Logic(s) of Imagination

Chris is a PhD student in Philosophy at the Ruhr University Bochum in Germany, writing a dissertation on logic(s) of imagination. His research interests include philosophical logic, formal semantics, and philosophy of fiction. In his free time, he likes to play board and card games.

Chris is a PhD student in Philosophy at the Ruhr University Bochum in Germany, writing a dissertation on logic(s) of imagination. His research interests include philosophical logic, formal semantics, and philosophy of fiction. In his free time, he likes to play board and card games.

A post by Christopher Badura.

What is the logic of imagination?  Which, if any, inferences involving the concept of imagination are valid?[1] Answering this contributes to understanding how and what we can learn from imagination, and also how imagination is constrained. In what follows, imaginative agents of interest are subjects that can be reasonably described as being capable of performing at least one step deductive inferences like conjunction introduction/elimination, modus ponens, and disjunction introduction.[2]

It's tempting to claim that there are no valid inferences involving imagination because imagination is “up to us”, meaning it's always possible for an agent to change the content of their imagining. So, regardless of what I have imagined just now, what I imagine next is entirely up to me, and thus unconstrained by what I have imagined first. This is partially true but targets only one aspect of imaginative episodes. While at any point in an imaginative episode we can  choose to imagine something new, we often “unfold” what we have already chosen to imagine (Langland-Hassan, 2016). Despite agentive connotations of the term, unfolding is not subject to the genuine choice of an agent, that is, agents can't willfully influence the unfolding directly. It is thus involuntary and not up to us.[3] I call what we choose to imagine the “initial content” and what we unfold to the “arrived content”.

With this at hand, we can classify two recent accounts of logics of imagination: Wansing (2017) accounts for imagining initial content, while Berto (2018) focuses on imaginative episodes as a whole.[4] 

On Wansing's account, an agent's “mental image” is a set of sets of worlds.[5] Contents are propositions, which are sets of worlds. In a world, agents can genuinely choose the propositions they have in their mental image at that world. So, an agent imagines A just in case they genuinely choose having A in their mental image. Virtuously, this account avoids “omni-imagination”:

  • agents don’t imagine all tautologies

  • if one imagines A, and A entails B, one doesn’t necessarily imagine B

  • if one imagines A and one imagines that A entails B, one doesn’t necessarily imagine B

  • agents can imagine contradictions

Whether we can imagine contradictions is debatable, and I won't take a stance here.[6] Is imagination closed under imagined implication? If we think of imagining B in terms of agentive imagination, like Wansing, then it is plausible that it is not. If, however, imagining the consequent is imagining the arrived content, which is somehow connected to the initial content, it seems plausible that imagination is closed under imagined implication. By introspection, if one initially imagines A and initially imagines A implies B, then one unfolds this into imagining B, unless one chooses not to unfold.

How could one choose to not unfold one's initial imaginings? As mentioned, claiming that one influences the unfolding directly is not an option because the unfolding itself is by definition not directly subject to the will of the agent. An indirect non-genuine choice not to unfold would be if one was unable to unfold because of, say, cognitive or computational limitations. I think that the subjects of interest are not so limited.

Another non-genuine choice would be if the goal of the imaginative projects was reached then we’d simply stop imagining (Dorsch, 2012).[7] If the imaginative project keeps going on, however, the question remains. An indirect genuine choice would be that one blocked the unfolding by either actively imagining that B is not the case, or added to the initial content something that forced us to imagine that A didn’t imply B. So, unless the agent initializes other imaginings, initially imagining A and initially imagining that A implies B suffice to unfold to imagining B. This suggests that imagination is non-monotonic.

One case of omni-imagination is valid in Wansing's logic: if A and B are equivalent, agents imagine A just in case they imagine B. So, if I genuinely choose A to be in my mental image, I genuinely choose to make B my mental image. Some (e.g., Berto 2018) argue that this ignores the fact that imagination is intentional. In imagining riding a horse, one is not also imagining riding a horse and that the earth is round or the earth is not round.[8] A second consequence of Wansing's logic is that imagining A & B does not entail imagining A (B). Again, if one takes seriously the idea that agents must have genuinely chosen the content, then this seems unproblematic. For choosing A & B genuinely does not mean that the agent genuinely chooses A (B). The agent thereby chooses A – it is a side-effect of their genuine choice. Given this argument, the aforementioned case of imagination being closed under logical equivalence is problematic: while one genuinely chooses imagining A, one only thereby imagines the equivalent B. If this suffices for saying that one didn’t genuinely choose imagining B, which it does in the case of conjunction, imagination should not be closed under logical equivalence.

As mentioned, Wansing (2017) models initial imaginings and so the approach is not meant to capture the unfolding component. Berto (2018) focuses on the episodic character and also takes into account that imagination is intentional.   His notation for the episodic imagining is [A]B: “in imagining A, the agent imagines B”. Informally, the idea for the truth condition for this is the following: in an imaginative episode with initial content A and arrived content B, we first mentally represent A, see whether B is then true, and also the content of B must already be part of the content of A.[9]

His semantics also invalidates cases of omni-imagination, including closure of imagination under logical equivalence. Moreover, [A]B does not entail that [A](B v C). This seems plausible, given we take the intentionality of imagination seriously. If I imagine that I am on the beach, then I don't thereby imagine that I am on the beach or Trump is meeting Merkel. The second disjunct is not part of the content of what I imagined first.

