Abstract
Possible worlds semantics for conditionals leave open the problem of how to construct models for realistic domains. In this paper, we show how to adapt logics of action and change such as John McCarthy’s Situation Calculus to conditional logics. We illustrate the idea by presenting models for conditionals whose antecedents combine a declarative condition with a hypothetical action.
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Notes
Both terms are in some ways unsatisfactory.
It has, however, been noted by the situation semanticists. See Barwise (1986).
States are not to be thought of as possible worlds, but as temporal points in a world history.
See especially Akman et al. (2004).
For instance, an adaptation of the Event Calculus similar to the one described here of the Causal Calculus, in connection with Shanahan’s solution to the Egg Cracking Problem Shanahan (2001), would deliver a semantics for conditionals in that domain.
Our use of a causal notion in our account of conditionals would introduce circularity only if all causal notions had to be defined in terms of conditionals. There are in fact conditional-based accounts of causality, but these are problematic in many ways. Other approaches to causality, relying on causal graphs or on causal mechanisms (as in Woodward (2003)), are available. We ourselves find these approaches preferable to analyses in terms of conditionals. Moreover, as we point out below, the causal apparatus we use is very circumscribed and can, if desired, be dispensed with.
Simultaneity can be an abstract relation—we do not need to resort to a temporal metric. For a general treatment of conditionals and time, see Thomason and Gupta (1980).
\( \mathtt{[} \mathsf{C} \mathtt{]} \) is the “caused” modality of Turner (1999).
Here, \(\lnot \phi \) is \(\phi \rightarrow \bot \).
Roughly speaking, these are transitions where there are good intuitions about how the condition could have been caused. There are conditionals where such intuitions don’t seem to be available: Lewis’ example ‘If kangaroos didn’t have tails’ is one of these. Our methods do not apply well to such examples. This is a limitation of our approach, which is more or less worrisome depending on how seriously you take these examples, which often seem vague and somewhat whimsical. We can claim, however, that our methods apply to a large body of conditionals which have played a prominent role in the literature on this topic.
This idea is similar to Pearl’s characterization of counterfactuals using “principled minisurgery” operators do \((X=x)\). See (Pearl (2000), Sect. 7.1).
See Lewis (1979) for a discussion of backtracking.
See Stalnaker (1980).
We assume that a match that has been struck will not light if struck again. This is an oversimplification, but we see no way to do better (in the context of a deterministic framework) without introducing a hidden variable.
This language is easier to work with than a more familiar language with conditional antecedents of the form \((\eta _1\,{\wedge }\,\,\ldots \,{\wedge }\,\,\eta _n)\,{\wedge }\,\,\small {\textsc {Do}}(a)\). Clearly, though, the frame we define could easily be adapted to that sort of language.
Disjunctive antecedents are often used by skeptics who question the meaningfulness of conditionals. Quine asks: “If Bizet and Verdi had been compatriots, would Verdi have been French or Bizet have been Italian?”
This is another simplification that has been addressed in the literature on reasoning about actions and time; see, for instance, Reiter (1996).
Lighting the stove when it is already on is, of course, pointless, but it is permitted.
This, of course, is a simplification. A better theory would separate natural processes, and the times required for them, from agent actions. The way we treat a kettle on the stove illustrates the simplification: bringing the water to a boil is treated as a one-step causal consequence of the action of a lit stove, whereas obviously the process takes an amount of time that will depend on the heat of the stove, the amount of water in the kettle, the atmospheric pressure, and other variables. See Reiter (1996) for ideas about how to deal with such automatic processes.
Waiting is a matter of not doing any of the actions in \(\mathcal{A}\), so that \(\small {\textsc {Do}}_{h,k}(\mathsf{wait})\) is the conjunction of all formulas \(\lnot \small {\textsc {Do}}_{h,k}(a)\), where \(a\in \mathcal{A}\).
A subtle but important point comes into play here: directionality would make no sense without the causal modality \( \mathtt{[} \mathsf{C} \mathtt{]} \). If the axiom for separating were
$$\begin{aligned}{}[\mathsf{together}_{h,k}\,{\wedge }\,\,\small {\textsc {Do}}_{h,k}(\mathsf{separate})]\rightarrow \lnot \mathsf{together}_{h,k+1}, \end{aligned}$$it would be logically equivalent to its contraposition
$$\begin{aligned} \mathsf{together}_{h,k+1}\rightarrow [\lnot \mathsf{together}_{h,k}\vee \lnot \small {\textsc {Do}}_{h,k}(\mathsf{separate})], \end{aligned}$$and so would have no intrinsic directionality.
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Lent, J., Thomason, R.H. Action Models for Conditionals. J of Log Lang and Inf 24, 211–231 (2015). https://doi.org/10.1007/s10849-015-9213-8
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DOI: https://doi.org/10.1007/s10849-015-9213-8