Bolzano on merely possible objects – A reply to Schnieder

– This post has been updated December 5th 2015 –

In the paper Mere Possibilities: A Bolzanian Approach of Non-Actual Objects, Benjamin Schnieder examines, as he puts it himself, Bernard Bolzano’s views on merely possible objects. Beyond doubt, Schnieder is right in arguing that Bolzano held that there are more objects than those inhabiting the actual world. Bolzano held for example that there are propositions, numbers, geometrical figures, etcetera, which will never enter into actuality. But Schnieder argues that Bolzano also accepts what he calls merely possible objects: objects which are not in the actual world, but could become actual.

Now, there is no passage in any of Bolzano’s works in which he plainly and unquestionably argues that there are merely possible objects in Schnieder’s sense. And in order to defend the view that Bolzano holds this, Schnieder claims that there are several passages in which Bolzano is contradicting his own theory: Schnieder argues that Bolzano sins against his own insight that properly understood, ‘merely possible’ is a modifying predicate by using it in a determining way (p. 537), and he argues that Bolzano misunderstands his own notion of impossibility (p. 541). These are, in my opinion, already reasons enough to doubt that Bolzano indeed had the view that Schnieder ascribes to him. For this reason I’d like to find out: isn’t there another viable interpretation of Bolzano’s writings on the possible, that does not require that we attribute such mistakes to him? Here’s an attempt.

My reconstruction of Bolzano’s views on the possible will start from his theory of ideas. Bolzano answers many metaphysical questions by means of his theory of ideas, and notably also formulates his account of necessity in terms of it. Therefore it seems to me that his theory of ideas is a good place to start.

As rightly explained by Schnieder, the question whether there are such-and such objects, comes down for Bolzano to the question whether the idea of such-and-such is objectual, i.e. has (an) object(s) falling under it. Ideas, in Bolzano’s theory, are abstract entites which are the parts of propositions that are not themselves propositions. Ideas make up the matter (Stoff) of their counterparts in the physical world, subjective ideas, which in turn are constitutive of our judgments and linguistic utterances. Importantly, for every object in Bolzano’s universe there is an idea that has this object in its extension, but not every idea that there is has an object in its exension. Some ideas have objects falling under them which are concrete, i.e. inhibiting the physical (actual) world, others have objects which are abstract, i.e. not inhabiting the physical world, but a popperian World 3-like abstract realm instead. To give an example saying that there is the number 2, means for Bolzano that the idea of the number 2, which is following a common use in Bolzanian scholarship denoted by [2], is objectual. Objects like the number 2 do in Bolzanian terms not exist, i.e. they do not inhibit the physical world. I will use the term abstract, as Schnieder does, for those objects that cannot exist, like propositions, ideas, and mathematical objects. Phrased in these terms, the question that we want to answer is whether for Bolzano there are non-existing objects which are not abstract; in particular, whether there are in Bolzano’s view what Schnieder calls merely possible objects.

Schnieder defines a merely possible object in Bolzano’s view (henceforth MPO) as follows (enumeration is mine):

(Df. MPO) x is a merely possible object ↔df.

  1. x is non-actual &
  2. x is possibly actual. 

We know what it means for an object to be (1.) non-actual: an object is non-actual iff it is there but is does not exist. Abstract objects are non-actual objects, for they are not in the physical world. Importantly however, abstract objects cannot exist. Thus, abstract objects are not (2.) possibly actual, and are consequently not MPO’s. A presumed example of an MPO would be a golden mountain, because there is no actual golden mountain (1.) but a golden mountain could be actual (2.).

Now, did Bolzano indeed hold that there are MPO’s? Are there in his view merely possible golden mountains? Let’s look a bit further into his theory of ideas.

