Abstract
A conclusion is a “material consequence” of reasons if it follows necessarily from them in accordance with a valid form of argument with content. The corresponding universal generalization of the argument’s associated conditional must be true, must be a covering generalization, and must be true of counterfactual instances. But it need not be law-like. Pearl’s structural model semantics is easier to apply to such counterfactual instances than Lewis’s closest-worlds semantics, and gives intuitively correct results.
Bibliographical note: This chapter was previously published in Virtues of Argumentation. Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), 22–26 May 2013, ed. Dima Mohammed and Marcin Lewiński, pp. 1–13. Windsor, ON: OSSA, 2014. The chapter is republished under the terms of a Creative Commons 4.0 International License, the terms of which are available at https://creativecommons.org/licenses/by/4.0/.
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Notes
- 1.
Added in the present republication: In this chapter, an asterisk in front of a sample argument indicates that intuitively its conclusion does not follow from the premiss.
- 2.
Correction in the present republication: The original article claimed falsely that the generalization in question is logically equivalent to the proposition that, if there is some kind K such that every K grows, then Mars is a planet. The proof of the logical equivalence of the generalization to the statement that Mars is a planet runs as follows:
Left to right: Suppose that, for every kind K, if Ks grow, then Mars is a planet. Then, by universal instantiation, if growing things grow, then Mars is a planet. But, as a matter of logic, growing things grow. So Mars is a planet.
Right to left: Now suppose that Mars is a planet. Then, by one of the paradoxes of material implication, if for some arbitrary kind X that Xes grow, then Mars is a planet. Hence, by universal generalization, for every K, if Ks grow, then Mars is a planet. QED
In general, a universal generalization over a content expression that occurs only in the antecedent of an argument’s associated conditional is logically equivalent to the argument’s conclusion, by the reasoning of the above proof.
- 3.
Correction in the present republication: The original article claimed falsely that the generalization in question is logically equivalent to the proposition that there are reptilian mammals if Mars is a star. In fact, the generalization is logically equivalent to the true proposition that Mars is not a star. The proof is as follows.
From left to right: Suppose that, for any kind K, if Mars is a star, then some Ks are reptilian mammals. Then, by universal instantiation, if Mars is a star, then some non-reptilian mammals are reptilian mammals. But, as a matter of logic, no non-reptilian mammals are reptilian mammals. Hence, by modus tollendo tollens, Mars is not a star.
From right to left: Suppose that Mars is not a star. Then, by one of the paradoxes of material implication, if Mars is a star. then for some arbitrary kind X, some Xes are reptilian mammals. Hence, by universal generalization, for any kind K, if Mars is a star, then some Ks are reptilian mammals.
In general, a universal generalization over a content expression that occurs only in the consequent of an argument’s associated conditional is logically equivalent to the contradictory of the conjunction of the argument’s premisses, by the reasoning of the above proof.
- 4.
http://www.weirdasianews.com/2010/02/16/japan-pours-worlds-largest-gold-bar/; accessed 2016 08 09.
- 5.
Added in the present republication: Not being a topologist, I don’t know how there could be such a space. But there seems nothing logically impossible in having a space with peculiar planes in which a figure in the plane whose boundary consisted of points equidistant from some point in the plane would be bounded by four straight lines of equal length whose adjacent sides were at right angles to each other.
- 6.
Added in the present republication: The last example is a counterfactual instance of the antecedent of the covering generalization of argument (14) that all descendants of theropod dinosaurs are products of evolution. It happens that all living descendants of theropod descendants are birds. But the evolutionary branch stemming from theropod dinosaurs might have divided into birds and some other major groups of animals, in the same way that early reptiles evolved into lizards, turtles and other large groups of animals.
- 7.
Added in the present republication: They are exogenous in virtue of what is known about how people come to live in the White House. The causal pathways leading to their taking up residence n that building do not include references to its geographical location or to the boundaries of Washington, D.C.
- 8.
Correction in the present republication: The original article had “from being two-legged to being four-legged” instead of “from being four-legged to being two-legged”.
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Hitchcock, D. (2017). Material Consequence and Counterfactuals. In: On Reasoning and Argument. Argumentation Library, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-53562-3_9
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