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As we mention in our previous chapter, our primary focus in this work has been Buri- dan’s analysis of divided modal propositions. Buridan distinguishes two kinds of modal propositions, composite and divided modals. The second half of Book Two of theTrea- tise on Consequences addresses inferences between composite modal propositions, and the relationship between composite and divided propositions. Our main interest here will to be to sketch enough of Buridan’s theory to allow for us to unpack the following conclusions:

from no affirmative composite of possibility does there follow a divided one of possibility with the mode affirmed, or conversely, except that from an affir- mative composite with an affirmed dictum there follows a divided particular affirmative. . . . from no composite affirmative of necessity does there follow a divided one of necessity with an affirmed mode, nor conversely, except that from a divided universal negative there does follow a composite universal with a negated dictum. . . . from no proposition, [whether] assertoric, of pos- sibility or of necessity does there follow one of contingency with both modes affirmed; similarly, from none of contingency does there follow an assertoric or one of necessity, but there does follow one of possibility. [51, pp. 55-57]25 Grammatically, composite modal propositions are ones where the modality occurs as one of the two terms in the proposition.26 In English these kinds of modals are usually translated with the presence of a ‘that’ clause. The other term in a composite modal proposition is normally a categorical proposition as well. For example, ‘that ‘Every A is B’ is necessary’ and ‘it is possible that ‘No B is A” are both composite modal propositions. In Latin such propositions are distinguished by the presence of an accusative-infinitive construction. Buridan offers the following two examples: “Hominem currere est possibile et haec: Necessarium est hominem esse animal”. [4, Bk2-C2] 25. These are conclusions 17–19 of Book Two.

26. For our purposes here, as well as Buridan’s, we will assume that a proposition is either composite or divided, and not both, i.e. we will not consider cases where one of the terms is a modal and the copula is modified by a modal term. For example, we will rule out cases like: that ‘Every A is B’ can be necessary.

Here the clauses ‘Hominem currere’ (a man runs) and ‘hominem esse animal’ (a man is an animal) are examples of the accusative-infinitive construction.

When referring to composite modal propositions, Buridan refers to the infinitive– accusative construction as the dictum. For example, in ‘it is possible that ‘No B is A” ‘No B is A’ is the dictum of the proposition. The truth conditions for these propositions are also very different from the ones for divided modal operations. Composite modals do not ampliate either the subject or the predicate to supposit for anything. Instead, Buridan discusses the signification of such propositions.

Here ‘possibility’ is taken not for what can be but for a possible proposition, which is said to be possible in so far as things can be altogether as it signifies. So in the examples above, saying ‘Every possibility is that B is A’ is the same as to say ‘Every possible proposition is that B is A’. . . It should also be noted that in the proposition ‘Every possibility is that B is A’, the predicate ‘that B is A’ supposits materially for the proposition ‘B is A’, and does not supposit for itself, since the phrase ‘That B is A’ is not a proposition.[51, p.49]

There are a number of points to pay attention to here. The main one that concerns us is Buridan’s point that the proposition which falls under the ‘that’ clause supposits for the proposition itself, and then the modal term is assessed based on if the proposition is in fact the way the term describes. For example, in ‘it is necessary that ‘Every man is an animal” the sentence ‘Every man is an animal’ supposits for that proposition, and it is true just in case it is necessary, which, according to Aristotle, it is.

After this, Buridan goes on to discuss an important grammatical difficulty that occurs because of the Latin in which the propositions are expressed. The main problem here has to do with propositions like: ‘Nullum B esse A est possibile’. Here ‘nullum’ could be either nominative or accusative, and so it is ambiguous as to whether the quantifier is part of the accusative–infinitive construction, and thus part of the dictum, or if it ranges over the whole proposition.

Notice that in some ways this analysis is much closer to the way modern modal logic relates to propositions. The modal operator binds to the truth of the entire proposition, not to the various terms in the proposition ampliating the supposition of various terms. In other ways, this is rather different. The modals here are not functioning as operators (as they do in our standard modal logics), but instead they function as terms that modify various propositions. These term-based operations expand the expressive power of the syllogistic logic in some very interesting ways. For example, if we add the modality ‘false’ to our language (as Buridan does), then we can define an operation that looks very similar to what we now think of as propositional negation.27

When it comes to thinking about truth conditions between these sorts of propositions, it is easiest if we resort to the usual kinds of relationships that we think of in an operator- based modal logic. We have already argued that Buridan’s account of modal logic is in the same spirit as possible worlds semantics, and as such, reading his operations this 27. The idea here is that, we equatenot−φwithφis false (or perhaps even more clearly ‘it is false that

way should not do too much damage. Strictly speaking, since Buridan does not offer us a reductive account of modality, it would be safer if we consider truth conditions in terms of the primitive notions of possibility and necessity. Those persuaded that Buridan is working with a kind of possible worlds semantics can then supply the needed truth conditions.

