Buridan’s Theory of Consequences

Abstract

Buridan endorses the basic idea that q follows from p iff it is impossible that p is true but q is false. Since he also accepts the law that, if p is impossible, the conjunction (p ∧ q) must be impossible, he comes to regard the principle ‘Ex impossibili quodlibet’ (EIQ) as basically correct. However, his logic is based on a ‘nominalist’ view according to which propositions are tokens of spoken, written or thought language existing in space of time, and the truth of such a proposition presupposes that it exists. This conception prompted Buridan to modify the definition of ‘consequence’, but none of his attempts was fully successful. Yet, a large part of his theory of consequences is based on the usual account of necessary truth-conservation. As regards principle EIQ, Buridan distinguishes two variants of impossibility. Proposition p is possible iff the state of affairs as described by p is possible; this doesn’t guarantee, however, that p is ‘possibly true’. E.g. ‘No proposition is negative’ is considered by Buridan as describing a possible state of affairs, but this proposition is not possibly true, because in order to be true, it would have to exist, and as soon as it comes into existence, it becomes false. Accordingly, EIQ requires that the antecedent p is really impossible and not only impossibly true.

Keywords:

• John Buridan
• theory of consequences
• ex impossibile quodlibet: possibility vs. possible truth

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We are using ‘¬’, ‘∧’, ‘∨’, ‘→’, and ‘↔’ as symbols for negation, conjunction, disjunction, implication, and equivalence. In addition, ‘⇒’ serves as a symbol for logical inferences. In Lenzen 2024, Buridan’s distinction between formally valid, materially valid, and as-of-now valid inferences was represented by ‘⇒_for’, ‘⇒_mat’, and ‘⇒_now’, respectively. In the present paper, these distinctions don’t really matter. We don’t have to consider as-of-now inferences at all, and as a default option, ‘⇒’ may be interpreted as an analytically valid inference (which Buridan calls ‘materially valid’). In section 3 below, various attempts to define this notion are distinguished as ‘⇒₁’, ‘⇒₂’, … .

2 Clearly, if p is impossible, then ¬p is necessary; but, according to Eccq, ((p ∧ ¬p) ⇒ q), hence, according to Buri 1, (p ⇒ q), i.e. Eiq holds.

3 Cf. McCall 2012, p. 416.

4 Like many other medieval logicians, Buridan doesn’t see any substantial difference between the conditional ‘if p, then q’ and the inference, ‘p, therefore q’; cf. section 3 below.

5 The terminology of connexive logic is explained in Wansing 2020. Martin 2004 spoke of ABEL 1, 2 (together with ARIST 1, 2) as ‘four connexive principles that Abelard makes “the centrepieces of his theory of conditionals”’.

6 According to Łukasiewicz 1935, p. 118, already the ancient theologian Origenes (185–253) considered the variant: ‘Wenn du weißt, daß du tot bist, so bist du tot (denn man kann nicht etwas Falsches wissen); wenn du weißt, daß du tot bist, so bist du nicht tot (denn der Tote weiß nichts); also weißt du nicht, daß du tot bist’. Another version is to be found in a 12^th century text: ‘Si Socrates scit se esse lapidem, Socrates est lapis. Si est lapis, nihil scit. Ergo si scit se esse lapidem, nihil scit. Sed etsi scit se esse lapidem, [ergo] aliquid scit’ (Iwakuma 1993, p. 143).

7 The proof is based on the principle of contraposition and the law of transitivity: ‘Now the fact that every proposition that includes opposites implies its contradictory is plain through two rules set out earlier. One of them is that if an inference is a good one, from the contradictory of the consequent there follows the contradictory of the antecedent. The second is that whatever follows from a consequent, the same thing follows from the antecedent’ (Burley 2000, p. 157).

8 Cf. Abaelardus 1970, p. 396: ‘“si omne ens est paternitas, nullum ens est filiatio” […] Unde et illud contingit: “si omne ens est pater, nulla paternitas est”; hinc quoque istud: “si omne ens est paternitas, nullum ens est paternitas”’.