Ideas can have other ideas as parts (i.e. be complex) according to Bolzano, and some complex ideas are such that their parts attribute properties that contradict each other, which means that they cannot be united in an object (WL §70). Consequently, such ideas are not objectual, i.e. they are objectless (cf. WL §352). An example of such an idea would be [triangle which has four angles]. Bolzano writes about these contradictory ideas that:

“Hitherto it was common to call them empty [leere], impossible [unmögliche], or imaginary [imaginäre] ideas. All the others one gave the name possible [möglich], actual [wirklich], or real [real] ideas.”  (WL §70)

And Bolzano adds that he will stick with this common use of the terms. Thus, possible seems to be in Bolzano’s view whatever is not impossible (cf. WL §182); impossible, in turn, simply means contradictory. Importantly, possibility, etc., is here predicated of ideas and not of the objects that (are supposed to) fall under them. Later we will come back at the question whether it is correct to see possibility as a property of ideas only in Bolzano’s view. I can already say that this will be exactly the interpretation that I propose: when Bolzano is attributing possibility, etc., to things, then he does this not at the level of the objects themselves, but at the level of the ideas that these objects (are supposed to) fall under.

The important thing to understand now is Bolzano’s view on contradictory ideas. Determining whether an idea is contradictory helps us in Bolzano’s view to determine whether or not this idea is objectual, as he describes in WL §352:

“Is the idea that we are considering complex, and has the form [something] a + b + c + d + …, then we should investigate whether the properties a, b, c, d, which are united here do not contradict each other, that is, whether one of the propositions: No A is B, No B is C, No C is D, etcetera, is provable from purely conceptual truths. If we can find such propositions, then our idea is imaginary; if we do not find them, then it is real.” (WL §352)

We will focus here on complex ideas considered as concretum (cf. WL §60), so that  the ideas that Bolzano discusses in the above quote are the relevant ones for us (extending the following to complex ideas of other forms will be easy; extending it to simple ideas will be highly complicated, since Bolzano’s views on simple ideas are so far hardly understood, so I’ll leave that aside here). According to this quote, these ideas are real (i.e. possible, cf. WL §70 above) if and only if they do not have ideas as parts that attribute properties which contradict each other (cf. WL §138). It is important to note that contradictory ideas are in Bolzano’s view not (only) ideas of the form [A, which is not A] or [something, which has b and not b]. Rather, as he puts it in WL §70, contradictory ideas are of the form [A, which is both B and P], where B and P are such that there is a property M such that all B are M and all P are not M. As an example, Bolzano gives the idea of a triangle which is equilateral and has a right angle. From pure concepts (i.e. from solely a priori truths) one can prove that all equilateral triangles do not have a right angle, and that all triangles with a right angle are not equilateral (WL §70).

Thus, so far we have:

(Df. POS) [x] is possible (actual, real)  ↔df.

it cannot be proven from purely conceptual truths that the properties that [x] attributes cannot be found in one object simultaneously.

Note that for all ideas (thus, also for empirical ideas), purely conceptual truths settle the question whether or not they are possible (cf. WL §182). Note further, that to say that [x] is possible is not yet to say that [x] is objectual (cf. WL §352). In order to establish whether [x] is objectual, we need to take into account whether [x] is what Bolzano calls an actuality-demanding (Wirklichkeitsfordernd) idea. Actuality-demanding ideas are such that if they have an object falling under it, then this object is in the physical world. An example of such an idea would be the idea of a horse: if [horse] is objectual, then a horse exists in the physical world.

Now, for not actuality-demanding ideas according to Bolzano it holds that if they are possible, then they are objectual (WL §352). This is not the case for actuality-demanding ideas: if they are possible, they can still be objectless because they contradict empirical truths (WL §§70, 352). 

Thus, we have:

(PI) If [x] is a possible idea, then exactly one of the following holds:

  1. [x] is not actuality-demanding and [x] is objectual;
  2. [x] is actuality-demanding and [x] is objectual;
  3. [x] is actuality-demanding and [x] is objectless.

(For simplicity I ignore here the temporal aspect; it will not be problematic to incorporate this.) Ideas that satisfy 1. are for example ideas of mathematical objects; ideas that satisfy 2. are ideas of objects in the physical world. Ideas that satisfy 3. are ideas such that if they are objectual then their object is in the physical world, but there are empirical truths that prohibit these ideas to be objectual.

Back to MPO’s, specifically to golden mountains. As might have been expected, Bolzano claims that the existence of a golden mountain is not prohibited by purely conceptual truths (WL §67, cf. WL §§70, 352). Notably, this means by (Df. POS) that the idea of a golden mountain is possible. Further, it seems that Bolzano would have agreed that [golden mountain] is actuality-demanding (since a golden mountain should be in the physical world), and that it is objectless (since no golden mountain is in the physical world). This means in terms of (PI) that 3. holds.