What we have said so far should be enough to help us understand the importance of Buridan’s final conclusions in Book Two. When we come to the four conclusions listed above, it will be helpful to take them each in turn. First,

From no affirmative composite of possibility does there follow a divided one of possibility with the mode affirmed, or conversely, except that from an affir- mative composite with an affirmed dictum there follows a divided particular affirmative.[51, p.110]

What this tells us that ‘it is possible that ‘some A is B” entails ‘Some A is possibly B’, but that in the other cases there is no valid inference.28 From what we have already seen, this makes sense. Reading, ‘it is possible that ‘some A is B” as telling us that there is some world where ‘Some A is B’ is true, we know from what Buridan has already said, that this is only true if there is some object, say D, such that ‘This D is A’ and ‘The same D is B’. But then, it is possible that ‘This D is A’ and it is possible that ‘This D is B’. As we have already seen, it follows by expository syllogism then that ‘Some A is possibly B.’ At first glance, this might seem to look like Buridan endorsing the Barcan formula and we can find such a view in the literature.29

However things are somewhat more complicated when it comes to translating Buri- dan’s logic into first-order logic, and ∃x◇ (Ax∧Bx)is not equivalent to AMi B, as the quantifier gets the ampliation of the terms wrong.30 The problem is that the quantifica- tion used here does not range over the specific world at which the formula is evaluated, but should range over all of the objects at all of the worlds.

So, at least here, it seems Buridan is not committed to either of the Barcan formulae. In fact, a counterexample to the Barcan formula is easily seen to follow from Buridan’s consideration of the definition of possibility modals. Recall that Buridan said: “Accord- ingly, it is true that air can be made from water, although this may not be true of any air which exists.31”[4][p.58]. Let us assume that this situation does indeed obtain, there is some air that can be made from water. Let us assume further that there is currently 28. This is equally clear since, even if something can be A and can be B, it does not entail that something

can be A and B at the same world. i.e. (◇A∧ ◇B) → ◇(A∧B)is not valid.

29. See [35, pp.158,160 fn. 56] In addition to Lagerlund’s own concerns about his proof, the comments we make here raise similar problems of Lagerlund’s formalisation of ‘Every B is necessarily not A⇒ That every B is not A is necessary.’

30. A proper spelling out of this would require a formal reconstruction of Buridan’s composite modal propositions and then a proof of the inference in a system that does not validate the Barcan and Con- verse Barcan formulae. Unfortunately, due to time and space constraints, this will not be attempted here.

no water but that there will be. Then, we have A Mi W (reading A for air and W for water) is true, and hence so is ◇∃x(Ax∧W x)but∃x◇ (Ax∧W x)is not true because no water currently exists at the world of evaluation ex hypothesi.

For the Converse Barcan formula, things are a little bit more tricky but Buridan will reject it, given the following sorts of remarks:

As to whether the proposition ‘A horse is an animal’ is necessary, I believe it is not, speaking simply of a necessary proposition, since God can annihilate all horses all at once, and then there would be no horse; so no horse would be an animal, and so ‘A horse is an animal’ would be false, and so it would not be necessary. But such [propositions] can be allowed to be necessary, taking conditional or temporal necessity, analysing them as saying that every human is of necessity an animal if he or she exists, and that every human is of necessity an animal when he or she exists.[51, p.141]

Informally, what Buridan is pointing out here is that it is entirely possible for all objects to cease existing. It is within the power of God to bring it about that no horses exist, or in fact ever existed. More to the point, such objects also lose all of the properties that they might have, upon ceasing to exist concretely. As such, it seems that Buridan allows for objects to pass out of existence.

As is well-known, we can use this to construct a counterexample to the Converse Barcan formula along the usual lines. Let us assume that some horse exists. Then, clearly given what Buridan has said above, it is clearly possible that this horse does not exist and hence, ∃x◇ ¬Ex, where Ex stands for ‘x exists’. However, since Buridan maintains that horses (and objects more generally) lose their properties once they cease to exist,◇∃x¬Ex, will turn out to be impossible on Buridan’s view, as it would require the existence of a non-existent object.

From what we’ve shown here, it seems then, that Buridan would be some sort of contingentist. What he says about his modal logic suggests he would deny the validity of the Barcan and Converse Barcan formulae. As further support for this, we should observe that to formalise the divided fragment of Buridan’s modal logic, we did not need to impose any sort of domain restriction on the models.32 This does raise some interesting questions, as Buridan does opt for a particularly broad reading of the modal operations and his definitions of quantification could be interpreted in nonstandard ways.