9 Burley 2000, p. 228. The original formulation ‘Tantum pater est’ would better be translated ‘There exist only fathers’ instead of ‘Only the father exists’. In general, ‘tantum A est B’ means that only A are B, i.e. whenever something, x, is not A, x can’t be B either. A short version of the paradox is also presented by ‘the Master of Abstractions’: ‘Si tantum pater est, pater est; et si pater est, filius est; et si filius est, non tantum pater est; ergo a primo: si tantum pater est, non tantum pater est’ (Ebbesen 2015, p. 336).

10 Cf. Buridan 2015, p. 80. Curly brackets mark additions either by the translator (Read) or the editor (Hubien) of Buridan’s Treatise; cornered brackets indicate changes that go on my account.

11 Cf. Neckham 1863, p. 290: ‘si verum est nihil esse, nihil est. Sed si hoc est verum, aliquid verum est, et, si aliquid verum est, aliquid est; ergo si verum [est] nihil esse, […] aliquid est’.

12 Buridan’s example is a bit strange because in order to obtain ‘No God is just, so no ass is running’ via contraposition, one would have to have ‘Some ass is running, so some God is just’, i.e. ‘God’ would have to function as a general term. Normally, however, medieval logicians consider ‘God’ as a singular term.

13 Leibniz defined the possibility of a concept C by requiring that C does not contain any conjunction of B and not-B: ‘B not-B is impossible; or, if B not-B = C, C will be impossible’ (Leibniz 1966, p. 58). As he further noticed, this definition can be immediately transferred from the logic of concepts to the logic of propositions. For details cf. Lenzen 2019.

14 The term ‘humble connexivity’ has been coined in Kapsner 2019.

15 The ‘passive’ counterparts of these ‘actively’ formulated principles say that no proposition q is entailed by its own negation, unless q is necessary, and that no proposition q is entailed by both of two contradictory propositions, unless q is necessary.

16 Cf. Paulus Venetus 1984, p. 167: ‘A solid material inference is that in which the contradictory of the consequent is materially repugnant to the antecedent. E.g. “God is not; therefore no man is”’. This inference, though not logically valid, may be regarded as theologically valid because if the Creator would not exist, none of his creatures could exist either. Read 1993, p. 252, points out to the example ‘si Deus non est, Deus est’ which had been considered, e.g. by Pseudo-Scotus and by Ralph Strode.

17 Dutilh Novaes 2005, pp. 280–281, pointed out to the danger of conflating the concepts ‘inference’ and ‘consequence’. In her opinion, an inference from p to q involves the assertion of the antecedent p (and henceforth also of the consequent q), while a ‘consequentia’ only concerns the logical relations between ‘unasserted contents’. For medieval logicians, however, the latter understanding prevails also when the consequence is formulated as ‘p, therefore q’. This is evident in particular in case of impossible antecedents!

18 Cf. Buridan 2015, p. 67. Read’s translation ‘if’ suggests that the impossibility of (p ∧ ¬q) is only meant as a sufficient condition. However, the original version may be interpreted as a sufficient and necessary condition: ‘Dicunt ergo multi quod propositionum duarum illa est antecedens ad aliam quam impossibile est esse veram illa alia non existente vera’ (Buridan 1976, p. 21).

19 Read uses this expression in his introduction to Buridan 2015, p. 10: ‘What Buridan’s definition of consequence amounts to in Book I, Chapter 3 is necessary truth-preservation’.

20 Thus Dutilh Novaes 2008, p. 472 points out that ‘[…] most authors of the 14th century accept at least as a necessary condition for a (valid) consequence that the antecedent cannot be true while the consequent is false; many accept this as a sufficient condition as well’.

21 Somewhat more exactly, EIQ only contradicts the ‘hardcore’ version of the connexive principles while it is well compatible with the ‘humble’ variant; cf. Lenzen 2022.

22 Cf. Abaelardus 1970, pp. 283–284, and the translation in Martin 2004, p. 181. A detailed discussion of Abelard’s attempt to ‘save’ the connexive principles can be found in Lenzen 2021.

23 Cf. Buridanus 1977, p. 125: ‘Quarta conclusio est quod impossibile est bonae consequentiae antecedens esse verum consequente existente falsum. Sic enim debet intelligi quod non potest sequi ex vero falsum, ut habetur in libro Priorum’.

24 Cf. Hughes 1982, pp. 34–35, or Buridanus 1977, p. 123: ‘Deus enim posset annihilare omnes negativas, dimittendo affirmativas, ideo tunc omnis propositio esset affirmativa. […] Et tunc illud consequens non esset verum, quia non esset’.

