A Picturing Theory of Conceivability

            The question of how conceivability can give us information about what is possible is recently popular and perplexing. On the one hand there are many cases like the celebrated Gettier example where conceiving of a certain possibility does seem to give us information that a state of affairs could obtain or a sentence could be true. On the other hand it seems mysterious that quasi-psychological facts about what we can conceive should give us access to the limits of metaphysical possibility. In this paper I will describe a ‘picturing’ theory of conceiving on which is a matter of constructing pictures which we antecedently know to correspond to genuine metaphysical possibilities. Thus conceiving is not a source of new modal knowledge but rather it allows us to mine out consequences of modal knowledge which we already have.

I suggest that we start trying to figure out how an activity like conceiving can be a guide to possibility by thinking about another question: why is it that other attitudes (imagining, believing, denying, supposing, refuting) are not a guide to possibility? That is, what allows us to assume these attitudes towards impossible propositions or states of affairs?

It is pretty uncontroversial that we can: suppose some contradictory state of affairs in the course of a reductio argument, reject certain hypotheses on a priori grounds, or –in some sense– believe the conjunction of two complicated sentences which turn out to be logically or mathematically impossible. But at the same time this ability to take such attitudes towards impossible sentences or states of affairs is quite interesting. It shows, for example, one way in which people’s belief-states aren’t perfectly represented as classes of centered possible worlds (the worlds which are compatible with everything they believe). For any belief state which includes at least one necessary falsehood will be compatible only with the same –empty – set of possible worlds. So this way of representing belief states can’t do justice to differences between belief states which include logically incompatible or necessarily false beliefs.

A complete account of the cases just mentioned would go far beyond the scope of this paper, but I want to suggest that it is relating to depictions (like English or mathematical sentences) rather than directly to the states of the world which would make these depictions true is what allows us to believe, suppose, deny etc. the impossible.

I’ll try to make all this clearer by a little digression on the subject of beliefs about the impossible.

It seems to me that that at least in some cases, people must believe things by coming in to the right relationship to descriptions of states of affairs, rather then to these states of affairs directly. Roughly speaking, the person who believes their town barber shaves everyone who doesn’t shave themselves, has a description in mind (perhaps this very sentence) and also an understanding of how the various components of this description make claims about reality (‘shaves’ refers to shaving, ‘everyone’ quantifies over everyone in town etc.); their belief is a matter of their having the right relationship to this description rather than to the (impossible) state of the world which would make it true. And this ‘right relationship’ is something like being willing to draw the ‘obvious’ linguistic consequences, act on the ‘obvious’ practical consequences of this representation (in this case, for example, to expect that no-one in town has a beard, and believe that their friends who don’t shave themselves must know the barber, to not to try to sell beard-grooming supplies in town) etc.

Notice two things. First, the notion of ‘obvious (practical or logical) consequences’ is something which applies to descriptions of states of affairs rather than to these states directly, since different ways of describing the same states of affairs have different obvious practical and logical consequences. For example, if we let M stand for a non-obvious mathematical falsehood, then the descriptions ‘my pants are on fire’ and ‘my pants are on fire or M’ correspond to the same state of affairs (are true in the same set of possible worlds) but have different obvious practical and logical consequences. So if we want to say that the relationship which a believer must have to their beliefs includes agreeing with and acting on (all or most of) the obvious consequences of what they believe, then what they believe had better be some kind of description rather than an unstructured entity such as a set of possible worlds. Second, this account explains why it seems like nothing would count as having certain clearly contradictory beliefs (e.g. believing that it’s raining and that it’s not raining). For if what you need to do to count as believing something is act on the obvious consequences, then in the case of clear contradictions acting on the obvious consequences of one conjunct precludes acting on the obvious consequences of the other. But in contrast this is not true for belief-sets whose incompatibilities (and hence whose incompatible practical and logical consequences) are hidden from view.

 One might give a similar account, in terms of descriptions and obvious consequences, of supposing (e.g. that a supposer readies themselves to draw the obvious consequences of some description sotto voce).