Note that at this point, we have all we need to account in a Bolzanian manner for the role of golden mountains in our worldly and scientific activities: we have [golden mountain] which enables us to think and speak about golden mountains, which is at the logical level constitutive of any truth that concerns golden mountains (cf. WL §67), and which can become objectual if for example at a certain point God decides to enrich this planet with a golden mountain. What we so far do not have, however, are merely possible golden mountains.

If there is a merely possible mountain in Bolzano’s universe, then, in all likelyhood and also according to Schnieder, Bolzano would have taken it to fall under the idea [merely possible golden mountain]. Let’s assume this for the moment, and let’s examine this idea. Importantly: in which respect is [merely possible golden mountain] different from [golden mountain], i.e. what does the [merely possible] do?

If we plug in “golden mountain” in (Df. MPO), we should get a merely possible golden mountain in Schnieder’s sense. Thus, [merely possible] seems to attribute to the golden mountain that it is not actual but could be actual. But as was noted above, this is already true of [golden mountain]: since it is actuality-demanding and possible there could be an actual golden mountain, and since it is actuality-demanding and objectless there is no actual golden mountain. This means that [merely possible] in [merely possible golden mountain] is wholly redundant.

One might even wonder whether Bolzano does not take [merely possible golden mountain] to be a contradictory idea. For [golden mountain] is an actuality-demanding idea, and [merely possible] attributes non-actuality to the object. As I see it, it might very well be that [merely possible golden mountain] is not a possible idea in Bolzano’s view at all, and for that reason there are no and cannot be merely possible golden mountains in Bolzano’s universe. 

My proposal is to understand Bolzano’s theory of the possible in the following manner:

(BP) Being possible is a second-order property.

Interpreted like this, being possible in Bolzano’s view thus closely resembles being there in his view. Just like according to Bolzano, sentences of the form “there are F‘s” expresses a proposition of the form [[F] is objectual], which means that the idea of an F is an objectual idea, according to (BP) in his view, sentences of the form “F‘s are possible” express a proposition of the form [[F] is possible], which means that the idea of an F is a possible idea.

Now, if a proposition of the form [[F] is possible] is true, then by (PI) [F] can be either objectual (option 1. and 2.) or objectless (option 3.). In this spirit I propose that we should interpret merely possible in Schnieder’s sense as expressing that option 3. holds: 

(Df. MP) [x] is merely possible ↔df.

0. [x] is possible &

1. [x] is actuality-demanding &

2. [x] is objectless.

Thus, merely possible according to (Df. MP) expresses that the idea is possible (0.), and furthermore not abstract (1.) and does not have an actual object falling under it (2.). Note that in virtue of (2.), (Df. MP) implies that there are no merely possible objects

Against my argument based on Bolzano’s theory of ideas, and in favor of Schnieder’s account, it could be argued that it might be that we do not have epistemic access to the merely possible and therefore fail to understand under which idea a merely possible golden mountain falls, and that therefore my argument is not decisive to show that in Bolzano’s universe there are no merely possible golden mountains. In reply to this, I would like to point out that with the exception of 1 passage, all the passages that Schnieder quotes to show that Bolzano does hold that there are merely possible objects, can be interpreted in such a way that Bolzano simply argued that there are non-actual objects (which is beyond doubt true, for Bolzano held that there are logical and mathematical objects, but these qualify as abstract objects). The only passage in which Bolzano indeed seems to argue that there are non-actual objects besides abstract objects is WL §483. But in my view this is meager evidence to attribute to Bolzano the view that there are non-actual and non-abstract objects. Especially since, as I have shown, he does not need these objects to be there: already the fact that the idea under which these objects are supposed to fall are there makes that his theory can account for all our worldly and scientific activities that concern objects that are not actual but could be actual. Furthermore, Bolzano developed and then employed his theory of ideas in order to deal with virtually all metaphysical questions. This makes it highly likely that if he had held that there are merely possible objects, we would have found somewhere in his works an account of the ideas under which these merely possible objects fall. Bolzano is way too precise a thinker just to “forget” to deal with merely possible objects.