25 Thus, Hughes 1982, p. 83 remarked: ‘Take any valid inference in which the premiss(es) can be true (and there are obviously many such inferences). Now it is always possible for the conclusion not to exist, and therefore not to be true. So in such an inference it is not impossible for the premiss(es) to be true but the conclusion not to be true’ so that INF 1 (which Hughes calls ‘Theory A’) cannot be correct.

26 As regards the laws of conjunction and of disjunction, cf. Buridan 2015, p. 113: ‘For from every conjunction each of the conjuncts follows and from any proposition there follows every disjunction disjoining it with another’. Buridan’s views concerning the laws of subalternation, conversion, etc. will be discussed in section 5 below.

27 Cf. Buridan 2015, p. 67; my emphasis. Again, it is not clear whether the ‘if’ shall be understood as a mere ‘if’ or as ‘iff’. The original formulation ‘illa propositio est antecedens ad aliam propositionem quam impossibile est esse veram illa alia non existente vera illis simul formatis’ doesn’t settle this issue.

28 Thus, also d’Ors 1993, p. 202 presumed that Buridan’s ‘second definition comes to correct this insufficiency by means of the clause “illis simul formatis”, which alludes in an immediate manner to the existence of the propositions in question, and tries to elude the problem posed by the possible nonexistence of a proposition’.

29 Cf. Hughes 1982, p. 35, or Buridanus 1977, p. 124: ‘Contra, quia si ex eo haec consequentia diceretur bona, sequeretur quod ista consequentia diceretur bona: “Nulla propositio est affirmativa, ergo baculus est in angulo”’.

30 This formalization has already been used in Lenzen 2024 as a second criterion for the validity of consequences, labelled ‘Cons 2’.

31 Cf. Hughes 1982, p. 35; Hughes introduces this passage by the phrase ‘To resume the argument against the sophism’ thus indicating that he thinks that Buridan’s explanation only refers to the main topic of Chapter 8, i.e. sophism AFFIRM 1. The original text, however, rather speaks in favor of the interpretation that both AFFIRM 2 and AFFIRM 1 are concerned: ‘Item illa non est bona consequentia ubi consequens, si apponeretur antecedenti vero, falsificaret ipsum, quia tale consequens ad tale antecedens magis videtur habere repugnantiam quam convenientiam, et sic est in proposito’ (Buridanus 1977, p. 124).

32 Let it be noted in passing that this consideration would not invalidate the related example ‘No proposition is affirmative, therefore no stick is standing in the corner’!

33 Since Buridan later makes a distinction between a proposition which ‘cannot possibly be true’ and one which is impossible, the provisional symbolization of p’s not being possibly true as ‘¬◊p’ will have to be modified; cf. the introduction of the new operator ‘◊_t’ below.

34 Principle Contra 1 is stated very clearly in Buridan 2015, p. 76: ‘In every good consequence, the contradictory of the antecedent must follow from the contradictory of the consequent’. In view of the general validity of this principle; it is not necessary to add any subscript to ‘⇒’. Note also that the propositional principle Contra 1 must not be confounded with the syllogistic principle Contra 2 which will be briefly discussed in section 5.

35 Cf. Burley 2000, p. 159: ‘For it is generally said that God can do everything that does not include a contradiction’. Burley uses this theological principle to support his own thesis that there exist propositions which ‘include opposites’, i.e. which refute themselves in the sense of DEF 1 above.

36 Cf. Hughes 1982, p. 37: ‘Something more is therefore required for the validity of an inference, and that is that the facts cannot be as the premiss says they are unless they are also as the conclusion says they are’. The original version in Buridan 1976, p. 22 says: ‘[…] illa propositio est antecedens ad aliam quae sic se habet ad illam quod impossibile est qualiter cumque ipsa significat sic esse quin qualitercumque illa alia significat sic sit ipsis simul propositis’. The Treatise contains the related definition: ‘saying that one proposition is antecedent to another, which is such that it is impossible for things to be altogether as it signifies unless they are altogether as the other signifies’ where a clause is added ‘when they are proposed together’ (Buridan 2015, p. 67). D’Ors 1993, pp. 203–204 considered this addition as redundant for since ‘[…] the appeal to the truth of propositions […] is replaced by the appeal to the disposition of the things signified by them […], a disposition with respect to which the existence or non-existence of the propositions […] is no longer taken into account ([…] in this case one does not understand why the clause “ipsis simul propositis” is retained)’. Cf., however, the discussion of criteria INF 5–7 in Section 3.7 below.