Now if this speculation is right, and beliefs, suppositions, denials of the impossible etc. relate us to depictions, then our original question (how do these attitudes not entail possibility?) reduces to the question of how we can depict what’s impossible. How can a possible sentence or picture make impossible or incompatible claims about the world? I suggest that the explanation lies in the fact that different parts of a sentence can talk about the same fact (hence one part can affirm it while the other denies it). If we imagine depiction in terms of an association between features (or parts) of a sentence or diagram on the one hand and features (or parts) of the world, or the thing being depicted on the other, then we can account for depiction of the impossible, roughly, as follows. Depiction of the impossible happens when a) distinct parts of the picture get associated with the same part of the world, or independent features of the picture are associated with non-independent features of the world and b) these different parts and features are then configured so as to make contradictory or incompatible claims about the relevant features of the world. We can make unsatisfiable descriptions by using kinds of pictures which have more or different ‘degrees of freedom’, so to speak, then the things they depict.

Now, I think this account of how activities like imagining, believing and supposing are not a guide to possibility points to a suggestion for how conceivability might be such a guide. Specifically, if we can adopt these attitudes towards the impossible because we relate to a depiction with a special property (it depicts in a way which ‘maps’ independent features of the picture to non-independent features of the world), then we can distinguish the depictions which have this property from those which don’t ( I will call the latter ‘depictions using a modally reliable method of picturing’). And it might be that conceiving involves assuming the same relation to a depiction as regular imagining does, but this depiction has to be made in a modally reliable method of picturing, i.e. one which does not allow depiction of the impossible.

Here is an example, by way of clarification: An interior decorator wants to determine whether it would be possible to arrange certain pieces of furniture in a room in such a way as to leave a 5-5 foot square of space empty. To do this, he draws a scale model of the room on a piece of graph paper, and he cuts out a tab of cardboard with the same shape as each of the pieces of furniture – also to scale (say, the things in the model are n centimeters long when the corresponding things are n feet long). Then he plays around with these tabs, moving them over the graph paper in accordance with certain rules (e.g. every tab has to be completely inside the lines representing the rooms walls, no two tabs can overlap) and manages to arrange them so as to leave a 5x5 cm space inside the lines on the graph paper uncovered. He then reasons 1) that the real furniture in the room could be arranged in a corresponding way and 2) that if it were so arranged there would be a 5x5 ft space in the room. So he concludes that it would be possible to arrange the room so as to leave such a space.

            As you can see, this kind of inference (though perfectly ordinary) involves a few different parts. I think that the parts are as follows.

1)      The picturing medium: a bunch of physical stuff, together with some conventional and/or physical constraints on how it can be configured. In this case the picturing medium is the drawing of the room on graph paper together with the tabs, and the constraints on how it can be arranged come partly from the physics of our world (there’s no way of moving a tab around on the paper so that it occupies less area) and partly from some conventions (you mustn’t put one tab over another, or fold them in half etc).

2)      The method of projection: a way of associating features of the picturing medium with features of the stuff pictured so as to generate a function from each allowed configuration (specified in terms of these features) of the picturing medium to a configuration (also so specified) of the stuff pictured. In this case, the method of projection is a little hard to give verbally, but it’s what tells us that e.g. configurations of the model on which the desk shaped tab is next to the rightmost wall correspond to configurations of the room in which the desk is against the east wall of the room.

3)      The method of picturing: a method of picturing together with a method of projection for it. In this case the tabs and rules for moving them together with the rules for how to associated configurations of the tabs with configurations of the room.

Given these terms, I will now restate what a modally reliable method of picturing is, and how it can figure in conceivability arguments.

A method of picturing is modally reliable iff for every permitted configuration of the picturing medium (where which configurations are permitted will be determined by the constraints that partially constitute the picturing medium) it takes (via the method of projection) the combination of features found in that configuration to a possible combination of features of the stuff pictured.

Thus a modally reliable method of picturing is a language in which, unlike in English, or the graphical conventions used in Escher drawings, it is not possible to depict anything impossible. If a picture can be made in such a language then its content could be true.

            A picture ensures the truth of a sentence if every possible world which is as the picture depicts is one in which the sentence is true.

Summarizing: to conceive of the truth of a sentence is to create or consider a picture which both depicts via a modally reliable method of picturing and ensures the truth of the sentence. If we understand conceiving as consisting in or requiring the satisfaction of these two conditions, then a sentence’s being conceived entails its possible truth as follows. Since the picture depicts via a modally reliable method of picturing the possibility of constructing the picture implies that things could possibly be as the picture depicts, i.e. that there is some possible world at which they are. But then since the depiction ensures the truth of the sentence this world is one in which the sentence is true, so the sentence could possibly be true.

