ambiguous claims, that may be interpreted both de rebus and de sensu. This English translation of Abelard's nominal propositions is introduced in Martin 2016.

2 Whether proposition (3) must be considered as an affirmative or a negative claim is not entirely clear. In Section 4, I shall propose that propositions about impossibility should be taken as being negative in meaning, even if they are affirmative with respect to their grammatical structure.

3 Not every modal proposition is apt to be interpreted in both ways, at least according to Abelard in the Logica Ingredientibus. As he says in Abaelardus 2010, pp. 403.310–408.425, there seem to be some modal propositions that—being entirely impersonal (i.e. impersonal with respect both to grammar and meaning)—can only be interpreted de sensu, not de rebus. Examples of propositions of this sort are ‘It is necessary that a chimera is not a man’ (Necesse est chimaeram non esse hominem) or ‘It is possible for what is not to be’ (Possibile est id quod non est esse).

4 Abelard is not always consistent in his use of the de sensu interpretation, and he refers, at least in the Logica Ingredientibus, to two different understandings of the de sensu reading. I will briefly address this point in Section 3.

5 Abaelardus 1956, pp. 173–84.

6 Abaelardus 2010, pp. 400.240–401.249. On the existential import of categorical propositions, see also Abaelardus 1956, pp. 176.20–4, where Abelard says that the proposition ‘Every man is a man’ is false if no human being exists. However, there is also some evidence contrary to this interpretation. In some places of the Dialectica and of the Logica, Abelard seems to think that some affirmative propositions might be true even if their subject term is empty. For instance, Tweedale notices that for Abelard, propositions like ‘the chimaera is a chimaera’ do not require the existence of any chimera in order to be true (see Tweedale 1976, p. 336). In other parts of his logical works, Abelard considers propositions such as ‘Homer is a poet’, ‘a chimaera is thinkable’, and ‘the chimaera does not exist’ and says that they are true propositions (Abaelardus 1956, pp. 135.9–138.26; 162.16–18; 167.6–169.28). As Jacobi has shown, Abelard justifies the truth of these last three claims by saying that propositions of this sort are ‘transformable’ into other sentences, the subjects of which refer to existing items. The proposition ‘Homer is a poet’, for instance, is transformed into a sentence about a poem written by Homer that still exists (Abaelardus 1956, pp. 135.24–5; 135.36–136.36; 168.11–7). The proposition ‘A chimaera is thinkable’ is transformed into a sentence about a person who imagines a chimaera (Abaelardus 1956, pp. 136.32–36; 168.21–25). In this way, Abelard attempts to justify the intuitive truth of propositions whose implicit import is not satisfied. See Jacobi 1986, pp. 157–8.

7 As Martin has argued, Abelard seems to conceive of extinctive negation as a truth-functional propositional operator. This interpretation is based on several different observations. First, Abelard's extinctive negation is applied to an entire proposition and renders a proposition as its value. Also, this sort of negation might be iterated or combined with other connectives: for instance, we may apply an extinctive negation to a proposition that already includes a separative negation, for example, by saying ‘It is not the case that some man is not an animal’ (Non quidam homo non est animal), or we may formulate a sentence that contains a double extinctive negation, like ‘It is not the case that not every man is an animal’ (Non non omnis homo est animal). On this point, see Martin 2004, p. 167.

8 On the different behaviour of separative and extinctive negation with respect to existential presupposition, see also Wilks 2008, pp. 108–9.

9 Abaelardus 1956, pp. 175.33–176.37.

10 Abelard's reformulation of Boethius’ square of opposition is discussed in Wilks 2008, pp. 108–9.

11 The rule of conversion by contraposition is also invalid for propositions containing terms that refer to everything that exist, such as the term ‘substantia’. For indeed, propositions containing terms of this sort convert into propositions that contain empty terms, as the reader may easily verify. Abelard is not the first to be aware of the incompatibility between Boethius’ rules of conversion and the presence of empty names. We may find traces of this idea also in the Dialectica of Garland the Computist. Here, Garland notices that if we admit conversions for those propositions that contain terms like ‘substantia’, which refer to everything that exists, some problems arise, for these propositions convert into propositions that contain empty terms, such as ‘non-substantia’, which do not have the same truth value of the initial proposition. On this topic see Garlandus Compotista 1959, pp. 57.20–63.34.