Another result concerning disjunction is that one can imagine a disjunction without imagining either disjunct, i.e. [A](B v C) doesn't entail [A]B v [A]C. Suppose, I initially imagine having rhinitis, and unfold to I have measles or a cold, [R](M v C). Does my initial imagining A unfold either way, so that [R]M or [R]C must be true? If one buys into the idea that we always imagine indeterminate situations, then no. If the goal of my imaginative project was to imagine the disjunction (Dorsch, 2012), then also no. What if the project continues? It is not guaranteed that I unfold to one of the disjuncts because I might choose some new initial content which I then unfold instead of the disjunction.

Another plausible result on Berto's account is that imagination is non-monotonic: [A]B doesn't entail [A & C]B. Suppose imagining drinking a coffee unfolds into imagining that I am happy. This doesn't guarantee that if I imagine drinking a coffee and that it is a poisoned coffee, this unfolds into imagining that I am happy.

On Berto's account, some inferences involving imagination become valid. In particular, [A](B & C) entails [A]B and [A]C and vice versa. While the former is plausible, the latter is questionable. As has been pointed out in the literature on fictions with impossible content, it might be that while reading Sylvan’s Box (Priest, 1997), one imagines an empty box, and one imagines a non-empty box, but one never imagines an empty-and-non-empty box.[10]

Since Berto focusses on imagination episodes, the agentiveness present in choosing the initial content is pragmatically internalized. Imagination is “up to us” because any formula can feature as expressing the initial content. Phrases like “agent a imagines (that) A” are difficult to model. The most plausible candidate is the tautology [A]A, “in imagining A, one imagines A”.  But that's not what “agent a imagines A” expresses. One problem is that the grammatical surface structure doesn't support the “conditional” understanding of imagination present in “in imagining A, one imagines A”. Secondly, “a imagines A” correlates with agents initializing imaginative episodes. These imaginings are subject to the will, that is, they require action on the agent's part. Thus, if “a imagines A” is to be understood as “in imagining A, a imagines A”, and the latter is a tautology, there is no action required to imagine A because logic dictates that the agent imagines A (given they imagine A).  The point is: while Berto focusses on imaginative episodes, he has no formal account of initializing imaginings.[11]

As pointed out above, Wansing's logic models initializing imaginings but lacks the unfolding component. Thus, combining the two such that the initial imagining is determined according to Wansing's logic but then unfolding happens along the ideas of Berto, seems promising.

Both approaches remain on the propositional level. Thus, extending them to first-order logics is a natural next step. This is interesting not only for technical but also philosophical reasons. For example, Niiniluoto (1985) discusses an intensional first-order logic that allows us to model phrases like “a imagines b” and “a imagines b as c”. A special case of this is imagining oneself in someone else’s shoes.

Summing up, when we speak of imagination, we sometimes mean imagining initial content, which involves genuinely choosing to mentally represent some content.[12] This is what we are after when we say that imagination is up to us. Other times, we mean an entire imaginative project composed of some initial imagining(s), unfoldings of these, and further initial imaginings along the way, all culminating in mentally representing some arrived content.  The former doesn’t validate any inferences and is not closed under any logical operations, while the latter is closed under conjunction elimination and imagined implication.


[1] In what follows, I speak of propositional, experiential, and objectual imagination, although the logics discussed model propositional imagination. Nothing depends on this. Everything I say can be phrased in propositional imagination terms only.

[2] They might not be able to iterate these into longer deductions.

[3] This doesn't mean that we can't influence it indirectly by choosing new initial contents that unfold differently.

[4] Casas-Roma (2018) provides an algorithmic model for imagination. See also Wansing & Olkhovikov (2018) and Olkhovikov & Wansing (forthcoming) for a proof system.

[5] It is actually sets of sets of moment-history pairs because they use branching-time models of stit-logic. But this detail need not concern us here. Also, the phrase “mental image” is not to be taken too seriously here.

[6] If need be, Wansing’s logic can be modified to disallow imagination of contradictions.

[7] An imaginative project is a sequence of imaginative episodes that all contribute to the goal of the project.

[8] Also, the question arises, what it means to imagine a tautology.

[9] He models imagination as a variably strict conditional operator constrained by a content filter.

[10] This can be fixed by using impossible worlds (Berto, 2017).

[11] Impossible worlds don't help with this problem.

[12] Plus some special phenomenology, maybe.


References

Berto, F. (2017). Impossible Worlds and the Logic of Imagination. Erkenntnis, 82(6), pp. 1277-1297.

Berto, F. (2018). Aboutness in Imagination. Philosophical Studies, 175(8), pp. 1871-1886.

Casas-Roma, J. (2018). Deeper Down the Rabbit-Hole. The Dynamics of Imagination Acts. (U. O. Catalunya, Ed.)

Dorsch, F. (2012). The Unity of Imagining. De Gruyter.

Langland-Hassan, P. (2016). On choosing what to imagine. In A. a. Kind (Ed.), Knowledge through Imagination (pp. 61-84). Oxford University Press.

Niiniluoto, I. (1985). Fiction and Imagination. Journal of Semantics, 4(3), pp. 209-222.

Olkhovikov, G., & Wansing, H. (forthcoming). Simplified Tableaux for STIT Imagination Logic. Journal of Philosophical Logic, pp. 1-21.

Priest, G. (1997). Sylvan's Box: A Short Story and Ten Morals. Notre Dame Journal of Formal Logic, 38(4), pp. 573-582.

Wansing, H. (2017). Remarks on the logic of imagination. A step towards understanding doxastic control through imagination. Synthese, 194(8), pp. 2843-2861.

Wansing, H., & Olkhovikov, G. (2018). An Axiomatic System and a Tableau Calculus for STIT Imagination Logic. Journal of Philosophical Logic, 47(2), pp. 259-279.