Balance General 21

In this section our aim will be to present a formal reconstruction of Buridan’s divided modal syllogism as contained in theTreatise on Consequence.7

5.2.1 Preliminaries and Semantics

Buridan Language. A Buridan Language

L= ⟨CON S, P RED, V AR, a, e, i, o, L, M, Q,Q¯⟩ where: CONS is a countable set of singular terms/objects.8 PRED is a countable set of (monadic) predicates.9

Well-Formed Formulae. If A, B∈P RED then:

AaB, AeB, AiB, AoB are well-formed.

If A×B is a well-formed formula where × is one of a,e,i, or o then:

∇A×B is a well-formed formula where ∇ is one of L,M,Q,Q¯, or -.10 Nothing else is a well-formed formula.

A formula is said to be categorical if it is any of the well-formed formulae above. In what follows we will introduce another set of formulae which will be called singular formulae and will make use of CONS. We include the full language here to avoid having to expand the language later in this chapter. A formula is said be modal if it contains the symbolL,M,Q, or ¯Q. In what follows, we will often contract ‘well-formed formulae’ and speak only of formulae. A brief comment on the notation conventions should be made here. As Buridan remarks, and as will be clear from our semantic definitions below, divided modal propositions are different from both composite propositions (which Buridan treats but will not be treated in this thesis) and from the usual semantics given 7. For a similar treatment of modal propositions by other medieval logicians see chapters 4 and 5 of [62]. 8. In what follows we will use terms from the beginning of the alphabet for objects and terms from the

end of the alphabet for variables or worlds.

9. To avoid confusion we will not include Q or R among our predicates.

10. To avoid ambiguity between divided and compounded senses of the modality, we will superscript the modal operation over the copula. Following standard convention, we will sometimes writeX instead of - for the assertoric proposition.

in modern logic for the operation◻. As such, and to help avoid confusing divided modal propositions with composite ones, we avoid the use of the notation ◻,◇, and ∇, which are often (thought not exclusively) used in the modern literature preferring L, M, Q, and ¯Q, which are used less frequently, and are also common in the historical literature about modal syllogisms.

Buridan Modal Model. A Buridan Modal Model is a tuple:

M= ⟨D, W, R, O, v⟩ such that:

DandW are non-empty sets. Dis the domain of objects andW is a set of worlds.

R⊆W2.

O∶W → P(D) s.t. O(w) ⊆D v∶W ×P RED→ P(D)

We require that R be an equivalence relationship.

Semantic Abbreviations. Let P be a term. Using the semantics we can define the following operations:

V(w, P) =O(w) ∩v(w, P) V(w,¬P) =D∖ (O(w) ∩v(w, P))

Informally, we can think ofV, (andM and L; see below) as giving the supposition or extension of a particular term. V(w, P) returns the extension of the predicate for the objects that exist atw. V(w,¬P)gives the anti-extension ofP atw. It should be noted that an object will fall into the anti-extension at a worldwif it either fails to exist atw

or if it is not in the valuation ofP atw. It is for this reason that we use the notation¬ as it is suggestive of the negation operation in a more familiar setting.

Moving into the modal context, we will want to consider situations where an object, saydcan fall under a predicate (sayP) and cases where an objectddoes not fall under a predicate. As such we will need to use both V(w, P) and V(w,¬P)in our definitions of modal operations. As such, we will abuse notation slightly and write, e.g. V(w, K)

with the understanding thatK ranges over terms and terms with¬in front of them. To that end we define our modal operations as follows:

M(w, K) = {d∈D∶ there is some z s.t wRz andd∈V(z, K)} L(w, K) = {d∈D∶ for all z if wRz thend∈V(z, K)}11

11. If one finds the use ofKunclear, it may be helpful to alternatively think of these as four operations defined as follows:

M(w, P) = {d∈D∶ there is some z s.t wRz andd∈V(z, P)} M(w,¬P) = {d∈D∶ there is some z s.t wRz andd∈V(z,¬P)} L(w, P) = {d∈D∶ for all z if wRz thend∈V(z, P)}

When we move to consider the operations M(w, K) and L(w, K), these are being used to encode the set of objects that are possible relative to a world and are necessary relative to a world. The basic idea is that an object (d) is possiblyK relative to a world (w) if there is some world v such that wRv and the object d is K at w. From this it should be clear that, as in the usual semantics for modal logic, the operationR is used to encode which worlds are accessible from a given world and which worlds are not. In practice, since we take R to be an equivalence relationship, we can dispense with R in our formulations of many of these principles. We retain mention ofRsince we will prove some results that do not depend on what kind of relationshipR is.