12 Abaelardus 2010, pp. 417.639–650: ‘Sed et, cum nullum filium habeam, propositio uera uidetur de sensu quae ait: ‘Possibile est filium meum uiuere’ id est: ‘Possibile est ita euenire ut haec propositio dicit “Filius meus uiuit”’, quia adhuc fortasse ita continget; nec tamen uera est de rebus quae ait: ‘Filius meus potest uiuere’ quippe per subiectum quod est ‘filius meus’ positionem existentiae filii mei facio et quasi ipsi existenti posse uiuere copulo. Unde nec, cum dico: ‘Filius meus non uiuit’ faciens negationem separatiuam, pro uera eam recipimus quia in subiecta oratione, filii <mei> scilicet, positio facta est et ab eo quasi existente uiuere separo; ideoque, cum non existat, falsa etiam est negatio. Ex quo multo magis falsa uidetur affirmatio quae dicit filium meum posse uiuere, cum de re ipsa accipitur’.

13 For examples of this sort see Abaelardus 1956, pp. 201.2–9; 201.12–17; 202.21–25 and Abaelardus 2010, pp. 417.639–641.

14 For this distinction see Martin 2016, p. 130 ff. and Thom 2003, pp. 45–47.

15 Evidence for this interpretation is provided by a passage of the Dialectica (Abaelardus 1956, pp. 192.26–28) in which Abelard says that, in propositions of this sort, the particle ‘non’ separates the predicate (i.e. the mode) from the subject (i.e. the verb of the infinitive clause): ‘Oportet enim ut in istis negatio praedicatum removeat, modum scilicet, ab eodem subiecto, sive scilicet “esse” sive “non esse”’. Another passage indicating that Abelard conceived negation in modal claims as separative and not as extinctive is Abaelardus 1956, pp. 199.13–14, where Abelard claims that the following inference is true, but its opposite is false: ‘if it is not possible for Socrates not to be white, then it is possible (for him) to be white’ (‘Si enim non est possibile Socratem non esse album, possibile est esse album, sed non conuertitur’). The validity of the inference given here is easily explained if we take negation to be separative, while the same inference should turn out to be invalid if we take negation to be extinctive. If Socrates does not exist, the antecedent would be true and the consequent false. I thank Christopher Martin for bringing these passages to my attention.

16 The equipollence between possibility and necessity will be considered in Section 5.

17 In his discussion on modal logic, Abelard often refers to the distinction between the syntactic form and the semantic, ‘logical’ form of a proposition, which he describes as a distinction between considering a certain term or proposition with respect to its grammatical structure (secundum constructionem), or with respect to its ‘meaning’ (secundum sensum, in sensu). On this point, see for instance Abaelardus 2010, pp. 394.86–90; 396.124–142.

18 See Abaelardus 1956, pp. 179.20–26; this passage is quoted and commented upon in Martin 2004, p. 167. In contrast to the relationship between (3.a) and (3.b), which is both a relation of ‘concomitance’ (i.e. they have the same truth value in all possible situations) and a relation of mutual consequence (i.e. the meaning of the first is ‘contained’ in the meaning of the second and vice versa), between propositions (1.a) and (1.b) there is only a relation of ‘concomitance’ but not a relation of meaning.

19 The same problem is observed in quantified modal propositions: the two propositions ‘It is necessary for every man to be an animal’ and ‘It is not possible for every man not to be an animal’, which are supposed to be equivalent according to Abelard's rules of equipollence, do not have the same truth value in all possible situations, for when no men exist the first proposition is false, while the second is true.

20 For the distinction between the notions of concomitance and inference in Abelard's logic see Martin 2004.

21 ‘Sed uidentur nobis huiusmodi aequipollentiae modalium propositionum tantum aequipollentiam custodire re subiecti termini permanente, uelut tantum dum Socrates permanet […] Quamdiu itaque Socrate permanente uera est: “Non possibile est Socratem esse album” (uel: “Impossibile est”) uera est etiam quae ait: “Necesse est Socratem non esse album” et e conuerso. Cum enim uelimus in ui affirmatiuae categoricae accipere: “Necesse est Socratem non esse albedinem” oportet ad hoc ut uerum sit rem manere sub subiecto uocabulo, ut supra meminimus. […] Nos uero in diuerso sensu eas quoque concedimus aequipollere ita ut rebus permanentibus nulla possit esse uera uel falsa sine aliis.’