Abstract
In his original semantics for counterfactuals, David Lewis presupposed that the ordering of worlds relevant to the evaluation of a counterfactual admitted no incomparability between worlds. He later came to abandon this assumption. But the approach to incomparability he endorsed makes counterintuitive predictions about a class of examples circumscribed in this paper. The same underlying problem is present in the theories of modals and conditionals developed by Bas van Fraassen, Frank Veltman, and Angelika Kratzer. I show how to reformulate all these theories in terms of lower bounds on partial preorders, conceived of as maximal antichains, and I show that treating lower bounds as cutsets does strictly better at capturing our intuitions about the semantics of modals, counterfactuals, and deontic conditionals.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bell, M., & Ginsburg, J. (1984). Compact spaces and spaces of maximal complete subgraphs. Transactions of the American Mathematical Society, 283(1), 329–338.
Chisholm, R. M. (1946). The contrary-to-fact conditional. Mind, 55, 289–307.
Copley, B. (2006). What should should mean? Ms., CNRS/Université Paris 8. http://copley.free.fr/copley.should.pdf.
Doignon, J.-P., & Falmagne, J.-C. (1985). Spaces for the assessment of knowledge. International Journal of Man-Machine Studies, 23(2), 175–196.
Doignon, J.-P., & Falmagne, J.-C. (1999). Knowledge spaces. Berlin: Springer.
Ginsburg, J. (1984). Compactness and subsets of ordered sets that meet all maximal chains. Order, 1, 147–157.
Goodman, N. (1947). The problem of counterfactual conditionals. Journal of Philosophy, 44(5), 113–128.
Grillet, P. A. (1969). Maximal chains and antichains. Fundamenta Mathematicae, 65, 157–167.
Hájek, A. (2009). Most counterfactuals are false. Ms., Australian National University.
Halpern, J. D. (1972). On a question of Tarski and a maximal theorem of Kurepa. Pacific Journal of Mathematics, 41(1), 111–121.
Hausdorff, F. (1914). Grundzüge der Mengenlehre. von Veit, Leipzig.
Higgs, D. (1985). A companion to Grillet’s theorem on maximal chains and antichains. Order, 1, 371–375.
Kelley, J. L. (1955). General topology. New York: van Nostrand.
Kolodny, N., & MacFarlane, J. (2010). Ifs and oughts. Journal of Philosophy, 107(3), 115–143.
Kratzer, A. (1976). Was ‘können’ und ‘müssen’ bedeuten können müssen. Linguistische Berichte, 42, 1–28.
Kratzer, A. (1977). What must and can must and can mean. Linguistics and Philosophy, 1, 337–355.
Kratzer, A. (1979). Conditional necessity and possibility. In R. Bauerle, U. Egli, & A. von Stechow (Eds.), Semantics from different points of view (pp. 117–147). Springer.
Kratzer, A. (1981). The notional category of modality. In P. Portner, & B. H. Partee (Eds.), Formal semantics: The essential readings (pp. 289–323). Oxford: Blackwell.
Kratzer, A. (1991). Modality. In A. von Stechow, & D. Wunderlich (Eds.), Semantics: An international handbook of contemporary research (pp. 639–650). Berlin: W. de Gruyter.
Kurepa, Đ. (1953). Über das Auswahlaxiom. Mathematische Annalen, 126, 381–384.
Lemmon, E. J. (1962). Moral dilemmas. Philosophical Review, 71(2), 139–158.
Lewis, D. K. (1973). Counterfactuals. Malden: Blackwell.
Lewis, D. K. (1975). Adverbs of quantification. In E. L. Keenan (Ed.), Formal semantics of natural language (pp. 3–15). Cambridge: Cambridge University Press.
Lewis, D. K. (1981). Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic, 10, 217–234.
Lonc, Z., & Rival, I. (1987). Chains, antichains, and fibres. Journal of Combinatorial Theory, Series A, 44, 207–228.
Maltby, R. (1994). When is every minimal cutset an antichain? Acta Scientiarum Mathematicarum, 59, 381–403.
Rescher, N. (1964). Hypothetical reasoning. Amsterdam: North Holland.
Sæbø, K. J. (2001). Necessary conditions in a natural language. In C. Féry, & W. Sternefeld (Eds.), Audiatur vox sapientiae: A festschrift for Arnim von Stechow. Studia Grammatica (Vol. 52, pp. 427–449). Berlin: Akademie.
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: Reidel.
Stalnaker, R. C. (1984). Inquiry. Cambridge: MIT Press.
von Stechow, A., Krasikova, S., & Penka, D. (2006). Anankastic conditionals again. In A. Grønn, D. Haug, & T. Solstad (Eds.), A festschrift for Kjell Johan Sæbø—In partial fulfillment of the requirements of the celebration of his 50th birthday (pp. 151–171). Oslo: Oslo University.
Swanson, E. (2010). Ordering supervaluationism, counterpart theory, and ersatz fundamentality. Ms., University of Michigan Ann Arbor.
van Fraassen, B. C. (1973). Values and the heart’s command. Journal of Philosophy, 70(1), 5–19.
von Fintel, K., & Iatridou, S. (2005). What to do if you want to go to Harlem: Anankastic conditionals and related matters. Ms., Massachusetts Institute of Technology.
von Fintel, K., & Iatridou, S. (2008). How to say ought in foreign: The composition of weak necessity modals. In J. Guéron, & J. Lecarme (Eds.), Time and modality. Studies in natural language and linguistic theory (Vol. 75, pp. 115–141). Dordrecht: Springer.
Veltman, F. (1976). Prejudices, presuppositions and the theory of conditionals. In J. Groenendijk, & M. Stokhof (Eds.), Amsterdam papers in formal grammar (Vol. 1, pp. 248–281). Amsterdam: Central Interfaculteit, University of Amsterdam.
Weatherson, B. (2001). Indicative and subjunctive conditionals. The Philosophical Quarterly, 51(203), 200–216.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Swanson, E. On the Treatment of Incomparability in Ordering Semantics and Premise Semantics. J Philos Logic 40, 693–713 (2011). https://doi.org/10.1007/s10992-010-9157-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-010-9157-z