Top

Journal of Logic, Language and Information

Published in:

01-06-2015

Action Models for Conditionals

Authors: Jeremy Lent, Richmond H. Thomason

Published in: Journal of Logic, Language and Information | Issue 2/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Logic Reduced To Bare (Proof-Theoretical) Bones

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Both terms are in some ways unsatisfactory.

For information about this topic, see Davis (1991) and Mueller (2006).

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.

Akman, V., Erdoǧan, S. T., Lee, J., Lifschitz, V., & Turner, H. (2004). Representing the zoo world and the traffic world in the language of the causal calculator. Artificial Intelligence, 153(1–2), 105–140.CrossRef

Barwise, K. J. (1986). Conditionals and conditional information. In E. Traugott, A. ter Meulen, J. Reilly, & C. Ferguson (Eds.), On Conditionals (pp. 21–54). Cambridge, England: Cambridge University Press.CrossRef

Bobzien, S. (2012). Dialectical school. In: Zalta EN (ed) The stanford encyclopedia of philosophy, http://plato.stanford.edu/archives/fall2011/entries/dialectical-school/.

Carnap, R. (1936). Testability and meaning. Philosophy of Science, 3, 419–471.

Carnap, R. (1937). Testability and meaning—continued. Philosophy of Science, 4, 1–40.

Davis, E. (1991). Representations of common sense knowledge. San Francisco: Morgan Kaufmann.

Fine, K. (1975). Critical notice: Counterfactuals. Mind, 84(335), 451–458.CrossRef

Giunchiglia, E., Lee, J., Lifschitz, V., McCain, N., & Turner, H. (2004). Nonmonotonic causal theories. Artificial Intelligence, 153(5–6), 49–104.CrossRef

Goodman, N. (1955). Fact, fiction and forecast. Cambridge: Harvard University Press.

Kowalski, R. A., & Sergot, M. J. (1986). A logic-based calculus of events. New Generation Computing, 4, 67–95.CrossRef

Lewis, D. K. (1973). Counterfactuals. Cambridge, Massachusetts: Harvard University Press.

Lewis, D. K. (1979). Counterfactual dependence and time’s arrow. Noûs, 13(4), 455–476.CrossRef

McCarthy, J., & Hayes, P. J. (1969). Some philosophical problems from the standpoint of artificial intelligence. In B. Meltzer & D. Michie (Eds.), Machine intelligence 4 (pp. 463–502). Edinburgh: Edinburgh University Press.

Mueller, E. T. (2006). Commonsense reasoning. Amsterdam: Elsevier.

Pearl, J. (2000). Causality: Models, reasoning, and inference. Cambridge, England: Cambridge University Press.

Reiter, R. (1996). Time in the situation calculus. In: S. Buvač & T. Costello (eds)., Working Papers: Common sense ’96. Computer Science Department, Stanford University, Stanford University, pp 176–185, consult http://www-formal.Stanford.edu/tjc/96FCS.

Reiter, R. (2001). Knowledge in action: Logical foundations for specifying and implementing dynamical systems. Cambridge, Massachusetts: The MIT Press.

Schneider, E. (1953). Recent discussions on subjunctive conditionals. Review of metaphysics, 6, 623–649.

Shanahan, M. (2001). An attempt to formalize a nontrivial benchmark problem in common sense reasoning. Artificial intelligence, 153(5–6), 141–165.

Shanahan, M. (2009). The frame problem. In: EN. Zalta (ed)., The Stanford encyclopedia of philosophy, winter 2009 edn, http://plato.stanford.edu/archives/win2009/entries/frame-problem/.

Stalnaker, R. C. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory (pp. 98–112). Oxford: Basil Blackwell Publishers.

Stalnaker, R. C. (1980). A defense of conditional excluded middle. In W. L. Harper, R. Stalnaker, & G. Pearce (Eds.), Ifs: Conditionals, belief, decision, chance, and time (pp. 87–104). Dordrecht: D. Reidel Publishing Co.CrossRef

Thomason, R., & Gupta, A. (1980). A theory of conditionals in the context of branching time. The Philosophical Review, 80, 65–90.CrossRef

Thomason, RH. (2007). Conditionals and action logics. In: E. Amir, V. Lifschiz, & R. Miller (eds)., AAAI 2007 Spring symposium on commonsense reasoning. American Association for Artificial Intelligence, American Association for Artificial Intelligence, Menlo Park, California, pp 156–161, available at http://web.eecs.umich.edu/rthomaso/documents/lai/Conditionals_and_Action_Logics.pdf.

Turner, H. (1999). A logic of universal causation. Artificial Intelligence, 113(1–2), 87–123.CrossRef

Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.

Title: Action Models for Conditionals
Authors: Jeremy Lent
Richmond H. Thomason
Publication date: 01-06-2015
Publisher: Springer Netherlands
Published in: Journal of Logic, Language and Information / Issue 2/2015
Print ISSN: 0925-8531
Electronic ISSN: 1572-9583
DOI: https://doi.org/10.1007/s10849-015-9213-8

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 2/2015

Not Only Barbara

A Double Team Semantics for Generalized Quantifiers

Logic Reduced To Bare (Proof-Theoretical) Bones

A Sequent Calculus for Urn Logic

Premium Partner