At this point we should mention two important caveats. As may have been clear from a moments reflection on the language in which we are working in, we do not have the ability to iterate modalities in our well-formed formulae. As such, our language is incapable of distinguishing betweenT and stronger modal systems where the characteristic formulae involve iterative modal operations (such as 4, 5, orB, to give three familiar examples). As such, the choice to work with a universal accessibility relation is less interesting then it otherwise might be. In some senses, this is, however, as it should be. As we shall see, none of Buridan’s conclusions require iterated modalities. In fact, as Read points out in his review of Thom’sMedieval Modal Systems:

But in fact, this appeal to theses of K4 is unnecessary and misleading. Ax- ioms 1.12 – 1.15 are only ever used [144, 152– 5, 163–4, 180–4] in conjunction with 1.11 (the characteristic T-thesis if ◻p thenp), but in each case the ap- peal to T+4 is unnecessary (for in each case, T simply cancels the Lwhich 4 preserves). . . What little modal labour there is, is carried out by 1.7 – 1.8 (matching the T-theses CLpp and CpMp) and the revised 1.12 – 1.15 (match- ing the K-theses CLpqCLpLq and CLpqCMpMq–they are also all valid in S2). [50, p.612]

Buridan’s modal logic is one of the logics treated there, and in our discussion of Buridan’s modal logic in the previous chapter at no point did we observe Buridan making use of iterative modal operations. As such, the choice to work with a universal accessibility relation simplifies some of the formal details.

With this in place, we can simplify the various collections we defined above as follows:

M(w, K) = {d∈D∶ there is some z such thatd∈V(z, K)} L(w, K) = {d∈D∶ for all z d∈V(z, K)}12

12. Again, if one finds the use of K unclear, the four operations are defined as follows: M(w, P) = {d∈D∶ there is some z such thatd∈V(z, P)}

M(w,¬P) = {d∈D∶ there is some z such thatd∈V(z,¬P)} L(w, P) = {d∈D∶ for all zd∈V(z, P)}

Using these operations we can define the truth for categorical propositions.

Assertoric Categorical Propositions.

M, w⊧AaB if and only if V(w, A) ⊆V(w, B) and V(w, A) ≠ ∅

M, w⊧AeB if and only if V(w, A) ∩V(w, B) = ∅

M, w⊧AiB if and only if V(w, A) ∩V(w, B) ≠ ∅

M, w⊧AoB if and only if V(w, A) ⊈V(w, B) or V(w, A) = ∅

Modal Categorical Propositions.

M, w⊧ALa B if and only if M(w, A) ⊆L(w, B) and M(w, A) ≠ ∅

M, w⊧ALe B if and only if M(w, A) ∩M(w, B) = ∅

M, w⊧ALi B if and only if M(w, A) ∩L(w, B) ≠ ∅

M, w⊧ALo B if and only if M(w, A) ⊈M(w, B) or M(w, A) = ∅

M, w⊧AMa B if and only if M(w, A) ⊆M(w, B) and M(w, A) ≠ ∅

M, w⊧AMe B if and only if M(w, A) ∩L(w, B) = ∅

M, w⊧AMi B if and only if M(w, A) ∩M(w, B) ≠ ∅

M, w⊧AMo B if and only if M(w, A) ⊈L(w, B) or M(w, A) = ∅

M, w⊧AQa B if and only if M(w, A) ⊆M(w, B) ∩M(w,¬B) andM(w, A) ≠ ∅

M, w⊧AQe B if and only if M, w⊧AQa B M, w⊧AQi B if and only if M(w, A) ∩M(w, B) ∩M(w,¬B) ≠ ∅ M, w⊧AQo B if and only if M, w⊧AQi B M, w⊧A ¯ Q a B if and only if M(w, A) ∩M(w, B) ∩M(w,¬B) = ∅ M, w⊧A ¯ Q e B if and only if M, w⊧A ¯ Q a B M, w⊧A ¯ Q i B if and only if M(w, A) ⊈ (M(w, B) ∩M(w,¬B))or M(w, A) ≠ ∅ M, w⊧A ¯ Q o B if and only if M, w⊧A ¯ Q i B

The ¯Qnotation is used to indicate that the negation occurs in front of the modal operator instead of after it.