            Here I want to note that when the boundaries around the permissible configurations of a picturing medium are drawn by convention rather than required by some kind of natural law, it may not be possible to (and certainly the conceiver need not be able to) state verbally what the convention in question is. We can have conventions (our conventions about how far away from people to stand, for example) that pick out a range of permissible options without being able to explicitly state what the rules in question are. And we can have modal knowledge which consists in a capacity to recognize rather than in propositional beliefs. Thus a person who is merely able to recognize which descriptions in a certain language correspond to genuine possibilities can be thought of as having a modally reliable method of picturing if we restrict the allowable configurations of the picturing medium to the configurations which he would recognize as possible[1].

Now let’s turn to some modifications to the basic idea of conceivability as modally reliable picturing. I think that if we add two more stipulations the resulting account will largely match ordinary linguistic usage and intuitions about words like ‘conceiving’ and ‘conceivability’, while preserving the entailment relationship between conceivability and possibility. In this way our theory of what justifies the inference from conceivability to possibility (conceivability requires modally reliable picturing) can form part of a conceptual analysis of conceiving.

First, we should say that in order to count as conceiving of the truth of a sentence S a person must not only be considering a depiction which is made in a modally reliable method of picturing and suffices to ensure the truth of S but must know that the picture they have in mind has both these features. Admittedly, this point isn’t obvious (especially not in these terms) but I think our intuitions about cases where people have an appropriate picture in mind but lack one or both of these pieces of knowledge ultimately support it. Suppose you and I are considering whether there can be a number which has a certain mathematical property P, and you discover that if a number was, say, divisible by say 3, 11 and 17 but not by 5 it would have property P in question.  You and I both consider this description. Now, suppose the fact that for every description of the form ‘n is divisible by prime factors S and not by prime factors P’ where S and P are distinct sets of prime numbers is true of some number (i.e. corresponds to a genuine possibility) occurs to you but not to me. Then I think it’s clear that we’d say you count as conceiving of a number which has property P by considering this description, but that I do not. Similarly, if we both know that there must be a number that satisfies the description above, but only you know that a number’s satisfying this description suffices to ensure that it has property P, you could be said to conceive of a number with property P by thinking of this description but I could not.

Secondly, consider the following kind of limiting case of the account of conceivability given so far. So far I have said that conceiving of a sentence means considering a depiction which one knows to a) depict via a modally reliable method of picturing and b) suffice to ensure the truth of a sentence in question. And I have described a method of picturing as a class of ‘permitted configurations’ of some stuff together with a way of projecting this which associates features of this configuration with features of the world. So a single copy of an English sentence together with the ‘method of projection’ which associates it with its propositional content counts as a degenerate or limiting case of a method of picturing. The curious result of this is that, according to what we have said so far it looks like, if you know that a sentence S is possibly true then you can conceive of the truth of S just by thinking of S itself. This is because if S is possibly true, then the degenerate method of picturing which includes only S will be modally reliable since it projects every possible configuration of the picturing mechanism (there is only one) to a possible configuration of the world. Thus if you reflect on this fact, by thinking of S itself you will be thinking of a depiction which you know to be both modally reliable and (of course) sufficient to ensure the truth of S.

So, if we want to give an account of the concept of conceiving as used in ordinary English, we should further stipulate that a person only counts as conceiving of the truth of a sentence if the depiction they have in mind is substantively different from that sentence. Intuitively, even if a person knows that a sentence expresses a possible truth, they only count as conceiving of the truth of the sentence if they consider a depiction which shows how that sentence could be true i.e. a depiction which is substantively different from the sentence in question which uses a method of picturing which they know to be modally reliable for reasons different and/or antecedent to any reasons they have for believing the sentence itself could be true.

These modifications give us the following account of conceivability. A person conceives of the truth of a sentence S if they are considering a depiction which is substantively different from S itself and which they know to be modally reliable and sufficient to ensure the truth of S.

In this section I have tried to articulate and motivate a picturing theory of conceivability. Consideration of how we can adopt other psychological attitudes like believing and imagining towards the impossible suggested an account on which conceivability involves relating to a special kind of depiction (one which is modally reliable) which ensured the possibility of what it depicted. I then briefly considered some ways in which this account of the features of conceiving which license the inference from conceivability to possibility could be modified to give a general analysis of conceivability.