37 In Lenzen 2024, this criterion, labelled ‘Cons 3’, was formalized as follows: (p ⇒ q) iff ¬◊({p} ∧ {¬q}), where ‘{p}’ symbolizes the state of affairs described by proposition p.

38 Cf. Hughes 1982, p. 37. The italics have been added by the editor of the English version; the original version just runs: ‘[…] quod haec est possibilis: “Nulla propositio est negativa”, licet non possit esse vera’ (Buridanus 1977, p. 125).

39 Normore 2015, p. 362, similarly paraphrased this conception by saying that ‘[…] the goodness of a consequentia […] requires […] that it be impossible that things be/have been/will be as the antecedent has it and not be as the consequent has it’.

40 Cf. Hughes 1982, p. 86: ‘[…] if we take any proposition and add to it the further proposition that it exists, then from these two premisses together we can validly infer that the original proposition is true’.

41 Cf. Klima 2016, p. 321: ‘[…] since this revision of the definition of the validity of a consequence had to be introduced only because of the possibility of a proposition token quantifying over itself in a natural language, once one keeps this possibility in mind the definition of validity need not be totally overhauled’.

42 Cf. Dutilh Novaes 2008, pp. 472–473; my emphasis. Similarly, Dutilh Novaes 2020, section 3.3 explains that ‘[c]ommitments to sentence-tokens aside, Buridan’s notion of consequence clearly has necessary truth preservation […] as its fundamental component’ (my emphasis).

43 # 8 deals with the number of ‘causes of truth’ of propositions; # 9 with the supposition of terms; # 10 with the logical relation between distributed and not distributed terms; # 11 with the relation between ‘merely confused’ supposition and determinate supposition; and ## 12, 13 with the distinction between the terms ‘B’ and ‘that which is B’.

44 A corresponding modification, of course, applies to NAQ. In order to be entailed by every proposition p, q must really be necessary in the sense of ¬◊¬q; the slightly weaker condition ¬◊_t¬q would not do.

45 Buridan tries to prove this, but his proof is unnecessarily complicated because it distinguishes between cases where the antecedent A is impossible and where A is possible.

46 As regards the quotation marks around the attribute ‘improper’, a referee objected that all ‘medieval logicians considered “true” and “false” to be modes in the proper sense’. Yet, from the perspective of modern modal logic, they are only improper elements.

47 This ‘meta-inference’ is closely related to modus ponens: If p implies q, and if p is true, then q must be true, too.

48 Read’s translation explicitly uses the expression ‘disjunctive syllogism’ (Buridan 2015, p. 79). Buridan’s original formulation was: ‘et iste syllogismus tenet per locum a divisione, quia duobus positis sub disiunctione si alterum interimatur reliquum concludetur’ (Buridan 1976, p. 37).

49 A good summary of Buridan’s theory of syllogisms is to be found in Read’s introduction to Buridan 2015, pp. 23–26; the theory of modal syllogisms is treated there on pp. 36–50.

50 Here ‘s’ and ‘p’ abbreviate the names ‘Socrates’ and ‘Plato’, and ‘R’ stands for the predicate ‘is running’. Furthermore, we adopt the usual symbols ‘∃’ and ‘∀’ for the existential and the universal quantifier.

51 Thus, also Hughes 1989, p. 107 remarked: ‘In spite of what later textbooks had to say about “obversion”, “Some A is-not a B” is not equivalent to “Some A is a non-B”. At least Buridan says this quite explicitly. The former is a negative proposition, and is therefore true if there are no A’s; but the latter is an affirmative proposition with a negative predicate, and is false if there are no A’s’.

52 Buridan’s proof of the (restricted) laws of obversion are based on the insight that ‘a finite and an infinite term do not supposit for the same; so of whatever either of them is truly affirmed it is necessary that the other is denied’ (Buridan 2015, p. 93). Accordingly, Buridan 2015, p. 94 paraphrases ‘B is non-A’ also as ‘B is other than A’, and thus the obversion of the particular affirmative proposition can take the form: ‘“Some B is A”; so some B is not other than A’.