As we have already observed Buridan tells us that modal propositions ampliate their subject to supposit for what is or can be the case. Because R is universal, we do not need to mention the union M(w, P) ∪V(w, P). It can be simplified to M(w, P). The reason for this is that since R is reflexive (which follows from R being universal), it is easy to show thatV(w, A) ⊆M(w, A). This is proven on 211.

5.2.2 Single Premise Inferences

Just as it is possible to visualize Aristotle’s assertoric propositions as a diagram with four points, it is possible to envision Buridan’s modal propositions in diagrammatic form. If

M(w, A) ⊆L(w, B),M(w, A) ≠ ∅ (La) M(w, A) ⊆M(w, B) and M(w, A) ≠ ∅ (Ma) M(w, A) ∩L(w, B) ≠ ∅ (Li) M(w, A) ∩M(w, B) ≠ ∅ (Mi) M(w, A) ∩M(w, B) = ∅ M(w, A) ∩L(w, B) = ∅ M(w, A) ⊈M(w, B)orM(w, A) = ∅ M(w, A) ⊈L(w, B) orM(w, A) = ∅ (Le) (Me) (Lo) (Mo) ? C C C C C C C CW C C CW A A U C C C C C C C CW ? H H H H H H H_H @ @ @ @ @ @ @ @ - - - -- - - -- - - -- - - - - - - - - - - - -_- -- -- -- -- -- -- -- - -_- -_- - -_- - -_- - -_- -

Key: contradictory _contrary

- - - subcontrary

- subaltern

Figure 5.1: Buridan’s Modal Octagon of Opposition

we limit ourselves to possibility and necessity, we obtain an octagon of opposition.13 Formally, the octagon gives rise to 24 distinct inferences. The proofs of these proper- ties are all obvious semantic consequences of the system. We present two examples.

1. ALa B contradicts AMo B

2. ALa B is a contrary of AMe B

Proof of 1: Normally two propositions are said to be contradictory if the truth of one entails the falsehood of the other and vice versa. In this context we say that two well-formed formulae are said to be contradictory if, in every model the truth of one of the formulae entails the falsity of the other, and the falsity of the one entails the truth of the other.

Proof. Assume that M, w ⊧ ALa B. Then M(w, A) ⊆L(w, B) and M(w, A) ≠ ∅. We claim that M, w ⊭ A Mo B. If this were not so, then either M(w, A) ⊈ L(w, B) or

M(w, A) = ∅is true. However, it is clear that both of these contradict our assumptions thatM(w, A) ⊆L(w, B) and M(w, A) ≠ ∅. The other direction is similar.

13. Adding contingency and negated contingency produces a hexadecagon of opposition. Figure 1 (seen below) is due to Stephen Read and can be found in his paperNon-Contingency Syllogisms in Buridan’s Treatise on Consequences [52, p.450].

Proof of 2: Normally two propositions are said to be contrary if they cannot both be true, but they can both be false. In this case, we interpret this in the following way: Two well-formed formulae are said to be contrary if there are no models in which both of the formulae are true, but there is a model where they are both false. Subcontraries are treated in a similar way. Two well-formed formulae are said to be subcontrary if there is a model where both formulae are true, but there is no model where both formulae are false.

To see thatALa B and AMe B cannot both be true:

Proof. Assume that 1)M, w⊧ALa B and 2)M, w⊧AMe B. From 1) we haveM(w, A) ⊆ L(w, B)andM(w, A) ≠ ∅. From 2) we haveM(w, A)∩L(w, B) = ∅. SinceM(w, A) ≠ ∅

we know that there is some d∈ M(w, A). From 1) it follows that d∈L(w, B) and so

M(w, A) ∩L(w, B) ≠ ∅, contradicting 2).

To see that they can both be false consider the following model:

D= {a, b} W = {w1} R=W2 O(w1) =D

a∈v(w1, A) a∈v(w1, B) b∈v(w1, A)

Since a∈v(w1, A) and a∈ v(w1, B) we have a∈M(w1, A) and a∈ L(w1, B). Hence

M, w1 ⊭ A

M

e B. Likewise, b∈ v(w1, A) and b ∉ v(w1, B), we have a ∈ M(w1, A) and

a∉L(w1, B), from which it follows thatM, w1⊭A

L

a B.

The results of conclusions three through eight in Book Two of the Treatise on Con- sequences are summarised in Table 4. As in the case of the modal octagon, the proofs and construction of the relevant countermodels are straightforward. The proofs of the contradictories can be found on page 239.