Abstract
Imagination has received a great deal of attention in different fields such as psychology, philosophy and the cognitive sciences, in which some works provide a detailed account of the mechanisms involved in the creation and elaboration of imaginary worlds. Although imagination has also been formalized using different logical systems, none of them captures those dynamic mechanisms. In this work, we take inspiration from the Common Frame for Imagination Acts, that identifies the different processes involved in the creation of imaginary worlds, and we use it to define a dynamic formal system called the Logic of Imagination Acts. We build our logic by using a possible-worlds semantics, together with a new set of static and dynamic modal operators. The role of the new dynamic operators is to call different algorithms that encode how the formal model is expanded in order to capture the different mechanisms involved in the creation and development of imaginary worlds. We provide the definitions of the language, the semantics and the algorithms, together with an example that shows how the model is expanded. By the end, we discuss some interesting features of our system, and we point out to possible lines of future work.
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Notes
In a nutshell, hybrid logic allows to uniquely identify possible worlds through a set of nominals \(\text{ NOM }\). Then, a formula \(@_{i}\varphi \) expresses that “at world i, it is the case that \(\varphi \)”. The addition of such operator to our logic can greatly increase its potential by allowing to express what is the case in different stages of an imagination act. For a thorough introduction to hybrid logic, we refer to Blackburn (2000), or Blackburn and Seligman (1995).
The way we define the set of actions \(\text{ ACT }\) is inspired by the way Propositional Dynamic Logic, or PDL, defines a set of atomic programs \(\varPi _{0}\). In PDL (see Harel 1984), these programs are also used to sign a modal operator, just as we do in our case; nevertheless, PDL also defines a set of operators over programs, which can be used to combine them in different ways.
In a nutshell, we only allow the antecedent and the consequent to belong to the propositional fragment of the language because we are interested in seeing how imaginary worlds are created and developed: modal and hybrid operators convey information about other worlds and the relations between different worlds. Leaving aside the technical complications that this would involve, imagining about other worlds or their relations falls outside the scope of the current goal of this work.
Each one of the four algorithms we define in the following pages is executed upon a model \(\mathcal {M}\). On the first step of their execution, there are certain initial conditions that each algorithm has to check; if any of those conditions is not fulfilled, the algorithm does not expand model \(\mathcal {M}\) in any way; in that case, we consider that \(\mathcal {M}^{+} = \mathcal {M}\).
Roughly speaking, this process aims to account for a ceteris paribus effect in the new imaginary world. As argued in Berto (2017b), conceived and imagined worlds are usually governed by such effect.
It is worth noting how this kind of hypothetical conditionals are somewhat similar to the kind of formulas David Lewis is interested in Lewis (1973). Nevertheless, Lewis’ use is different from ours: in his work, Lewis evaluates a formula of the kind at a world aimed to represent the real world, and the operator moves the whole evaluation point to an accessible counterfactual world in order to assess whether the conditional \(\varphi \rightarrow \psi \) holds in there. His way of evaluating hypothetical conditionals, therefore, is by moving the whole conditional to an alternative world. Our way of understanding them, however, will be to asses whether the antecedent \(\varphi \) holds in the current world of evaluation, and if so, then create a new world fulfilling \(\psi \) and defined by taking the current one as the reference. Our understanding of this kind of conditionals, therefore, will be used to determine the way a certain imaginary world could change, given the information provided by the specific formula being evaluated.
We can draw a parallelism with the way variables are usually handled in programming languages. In there, it is typical to override the value of a variable by using its own value; for instance, one can increase the value of an integer index i by saying \(i = i+1\).
In a nutshell, we process \(\langle \rightarrow \rangle ^{\alpha }\) formulas first to give priority to \([ \rightarrow ]^{\alpha }\) formulas. Last evaluated formulas could override something already added by previous formulas, and we claim that those scripts detailing necessary consequences of \(\alpha \) should have priority over scripts detailing possible consequences of it.
The idea behind this loop is that, conversely to what happened with the Description process (in which the agent elaborated the scenario step by step by picking a single factual rule each time), in this case the agent imagines performing an action. Therefore, the agent must check for all the consequences of such action, which are described (according to their preconditions) by the formulas in \(\text{ SCRIPT }\); as we have already argued by the beginning of the current section, the reason for doing so is that one cannot imagine that she performs an action, and then that only some of its consequences happen.
In order to clarify the intuitions behind this step, suppose that the algorithm is evaluating a box-formula that represents m alternative outcomes: by following the way our other algorithms have been working, we should create m new possible imaginary worlds to account for each one of those outcomes. However, if there already exists an imaginary possible world (possibly as a result of evaluating a diamond-formula), then one of such outcomes should be represented in the world that already exists—for, otherwise, if we created m new imaginary worlds, aside from the already existing one, we would have \(m+1\) new possible worlds to account for just m alternative outcomes represented in the current box-formula. Therefore, when evaluating a box-formula upon an already existing imaginary world, we only have to create \(m-1\) new worlds.
The relation between this step and step 3(b).ii could be understood as an “if ...else” statement, in terms of programming languages. Namely, the algorithm first checks whether there already exists any imaginary world and, if it does, goes through the corresponding “if” branch; otherwise (or “else”), if there are no already existing imaginary worlds, the algorithm must go through this current branch. Note, therefore, that both branches are never going to be executed for the same script, but rather just one of the two branches.
It is worth mentioning that, in modal logic, the box operator \(\square \) has a sort of “vacuous” or “trivial” truth-condition: namely, if a world w has no accessibility relations at all, then every formula of the form \(\square \varphi \) would be vacuously true in there; as there are no worlds accessible from w, then every world accessible from w satisfies \(\varphi \). That being said, we do not want our EvoAlg to conform to this fact, when evaluating a box-formula. One may argue that, if no diamond-formula about \(\alpha \) has been previously evaluated, and so no new worlds have been created, then every box-formula about \(\alpha \) could be true without the need of creating any world at all as a consequence of the agent imagining it. Nevertheless, this is not the way we want these formulas to behave, regarding the development of imaginary worlds, and so we still require our EvoAlg to create, at least, one witness world for a box-formula, in case there exist none yet.
Also considered by Berto (2017a).
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This work is funded by EC FP7 grant 621403 (ERA Chair: Games Research Opportunities), by the Project “Hybrid Intensional Logic” (Ref. FFI2013-47126P) given by the Spanish MINECO, the Project 2018–2020: Traducciones, lógicas combinadas, descripciones, lógica intensiva, teoría de tipos, lógica híbrida, identidad, lógica y educación (Ref. FFI2017-82554), given by the Spanish MICINN, and a doctoral grant from the Universitat Oberta de Catalunya (UOC).
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Casas-Roma, J., Huertas, A. & Rodríguez, M.E. The Logic of Imagination Acts: A Formal System for the Dynamics of Imaginary Worlds. Erkenn 86, 875–903 (2021). https://doi.org/10.1007/s10670-019-00136-z
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DOI: https://doi.org/10.1007/s10670-019-00136-z